learning process control mtsn 1 medan titin rahmayanti rambe1, fasti rola2, siti habsari pratiwi3, anggili pratama4, jhohas dongoran5, hasratuddin6 1,2,3,4,5 doctoralstudent at medan state university 6doctoral lecturer at medan state university 1e-mail:titinrahmayanti@stkipalmaksum.ac.id 2e-mail:fastirola@usu.ac.id 3e-mail:shihabpratiwi@iainlangsa.ac.id 4e-mail: anggilipratama@gmail.com 5e-mail: dongoran231089@gmail.com 6email:siregarhasratuddin@yahoo.com abstract.abstract. the purpose of this study is to analyze whether the product produced by the process meets the initial expectations or meets the standards expected beforehand. the learning process is one that must be tested for execution and outcomes to ensure that the outputs meet the objectives and are of the appropriate quality. it would be very interesting to evaluate and explain the original incident on the basis of the authentic data collected, as well as how to present that data to interested parties and how the data is processed using appropriate statistics; this will give us an idea of how a process occurs. therefore, the sample for this study consisted of valid national exam (ne) results data for mtsn 1 medan students from the academic year 2013/2014 to 2017/2018 in four subjects (indonesian, english, mathematics, and science). the results of the study showed that the learning process that took place at the medan mtsn 1 school in the academic year 2013/2014 to 2018/2019 went smoothly. keywords : control, process, learning, ne, medan introduction the multivariate statistical approach is an analytical methodology that considers the correlation of a set of correlated criterion variables as a system. researchers can use multivariate analysis to solve more generic or difficult issues, as well as challenges that more properly mirror the actual world. one of the goals of multivariate analysis is to discover and explain the underlying structure or properties of the data. multivariate analysis is also used to find new variables that are less in number than the original variables but may explain fluctuations in the original variables. the control analysis of mtsn 1 medan's learning processes and results is discussed in this article. the purpose is to establish if the product generated by the process satisfies its initial expectations or continues to satisfy the expected standards or original design for which it was built. the learning process is one that must be monitored for execution and results to ensure that the outputs fulfill the objectives and are of appropriate quality. it would be fascinating to explore the real occurrence of legitimate data, how to establish the data's significance, and how to analyze the data using relevant statistics to demonstrate how the process works in this context. as a consequence, valid student national exam results (mtsn 1 medan 2013/2014-2017/2018) are used in four courses (indonesian, english, mathematics, and science). ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 1 mailto:titinrahmayanti@stkipalmaksum.ac.id mailto:fastirola@usu.ac.id mailto:sihabpratiwi@iainlangsa.ac.id mailto:anggilipratama@gmail.com mailto:dongoran231089@gmail.com how may multivariate statistical approaches be used to investigate the links and interrelationships between the topics of indonesian (ind), english (ing), and science with mathematics (mat)? what are the consequences of the continuing learning process control? the goal of this article is to investigate and describe the relationships and connections between the subjects covered in the national examination, as well as to provide interested parties with information on how to achieve learning process control results in mtsn 1 medan by using multivariate statistical methods. result and discussion the sample for the analysis in this article is students' national examination (mtsn 1 medan 2013/14-2017/18) results in four disciplines (indonesian, english, mathematics, and science). over the course of five years, the total number of data points on student scores from 1,734 people was discovered to be distributed as follows: in 2013/2014, there were 294 people; in 2014/2015, there were 308 people; in 2015/2016, there were 378 people; in 2016/2017, there were 355 people; and in 2017/2018, there were 399 people. the data analysis in this paper focuses on 1) the relationship between indonesian, english, mathematics, and science training. the author chose the 2013/2014 implementation year because he wanted to investigate and disclose more linked themes. the author chose the 2013/2014-2017/2018 national examination with indonesian, english, mathematics, and ipas for the whole five-year period. 2). concerning the control of the fiveyear learning process, the un value data is sent through the control model upercontrollimit(ucl), data analysis for 2013/2014 this section analyzes actual data from indonesiamtsn 1 medanin 2013/2014 test results, especially four subject tests: indonesian (x1), english (x2), mathematics (x3), and science (x4), with a total of 294 pupils. we acquired the following results by calculating the average of each: the following results were obtained from calculating the average of each: table 1. average of four subjects in comparison subject eng ing matt ipa jl 2405,400 2586,500 2414,300 2451,000 means 8,182 8,798 8,212 8,337 var 0.862 0.279 1,111 0.723 range 4,300 3,800 4,700 4,200 source: results of data processing by researchers, 2022 based on the analytical results in table 1, the assumption is that the data is normally distributed and that the learning is done by the same teacher. the highest average score in english (ing) courses was 8.798; the lowest average score in indonesian (ind) subjects was 8.182. this demonstrates that the english subject has the highest student accomplishment, whereas the indonesian language subject has the lowest. this section examines actual data from indonesiamtsn 1 medanin test results from 2013/2014, especially four subject tests: indonesian (x1), english (x2), mathematics (x3), and science (x4), with a total of 294 pupils. we acquired the following results by calculating ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 2 the average of each: the following results were obtained from calculating the average of each: to see the relationship or correlation between topics, the following correlation matrix calculation results are presented: table 2. matrix of variance-covariance correlation bind 1.0000 0.2185 0.2952 0.1256 bing 0.2185 1.0000 0.3581 0.3487 mate 0.2952 0.3581 1.0000 0.2867 ipa 0.1256 0.3487 0.2867 1.0000 eng ing matt ipa source: results of data processing by researchers, 2022 the connection between english (ing) and mathematics (mat) topics has the greatest value of 0.3581, while the correlation between ind and science subjects has the lowest value of 0.1256, according to the study shown in table 2. this suggests that the strongest relationship between disciplines is between english and mathematics, whereas the poorest link is between indonesian and science. when the whole correlation value is taken into account, the correlation value is positive. this proves that pupils do admirably in all courses. the following are the results of constructing the inverse correlation matrix to see which topics can best predict other subjects. table 3. matrix of inverse correlation eng 1.1153 -0.1351 -0.2813 -0.0124 ing -0.1351 1.2657 -0.3283 -0.3314 matt -0.2813 -0.3283 1.2660 -0.2135 ipa -0.0124 -0.3314 -0.2135 1.1788 eng ing matt ipa source: results of data processing by researchers, 2022 each diagonal member of the inverse correlation matrix is proportionately connected to the correspondence variable specified by regression, according to the study shown in table 3. each diagonal element is obviously equivalent to to � ���� ,, where r is the multicorrelation coefficient between the other variables. according to the aforementioned computation, the greatest percentage value for mat classes is 21% ( �.������ �.���� ), while the lowest percentage value for ind classes is 10.3% ( �.������ �.���� ). this means that mat is the topic most anticipated by other subjects, whereas ind is the subject least predicted by other subjects. the results of the predictive analysis of subject values using regression analysis, .�� = �� + ���(�, �). ���(�)��(� − ��). 1. the magnitude of the expected value provided by each indonesian value to the value of mathematics is is����� = 5,471 + 0,335���� table 4. indonesian language scores on mathematics scores coefficient ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 3 coefficientsa model unstandardized coefficients standardize d coefficients t sig. b std. error beta 1 (constan) 5,471 .523 10,470 .000 bind .335 063 .295 5,279 .000 a. dependent variable: mathematics 2. the magnitude of the anticipated value provided to the math score by each english score is����� = 1,924 + 0,715���� table 5. english scores on mathematics scores coefficient coefficientsa model unstandardized coefficients standardize d coefficients t sig. b std. error beta 1 (constan) 1924 .961 2001 046 bing .715 .109 .358 6,553 .000 a. dependent variable: mathematics 3. the magnitude of the expected value for the value of mathematics supplied by each value of indonesian (ind), english (ing), and ipa is����� = 5,250 + 0,355���� table 6. coefficient of natural science scores on mathematics values coefficientsa model unstandardized coefficients standardized coefficients q sig. b std. error beta 1 (constant ) 5,250 .582 9017 .000 ipa .355 .069 .287 5.113 .000 a. dependent variable: mathematics 4. the magnitude of the predicted value given by each value of indonesian (ind), english (ing), and ipa for the value of mathematics is����� = 0,008 + 0,248���� + 0,500���� + 0,213���� table 7. coefficient of ind, ing, ipa values on mathematical values coefficientsa model unstandardized coefficients standardized coefficients q sig. b std. error beta 1 (constant) 008 .992 008 .994 son .248 061 .219 4,070 .000 ing .500 .114 .250 4,396 .000 ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 4 ipa .213 .069 .172 3,070 002 a.dependent variable: mathematics learning process control analysis the data utilized for analysis in this topic runs from 2013/2014 to 2017/2019 for the subjects of indonesian, english, mathematics, and science on national exam scores.mtsn 1 medan. the first stage of process control analysis is to determine the determinant value of the covariance matrix, and the following data are obtained: table 8. determinants of the covariance matrix and ucl values year mdk0 ucl2 ucl3 2013/2014 0.128366 44.62798 99.86698 2014/2015 3.680986 44.62798 99.86698 2015/ 2016 17.2321 44.62798 99.86698 2016/ 2017 1.01308 44.62798 99.86698 2017/ 2018 1.675714 44.62798 99.86698 source: results of data processing by researchers, 2022 figure 1. pbm variability control table 9. values – chart for future sub samples means and ucl values�� year t^2 ucl2 ucl3 2013/2014 78,459 223.1399 499.3349 2014/2015 74019 223.1399 499.3349 2015/ 2016 24,564 223.1399 499.3349 2016/ 2017 24,862 223.1399 499.3349 2017/ 2018 24.76 223.1399 499.3349 source: results of data processing by researchers, 2022 0 20 40 60 80 100 120 2013/2014 2014/2015 2015/2016 2016/2017 2017/2018 kontrol variabilitas pbm mdk0 ucl2 ucl3 ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 5 figure 2. variability control of pbm both the variability control lines and the process achievement control lines (t2) are below the ucl(2) and ucl(3) lines, according to the examination of the preceding tables and figures. this shows that the learning process has been running efficiently during the last five years in accordance with the learning outcomes provided. thus, without lowering the amount of time variables required, the national examination score data from 2013/2014 to 2017/2018 may be used as a controller for the learning process. from the graph above, it can be concluded that the learning process that took place at the mtsn 1 medan school for the 2013/2014 school year to 2017/2018 went smoothly as expected. conclusion 1. in general, the highest average student achievement score in the four subjects, namely indonesian, english, mathematics, and natural sciences, which were in the 2013/2014 national examination in mts n 1 medan, is 8.798 in the english (ing) subject, with a smaller standard deviation of 279; the lowest average student achievement score was 8,182 in indonesian (ind) lessons, with a standard deviation of 0.862. 2. the weakest subject correlation, with a magnitude of 1256, is between indonesian and ipa.the strongest correlation between these subjects, with a magnitude of 0.3581, is between english and mathematics.indonesian and english have little or no correlation, as do science and mathematics.however, when the overall correlation value is considered, the data shows a positive correlation value. 3. mathematics is the subject most predicted by other subjects (21%), while indonesian is the least predicted (10.3%). 4. one subject's estimated value for another lesson (assuming there are no predictable lesson provisions). the results of the regression analysis used to forecast subject value,.�� = �� + ���(�, �). ���(�)��(� − ��) a) the magnitude of the expected value provided by each indonesian value to the value of mathematics is����� = 5,471 + 0,335���� b) the magnitude of the anticipated value provided to the math score by each english score is is����� = 1,924 + 0,715���� c) the magnitude of the expected value assigned by each scientific value to the mathematical value is����� = 5,250 + 0,355���� 0 100 200 300 400 500 600 2013/2014 2014/2015 2015/2016 2016/2017 2017/2018 kontrol capaian pbm t^2 ucl2 ucl3 ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 6 d) the magnitude of the expected value provided to the value of mathematics by each bin, ing, and ipa value is is����� = 0,008 + 0,248���� + 0,500���� + 0,213���� 5. based on the seven years of national examination score data examined, from 2013/2014 to 2017/2018, both the variability control lines (mdkov) and the pbm achievement control lines (t2) are lower than the ucl(2) and ucl(3) lines. this shows that the learning process has been running efficiently during the last five years in accordance with the learning outcomes provided. thus, without lowering the amount of time variables required, the national examination score data from 2013/2014 to 2017/2018 may be used as a controller for the learning process. bibliography djauhari, maman a. 2005. improved monitoring of multivariate process variability. journalsof quality technology.wisconsin: american society for quality djauhari, m.a., dan dyah e. herwindiati. 2022. kontrol kualitas proses kompleks. itb press: bandung. hasratuddin. 2019. weakness analysis learning mathematics junior high school in medan. journal international of mathematca. issn: 2456-8538volume 02 |issue 08 |august 2019 p. 1-18. johnson, richard a. 2002. applied multivariate statistical analysis (5th). new jersey: personeducation international. whittaker, joe. 1996. graphical models in applied multivariate statistics. new york: john wiley & sons ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 7 learning process control mtsn 1 medan figure 2. variability control of pbm distinction between spepectral analysis and harmonic analysis in time series analysis: experience from the number cruncher statistical system (ncss) software. authors okafor uchenwa linusa, m.o. oladejoa, c.o.uwab, d.t.chineyob and jighjigh,a.tc. a. department of mathematical sciences, faculty of science nigerian defence academy, kaduna b.department of computer science, faculty of military science and interdisciplinary studies ,nigerian defence academy, kaduna. c department of mathematics/computer science. benue state university, makurdi abstract this paper shows a clear distinction between spectral analysis and harmonic analysis and highlights what is common amongst these two terms. the distinction and commonalities are illustrated using the data from sales of cement obtained from a sales record. the work identified the role of spectral analysis as that of showing the periods /wavelengths having containing more energy with the sine and cosine components but without stating whether the components are significant or not. spectral analysis helps to identify hidden periodicities but does not form the model equation. harmonic analysis on the other hand makes use of the periods with higher energies identified by the spectral analysis to form a model equation, it also shows whether the frequencies are significant or not by displaying the t-values for each kth harmonic value used to form the model equation. from the giving data, harmonic analysis makes predictions /forecasts, performs model parameter estimation, analysis of variance, displays the asymptotic correlation matrix of parameters. in harmonic analysis, the graphs of errors against time and against the predicted values are also plotted. keywords: harmonic analysis, spectral analysis, period, wavelength, model, forecasting. corresponding author’s address: email-linusokafor @gmail.com,phone;+234-9022982454 co-authors address: (1)mikeoladejo2003@yahoo.com.(2)ozocel2014@gmail.com.(3)atamber@bsum.edu.ng.(4)dtchinyio @yahoo.com ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 1 1.0 introduction most text book writers take spectral analysis as harmonic analysis and vice versa.brillinger(2001) considered harmonic analysis as a tool for search of hidden periodicities. but rebeca (2004) unlike most of the other authors explained that harmonic analysis works on the assumption that the modeler already knows the period, one can easily set up a regression equation and find out the amplitude, the phase and the mean of the signal. in the analysis of time series data, etc, it is often necessary to identify the periodicities in the data by spectral analysis which is concerned with the partition of the variation in a time series between the components at different frequencies or periods. in spectral analysis, if the data is not stationary it should first be detrended by fitting a proper curve to the data set and then using that to remove the trend. but if the data does not show obvious trend, one can then remove the mean and proceed to carry out spectral analysis. differencing can also be employed to make the data set stationary. the number cruncher statistical system (ncss) has six portions on carrying out spectral analysis as shown in table 1. table 1. fourier analysis (p,q,s,t,u) (a) (b) (c (d) (e ) (f) frequencies wavelength period cosine(�� ) sine(�� ) spectrum but we recommend a sevencolumn portion as given in table 2. so as to make room for the harmonic number (k) table 2. fourier analysis (p,q,s,t,u) (a) (b) (c (d) (e ) (f) (g) harmonics (k) frequencies(f) (f� ) wavelength l = � � period cosine(�� ) sine(�� ) spectrum where, �� = � � ∗ ∑ �� � � ��� ∗ cos(ɵ� ) �� = � � ∗ ∑ �� � � ��� ∗ sin(ɵ� ) ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 2 the data points are: �� =(�� , �� ,…�� ) k = � � for even number of data points, or ��� � for odd numbered data points. when the data has a clear seasonal length, l, then k = � � the column headed spectrum shows the amount of energy or proportion of the contribution of each of the frequencies to the variance of the series. spectral analysis also makes the plots of the spectrum against each of the following items: time, wavelength, and, frequency it is the spectral analysis that helps one to begin to think of the frequencies /wavelength that will enter into the model building process. spectral analysis does not give the model, the phase angels and makes no prediction from the data. in harmonic analysis, one uses the frequencies identified from spectral analysis to form a model and harmonic analysis gives about eight different specific results as shown through table 3 to table 10.fourier analysis is part of harmonic analysis, but in literature, most authors’ mistake spectral analysis for harmonic analysis and most statistical software’s also fall into the same misconception, but the use of the number cruncher statistical system(ncss), illustrates the role of either spectral analysis or harmonic analysis. 2. statement of the problem. in order to carry out a fourier series analysis of any time series data, one must use the periodic components and time series data are seasonal or periodic and more often than not these periodicities are hidden. to unmask the hidden periodic components spectral analysis needs to be performed on the data set. results from spectral analysis are then used to perform harmonic analysis through fourier series analysis. there is therefore the need to make a clear distinction between spectral analysis and harmonic analysis in the fourier analysis of time series data . 3. aim the aim of this study is to show the difference between spectral analysis and harmonic analysis in the analysis of time series data through a worked example using the number cruncher statistical system. 4. data source and materials used. the data set used in this work is the sales of bags of cement obtained from ogwumabiri market in obowo local government area of imo state ,nigeria and this is shown graphically in figure 1 and the data set is shown in table 1.the software used to do the analysis of date was the number cruncher statistical system (nccs2019) by chris hintz ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 3 fig.1.monthly sales of cement 2010 to 2018 table 1.sales of cement 2010 to 2017 year month/ 2010 2011 2012 2013 2014 2015 2016 2017 2018 jan 300 330 310 350 420 480 530 490 653 feb 345 355 345 368 530 560 590 540 640 marc 420 450 460 480 590 610 760 690 670 apr 350 370 390 410 500 520 580 560 690 may 290 350 360 390 480 490 560 530 610 jun 320 330 345 360 450 460 530 500 570 jul 280 300 320 340 390 400 500 490 600 aug 340 360 370 390 490 530 580 560 640 sep 370 390 410 450 510 590 630 640 662 oct 390 420 450 470 570 635 690 695 735 nov 430 470 490 560 630 690 740 760 790 dec 500 540 690 675 740 730 800 820 830 5. methodology the data was first subjected to outlier test by the grubbs test for outliers and the (ncss 2019) found are no outliers. 51. spectral analysis the software (ncss2019) gave the spectral analysis results in two folds as will be shown in the sequel, it removed the trend as the data displayed in table 1. depicts trending pattern. note that if the data has no trend, the mean will be then removed from the data.the spectral plots are displayed in figures 2 and 3. fig.2 .plot of periodogram against frequency. 0 200 400 600 800 1000 1 5 9 131721252933374145495357616569737781858993 series1 ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 4 fig.2.plot of periodogram against wavelength table 2.spectral analysis results sales of cement (k)harmonic frequency wavelength period cosine(a's) sine(b's) spectrum 1 0.129536 48.50526 16617.3 32.59697 -33.6977 1.00e+33 2 0.193623 32.4507 43948.08 73.84023 18.99913 1.00e+33 3 0.257709 24.38095 29364.56 -53.4418 32.06624 1.00e+33 4 0.321795 19.52542 38437.91 -71.1648 4.476748 1.00e+33 5 0.385881 16.28269 16091.53 -32.3992 -32.8458 1.00e+33 6 0.449968 13.96364 103884.1 -53.0957 -104.51 1.00e+33 7 0.514054 12.22281 2.15e+07 -1541.93 -681.31 1.00e+33 8 0.57814 10.86792 390103.1 206.1583 95.39722 1.00e+33 9 0.642227 9.78344 254522.4 173.7618 58.94443 1.00e+33 10 0.706313 8.895753 60140.59 -88.3929 11.91382 1.00e+33 11 0.770399 8.155752 613223.8 227.667 -171.124 1.00e+33 12 0.834486 7.529412 1110957 -370.889 -96.9354 1.00e+33 13 0.898572 6.992413 116157.1 -103.743 67.841 1.00e+33 14 0.962658 6.526912 1615692 410.1017 213.3926 1.00e+33 15 1.026744 6.119522 84240.29 96.81339 42.07458 1.00e+33 16 1.090831 5.76 11582.41 -29.9363 25.21726 1.00e+33 17 1.154917 5.440378 1979010 -438.565 -263.514 1.00e+33 18 1.219003 5.154362 134286 106.6879 -79.8794 1.00e+33 19 1.28309 4.896918 617032.6 46.99103 -281.8 1.00e+33 20 1.347176 4.663968 290284.8 -186.796 -59.206 1.00e+33 21 1.411262 4.452174 283300.5 181.0477 68.52771 1.00e+33 22 1.475349 4.25878 787854 197.8401 255.0974 1.00e+33 23 1.539435 4.081488 2601302 338.3845 -479.155 1.00e+33 24 1.603521 3.918367 3221233 -652.678 -10.3855 1.00e+33 ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 5 (k)harmonic frequency wavelength period cosine(a's) sine(b's) spectrum 25 1.667608 3.767784 919057.6 4.082906 -348.646 1.00e+33 26 1.731694 3.628346 1046506 -294.5 -227.374 1.00e+33 27 1.79578 3.498861 10701.19 37.52555 2.713192 1.00e+33 28 1.859866 3.378299 699923.8 -265.652 148.3689 1.00e+33 29 1.923953 3.265769 3044655 107.2807 625.4841 1.00e+33 30 1.988039 3.160494 1704203 435.7885 -188.458 1.00e+33 31 2.052125 3.061794 4084301 626.947 -383.665 1.00e+33 32 2.116212 2.969072 186455.8 -9.93235 -156.733 1.00e+33 33 2.180298 2.881801 72246.95 -97.085 11.45203 1.00e+33 34 2.244384 2.799514 745561.3 -198.049 -243.716 1.00e+33 35 2.308471 2.721796 512814.4 -169.859 -197.438 1.00e+33 35 2.308471 2.721796 512814.4 -169.859 -197.438 1.00e+33 36 2.372557 2.648276 541663.8 -202.375 175.1979 1.00e+33 37 2.436643 2.578623 951056.5 210.9437 285.1424 1.00e+33 38 2.50073 2.512541 455378.2 147.9449 195.828 1.00e+33 39 2.564816 2.449761 895302.3 -246.868 -239.76 1.00e+33 40 c 2.390042 4245736 422.5156 618.9477 1.00e+33 41 2.692988 2.333164 3190907 -113.368 -639.713 1.00e+33 42 2.757075 2.278932 1882224 -413.898 278.6835 1.00e+33 43 2.821161 2.227163 1877624 441.369 231.4324 1.00e+33 44 2.885247 2.177694 1599362 230.5599 -397.997 1.00e+33 45 2.949334 2.130374 485149.8 211.2075 -139.878 1.00e+33 46 3.01342 2.085068 1806633 -459.92 165.6836 1.00e+33 47 3.077506 2.041648 2362332 154.5963 -537.2 1.00e+33 48 3.141593 2 1065080 -375.348 1.17e-10 1.00e+33 the plots of periodogram against frequency and that of periodogram against wavelengths are shown as fig.2. and fig.3 and each of them show that there is only one dominant frequency at the harmonic number k =7 ,with a frequency f= 0.514054 and wavelength 12.22281. . table 2 shows the fourier analysis result indicating the highest energy content at the 7th harmonic k=7 as 2.15*107 ,the cosine component is -1541.93and the sine component is -681.31 7 0.514054 12.22281 2.15e+07 -1541.93 -681.31 1.00e+33 spectral analysis results end after identification of wavelength, sine term, cosine term, period, and the spectrum. using the ncss,the harmonic numbers are not shown, hence we invoke the use of excel to supplement the analysis in order to add the harmonic numbers as shown in colum (a) in table 1. the use of harmonic analysis harmonic analysis is used to obtain the various section as explained here under. ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 6 below are the steps involved in carrying out the harmonic analysis: after performing the spectral analysis as discussed above, when the harmonic section of the ncss2019 was now employed the following results were obtained: table 3.run summary section ─── item value item value dependent variable yt total rows 156 time variable t rows with missing values 60 r² 0.96 rows used 96 estimated model ft =307.273+(4.317)*t-15.1354)*sin((0.517*t+ 66.998*cos(0.517t)) the amplitude value of 68.68638 is obtained from the squares of the sine and cosine coefficients 0f; 68.68638 = �(−15.1354)� + (66.998)� table 4.regression coefficients section regression standard tstatistic upper 95% independent coefficient error to test prob conf. limit variable b(i) sb(i) h0: β(i)=0 level of β(i) intercept 307 5.59439 54.93 0 318.38 trend 4.32 0.10026 43.06 0 4.5164 sin(12.14371) -15 3.90635 -3.87 0.9999 -7.3771 cos(12.14371) 67 3.93496 17.03 0 74.813 in this calculation, the software uses the wavelength value of 12.44 ,instead of the the frequency value. wave length frequency amplitude phase 12.144 0.52 68.68638 0.22218 table 5.analysis of variance table ─── sum of mean f-ratio df squares squares intercept 1 25546130 25546130 total (adjusted) 95 1685138 mse/mse model 4 27163716 msr=6790928.96 9248.64 error 92 67552.16 mse=734.2626 total 96 27231268 734.2626 ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 7 table 6.correlation matrix of regression coefficients intercept trend sin(12.14371) cos(12.14371) intercept 1 0.869221 -0.083666 0.005182 trend -0.9 1 0.095877 0.003687 sin(12.14371) -0.1 0.095877 1 -0.000566 cos(12.14371) 0.01 0.003687 -0.000566 1 table 1.sales of cement 2010 to 2017 month/year 2010 2011 2012 2013 2015 2016 2017 2018 jan 365 420 410 511 630 630 750 735 feb 345 380 430 525 650 680 640 743 marc 290 392 400 480 590 700 690 700 apr 267 380 390 430 520 650 710 670 may 280 365 360 410 530 560 630 660 jun 275 320 355 420 500 570 610 625 jul 280 300 340 490 580 550 680 630 aug 320 360 390 480 530 580 630 640 sep 350 390 430 520 640 640 680 675 oct 390 420 490 540 635 690 725 730 nov 430 470 512 560 690 730 765 760 dec 450 480 560 590 700 745 780 800 table 7.forecasts sales of cement 2010 to 2017 month/year 2010 2011 2012 2013 2015 2016 2017 2018 jan 362 417.4384 472.2456 526.7375 634.72 688 741 741 feb 337 393.712 450.1702 506.459 618.43 674 729 729 marc 306 363.18 420.0715 476.9838 590.76 648 704 704 apr 279 334.9653 390.9591 447.1585 560.08 617 674 674 may 264 317.5843 371.5844 425.921 535.55 591 646 646 jun 266 316.7174 368.1497 419.9612 524.72 578 631 631 jul 285 333.7216 382.6842 431.9696 531.56 582 633 633 aug 318 365.2758 412.5133 459.9327 555.41 604 652 652 sep 358 404.2496 450.9583 497.6603 591.16 638 685 685 oct 394 441.5705 489.0851 536.4061 630.57 677 724 724 nov 419 468.5985 518.0427 567.1571 664.46 713 761 761 dec 427 479.3884 531.3807 582.9934 685.09 736 786 786 ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 8 table 3. forecasts errors of sales of cement 2010 to 2017 month/year 2010 2011 2012 2013 2015 2016 2017 2018 jan 2.67 2.56165 -62.24556 -15.73755 4.7183 -58 8.72 6.28 feb 7.89 13.71204 -20.1702 18.54102 31.57 5.93 -89 13.6 marc -16 28.82 -20.0715 3.016177 0.7581 52.4 -14 4.28 apr -12 45.03472 0.9590976 -17.15847 40.076 33.3 36.5 3.51 may 16.1 47.41569 -11.58441 -15.92099 5.5459 -31 -16 13.7 jun 9.34 3.282614 -13.14965 0.03875451 -24.72 -7.7 -21 5.97 jul -5.1 33.72165 -42.6842 58.03038 48.437 -32 47.4 2.59 aug 1.81 5.275815 -22.51331 20.06731 25.413 -24 -22 11.9 sep -7.5 14.24964 -20.95833 22.33971 48.843 1.99 -5 9.96 oct -3.8 21.57052 0.914923 3.593923 4.4325 12.5 0.74 5.74 nov 11.2 1.401448 -6.042702 -7.157125 25.542 17.3 4.36 0.64 dec 23 0.611624 28.61931 7.006566 14.913 9.42 -5.7 14.3 fig.3.predicted and actual values values against time 0 100 200 300 400 500 600 700 800 900 1 7 1319253137434955616773798591 actual value predicted value ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 9 fig.4.graph of residuals against time the plot of residuals against time shown in fig.4 demonstrates that the errors are random in nature, and the plot of actual and predicted values against time shown in fig.3. is an evidence that that the actual and predicted values are reasonably close to each other. fig.5.plot of autocorrelation functionsof errors(acf). fig .6. plot of partial autocorrelation coefficients(pafc) -100 -50 0 50 100 0 50 100 150 residual residual ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 10 table 4.autocorrelations of c6 (0,0,0,0,1) ───────────────────────────────────────────────── lag correlation lag correlation lag correlation lag correlation 1 0.017623 11 -0.206787 21 0.048251 31 -0.009072 2 -0.198659 12 0.126689 22 -0.023265 32 0.103056 3 -0.033041 13 -0.023120 23 -0.068452 33 0.137523 4 0.037400 14 -0.141602 24 0.078515 34 -0.077645 5 -0.024658 15 -0.041887 25 -0.061904 35 -0.112095 6 0.096889 16 0.149367 26 -0.114473 36 0.028800 7 0.186567 17 -0.122003 27 0.067000 37 0.010588 8 0.027944 18 -0.154647 28 0.087605 38 -0.094329 9 0.019432 19 -0.086806 29 -0.046813 39 0.058022 10 -0.094473 20 -0.017866 30 -0.097414 40 0.082354 significant if |correlation|> 0.204124 table 5.partial autocorrelations of yt (0,0,0,0,1) ──────────────────────────────────────────── lag correlation lag correlation lag correlation lag correlation 1 0.017623 11 -0.210698 21 0.062361 31 -0.099679 2 -0.199031 12 0.092510 22 -0.052010 32 0.070056 3 -0.026389 13 -0.178303 23 -0.026525 33 0.106009 4 -0.001082 14 -0.177207 24 0.089137 34 -0.127389 5 -0.038603 15 -0.094584 25 -0.089422 35 0.004287 6 0.108780 16 0.074532 26 -0.020151 36 -0.062159 7 0.181286 17 -0.082308 27 0.110811 37 -0.065611 8 0.067983 18 -0.036400 28 -0.087167 38 -0.049178 9 0.109374 19 -0.129148 29 -0.030008 39 -0.049802 10 -0.075281 20 -0.000654 30 -0.124178 40 -0.092858 significant if |correlation|> 0.204124 the results from both tables 5 and 6 show that that the pac f,and acf have absolutes values that are less than the table value of 0.204124 as indicated by the software.this further confirms the adequacy of the adopted model.the plots of acf and pacf shown in figures 4 and 5 each did not exhibit spike at any lag,demonstrating that the correct model has been obtained. conclusion. the paper has explained briefly the difference between spectral analysis and harmonic analysis given the results in each case using an example. it has also shown that to carry out harmonic analysis, spectral analysis must be performed to obtain significant frequency that will be used to get an appropriate model. ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 11 references 1.david r. brillinger ((2 numberz001) time series data analysis and theory.holden day.inc,san francisco. 2.chris hintz( 2019 ) .number cruncher statistical system (ncss 2019) 3. ijo international journal of mathematics volume 03 |issue 01 | january 2020 www.ijojournals.com 12 word bookmarks 12partial_autocorrelations_of_c6_(0,0,0, 12autocorrelations_of_c6_(0,0,0,0,1)12/1 modeling and analysis of the interaction of neutral and protester populations: a competing species model a. kazmierczak department of computer science oklahoma state university stillwater, ok akazmie@okstate.edu abstract the rise of radicalized terrorist groups, such as the islamic state of iraq and syria (isis), throughout the world have brought concern, debate, and contention to the modern world. the recruitment strategies of terrorist networks are global and are no longer concentrated in a particular location. in this paper, we present a dynamical model of the interaction and recruitment between a non-radicalized or neutral and radicalized population. the formulation is based on models of interactions between competing species [3] type dynamics. an exploration of the long term dynamics and stability of homogeneous equilibrium solutions and their stability is given. the paper is given in two parts. part one analyzes the current populations. part two analyzes the situation when an additional number of radicals are introduced into the radicalized population. keywords: terrorism, competing species model, equilibrium solutions, stability at equilibrium solutions. mathematica subject classification: 62j12, 62g99 computing classification: i.4 1. introduction protestorsare not a new phenomena. however, there is a marked and exponential increase in the frequency of protests since the inception of the civil rights act. protestors can wreak havoc and spread fear and panic to native citizens. in addition, the strength and presence of protestor activities create emigration issues. consequently, countries are faced with extremely difficult, complex, and contentious political and social decisions on the activities that cause protest situations. despite these ongoing protests, there is not much literature that takes a dynamical systems approach to understanding the spread of terrorism, at a population. our primary objective is to bridge the gap. in our framework, we let n represent the neutral population. the protestor is denoted by p: p can be viewed as the protestor population of a certain situation. this paper is a first step in providing a mathematical modeling framework to study the evolution and interaction between this cop and protestor population. the protestor population is modeled by standard population growth models. also, we consider the addition to the protestor population from increased protestors. the paper is organized as follows. in section two, we develop and analyze the time-dependent autonomous protestor ordinary differential ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 16 mailto:akazmie@okstate.edu equation (ode) modeland the effect of a current populations. we examine the equilibrium solutions, the stability of the equilibrium solutions and investigate the dynamics numerically. in section three, we consider the situation when more protesters are in the system.we examine the equilibrium solutions, the stability of the equilibrium solutions and investigate the dynamics numerically for this situation also. in section four, we consider the scenario when the protestor population declines. in section 5 we present ur conclusions. 2. neutral protester (n, p) ode model consider the mathematical model n = a1n/(1+d1c) – anrnp/(1+d2n) – b1n 2 = 0 = fn(n, p) (1) p = a2p/(1+d3n) anrnp/(1+d2n) – b2p 2 = 0 = fr(n, p) (2) the populations n(t) and p(t) represent the populations of the neutral and protester populations.. the parameters are all assumed to be positive and their descriptions are given in table 1a. table 1a: list of parameters used in the differential equation model symbols meaning a1 growth rate of the protestor population a2 growth rate of the police population b1 population loss in n due to intra-species competition and natural mortality b2 population loss in p due to intra-species competition and natural mortality anr maximum per capita loss in n due to recruitment by protester groups d1 measures the effectiveness of n in disrupting the growth rate of n d2 measures the resilience of n to recruitment strategies by p d3 measures the effectiveness of nin disrupting protester activities in the case of di = bi = 0, the mathematical model becomes similar to the competing species model. the parameters di influence the carrying capacity of the individual populations. for instance, if d1>> 1 then the growth rate of n is reduced. this is interpreted as: a highly effective protester population can greatly hinder the growth rate of n. the growth rate of the protester population depends on the successful recruitment from the neutral. notice, that if d2>> 1 then the recruitment by p is small, also, if d3>> 1, new protester are introduced into the protester population is smaller. the values chosen for the variables in this model are listed in table 1b. table1b: values of parameters a1 a2 b1 b2 anr d1 d2 d3 2 2 0.5 0.5 2 2 2 3 2.1neutral protester (n, p) ode model consider the mathematical model fn(n. p) = ( a1/(1+d1p) – anrp/(1+d2n) – b1n) n = 0 (3) fr(n, p) = ( a2/(1+d3n) –anrn/(1+dpn) – b2p ) p = 0 (4) ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 17 since this system is nonlinear, the first step is linearization using the jacobian. the jacobian for this system is defined as │ ∂f/∂n ∂f/∂p │ j = │ │ │∂g/∂n ∂g/∂p │ taking the partial derivatives, simplifying and using the values in table for the parameters, the jacobian becomes. │2/(1+2p)-2p/(1+2n)^2-n -2/(1+2p)^2-2n/(1+2n) │ j = │ │ │ -6p/(1+3n)^2-2p/(1+2n)^2 2/(1+3n)-2n/(1+2n)-p │ 2.2 equilibrium points using maple cas, on (3) and (4) weobtained the following real valued equilibrium points: {n = 0., p = 0.}, {n = 0., p = 4.}, {n = 4., p = 0.}, {n = .4301871556, p = .8213492010}, {n = -.4311081397, p = -1.121275136}, {n = -.4346164212, p = .1299378971}, {n = -3.952306486, p = -2.658090053} 2.3analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. substituting equilibrium points into the jacobian and solving for eigenvalues, we get the results in table 2. ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 18 2.4summarization table 2 summarizes the results for the current population levels. table 2 – results for current population levels equilibrium point eigen values node type stability {n = 0., p = 0.}, 2.00, 2.00 repelling unstable {n = 0., p = 4.} -44/9+(2/9)*sqrt(185), -44/9-(2/9)*sqrt(185) attracting asymptotically stable {n = 4., p = 0}, -2, -86/117 attracting asymptotically stable {n = .4301871556, p = .8213492010}, .757984137794684, -1.31667584619468 saddle unstable {n = -.4311081397, p = -1.121275136}, 124.789757665452, -7.28136719345222 saddle unstable {n = -.4346164212, p = .1299378971} -6.620132656550+9.18652446854370*i, -6.620132656550-9.18652446854370*i attracting spiral asymptotically stable {n = -3.952306486, p = -2.658090053} 3.45507685676904, 1.47441380823096 repelling unstable 3. growth of the protester population in this section, we consider the situation where a there is a 25% increase in the protester population. the mathematical model now becomes fn(n, p) = ( a1/(1+d1(1.25p)) – anr(1.25p)/(1+d2n) – b1n ) n = 0 (5) fr(n, p) = ( a2/(1+d3(n) anrn/(1+d2n) – b2(1/+1.25p ) (1.25p) = 0 (6) using the maple cas, on (5) and (6) we obtained the following real valued equilibrium points {n = 0., p = 0.}, {n = 0., p = 3.200000000}, {n = 4., p = 0.}, {n = .3156552235, p = 1.024389733}, {n = -.4403859177, p = 1.855697361}, {n = -.4325472689, p = -.4910205568}, {n = 1.304975560, p = -.5057028313}, {n = -5.192142042, p = -1.990030373} ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 19 3.1analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. 3.2 summarization table 3 summarizes the results for an increased police population level. table 3 – results for increased protesterpopulation levels equilibrium point eigen values type of node stability (n = 0., p = 0.) 2.00, 2.00 repelling unstable (n = 0., p =3.200000000) -6.31260878122594, -1.01712094877406 attracting asymptotically stable (n = 4., p = 0.) -2, -86/117 attracting asymptotically stable (n = .3156552235, p = 1.024389733) -1.60703900385485, .793359729354847 saddle unstable (n = -.4403859177, p = 1.855697361) -249.390466169796, -11.5240335472042 attracting asymptotically stable (n = -.4325472689, p = -.4910205568) 82.9709967044000+730.356706992694*i, 82.9709967044000-730.356706992694*i repelling unstable (n = 1.304975560, p = -.5057028313) -156.658811051449, -19.7304049866513 attracting asymptotically stable (n = -5.192142042, p = -1.990030373) 4.53052768941096, .781943173589040 repelling unstable 4. decline of the protester population in this section, we consider the situation where there is a 25% in the protester population. the mathematical model now becomes fn(n, p) = ( a1/(1+d1(0.75p)) – anr(p)/(1+d2(n) – b1(n ) n = 0 (7) fr(n, p) = ( a2/(1+d3(n)) anrn/(1+d2 n) – b2(0.75p) ) (0.75p) = 0 (8) using the maple cas, on (7) and (8) we obtained the following real valued equilibriumpoints: {n = 0., p = 0.}, {n = 0., p = 5.333333333}, {n = 4., p = 0.}, {n = .4301871556, p = 1.095132268}, {n = -.4311081397, p = -1.495033515}, { {n = -.4346164212, p = .1732505294}, {n = -3.952306486, p = -3.544120071} ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 20 4.1 analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable . 4.2 summarization table 4 summarizes the results for a decreased protester population. table 4 _ results for decreased protestor population equilibrium point eigen values type of node stability (n = 0., p = 0.), 2, 2 repelling unstable (n = 0., p = 5.333333333) -10.5817315235956, -3.24683990940437 attracting asymptotically stable (n = 4., p = 0.) -2, -86/117 attracting asymptotically stable (n = .4301871556, p = 1.095132268) .561198866339937, -1.68177931653994 saddle unstable (n = -.4311081397, p = -1.495033515) 166.047756962696, -8.18559401169599 saddle unstable (n = -.4346164212, p = .1732505294), -9.22574755770000+9.57258822194460*i, -9.22574755770000-9.57258822194460*i attracting asymptotically stable (n = -3.952306486, p = -3.544120071) 3.45408293768524, 2.53347890031476 repelling unstable 5. conclusions in this paper we modeled and analyzed the interaction of protestor and neutral populations. a comparison of the results in table 2, table 3,and table 4 seem to indicate that no matter the relative sizes of the populations, there will besome level of instability in the system. references: 1. "thousands march against nuclear power in tokyo". usa today. september 2011. 2. st. john barned-smith, "how we rage: this is not your parents' protest," current (winter 2007): 17-25. 3. adam roberts, introduction, in adam roberts and timothy garton ash (eds.), civil resistance and power politics: the experience of non-violent action from gandhi to the present, oxford university press, 2009, pp. 2-3, where a more comprehensive definition of "civil resistance" may be found. 4. daniel l. schofield, s.j.d. (november 1994). "controlling public protest: first amendment implications". in the fbi's law enforcement bulletin. retrieved 2009-12-16. ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 21 https://www.usatoday.com/news/world/story/2011-09-19/japan-anti-nuclear-protest/50461872/1 https://en.wikipedia.org/wiki/adam_roberts_(scholar) https://books.google.com/books?id=bxoqkrce7uuc&dq=civil+resistance+and+power+politics&source=gbs_navlinks_s https://books.google.com/books?id=bxoqkrce7uuc&dq=civil+resistance+and+power+politics&source=gbs_navlinks_s http://www.thefreelibrary.com/controlling+public+protest%3a+first+amendment+implications.-a016473804 http://www.thefreelibrary.com/controlling+public+protest%3a+first+amendment+implications.-a016473804 https://en.wikipedia.org/wiki/federal_bureau_of_investigation https://en.wikipedia.org/wiki/fbi_law_enforcement_bulletin 5. kruszewski, brent baldwin, jackie. "why they keep fighting: richmond protesters explain their resistance to trump's america". style weekly. retrieved 29 march 2017. 6. global nonviolent action database 7. dynamics of collective action project 8. ratliff, thomas (2014). "practicing the art of dissent: toward a typology of protest activity in the united states". humanity & science. 38 (3): 268–294. 9. mcgrath, ben (november 13, 2006). "holy rollers". 10. "critical mass london". urban75. 2006. 11. "pittsburgh critical mass". 12. "critical mass: over 260 arrested in first major protest of rnc". democracy now!. august 30, 2004. 13. seaton, matt (october 26, 2005). "critical crackdown". london: the guardian. retrieved may 22, 2010. 14. rosi-kessel, adam (august 24, 2004). "[*bcm*] hong kong critical mass news". 15. https://www.flickr.com image of black bloc members during iraq war protest in washington, d.c., march 21, 2009. 16. d. parvaz, iran's silent protests 17. adam roberts and timothy garton ash (eds.), civil resistance and power politics: the experience of non-violent action from gandhi to the present, oxford: oxford university press, 2009. isbn 978-0-19-955201-6.[1] 18. newman, lily hay. "how to use social media at a protest without big brother snooping". wired. retrieved 2017-02-09. 19. deseret morning news, 13 nov. 2007 issue, p. e3, coverage of protests hurts firms, cornell-y. study says, angie welling ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 22 http://www.styleweekly.com/richmond/why-they-keep-fighting-richmond-protesters-explain-their-resistance-to-trumps-america/content?oid=2846422 http://www.styleweekly.com/richmond/why-they-keep-fighting-richmond-protesters-explain-their-resistance-to-trumps-america/content?oid=2846422 http://nvdatabase.swarthmore.edu/ http://www.stanford.edu/group/collectiveaction/cgi-bin/drupal/node/3 http://www.newyorker.com/printables/fact/061113fa_fact http://www.urban75.org/photos/critical https://en.wikipedia.org/wiki/urban75 http://pghcriticalmass.org/ http://www.democracynow.org/article.pl?sid=04/08/30/1453256 https://www.theguardian.com/g2/story/0,3604,1600570,00.html http://www.bostoncriticalmass.org/pipermail/bostoncriticalmass/2004-august/000146.html https://www.flickr.com/ http://www.aljazeera.com/news/middleeast/2011/02/2011220125132363934.html https://en.wikipedia.org/wiki/adam_roberts_(scholar) https://en.wikipedia.org/wiki/timothy_garton_ash http://www.oup.com/uk/catalogue/?ci=9780199552016 http://www.oup.com/uk/catalogue/?ci=9780199552016 http://www.oup.com/uk/catalogue/?ci=9780199552016 https://en.wikipedia.org/wiki/international_standard_book_number https://en.wikipedia.org/wiki/special:booksources/978-0-19-955201-6 https://books.google.com/books?id=bxoqkrce7uuc&dq=civil+resistance+and+power+politics&source=gbs_navlinks_s https://www.wired.com/2017/01/use-social-media-protest-without-big-brother-snooping/ https://www.wired.com/2017/01/use-social-media-protest-without-big-brother-snooping/ https://en.wikipedia.org/wiki/deseret_morning_news procrastination of students in working mathematics lina rihatul hima, lilin nur indah sari universitas nusantara pgri kediri linarihatul@unpkediri.ac.id abstract --this research is motivated by the existence of one behavior that can have a negative impact that harms students. one of them is manifested in the form of doing the assignments given by the teacher not on time. the tendency to delay completing tasks by doing other activities that are not useful results in tasks not being completed on time in completing math tasks, which is commonly referred to as procrastination. based on these problems, the formulation of the problem in the study is (1) how is academic procrastination in doing math assignments for male students? (2) how is academic procrastination in doing math assignments for female students? (3) is there a difference between the academic procrastination of male students and female students?. this study uses a quantitative method with a research population of class xi high school. the research sample was taken from 10 classes. there are 250 students in 7 classes, 120 male students and 130 female students. this research was carried out by distributing questionnaires to students, a questionnaire containing 40 questions. the results of this study are (1) academic procrastination in doing math assignments for male students is classified as moderate with a percentage of 35.53%. (2) academic procrastination in doing math assignments for female students is low with a percentage of 33.33%. (3) there are differences in academic procrastination in doing math assignments between male and female students. the conclusion of this study is that students' academic procrastination in doing math assignments between male and female students are at different levels. one of the reasons for the difference in procrastination is the motivation to learn mathematics. keywords: academic procrastination, mathematics tasks introduction article 31 paragraph (1) of the 1945 constitution states that every indonesian citizen has the right to education. in article 2, pip (regulation of the minister of education and culture of the republic of indonesia concerning the smart indonesia program), the government aims to support the implementation of universal secondary education or pilot 12-year compulsory education (pip). where this program requires indonesian children to get compulsory education starting from the elementary school, junior high school, high school level. of the three levels of education above, the most important level of education is senior high school. this is because high school is the final level of formal education. in law no. 20 of 2003 concerning the national education system, article 14 states that secondary school is one of the levels of formal education in addition to basic education and higher education (sisdiknas, 2003). article 18 paragraphs 1 to 3 of the national education system states that, secondary education is a continuation of basic education, secondary education consists of general secondary education and vocational secondary education, secondary education is in the form of senior high school, vocational high school. high school is one of the educational institutions that aims to support the realization of national education goals. senior high schools are held to increase competitiveness in the face of continuing levels, namely higher education levels in all fields. in the teaching and learning process at the high school level, it does not only get explanations and materials from the teacher, ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 9 but the teacher gives assignments to students and demands students to do them. students as subjects who study in high school will not be separated from learning activities, doing assignments from teachers and other activities. many school activities are carried out by students in addition to studying and doing assignments given by the teacher, there are also some students who become activists who take part in extra activities in school, so it takes the ability to manage good time so that all activities can run well and in balance. between one another. the fact that it was found that not all high school students have the ability to manage time well (jannah and muis, 2014), especially this usually happens to students who take part in activities or organizations both at school and outside of school. often face school assignments reluctant or lazy to do it. students who delay assignments will tend to do it in a hurry. procrastination is one of the behaviors that can have a very negative impact on oneself. this can be realized in the form of doing the assignments given by the teacher on time. the tendency to delay starting to complete tasks by doing other activities that are not useful so that the task becomes slow, does not finish on time, and is often late is called procrastination. people who do delaying behavior are called procrastinators (jannah and muis, 2014: 2). while academic procrastination can also be interpreted as an attempt to complete academic tasks but in a period that is not in line with expectations. the behavior of procrastinating work related to academics in psychology is termed academic procrastination. academic procrastination itself occurs because of the irrational beliefs that students have. irrational beliefs are beliefs that in their way of thinking instill suspicion. regarding procrastination, gender plays an important role. gender is the difference in roles, functions, status and responsibilities of men and women as a result of socio-cultural construction that is embedded through the process of socialization from one generation to the next. thus, gender is the result of an agreement between humans that is not natural (puspitawati, 2012: 1). we know that there are biological and social differences, they also have differences in the way they think and solve the problems they face. moreover, in the level of procrastination, there may be differences between the two. dagun in pramesti (2014) states that men and women have their respective advantages, men excel in the visual spatial field while women excel in the verbal field. from the results of several studies, it proves that someone doing academic procrastination is caused by several factors, and every individual must also have their own reasons for doing so. incidents like this of course also experienced by every school. likewise, what happened in high school, in this study the researchers focused on class xi high school students. this school is one of the educational institutions that highly values success in learning, as evidenced by the fact that its students qualify for several state universities which can be said to be favorites, although not so many, besides that this school has also received many achievements in the non-academic field. in this study, the researcher took class xi because of this class. in addition, the researcher also found a case when conducting observations in high school, that class xi students were not ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 10 punctual in collecting assignments. the delay in collecting the assignments indicated that there was procrastination in class xi high school students. after this research is completed, the researcher hopes that students can develop a sense of self-awareness in completing assignments, influencing teacher performance in teaching students, and increasing learning motivation. motivation is also expected to be given with a larger portion both from teachers, parents, the environment around students or the students themselves. so all of that can be increased to the next class so that the procrastination level of students is reduced or even non-existent. based on the background above, in doing procrastination each student must have different reasons. research method based on the problems and objectives to be achieved, this research uses a quantitative approach. in this study, the researcher wanted to compare students' procrastination in doing math assignments in terms of gender by conducting a comparative study. these independent variables include: (a) male students, which are expressed in l, (b) female students, which are expressed in p. in this study, the dependent variable is the tendency of academic procrastination behavior, which is expressed in y. this type of research technique is comparative. this study uses a quantitative research method approach. this study uses a comparative model between two independent samples, i.e. the samples are strictly separated from each other where one sample member is not a member of the other sample. this research was carried out in a high school in class xi with approximately 25 students in each class. the population in this study were all students of class xi high school from 10 classes with a total of 250 students. this study uses a research instrument in the form of a closed questionnaire, where the respondent only needs to choose the answer from the question or statement that has been stated in the questionnaire according to his situation. the instrument is an academic procrastination questionnaire. for scoring the questionnaire, the researcher used a likert model scale with 4 alternative answers because the researcher wanted students to have an opinion on the level of agree and disagree. the next instrument is an academic procrastination questionnaire. the procrastination scale in this study refers to the academic procrastination theory of ferrari in (ghufron and risnawita, 2012: 158). each question item in this questionnaire is given 4 alternative answers, namely: strongly agree, agree, disagree, strongly disagree. for scoring guidelines as follows: table 1 scoring guidelines answer item favourable unfavourable strongly agree 4 1 agree 3 2 disagree 2 3 ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 11 strongly disagree 1 4 validation of the instrument by testing the validity and reliability. the process of data analysis in this study includes normality test, homogeneity test and hypothesis testing. the procedures for collecting data using the lifting method are preparation and implementation.the following decision norms are used to answer the research hypothesis. a. the level of academic procrastination in doing math assignments for male class xi students can be categorized as very high, high, middle, low, very low. b. the level of academic procrastination in doing math assignments for female class xi students can be categorized as very high, high, middle, low, very low. to find out whether there is a significant difference or not in the level of academic procrastination, which is viewed from gender as follows. because the statistical test uses an independent sample t test, the conditions are as follows. a. if a significance > 0.05 is obtained, then ho is accepted, which means that there is no difference in the level of academic procrastination in doing math assignments between men and women. b. if a significance <0.05 is obtained, then ho is rejected, which means that there are differences in the level of academic procrastination in doing math assignments between men and women. research results and discussion research result 1. analysis of academic procrastination in doing mathematics tasks for class xi male students after testing the validity and reliability of the instrument, then removing invalid and unreliable item items, the researcher obtained a table of interpretations of the male procrastination score conversion. table 2 guidelines for conversion of male values conversion guidelines calculation result mean+(1,5x sd) 79,41 + (1,5 x 11,062) 96,003 mean+(1,5 x sd) 79,41 + (0,5 x 11,062) 84,941 mean-(1,5 xsd) 79,41 (0,5 x 11,062) 73,879 mean-(1,5 xsd) 79,41 (1,5 x 11,062) 62,817 ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 12 with the conversion guidelines above, it can be grouped into the following categories: table 3 interpretation of male procrastination score conversion score category x1≥ 96 very high 84≤ x1≤ 95 high 73≤ x1≤ 84 middle 62≤ x1≤ 73 low x1≤ 62 very low from table 3 we can use it to find out how many frequencies and percentages of each category are in male procrastination. table 4 number of respondents in each category of male procrastination kategori frequency percentage very high 6 4,42 % high 35 31,23 % middle 43 35,53 % low 27 21,37 % very low 9 7,45 % total 120 100 % based on table 4 it can be concluded that male students who have very high procrastination are 4.42%, high category is 31.23%, middle procrastination was 35.53%, low procrastination was 21.37% and very low procrastination was 7.45 %. so it can be concluded that academic procrastination in doing math assignments for male class xi students is in the moderate category with a frequency of 39 students with a percentage of 35.53%. this is because many male students choose the agree and disagree answers for both favorable and unfavorable statement items. 2. analysis of academic procrastination in doing mathematics tasks for class xi female students after we know about the percentage and frequency of the variable procrastination in the male gender, the researchers here will continue to calculate the percentage and frequency level of the variable procrastination in the female gender. for female gender data, the number of students is 130 students with a standard deviation of 76.62 with a standard deviation of 9.136. from the data we know above, we can create a guideline for converting values which will then be used for the conversion of women's scores. ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 13 table 5 guidelines for conversion of female's values conversion guidelines calculation result mean+(1,5x sd) 76,62 + (1,5 x 9,136) 90,324 mean+(1,5 x sd) 76,62 + (0,5 x 9,136) 81,188 mean-(1,5 xsd) 76,62 (0,5 x 9,136) 72,052 mean-(1,5 xsd) 76,62 (1,5 x 9,136) 62,916 with the conversion guidelines at table 5, it can be grouped into the following categories table 6 interpretation of female procrastination score conversion score category x1≥ 90 very high 80≤ x1≤ 89 high 71≤ x1≤ 80 middle 62≤ x1≤ 71 low x1≤ 62 very low from table 6 we can use it to find out how many frequencies and percentages of each category are in female gender procrastination. table 7 number of respondents in each category of female's procrastination kategori frequency percentage very high 8 6,67 % high 35 29,16 % middle 30 25 % low 40 33.33 % very low 7 5,83 % total 120 100 % based on table 7 it can be concluded that female students who have very high procrastination are 6.67%, high category is 29.16%, middle procrastination was 25%, low procrastination was 33.33% and very low procrastination was 5.83%. so it can be concluded that academic procrastination in doing math assignments for female class xi students is in the low category with a frequency of 40 students with a percentage of 33.33%. this is because many female students chose the agree and disagree answers for both favorable and unfavorable statement items. ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 14 3. analysis of are there differences between academic procrastination of male and female students after the data is tested for normality and homogeneity, the data will be analyzed by using the t test. this t test is assisted by spss 16.0. based on the calculation results of the independent sample t test with a significance level of 5%, it was obtained sig (2-tailed) 0.033. because sig (2-tailed) < 0.05, there is a significant difference. then ha is accepted. it can be concluded that there is a significant difference in academic procrastination in doing math tasks between men and women. interpretation of data analysis results based on the results of testing the analysis data, it can be interpreted as follows. a comparative study of academic procrastination in doing math assignments in terms of gender, it was found that based on the average score of male and female students, male students got a higher average score than female students. judging from the results of data analysis answering the formulation of problems 1 and 2 that the procrastination of male students is at a moderate level with a total frequency of 43 male students with a percentage of 35.53%, while for female students as many as 40 students with a percentage of 33.33%. based on the results of the t test analysis, sig. (2-tailed) 0.033, therefore 0.033 < 0.05 so ha is accepted. it can be concluded that there is a significant difference in academic procrastination in doing math tasks between boys and girls. hypothesis test a. submission of hypothesis 1 the hypothesis is that academic procrastination in doing math assignments for male class xi students can be categorized as very high, high, medium, low, very low. from the results of data analysis conducted by researchers on procrastination, it was stated that academic procrastination in doing math assignments for male class xi students was in the moderate category with a frequency of 43 students with a percentage of 35.53%. b. submission of hypothesis 2 the hypothesis is that academic procrastination in doing math assignments for female class xi students can be categorized as very high, high, medium, low, very low. from the results of data analysis conducted by researchers on procrastination, it is stated that academic procrastination in doing math assignments for female class xi students is in the moderate category with a frequency of 40 students with a percentage of 33.33%. c. submission of hypothesis 3 the hypothesis, "there is a difference in academic procrastination in doing math assignments between male and female" was accepted. based on the results of the independent ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 15 sample t-test, it is known that tcount is 5.358. the t distribution table is searched at = 5% : 2 = 2.5% (2-sided test) with degrees of freedom (df) n-2 or 244 – 2 = 242. with 2-sided test (t = 0.025) the results obtained for ttable of 1,971. it is explained that tcount 2.148 > ttable 1.971, so ha is accepted. then compare tcount with ttable and the probability obtained by the value of tcount < ttable and on sig. (2-tailed) that is 0.033 <0.05 then ha is accepted. it can be concluded that from the results of the 2-sided test (t = 0.025) compared to the 5% significant level, it is stated the same, which means ha is accepted. discussion academic procrastination in doing math assignments for male class xi students is classified as moderate when viewed from the number of respondents. the number of respondents was 43 with a percentage of 35.53%. this is supported by research conducted by akmal (2013: 11) which states that based on academic procrastination scores, the majority of subjects are in the medium category, namely 76.786%. judging from the level of trust, men have a lower level of trust than women in completing their learning tasks. then for academic procrastination in doing math assignments for female class xi students, it is relatively low when viewed from the number of respondents. the number of respondents was 40 with a percentage of 33.33%. this is supported by research conducted by akmal (2013: 28) which states that female students are superior in communication skills (verbal), mathematical, more motivated, organized in learning. it can be concluded that the main cause of student ability in completing school assignments is higher for female students compared to male students because female students are organized in studying or doing assignments compared to male students who are not necessarily organized like female students. from the explanation above, the researcher will discuss about the differences in academic procrastination in doing math tasks between men and women. from the results of hypothesis testing using an independent sample t test, the significance value or probability value is 0.033 <0.05. so it can be concluded that there are differences in academic procrastination in doing math assignments between male and female students. by using a gender differentiation scale, researchers can see that there are differences in procrastination between the two. the results of this study are supported by research conducted by akmal (2013) which concludes that there are differences in procrastination between female students and male students by controlling time management. female students have lower procrastination than male students. however, when compared with the research conducted by handaru (2014), this research is not appropriate, as seen from the results of handaru's research which states that female students have the same tendency and there is no difference in the level of procrastination between the two. based on the results of research supported by expert opinions and also strengthened by the consistency of the results of this study with the results of previous relevant studies, it is increasingly clear that there are differences in the level of academic procrastination in doing math tasks between men and women. ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 16 conclusion the conclusions of this study are academic procrastination in doing math assignments for male class xi students is classified as moderate in terms of the number of respondents. the number of respondents was 43 with a percentage of 35.53%.academic procrastination in doing math assignments for female class xi students is low in terms of the number of respondents. the number of respondents was 40 with a percentage of 33.33%. after analyzing using the independent sample t test, the significance value or probability value was 0.033 < 0.05, so ha was accepted. so it can be concluded that there are differences in academic procrastination in doing math assignments between male and female students. this statement has also been proven by the calculation of the average male student having a higher level of procrastination than female. reference akmal, vika elvira. 2013. perbedaan prokrastinasi akademik bedasarkan jeniskelamindenganmengontrolmanajemenwaktupadamahasiswayangkuliah sambil sambil bekerja di yogyakarta. empathy jurnal fakultaspsikologi,vol.2,no.1:12. ghufron,m.nurdanrinirisnawati.s.2010.teori-teoripsikologi.jogjakarta:ar-ruzz media. handaru,agungwahyu.2014.analisisperbedaantingkatprokrastinasiditinjaudarigender,soci o-personal,locusofcontrol,sertakecerdasanemosional: studi pada mahasiswa program studi manajemen fe unj.jurnal riset manajemen sains indonesia (jrmsi), (online), vol. 5, no. 2:247. jannah,m.,dandr.tamsilmuis.2014.prokrastinasiakademik(perilakupenundaanakademik)mah asiswafakultasilmupendidikanuniversitasnegerisurabaya.jurnalbkunesa,(online),vol.04, no.03:2. peraturanmenteripendidikandankebudayaanrepublikindonesiano.19tahun2016 tentang program indonesia pintar (online). pramesti,r.a.2014.prosesberfikirsiswadalampeneyelesaiansoalceritatentangkelilingdanlu aspersegipenjangditinjaudarigender.jurnalilmiahpendidikanmatematika,(online),vol.3,no.3 :190. undang – undang ri no. 20 tahun 2003 tentang sistem pendidikan nasional.jaringan dokumentasi dan informasi hukum badan pemeriksa keuanganrepublikindonesia(online). ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 17 predicting students’ study habits through attitude to mathematical mistakes in alimosho local government area, lagos state. by ruth folake lawal (ph.d.) femifolawal256@gmail.com;lawal.ruth@fcet-akoka.edu.ng deborah oyeronke oluwole (mrs) deboronke@yahoo.com;deboronke65@gmail.com and inioluwa damilola olaniyi-ojoawo (ms) email:inioluwaojoawo@gmail.com mathematics/statistics department, federal college of education (technical), akoka, lagos abstract discoveries enhancing societal development are sometimes products of repeated mistakes turned inventions. young students who study mathematics often face the pitfall of computational errors which may possess adverse influence on their interest in the subject. this study therefore explored the attitude of upper basic students to mistakes made during the learning of mathematics and its influence on the prospect of functional study habits capable of predicting success at current and higher levels. the cross-sectional survey involved three hundred junior secondary school students from alimosho local government area of lagos state and was guided by three research questions and two hypotheses. a validated and reliable questionnaire comprising of attitude towards mistake and study habits inventories was engaged for data collection in the study. findings from analysis of data gathered indicated that students’ attitude towards mistakes is positive with female students’ mean attitude slightly greater than male students. as well, study habits was observed to increase significantly with students’ advancement in class level and a positive attitude towards mathematical mistakes. the study findings led to the recommendation that teachers of mathematics should assist in constructive and beneficial analysis of students’ mathematical mistakes to help them develop and sustain effective study habits important for immediate and future self and societal development. keywords: study habits, attitude, mistakes, mathematics ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 1 mailto:femifolawal256@gmail.com mailto:deboronke@yahoo.com mailto:deboronke65@gmail.com mailto:inioluwaojoawo@gmail.com introduction individuals are daily faced with circumstances, concepts and situations to which a person mustrespond positively or negatively. this reaction describes the attitude of the individual to that occurrence. attitude mayalso be regarded as a positive or negative belief held by individuals which reflects opinions or feelings and sometimes manifest in behaviour (joseph, 2013).while several studies have been geared towards improvement of students’ achievement in mathematics quite few have addressed the problem of attitude in its various facets. this could be due to the fact that mathematics assessments are often directed toward the measurement of cognitive skills acquired while affective skills attract little or no attention (awofala, 2013). the indisputable importance of mathematical knowledge suggests that both cognitive and affective strengths are needed by learners to study the concepts within the subject. while few studies bother on students’ attitude towards mathematics, fewer research works focus on attitude to mathematical mistakes which cannot be overlooked because of the problemsolving nature of mathematics. although researchers (lawal, 2009;lai, 2012;rong&mononen, 2022) have studied error patterns in mathematics, students’ attitude towards these errors is seldom the point of focus. since attitude is capable of reflecting within individuals’ behaviour or learned tendencies to concepts the important of this investigation is hereby established. mathematical concepts continually evolve with development thus advancing the need for comprehension which is a basis for the study of mathematics (rushton, 2018). the level of students’ understanding of mathematical concepts and procedures is easily measured through display of accuracy and precision evident by their frequency of mathematical errors. mathematical errors can be regarded as misunderstanding of mathematical facts, procedures or terms which are obvious in students’ attempts of mathematical exercises (carter, 2013). every mistake made by students in mathematics should not be attributed to chance or ignorance because there is a possibility of carelessness by not paying enough attention to details or background gaps in the student’s knowledge of mathematics (lai, 2012). errors arise due to difficulties students experience while handling mathematical problems. these could be as a result of comprehension, procedural, measurement, presentation and transformation errors (lawal, 2009;rong &mononen, 2022) among others. one plausible way to handle these errors is ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 2 through students’ study habits which has been documented as a good predictor of students’ mathematics achievement. musa and garba (2019) explain study habits as the degree to which students engage typically denotes the degree to which students engage in consistent study inclusive of appropriate routines such as reviews of material, comprehensive note taking and conducive environment. in this study, it is meant to refer to a students’ habitual way or plan of studying encompassing both personal, formal (school) and informal (peers and siblings)reading. successful implementation of study routines by students determine how well they progress and outgrow their weakness (odiri, 2015; okesina,2019).when students’ habitual mode of study are proactive, consistent, environment friendly and health promoting, it is referred to as good study habits while actions devoid of this are bad study habits. odiri (2015) as well as ebele and olofu (2017) confirmed that study habits play a significant role in determining the quality of education and achievement of students in mathematics as students cannot grasp all the learning needed on the subject from teachers inside the class. study habits therefore subsumes both within and without classroom learning activities in mathematics. this links up with the importance of students’ attitude to mistakes whether during personal study or classroom learning processes. ranjana and kumar (2012) examined the influence of attitude towards mathematics and study habit on the achievement in mathematics at the secondary school stage and found that relationship among achievement in mathematics was most closely related with attitude towards mathematics and study habit. the regression equation obtained showed that study habit contributed more than a quarter to achievement in mathematics. this result was corroborated by ijadunola and lawal (2016) who also found out that study habits was the highest predictor of students’ mathematics achievement when compared with age and gender of students. theoretical background piaget’stheory of cognitive development (1952) includes four phases which are sensori motor (from birth to two years), preoperational (two to seven years), concrete operational (seven to ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 3 eleven years), and formal operational (twelve years and above). the phases are characterised by acquisition of knowledge through sensory and motor abilities, then language, perceptual images, and symbolic thought after which humans can engage in rational thinking through concrete objects and finally through abstractions and logical deductions. piaget opined that individuals begin to adjust to their environment (human and non-human) from birth so cognition is a product of the interaction between heredity and environment. cognitive structures change through processes known as assimilation, accommodation and equilibrium. assimilation occurs when new objects or situations are combined or incorporated into previous knowledge or structures. when a learnerin fuses a new mathematical knowledge into an already acquired one, assimilation becomes evident. during problem solving in mathematics, students often encounter problems with varying degrees of difficulties which necessitates assimilation. pitfalls are often woven into mathematical problems to help students learn, observe differences and acquire higher problem solving skills to achieve the objectives of mathematics education. these blocks of learning should then form stepping stones and not stumbling blocks to learning. accommodation is a modification of a scheme that occurs due to learners’ response to new situations (von glasersfeld, 1995). construction of knowledge therefore thrives on previous experience and development. when a learner attempts to assimilate a new concept unsuccessfully, disequilibrium occurs. the learner there after at tains equilibrium (balance) by modifying the concept in several ways to aid its assimilation. students are frequently faced with accommodation problems during mathematics instruction but need to be persistent so as to adequately interpret the new experience is a personal and uniquely understandable way to them as individuals. by implication, learners should be allowed to solve mathematical problems in different ways to allow for exploration (ojose, 2008)while allowing them to make mistakes without being punished for them but to learn from their errors. this will allow mathematics instructors to identify students’ mistakes in order to help low achievers or mathematically weak students (lai, 2012). by nature, mathematical concepts are hierarchical and cumulative, hence assimilation and accommodation are imperative in the formation of these concepts as postulated by piaget. thus, ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 4 constant practice and consistency in mathematical processes leads to unlearning, relearning and modification of already known concepts. students’ inability to form the concepts lead to repeated mistakes which may pave way for poor study habits, low achievement, lack of personal discoveries or lead to vices such as examination malpractice in the present and negative attitude to mathematics in the future. this could bring about career incapacitation or limited opportunities. the importance of study habits to the mathematics performance of students necessitates a study which proposes determinants of students’ development of functional study habits. a number of factors could make or mar students’ study habits including frequency of mistakes during mathematical problem solving, lack of textbooks and unconducive environment. these mistakes areeither persistent or incidental deviations from the right track. learners’ attitudes to these mistakes as they solve mathematical problems can either encourage them to learn more about mathematics or resultin an abandonment of mathematics. a credible cause of students’ poor achievement in mathematics is negative attitude towards the subjectarising from repeated mistakes for which they tend to avoid mathematics(regional educational laboratory northwest, 2017). to this end, the attitude of students toward mathematical mistakes should be a major concern tostake holders in the educational sector. this makes it viable to investigate the attitude of students toward mathematical mistakes as a basis for functional study habits. research questions 1. what is junior secondary school students’ attitude to mathematical mistakes? 2. what is the relationship among students’ class level, attitude towards mathematical mistakes and study habits? 3. what is the combined influence of attitude to mistakes and gender on students’ study habits? hypotheses on the basis of the stated research questions above, the following null hypotheses were formulated to guide the study: ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 5 h01: there is no significant relationship between students’ class level, attitude to mistake and their study habits. h02: there is no significant influence of attitude to mistakes and demographic variableson students’ study habits. methodology the study adopted a cross-sectional survey design centered on all junior secondary school students within the alimosho local government area of lagos state as population. this local government was selected purposively being the most populated local government in lagos state. five schools were selected by simple random sampling technique after which the stratified random sampling technique with students’ class as strata was used to select a sample of three hundred (300) students consisting of 146 male (48.7%) and 154 (51.5%) female students was drawn. the instrument for data collection is a questionnaire validated by two experts with an internal consistency of 0.60.the questionnaire is sectioned into three: biodata, attitude to mathematical mistakes inventory (atmmi) and study habits. the atmmi was adapted from leighton (2015)while the study habits inventory was adopted from charles-ogan and alamina (2014) with an internal consistency of 0.80. the atmmi is made up of 22 items set on a four-point scale of never, rarely, sometimes and always. the study habits inventory was made up of 18 items set on a similar scale with that of the atmmi. data was collected manually after due permission was obtained from the schools sampled. responses were analysed using mean, frequency, standard deviation, pearson product moment correlation and multiple linear regression with the aid of statistical package for social sciences. all hypotheses were tested at α = 0.05 ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 6 analysis of data research question 1:what is junior secondary school students’ attitude to mathematical mistakes? table 1: jss students’ attitude to mathematical mistakes by gender gender mean n std. deviation male 62.08 146 7.30 female 63.58 154 7.17 total 62.84 300 7.26 students’ attitude to mathematical mistakes as presented in table 1 shows that female students on the overall have a better attitude to mistake (m = 63.58, sd = 7.17) than male students in mathematics (m = 62.08, sd = 7.30). generally speaking, the students have a good attitude to mistakes in mathematics (m = 62.84, sd = 7.26). this mean value translates to 71.41% of the maximum points obtainable on the atmm inventory which is quite high. research question 2:what is the relationship among students’ class level, attitude towards mathematical mistakes and study habits? table 2: relationship among students’ class level, atmm and study habits class att to mistakes study habit class pearson correlation 1 sig. (2-tailed) att to mistakes pearson correlation .439** 1 sig. (2-tailed) .000 study habit pearson correlation .170* .228** 1 sig. (2-tailed) .012 .002 **. correlation is significant at the 0.01 level (2-tailed). *. correlation is significant at the 0.05 level (2-tailed). table 2 reveals the relationship among class, and attitude of junior secondary school students’ attitude to mathematics mistakes and study habits. data analysis illustrates that all relationship magnitude among the variables and study habits are positive indicating progress along same direction. the p-value for each correlation magnitude is less than 0.05 depicting statistical significance of all the relationship among the variables we therefore reject the null hypothesis and conclude that ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 7 there is a statistically significant relationship between class atmm and study habits. research question 3: what is the combined influence of attitude to mistakes, classlevel and gender on students’ study habits? table 3: linear regression of class, gender and atmmi on study habits b std. error beta t sig. 1 (constant) 31.381 5.027 6.242 .000 gender 3.378 1.100 .189 3.070 .002 class 2.045 .731 .188 2.800 .006 attitude to mistake .521 .114 .422 4.566 .000 r2 = 0.201, f(4, 296) = 13.43 the summary of regression analysis puts the adjusted r square value at 0.201 signifying that gender and attitude to mistakes in mathematics jointly account for 20.1% of the variations observed in students’ study habits. anova statistics as presented in the table reveals that the predictors class level, gender and atmmi significantly predicts students’ study habits with f(4, 296) = 13.43; p = 0.000. thus, the null hypothesis is rejected and we conclude that the influence of attitude to mistakes and demographic variableson students’ study habits is statistically significant. the standardized coefficients show that the highest influence on students’ study habits is exerted by attitude towards mathematical mistake (b = 0.422) followed by gender (b = 0.189). class level comes last withb= 0.188 discussion of findings students’ atmmi analysis by gender showed a faintly higher value for female students than males although students’ points on the inventory was adjudged as high indicating a positive attitude towards mathematical mistakes. junior secondary school students’ study habits was observed to progress as their class level advances indicating that as these students moved higher academically, their study habits increased. this could be due to their exposure to higher concepts in mathematics which leads into a more rigorous involvement in the subject. a close observation of the relationship magnitude and direction among the variables showed significant positive progress between the variables which is reflective of how students’ class ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 8 associates with their study habits in a positive manner. similarly, atmm was found to correlate momentously with students’ study habits as well. predictive investigation indicate that together, the variables explain more than one-fifth of the disparity observed in students’ study habits. the quantification of beta coefficients gave a comparative view of the influence of each variable on study habits with atmm as the highest followed by gender and lastly students’ class level. this result in is consonance with ranjana and kumar (2012) as well as ijadunola and lawal (2016) it therefore appears that atmm encourages jss students to study hardest and leads to a better achievement in the subject. this solution to dwindling mathematical proficiency will help to develop and sustain functional study habits in students so that distractions posed by phones, social media and social vices are conquered leading to national development. recommendations upon the premise of the study findings, the following recommendations are proposed 1. mathematics teachers rather than emphasize students’ mistakes should assist in constructive analysis of students’ mathematical mistakes to help them develop and sustain effective study habits important for immediate and future societal development. 2. students should be encouraged to make attempts without fear of mistakes as this will improve their study habits and aid personal discoveries and mathematics achievement. 3. a more robust exploration of students’ attitude towards mathematical mistakes and study habits should be investigated using higher class level of learners for a more comprehensive assessment of the relationship between the variables. ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 9 references awofala, a. o. a., arigbabu, a. a. &awofala, a. a. (2013). effects of framing and team assisted individualised instructional strategies on senior secondary school students’ attitudes toward mathematics, actadidacticanapocensia, 6 (1) 1-22. charles-ogan, g., &alamina, j. (2014). students’ study habits and performance in public and private schools mathematics in port harcourt area, rivers state. journal of international academic research of multidisciplinary studies, 2 (7), 258 – 265. demir, s., kilinc, m., &dogan, a. (2012). the effect of curriculum for developing efficient study skills on academic achievement and studying skills of learners. international electronic journal of elementary education, 4(3), 427-440. dorko, a. (2019). generalization, assimilation, and accommodation. the mathematics educator, 28 (2) 33–51. ijadunola, k. t. &lawal, r. f. (2016). age, gender, study habits and students’ performance in mathematics. akoka journal of pure and applied science education, 14 (1), 25 – 35. joseph, g. (2013). a study on school factors influencing students’ attitude towards learning mathematics in the community secondary schools in tanzania: the case of bukoba municipal council in kagera region.retrieved from http://repository.out.ac.tz/919/. kibrislioglu, n. (2015). an investigation about 6th grade students’ attitudes towards mathematics. procedia-social and behavioral sciences, 186, 64-69. https://doi.org/10.1016/j.sbspro.2015.04.024. lai, c. f. (2012).error analysis in mathematics. behavioural research and teaching, university of oregon. lawal, r. f. (2009). an investigation into the problems encountered by senior secondary school students in the acquisition of algebraic problem solving skills. a paper presented at the 5th national conference of school of science education, federal college of education (technical), akoka, lagos, may 13-16, 2009. leighton, j. p. (2015). developing and validating the attitudes towards mistakes inventory (atmi): a self-report measure. paper presented at the annual meeting of the national council on measurement in education (ncme), chicago, illinois. loibl, k., &rummel, n. (2014). knowing what you don’t know makes failure productive. learning instruction, 34, 74–85. mendezabal, m. j. n. (2013). study habits and attitudes: the road to academic success. open science repository education, doi: 10.7392/education.70081928. musa, d.c. &garuba a., (2019). attitude to mathematics, study habit and academic performance of selected secondary schools in makurdi metropolis. journal of advance research in mathematics and statistics, 6 (7). ojose, b. (2008). applying piaget’s theory of cognitive development to mathematics instruction. the mathematics educator, 18 (1) 26–30 okesina, a.f. (2019). causes of poor study habits of students as expressed by primary school teachers in nigeria. mimbarsekolahdasar,6(1)1-10 doi: 10.17509/mimbar-sd. v6i1.15224. piaget, j. (1952). the origins of intelligence in children. international universities press. ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 10 http://repository.out.ac.tz/919/ https://doi.org/10.1016/j.sbspro.2015.04.024 ranjana, c.&kumar d. d. (2012). influence of attitude towards mathematics and study habit on the achievement in mathematics at the secondary stage. international journal of engineering research and applications, 2(6), 192-196. regional educational laboratory northwest (2017). improving students’ attitudes and belief about mathematics: a literature summary of research-based practices and strategies. rong, l.&mononen, r. (2022). error analysis of students with mathematics learning difficulties in tibet. asian journal for mathematics education, 1(1) 52 – 65. vonglasersfeld, e. (1995). radical constructivism: a way of knowing and learning. falmer press ijo international journal of mathematics volume 5 | issue 12 | december 2022 | https://www.ijojournals.com/index.php/m/index 11 suppose that α ∈r and 0 <α <n. the fractional integral operator or potential riesz iα is for every x ∈ rn. size µ which satisfies the condition of growth, ie there are c> 0 and 0 <n ≤ d so that µ (b (x, r)) ≤ crn (2)for each ball centered on x µ) is called a non-homogeneous space. in nonhomogeneous space, fractional integr are defined as with for 0 <α <n ≤ d and x ∈ sizes then they are obtained. i. introduction r and 0 <α <n. the fractional integral operator or potential riesz iα is ) −α dy (1) n rn. size µ which satisfies the condition of growth, ie there are c> 0 and 0 ≤ d so that µ (b (x, r)) ≤ crn (2)for each ball centered on x ∈rd and has radius r> 0 then (rd, homogeneous space. in nonhomogeneous space, fractional integr rd. it can be seen that if n = d and µ are lebesgue sizes then they are obtained. boundedness of generally fractional integral operator on general morrey space lina nurhayati1, hendra gunawan2, iwan gunawan3, haryono edi hermawan4 , universitas sangga buana1, istitut teknologi bandung2, universitas langlang buana3 abstract. in this study i will discuss the limits of fractional integral operators in the homogeneous and nonhomogeneous lebesgue space, the morrey space and the general morrey space. in particular, in this study it will be proven that the fractional integral boundaries formulated in the morrey space are generally not homogeneous. evidence of integral fractional boundaries formulated in the morrey space is generally not homogeneous using the specified maximum operator properties in space and using hedberg's inequality. this evidence is an extension of hardy-littlewood-sobolev's inequality [11, 22]. my research related to boundedness of generally fractional integral operator on general morrey space as a scientific work that must be published in an international journal, as for the results i present in this journal, is the result of research r and 0 <α <n. the fractional integral operator or potential riesz iα is rn. size µ which satisfies the condition of growth, ie there are c> 0 and 0 rd and has radius r> 0 then (rd, homogeneous space. in nonhomogeneous space, fractional integral operators rd. it can be seen that if n = d and µ are lebesgue ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 1 in [5], it is proven that, if 1 and, for 0 <α <n then it is limited from lebesgue non in the wider space of the limited lebesgue space from the non space which is generally lp, φ (µ) to lq, ψ (µ). for any function f measured borel size on rd that satisfies the condition of growth (2), for 1 (0, ∞) morrey space is generally lp, || f || lp, φ (µ) <∞} with for 0 <α <n ≤ d and x ∈ rd. it can be seen that if n = d and µ are lebesgue sizes then they are obtained.in [5], it is proven that, if 1 and, for 0 <α <nthen it is limited from lebesgue non homogeneous space lp (µ) to lq (µ). furthermore, in the wider space of the limited lebesg space from the non-homogeneous morrey space which is generally lp, φ (µ) to lq, ψ (µ). for any function f measured-µ with µ borel size on rd that satisfies the condition of growth (2), for 1 ≤ p <∞ and φ: (0, ∞) → (0, ∞) morrey space is generally lp, φ {f ∈ lploc (µ): || f || lp, φ (µ) <∞} with with the function φ is a positive function where φ: (0, ∞) → (0, ∞) which must fulfill the following two conditions, 1. the function φ (r) is almost down, namely there is a cons applies φ (r) ≥ cφ (s). 2. the function rαφ (r) p almost rises, that is, there is a constant c> 0 such that for each r≤s applies rαφ (r) p ≤ csαφ (s) p. because both of these requirements must be fulfilled by the function doubling condition, namely there is a constant c> 0 such that if proposition 1 and lemma 2 below. proposition 1. suppose that ω is a non f is neutralized locally at rd, for z z |mµf(x)|pω(x)dµ(x) ≤ c |f( rd rd the above inequality is called the fefferman page 29. lemma 2. if the function φ: (0, ∞) → (0, ∞) satisfies the in [5], it is proven that, if 1 and, for 0 <α <n hen it is limited from lebesgue non-homogeneous space lp (µ) to lq (µ). furthermore, in the wider space of the limited lebesgue space from the non-homogeneous morrey space which is generally lp, φ (µ) to lq, ψ (µ). for any function f measured l size on rd that satisfies the condition of growth (2), for 1 ≤ p <∞ and ∞) morrey space is generally lp, φ (µ ) = lp, φ (rd, µ) is lp, φ (µ) = {f . rd. it can be seen that if n = d and µ are lebesgue sizes then they are obtained.in [5], it is proven that, if 1 and, for 0 <α <nthen it is limited from lebesgue non homogeneous space lp (µ) to lq (µ). furthermore, in the wider space of the limited lebesg homogeneous morrey space which is generally lp, φ (µ) to lq, ψ (µ). for µ with µ borel size on rd that satisfies the condition of growth (2), for 1 φ: (0, ∞) → (0, ∞) morrey space is generally lp, φ (µ ) = lp, φ (rd, µ) is lp, φ (µ) = lploc (µ): || f || lp, φ (µ) <∞} with with the function φ is a positive function where φ: (0, ∞) → (0, ∞) which must fulfill the 1. the function φ (r) is almost down, namely there is a constant c> 0 such that for each r r s 2. the function rαφ (r) p almost rises, that is, there is a constant c> 0 such that for each r≤s applies rαφ (r) p ≤ csαφ (s) p. because both of these requirements must be fulfilled by the function φ this function fulfills the doubling condition, namely there is a constant c> 0 such that if 2 then, for each r, s> 0. note proposition 1 and lemma 2 below. proposition 1. suppose that ω is a non-negative function and f is neutralized locally at rd, for 1 <p <∞, then there is c> 0 so that (x)|pmµω(x)dµ(x). (3) d the above inequality is called the fefferman-stein inequality and the proof can be seen in [22] lemma 2. if the function φ: (0, ∞) → (0, ∞) satisfies the doubling condition then homogeneous space lp (µ) to lq (µ). furthermore, homogeneous morrey space which is generally lp, φ (µ) to lq, ψ (µ). for any function f measured-µ with µ ≤ p <∞ and φ: (0, ∞) → φ (µ ) = lp, φ (rd, µ) is lp, φ (µ) = {f ∈ lploc (µ): rd. it can be seen that if n = d and µ are lebesgue sizes then they are obtained.in [5], it is proven that, if 1 and, for 0 <α <nthen it is limited from lebesgue nonhomogeneous space lp (µ) to lq (µ). furthermore, in the wider space of the limited lebesgue homogeneous morrey space which is generally lp, φ (µ) to lq, ψ (µ). for µ with µ borel size on rd that satisfies the condition of growth (2), for 1 (µ ) = lp, φ (rd, µ) is lp, φ (µ) = with the function φ is a positive function where φ: (0, ∞) → (0, ∞) which must fulfill the tant c> 0 such that for each r r s 2. the function rαφ (r) p almost rises, that is, there is a constant c> 0 such that for each r≤s φ this function fulfills the 2 then, for each r, s> 0. note negative function and stein inequality and the proof can be seen in [22] doubling condition then ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 2 for every r> 0 and k positive integers. based on proposition 1 and lemma 2 it can be shown that the maximum operator mµ is as for x ∈rd and f ∈ l1loc (rd), limited to lp, φ (µ) for 1 <p <∞ (see [16], page 8) stated in the following theorem. theorem 3. suppose f is integrally localized at rd, φ: (0, ∞) → (0, ∞) satisfies doubling conditions and for a c1> 0 for every r> 0 and 1 ≤ p <∞, then for a c> 0. evidence. take any f ∈ lp, φ (µ) and b (a, r) are open balls centered on a ω = χb (a, r) is a non-negative function. then according to equality (3) is obtained, |mµf(x)|pdµ(x) b(a,r) z ≤ |mµf(x)|pχb(a,r)dµ rd z ≤ c |f(x)|pmµχb(a,r)dµ rd " ∞ # z z ≤ c |f(x)|pdµ(x) + x |f( b(a,r) k=1 b(a,2k+1 for every r> 0 and k positive integers. based on proposition 1 and lemma 2 it can be shown that the maximum operator mµ is l1loc (rd), limited to lp, φ (µ) for 1 <p <∞ (see [16], page 8) stated in the theorem 3. suppose f is integrally localized at rd, φ: (0, ∞) → (0, ∞) satisfies doubling conditions and for a c1> 0 ≤ p <∞, then||mµf|| lp,φ(µ) ≤ c ||f|| lp,φ(µ) (4) lp, φ (µ) and b (a, r) are open balls centered on a ∈rd and radius r> 0 so negative function. then according to equality (3) is obtained, dµ(x) dµ(x) (x)|pmµχb(a,r)dµ(x) . +1r)−b(a,2kr)(5) for every r> 0 and k positive integers. based on proposition 1 and lemma 2 it can be shown that the maximum operator mµ is defined l1loc (rd), limited to lp, φ (µ) for 1 <p <∞ (see [16], page 8) stated in the theorem 3. suppose f is integrally localized at rd, φ: (0, ∞) → (0, ∞) satisfies doubling rd and radius r> 0 so negative function. then according to equality (3) is obtained,z ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 3 next, for based on (5) obtained, z |mµf(x)|pdµ(x)b(a,r so, got it ||f|| lp,φ(µ)maximum operator limitation mµ above is needed in proving the fractional integral operators and fractional integral operators commonly from the morrey space are generally lp, φ to the morrey space is generally for 1 <p <q <∞ with 5]. , we are estimatedmµχb(a,r)(x) as follows a,r) there formµf|| l maximum operator limitation mµ above is needed in proving the boundedness fractional integral operators and fractional integral operators commonly from the morrey space are generally lp, φ to the morrey space is generally for 1 <p <q <∞ with as follows. lp,φ(µ) ≤ c boundedness of fractional integral operators and fractional integral operators commonly from the morrey space are generally lp, φ to the morrey space is generally for 1 <p <q <∞ withp ψ = φq [3, ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 4 fractional integral operators here are generally fraction function ρ, which is a non and satisfies doubling conditions. for 0 <n fractional integrals are generally iρµ in nonhomogene lemma 4. suppose φ: (0, ∞) → (0, ∞) with lim φ (r) = ∞ and lim φ (r) = r → 0 + r → ∞ 0 and fulfills doubling conditions so for every t . (6) theorem 5. suppose φ doubling and fulfilling 1. and inequality 2. where 1 < p < q <∞, then evidence. for each x ∈rd and r> 0, we write note for i1 (x), obtained ii. discussion fractional integral operators here are generally fraction (integral) integrals using the function ρ, which is a non-negative function, namely ρ: (0, ∞) → (0, ∞) (also φ and ψ) and satisfies doubling conditions. for 0 <n ≤ d and the function ρ: (0, ∞) → (0, ∞) fractional integrals are generally iρµ in nonhomogeneous space defined as . suppose φ: (0, ∞) → (0, ∞) with lim φ (r) = ∞ and lim φ (r) = 0 and fulfills doubling conditions so for every t ∈ r, t> 0 there is r> 0 so that (6) theorem 5. suppose φ doubling and fulfilling then ||iρ µf|| q,φpq ≤ c||f|| lp,φ(µ). l (µ) rd and r> 0, we write (integral) integrals using the negative function, namely ρ: (0, ∞) → (0, ∞) (also φ and ψ) ρ: (0, ∞) → (0, ∞) ous space defined as suppose φ: (0, ∞) → (0, ∞) with lim φ (r) = ∞ and lim φ (r) = ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 5 next, for i2 (x) is obtained, by adding i1 and i2, obtained next, assuming f 6 = 0, suppose as a result, next, for i2 (x) is obtained, by adding i1 and i2, obtained next, assuming f 6 = 0, suppose 0. based on (4), . . (7) ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 6 for every x. thus obtained, z z |iρµf b(a,r) b( so, as a result, thus, it is evident that the morrey space which is generally not homogeneous. it can be seen that if the function ρ (t) = tα is chosen then for each x, y y |) = | x y | α, consequently for every x. thus obtained, iρµf(x)|qdµ(x) ≤ ||f||lq−p,φp (µ)mµf(x)pdµ(x (a,r) z ≤ c||f||ql−p,φp (µ) mµf(x)pdµ(x b(a,r) p ≤ c||f|| lp,φ(µ). q (µ) thus, it is evident that the generalized integral fractional operator is also bounded in the morrey space which is generally not homogeneous. iii. conclusion it can be seen that if the function ρ (t) = tα is chosen then for each x, y ∈ y | α, consequently x) x). generalized integral fractional operator is also bounded in the ∈rd applies ρ (| x ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 7 thus, the boundedness of the fractional integral operators that are generally formulated in the morrey space are not homogeneous resulting in the boundedness of fractional integral operators in the morrey space which are generally not homogeneous. in ad in the limitation of the fractional integral operator iα in the morrey space. next, with the selection of functions, for each f thus, if then lp, φ (µ) = lp, λ (µ). also, if selected φ (t) = whereas if dµ = dx, for lp, φ (µ) = lp, λ (rn) and for lp, φ ( µ) = lp (rn). [1] adams, d. r. dan l. i. hedberg, (1975), ”a note on riesz potentials”, duke math. j., 42, 765-778. [2] chiarenza, f dan m. frasca, (1987), ”morrey space and hardy function”, rend. mat. 7, 273 [3] eridani, (2002), ”on the boundedness of generalized fractional integral on generalized morrey spaces”, tamkang j. math. 33, 335 [4] eridani, h. gunawan dan e. nakai, (2004), ”on generalized fractional integral operators”, sci. math. jpn. 60, 539 [5] eridani, h.gunawan, (2006), ”fractional integral and generalized olsen inequalities”, itb research grant. no. 0004/ k01.03.2/ pl 2.1.5/ i. thus, the boundedness of the fractional integral operators that are generally formulated in the morrey space are not homogeneous resulting in the boundedness of fractional integral operators in the morrey space which are generally not homogeneous. in addition, if dµ = dx then it results in the limitation of the fractional integral operator iα in the morrey space. next, with the selection of functions, for each f ∈ lp, φ (rd) is obtained, thus, if then lp, φ (µ) = lp, λ (µ). also, if selected φ (t) = then lp, φ (µ) = lp (µ), whereas if dµ = dx, for lp, φ (µ) = lp, λ (rn) and for lp, φ ( µ) = lp (rn). bibliography adams, d. r. dan l. i. hedberg, (1975), ”a note on riesz potentials”, duke math. chiarenza, f dan m. frasca, (1987), ”morrey space and hardylittlewood maximal function”, rend. mat. 7, 273-279. eridani, (2002), ”on the boundedness of generalized fractional integral on generalized morrey spaces”, tamkang j. math. 33, 335-340. h. gunawan dan e. nakai, (2004), ”on generalized fractional integral operators”, sci. math. jpn. 60, 539-550. eridani, h.gunawan, (2006), ”fractional integral and generalized olsen inequalities”, itb research grant. no. 0004/ k01.03.2/ pl 2.1.5/ i. thus, the boundedness of the fractional integral operators that are generally formulated in the morrey space are not homogeneous resulting in the boundedness of fractional integral operators dition, if dµ = dx then it results in the limitation of the fractional integral operator iα in the morrey space. next, with the then lp, φ (µ) = lp (µ), whereas if dµ = dx, for lp, φ (µ) = lp, λ (rn) and for lp, φ ( µ) = lp (rn). adams, d. r. dan l. i. hedberg, (1975), ”a note on riesz potentials”, duke math. littlewood maximal eridani, (2002), ”on the boundedness of generalized fractional integral on generalized h. gunawan dan e. nakai, (2004), ”on generalized fractional integral operators”, eridani, h.gunawan, (2006), ”fractional integral and generalized olsen inequalities”, ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 8 [6] garcia-cuerva, j dan j. m . martell, (2000), ”two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous space”, departamento de matematicas, c-xv universidad autonoma de madrid 28049 madrid, spain. [7] gunawan, g., (2006), ”boundness of fractional integral operator in lebesgue space and morrey space”, penelitian program magister, institut teknologi bandung. [8] gunawan, h, (2000), ”generalized fractional integral operators and their modified versions”, department of mathematics, bandung institute of technology, bandung. [9] gunawan, h, (2003), ”a note on the generalized fractional integral operators”, j. indonesia. math. soc. 9, 39-43. [10] gunawan, h., y. sawano dan i. sihwaningrum, (2009),” fractional integral operators in non homogeneous spaces”, bull, austral. math. soc. 80, 324-334. [11] hardy, g. h dan j.e. littlewood, (1927), ”some properties of fractional integral i”, math. zeith. 27, 565-606. [12] lib, e. h dan m. loss, (1997), ”analysis”, american mathematical society. [13] morrey, c. b., (1938), ”on the solutions of quasi-linear elliptic differential equations”, trans. amer. math, soc. 43, 126-166. [14] nakai, e., (1994), ”hardy-littelwood maximal operator, singular integral operators and the riesz potentials on generalized morrey space”, math, nachr. 166, 95-103. [15] nakai, e., (2001), ”on generalized fractional integrals”, taiwanese j. math. 5, 587 602. [16] nakai, e., (2007), ”recent topics of fractional integrals”, sugaku exposition, 20. [17] nazarov, f, s. treil dan a. volberg, (1997), ”cauchy integral and calderon-zygmund operators on non homogeneous spaces”, internat. math. notices (15),703-726. [18] nazarov, f, s. treil dan a. volberg, (1998), ”weak type estimetas and cotlar inequalities for calderon-zygmund operators on non homogeneous space”, internat, math.res.notices, 463487. [19] nazarov, f, s.treil dan a. volberg, (2003), ”the tb-theorem on non homogeneous space”, acta math. 190(2),151-239. [20] p. s, herry, (2008), ”keterbatasan operator integral fraksional di ruang lebesgue tak homogen”, universitas sanata dharma yogyakarta. [21] sawano, y and h. tanaka, (2006), ”morrey space for non-doubling measure”, acta math. sinica,1, 153-172. [22] sobolev, s.l., (1938), ”on a theorem in functional analysis ”, math. sob. 46,471-497. [23] stein, e. m., (1993), ”harmonic analysis : real variable methods, orthogonality and oscilatory integrals”, princenton university university press, princenton, new jersey. ijointernational journal of mathematics (issn: 2805-413x) volume 02 |issue 07 | july 2019 www.ijojournals.com 9 word bookmarks _goback bayesian estimation of shape parameter of topp-leone dagum distribution arun kumar rao1, himanshu pandey2* 1department of statistics, mppg college, jungle dhusan, gorakhpur, india 2*department of mathematics & statistics, ddu gorakhpur university, gorakhpur, india e-mail: himanshu_pandey62@yahoo.com abstract in this paper, topp-leone dagum distribution is considered for bayesian analysis. the expressions for bayes estimators of the parameter have been derived under squared error, precautionary, entropy, k-loss, and al-bayyati’s loss functions by using quasi and gamma priors. keywords bayesian method, topp-leone dagum distribution, quasi and gamma priors, squared error, precautionary, entropy, k-loss, and al-bayyati’s loss functions. 1. introduction rasheed, n., [1] introduced the topp-leone dagum distribution. he obtained some basic statistical properties, incomplete rth moments, mean deviation from mean, and reliability measures of the distribution. the probability density function of topp-leone dagum distribution is given by                 1211 2 1 1 1 1 1 1 0 a a a f x; a x x x x ; x .                               (1) the joint density function or likelihood function of (1) is given by               11 1 2 1 1 1 n a an i i i i f x; a x x x                              1 12 2 11 1 1 1 1 1 1 n na a i i ii x exp log x                                      (2) the log likelihood function is given by               11 1 2 1 1 1 n a a i i i i log f x; nlog a log x x x                               1 12 2 11 1 1 1 1 1 1 n na a i i ii log x log x                                (3) differentiating (3) with respect to θ and equating to zero, we get the maximum likelihood estimator of θ which is given as    12 1 1 1 1 n a i i n log x                 . (4) ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 1 2. bayesian method of estimation the bayesian inference procedures have been developed generally under squared error loss function 2 l ,                  . (5) the bayes estimator under the above loss function, say, s  is the posterior mean, i.e,  s e    . (6) zellner [2], basu and ebrahimi [3] have recognized that the inappropriateness of using symmetric loss function. norstrom [4] introduced precautionary loss function is given as 2 l ,                    . (7) the bayes estimator under this loss function is denoted by p  and is obtained as   1 22 p e       . (8) calabria and pulcini [5] points out that a useful asymmetric loss function is the entropy loss     1p el p log        where ,      and whose minimum occurs at .    also, the loss function  l  has been used in dey et al. [6] and dey and liu [7], in the original form having 1p . thus  l  can written be as     1el b log ; b>0.        (9) the bayes estimator under entropy loss function is denoted by e  and is obtained by solving the following equation 1 1 e e .              (10) wasan [8] proposed the k-loss function which is given as 2 l ,                     . (11) under k-loss function the bayes estimator of θ is denoted by k  and is obtained as     1 2 1 k e e            . (12) al-bayyati [9] introduced a new loss function which is given as ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 2 2 cl ,                   . (13) under al-bayyati’s loss function the bayes estimator of θ is denoted by al  and is obtained as     1c al c e e       . (14) let us consider two prior distributions of θ to obtain the bayes estimators. (i) quasi-prior: for the situation where we have no prior information about the parameter θ, we may use the quasi density as given by  1 1 0 0 d g ; , d ,      (15) where d = 0 leads to a diffuse prior and d = 1, a non-informative prior. (ii) gamma prior: generally, the gamma density is used as prior distribution of the parameter θ given by     1 2 0g e ; .            (16) 3. posterior density under  1g  the posterior density of θ under  1g  , on using (2), is given by                                   11 1 1 12 2 11 11 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 n a an i i i i n na a d i i ii n a an i i i i i a x x x x exp log x f x a x x x x                                                                                                         1 12 2 0 11 1 1 1 n na a d i ii d exp log x                                                      12 1 12 1 1 1 1 1 1 1 0 n a i i n a i i log x n d log x n d e e d                                                           12 1 112 1 1 1 1 1 1 1 1 n a i i n d n a i log x i n d log x e n d                                             (17) ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 3 theorem 1. on using (17), we have          12 1 1 1 1 1 1 c n ac i i n d c e log x n d                          . (18) proof. by definition,    c ce f x d              12 1 112 1 1 1 1 0 1 1 1 1 n a i i n d n a i log x i n d c log x e d n d                                                          112 1 112 1 1 1 1 1 1 1 1 1 n d n a i i n d c n a i i log x n d c n d log x                                                    12 1 1 1 1 1 1 c n a i i n d c log x n d                         . from equation (18), for 1c  , we have        112 1 1 1 1 1 n a i i e n d log x                     . (19) from equation (18), for 2c  , we have          212 2 1 2 1 1 1 1 n a i i e n d n d log x                           . (20) from equation (18), for 1c   , we have      12 1 1 1 1 1 1 n a i i e log x n d                    . (21) from equation (18), for 1c c  , we have           112 1 1 2 1 1 1 1 c n ac i i n d c e log x n d                            . (22) 4. bayes estimators under  1g  from equation (6), on using (19), the bayes estimator of θ under squared error loss function is given by      112 1 1 1 1 1 n a s i i n d log x                      . (23) ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 4 from equation (8), on using (20), the bayes estimator of θ under precautionary loss function is obtained as        111 2 2 1 2 1 1 1 1 n a p i i n d n d log x                            . (24) from equation (10), on using (21), the bayes estimator of θ under entropy loss function is given by      112 1 1 1 1 n a e i i n d log x                     . (25) from equation (12), on using (19) and (21), the bayes estimator of θ under k-loss function is given by       1 2 112 1 1 1 1 1 n a k i i n d n d log x                           . (26) from equation (14), on using (18) and (22), the bayes estimator of θ under al-bayyati’s loss function comes out to be      112 1 1 1 1 1 n a al i i n d c log x                       . (27) 5. posterior density under  2g  under  2g  , the posterior density of θ, using equation (2), is obtained as                                  1211 1 12 1 1 11 2 1 1 1 1 1 1 1 1 1 2 1 1 n a a an i i i i i n a i i an i i a x x x x exp log x e f x a x x                                                                                             12 1 120 1 1 1 1 1 1 1 1 1 n a a i i i n a i i x x d exp log x e                                                                       12 1 1 12 1 10 1 1 1 1 1 1 n an i i n an i i exp log x exp log x d                                                            ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 5        12 1 1 1 1 1 12 1 1 1 1 n a i i log x n n n a i i e n log x                                                              12 1 12 1 1 1 1 1 1 1 1 n a i i n n a i log x i n log x e n                                                       (28) theorem 2. on using (28), we have          12 1 1 1 1 c n ac i i n c e log x n                            . (29) proof. by definition,    c ce f x d            12 1 12 1 1 1 1 1 0 1 1 1 n a i i n n a i log x i n c log x e d n                                                                     12 1 12 1 1 1 1 1 1 1 n n a i i n c n a i i log x n c n log x                                                         12 1 1 1 1 c n a i i n c log x n                           . from equation (29), for 1c  , we have        112 1 1 1 1 n a i i e n log x                       . (30) from equation (29), for 2c  , we have         212 2 1 1 1 1 1 n a i i e n n log x                              . (31) from equation (29), for 1c   , we have      12 1 1 1 1 1 1 1 n a i i e log x n                            . (32) from equation (29), for 1c c  , we have ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 6           112 1 1 1 1 1 1 c n ac i i n c e log x n                               . (33) 6. bayes estimators under  2g  from equation (6), on using (30), the bayes estimator of θ under squared error loss function is given by      112 1 1 1 1 n a s i i n log x                        . (34) from equation (8), on using (31), the bayes estimator of θ under precautionary loss function is obtained as        1 2 112 1 1 1 1 1 n a p i i n n log x                               . (35) from equation (10), on using (32), the bayes estimator of θ under entropy loss function is given by      112 1 1 1 1 1 n a e i i n log x                         . (36) from equation (12), on using (30) and (32), the bayes estimator of θ under k-loss function is given by       1 2 112 1 1 1 1 1 n a k i i n n log x                               . (37) from equation (14), on using (29) and (33), the bayes estimator of θ under al-bayyati’s loss function comes out to be      112 1 1 1 1 n a al i i n c log x                         . (38) conclusion in this paper, we have obtained a number of estimators of parameter of gompertz fréchet distribution. in equation (4) we have obtained the maximum likelihood estimator of the parameter. in equation (23), (24), (25), (26) and (27) we have obtained the bayes estimators under different loss functions using quasi prior. in equation (34), (35), (36), (37) and (38) we have obtained the bayes estimators under different loss functions using gamma prior. in the above equations, it is clear that the bayes estimators depend upon the parameters of the prior distribution. we therefore recommend that the estimator’s choice lies according to the value of the prior distribution which in turn depends on the situation at hand. references [1] rasheed, n, (2020): “topp-leone dagum distribution: properties and its applications”. research journal of mathematical and statistical sciences, vol. 8(1), 16-30. ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 7 [2] zellner, a., (1986): “bayesian estimation and prediction using asymmetric loss functions”. jour. amer. stat. assoc., 91, 446-451. [3] basu, a. p. and ebrahimi, n., (1991): “bayesian approach to life testing and reliability estimation using asymmetric loss function”. jour. stat. plann. infer., 29, 21-31. [4] norstrom, j. g., (1996): “the use of precautionary loss functions in risk analysis”. ieee trans. reliab., 45(3), 400-403. [5] calabria, r., and pulcini, g. (1994): “point estimation under asymmetric loss functions for left truncated exponential samples”. comm. statist. theory & methods, 25 (3), 585-600. [6] d.k. dey, m. ghosh and c. srinivasan (1987): “simultaneous estimation of parameters under entropy loss”. jour. statist. plan. and infer., 347-363. [7] d.k. dey, and pei-san liao liu (1992): “on comparison of estimators in a generalized life model”. microelectron. reliab. 32 (1/2), 207-221. [8] wasan, m.t., (1970): “parametric estimation”. new york: mcgraw-hill. [9] al-bayyati, h.n., (2002): “comparing methods of estimating weibull failure models using simulation”. ph.d. thesis, college of administration and economics, baghdad university, iraq. ijo international journal of mathematics volume 4 | issue 12 | december 2021 | http://www.ijojournals.com/index.php/m/index 8 the great gun grab a. kazmierczak t. h. e. institute 1111 e brooks st. norman ok 73071 akazmierczak1949@gmail.com abstract in recent years in the us there have been a number of mass shootings in middle schools, high schools and colleges. these shootings have cost a large number of fatalities of americans young people. after each shooting, the liberals come out with a call to ban guns, or certain types of guns. americans have been in love with their guns since the american revolution. in fact, the colonists only won because they possessed firearms. gun owners will not willingly give up their guns. in this paper, we want to look at various ways liberals can get their way and the potential consequences. keywords: gun control, confiscation, liberals, nato 1.0 introduction in recent years in the us there have been a number of mass shootings in middle schools, high schools and colleges. these shootings have cost a large number of fatalities of americans young people. after each shooting, the liberal politicians come out with a call to “ban guns”, or certain types of guns. they are pushing for new legal measure to implement gun control. americans have been in love with their guns since the american revolution. very many gun owners will not willingly give up their guns. in this paper, we want to look at various ways liberals can get their way and the potential consequences. there have already been groups, including many law enforcement agencies, that have come flat out and said “no! we will not obey those laws.” in response, the liberals have come up with the idea of confiscating guns if necessary. the same people have said the same thing “no!” the liberals’ next idea was to have law enforcement officials confiscate guns. law enforcement agencies across the country have expressed their feelings about this, “no! law enforcement will not act against the american people.” unfortunately, in a few northeastern states, laws have been passed to ban guns and gun confiscation has been done in those states. liberals are trying to pass such laws in other states. we have to expect that many local law enforcement personnel will quit police work rather than trying to force their neighbors to give up their guns. ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 1 mailto:akazmierczak1949@gmail.com the liberals’ next idea may well try to use the military to confiscate guns. there is a clause in the constitution that expressly forbids the use of american military forces to take action against us citizens. if law enforcement can’t or won’t do the job, perhaps other alternatives will be tried. try using the state militia to confiscate guns. this will be like the american civil war all over again, brother against brother. we can expect some militia members to resign rather than fight against american citizens. there is potentially one other source that could come in and confiscate americans guns. that would be to call in nato forces to carry out the confiscation. this will not be as simple as a nato truck pulling up at your door and saying “guns please”. with the american attitude that they will not give up their guns, there is a very great chance that shots will be fired between nato forces and american gun owners. many believe that the first shot fired will be the first shot of the second american revolution. let us explore how that option might work out. 2.0 gun owner – nato ode model consider the mathematical model: g = a1g/(1+d1n) agngn/(1+d2n) b1g2 = 0 = g(gn) (1) n = a2n/(1+d3g) agngn/(1+d2n) b2n2 = 0 = n(gn) (2) the populations g(t) and n(t) represent the number of gun owners and nato forces respectively. the parameters are all assumed positive and their description is given in table 1a. table 1a list of parameters symbol meaning a1 growth rate of gun owners a2 growth rate of nato forces agn maximum per capita loss in n b1 population loss in g b2 population loss in n d1 effectiveness of n in disrupting g d2 resilience of g to n d3 effectiveness of g in disrupting n the values chosen for the parameters in the model are shown table1b. ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 2 table 1b values of parameters a1 a2 agn b1 b2 d1 d2 d3 2 2 2 .5 .5 2 2 3 2.1 gun owner nato equal populations the first question to be asked is how can you tell what the population of gun owners is? because every time a gun owner buys a gun from a gun dealer, their name goes into a federal data base of registered gun owners. reconsider our original mathematical model. g = a1g/(1+d1n) agngn/(1+d2n) b1g2 = 0 = g(gn) (3) n = a2n/(1+d3g) agngn/(1+d2n) b2n2 = 0 = n(gn) (4) since these are second order equations, to solve them we need to form the jacobian and solve the jacobian for equilibrium points. the jacobian is formed as: | ∂g/∂g ∂g/∂n | j | | | ∂n/∂g ∂n/∂n | taking the partial derivatives, substituting values for parameter, the jacobian becomes: | g 4g 2n/(1+2n)2 | j | | | -6n – 2n/(1+2n) 2/(1+3g) -2n/(1+2n) –n| now we solve the jacobian for equilibrium points. we do that using the maple cas. 2.1.1 equilibrium points the real valued equilibrium points are: {g = 0., n = 0.}, {g = 0., n = 4.}, ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 3 {g = 4., n = 0.}, {g = .4891955799, n = .6319394087} {g = -.4325627635, n = -.6082709305}, {g = -.4345884397, n = .1197573734}, {g = -3.074988235, n = -2.874675564} the eigenvalues for these equilibrium points are: (0, 2) (-13/9+(1/27)*sqrt(3313), -13/9-(1/27)*sqrt(3313)) (4, 2/13) (0.547422083000000e-1+2.69265409004048*i, 0.547422083000000e-1-2.69265409004048*i) (-1.20049095525000+6.86103818723393*i, -1.20049095525000-6.86103818723393*i) (.662503816667346, -2.00886916896735) (6.48125503250000+10.0921214207669*i, 6.48125503250000-10.0921214207669*i) 2.1.2 stability in this section we use the eigenvalues to test for stability. equilibrium points eigen values node type stability {g = 0., n = 0.}, (0, 2) repelling unstable {g = 0., n = 4.}, (-13/9+(1/27)*sqrt(3313), -13/9-(1/27)*sqrt(3313))} attracting asymptotically stable {g = 4., n = 0.}, (4, 2/13) repelling unstable {g = .4891955799, n = .6319394087} (.0.547422083000000+ 2.69265409004048*i, 0.547422083000000e-12.69265409004048*i) repelling spirals unstable {g = -.4325627635, n = -.6082709305}, (-1.20049095525000+ 6.86103818723393*i, -1.20049095525000attracting spiral asymptotically stable ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 4 6.86103818723393*i) {g = -.4345884397, n = .1197573734}, (.662503816667346, -2.00886916896735 saddle unstable {g = -3.074988235, n = -2.874675564} (6.48125503250000+ 10.0921214207669*i, 6.4812550325000010.0921214207669*i) repelling spirals unstable as can be expected, the introduction of foreign troops on american soil is having a destabilizing influence. 2.2 nato forces outnumber gun owners since there may be hundreds of thousands gun owner, it may be necessary to call in a few more nato troops. fifty percent more nato may help. g = a1g/(1+d1n*1.5) agngn*1.5/(1+d2n*1.5) b1g2 = 0 = g(gn) (5) n = a2n*1.5/(1+d3g) agngn*1.5/(1+d2n*1.5) b2(n*1.5)2 = 0 = n(gn) (6) this of course changes the jacobian to: | 2/(1+3n) – 3/(1+3n) – g -6g – 3g/(1+3n) | j | | | 9n -3n/(1+3n) 3/(1+3g) – (3g + 4.5gn)/(1+3n) – 1.5n| 2.2.1 equilibrium points using the maple cas the real valued equilibrium points are: {g=0.,n=0.}, {g=0.,n=4.}, {g=4.,n=0.}, {g=0.5645952421,n=0.4242232302}, {g=-0.4331330963,n=-0.4050035932}, ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 5 {g=-0.4344789342,n=0.07995039023} {g=-3.409327263,n=-2.777186302}, 2.2.2 eigenvalues maple also gives the eigenvalues: 0, 0 -112/9+(128/27)*sqrt(7), -112/9-(128/27)*sqrt(7) 4, 0 -1.21987464895000+1.68955160788912*i, -1.21987464895000-1.68955160788912*i -9.17015494741073, 15.4304003821107 .548710843003933, -1.60074941050393 6.01711066850000+10.8450561032619*i, 6.01711066850000-10.8450561032619*i 2.2.3 stability equilibrium points eigen values node type stability {g=0.,n=0.}, 0, 0 attracting asymptotically stable {g=0.,n=4.}, -112/9+(128/27)*sqrt(7), -112/9-(128/27)*sqrt(7) attracting asymptotically stable {g=4.,n=0.}, 4, 0 repelling unstable {g=0.5645952421, n=0.4242232302}, -1.21987464895000+ 1.68955160788912*i, -1.21987464895000 1.68955160788912*i attracting spiral asymptotically stable {g=-0.4331330963, n=-0.4050035932}, -9.17015494741073, 15.4304003821107 saddle unstable {g=-0.4344789342, n=0.07995039023} .548710843003933, -1.60074941050393 saddle unstable ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 6 {g=-3.409327263, n=-2.777186302}, 6.01711066850000+ 10.8450561032619*i, 6.01711066850000 10.8450561032619*i repelling spiral unstable adding fifty percent more foreign does not seem to affect the overall stability of the system. 3.0 even more nato forces perhaps fifty percent more nato will not get the job done or will not get it done fast enough. so, double the original number of nato forces g = a1g/(1+d1n*2) agngn*2/(1+d2n*2) b1g2 = 0 = g(gn) (7) n = a2n*2/(1+d3g) agngn*2/(1+d2n*2) b2(n*2)2 = 0 = n(gn) (8) this changes the jacobian to: | 2n/(1+4n –g -8g – (16gn2 – 12gn)/(1+4n)2 | j | | | -12 4n/(1+4n) -2n 4/(1+3g) – 16gn2 – 4gn)/(1+4n)2 -2n| 3.1.1 equilibrium points the real valued equilibrium are: {g = 0., n = 0.}, {g = 0., n = 4.}, {g = 4., n = 0.}, {g = .6081201589, n = .3186790244} {g = -.4334163760, n = -.3035622451}, {g = -.4344240097, n = 0.6000496273e-1}, {g = -3.740679177, n = -2.699888708} ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 7 3.1.2 eigenvalues the eigenvalues for this are: 0, 0 -112/9+(128/27)*sqrt(7), -112/9-(128/27)*sqrt(7) 4, 0 -.846607010050000+1.71306981470163*i, -.846607010050000-1.71306981470163*i -1.85240411231455, 4.78569675231455 .474822350546164, -1.37642690614616 5.61568442150000+11.5023751979819*i, 5.61568442150000-11.5023751979819*i 3.3.3 stability equilibrium point eigen values node type stability {g = 0., n = 0.}, 0, 0 attracting asymptotically stable {g = 0., n = 4.}, -112/9+(128/27)*sqrt(7), -112/9-(128/27)*sqrt(7) saddle unstable {g = 4., n = 0.}, 4, 0 attracting stable {g = .6081201589, n = .3186790244} -.846607010050000 +1.71306981470163*i, -.846607010050000 -1.71306981470163*i attracting stable {g = .4334163760, n = .3035622451}, -1.85240411231455, 4.78569675231455 saddle unstable {g = .4344240097, n = .06000496273}, .474822350546164, -1.37642690614616 saddle unstable {g = 3.740679177, 5.61568442150000 +11.5023751979819*i, repelling spiral unstable ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 8 n = 2.699888708} 5.61568442150000 -11.5023751979819*i as we add one-hundred percent more nato forces, stability of the system remains constant, no more unstable nodes than before and no more stable nodes than before. 4.0 conclusion in this scenario where we call in nato forces to confiscate the guns owned by registered gun owners, the increased number of nato forces make no difference. this is not all that surprising. there are millions of americans that, even though they are not gun owners, would see this as an invasion by foreign troops. as more and more nato troops become involved, so will many more of the people that love america becomeinvolved. it is not inconceivable that such a war would wreak havoc on the us, and turn the us into a second rate country. 5.0 references 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et al. 2014. 110. villaveces et al. 2000. ijo international journal of mathematics volume 03 |issue 02 | february 2020 www.ijojournals.com 13 word bookmarks _goback geometry learning based on cognitive conflict in the department of mathematics universitas negeri medan hasratuddin1, kms. m. amin fauzi 2, nurhasanah siregar 3 1,2,3 lecturer at the department of mathematics, fmipa unimed medan email1: siregarhasratuddin@yahoo.com email2: amin_fauzi29@yahoo.com email3: nurhasanahsiregar@yahoo.com abstract. geometry is one of the compulsory courses for students majoring in mathematics at the state university of medan. geometry has a deductive axiomatic character, which is a science whose truth leads to previous truths. conflict is an action from the existence of a gap that arises and is felt from an information or incident. cognitive conflict occurs when students become aware of a mismatch between what is on their mind and information from outside. the concept of learning geometry occurs as a result of the information or system received by a person that causes action, interaction and reflection. thus for action to occur, it is necessary to have contextual problems that challenge as a source of conflict. with cognitive conflict, which results in an awareness of incongruity with prior knowledge, this can spur emotions or motivate students to seek the real truth. this motivation can encourage the integration of efforts to find the real truth as a result of one's rational thinking ability and will be arranged in a cognitive structure and occupy long-term memory. the purpose research is to improvement of students' mathematical reasoning abilities and creative thinking in learning geometry through cognitive conflict approaches in the mathematics. this type of research is a semi-experimental study with a two-class control and pretest-posttest experimental design. the location of this research is the department of mathematics, universitas negeri medan. while the subjects in this research were 35 students in the 2019 f pspm and 32 pspm 2019 a classes. the object in this study is a learning geometry based on a cognitive conflict approach. the research result that the role of cognitive conflict in learning geometry will be able to improve students' rational thinking and emotional intelligence. keywords: cognitive conflict, learning, geometry, rational thinking, emotional. preliminary a new paradigm of learning geometry is connecting learning and thinking and developing personality attitudes. hasratuddin (2018) said that teaching geometry now is the time to focus on thinking skills and learning reflection, interaction and development of specific thinking concepts, and developing interactive social attitudes and behavior. this becomes the basis for and consideration of changes in the geometric learning process, no longer only emphasizing the development of the cognitive realm alone, but need to involve attitudes or emotional intelligence. naturally, students have a tendency to understand the world around them. students construct their concepts as a result of their observations and investigations of the world ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 1 mailto:siregarhasratuddin@yahoo.com mailto:amin_fauzi29@yahoo.com mailto:nurhasanahsiregar@yahoo.com around them (bayer, 2016). therefore, studying geometry is a concept construction process based on observing a conflict or real world context. the existence of a conflict with someone will lead to dissatisfaction. learning through conflict is learning that raises dissatisfaction in students, because students realize that what they have is not in accordance with the real truth. naturally, students tend to nullify dissatisfaction. the desire to minimize conflict is a very strong human motivation (kang et al., 2015). this characteristic states that learning is a construction activity, where students discover their own concepts, principles or procedures for themselves. according to this principle, studying geometry is a construction activity. learners construct internally, mental representations that can concrete images, schemata, procedures, work methods at the level of abstract symbols, intuitions, contexts, schemata settlement, or through experiments. the characteristic of this construction, among others, is that students find themselves solving procedures of a contextual problem. in the framework of learning geometry, conflicts can occur if students realize that their understanding or perception of mathematical objects is incompatible with the nature of the true geeometric object.students will consciously try to eliminate this conflict by accepting the truth they should. this realization made him not repeat the mistakes he had made again. the process of eliminating conflict will strengthen the true understanding of a concept. therefore, learning through the process of eliminating conflict can lead to strong and holistic understanding for students. one alternative learning that can spur the mastery of student conceptions is to use cognitive conflict strategies, which is the application of constructivism. osborne (1993) suggests that cognitive conflict strategies have a general pattern, namely: exposing alternative frameworks, creating conceptual cognitive, encouraging cognitive accommodation. lee and kwon (2003) state that there are three phases in the cognitive conflict learning process, namely the preliminary stage, the conflict stage and the resolution stage. in the preliminary stage, the lecturer can explore the initial conceptions of students and create anomalous situations. that is, a situation that is contrary to the student's prior knowledge. in the conflict stage, the lecturer observed the student's response to the anomalous situation given. recognition of anomalous situations can be in the form of attraction or anxiety. the resolution phase, with lecturer conditioning, students will attempted resolve conflicts in cognitive structures to gain knowledge in accordance with scientific concepts. the stages of cognitive conflict-based geometry learning carried out in this study are as follows: 1. cognitive conflict orientation. in this case, the lecturer provides a conflict context that contains conceptual barriers to students. students understand and discuss in groups. 2. lecturers facilitate classical discussions. students present alternative solutions as a comparison. 3. lecturers provide opportunities for students to reflect on conflict resolution. 4. students find complete and applicable knowledge. from the cognitive conflict-based learning stage above, the problem of cognitive conflict context is the starting point of the learning process, and will end by eliminating the conflict that is presented. learning through conflict elimination will not occur if there is no conflict to condition it. so that learning through cognitive conflict can facilitate students to gain a strong understanding.therefore, to build conflict, the lecturer should create events ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 2 that are not in accordance with the student's final conception. the event in question is a symptom or situation that cannot be explained by the conception captured by the student, but can be explained by the actual concept. for example, students would know about original numbers, a = {1,2,3,4, ...} and whole numbers, c = {0,1,2,3, ...}. then, if students are asked "which has more natural numbers with whole numbers? what is their answer? also they certainly know about rectangles. then, do they know about the rectangle? learning mathematics through cognitive conflict is cognitive conflict conditioning during the mathematical process. conflict conditioning can occur when introducing concepts (as a preventive measure) or it can also occur to straighten misconceptions that occur to students. the delivery of procedural information or the provision of "drill" alone cannot achieve optimal cognitive conflict learning. a learning approach that demands a sense of dissatisfaction is more appropriate for learning through cognitive conflict. this feeling of dissatisfaction will lead to long duration retention. the emotional intelligence referred to in this research is a series of non-cognitive abilities and skills that affect a person's ability to successfully cope with environmental demands and pressures such as the ability to recognize feelings of oneself and others and interpret them, reach and generate feelings to help critical thinking and emotional development. . emotional intelligence includes interpersonal and intrapersonal intelligence. the application of cognitive conflict as a preventive action can be started by predicting student conceptions of the given topic. predictions about student conceptions can be made based on previous student experiences or the lecturers' personal experiences with the subject matter. research methods 1. this type of research is a semi-experimental study with a two-class control and pretestposttest experimental design. 2. the location of this research is the department of mathematics, fmipa unimed medan, jl. willem iskandar psr. v medan estate. 3. while the subjects in this research were 35 students in the 2019 f mathematics education study program (pspm) and 32 pspm 2019 a classes. 4. the object in this study is a learning geometry based on a cognitive conflict approach. 5. implementation of learning the learning action stage with the cognitive conflict approach to be carried out in this study are; a. conflict situation. at this stage, the lecturer presents problems in the form of challenges or conflicts to be resolved by students either individually or in groups. b. organizing learning. at this stage the lecturer organizes assignments related to problem conflicts. c. investigation guidance. at this stage the lecturer encourages students to collect information in carrying out experiments to get explanations and problem solving. presentation and sharing. at this stage the lecturer gives the opportunity to convey student ideas or solutions to the conflicts given. students report their own or group problem solving or activity results or discuss their answers to conflict solutions. process analysis and evaluation as a reflection stage. at this stage the lecturer fosters student analysis and evaluation as a reflection of thinking, so that students find concepts or knowledge. ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 3 research results and discussion a. online learning process. 1. meeting 1. learning begins with providing "online lecture rules", namely; 1)prepare equipment and special rooms for your meeting, 2) straighten your intention to learn, 3) dress politely and neatly, 4) login on time, 5) leave 10 minutes before the class meeting starts, 6) turn off the mic (mute) when not talking , 7) raise your hand when you want to talk, 8) focus and not leave the meeting, 8) do not share the meeting link with others. this thickest system recognition aims to form students who are orderly, civilized and have good emotional intelligence. in discipline 2) straighten the intention, this is harmonious in studying. this means that it is not enough for them to just study online or join online courses. if you learn, it's true, it can be done anywhere and anytime. thus, it is not wrong that ngadim makarim, minister of education and culture in 2020 said the term "independent learning". obviously, learning does not form morals or good behavior, because learning while squatting, smoking in the bathroom can also be done. however, if you are studying, it is clear that there must be a teacher, it cannot be done anywhere, you can ask questions if you don't know, and there are ethics as manners that must be maintained. this is something that must be done by teachers so that a culture of good character for students is formed. after that, the teacher / lecturer can provide useful knowledge for students. al'adaabul fauqol 'ilmu =: adab is more than science. student response. s1: about the intention. initially when asked about their answer to "learning" of course, the intention to learn this must be changed to learning, because we will provide knowledge when they have morals. this is called the partial formation of emotional intelligence. 2. second online learning. the lecturer gives a problem in the form of a conflict in mathematics study program students "what is mathematics". the purpose of giving this conflict problem is so that students have a consistent attitude towards the application and practice of mathematical values in the form of consistent, rational, honest, obedient, critical and creative thinking. (hasratuddin, 2018). as for the student response to the conflict "what is mathematics?", the researchers grouped them into 4 parts, namely; a. s1: mathematics is a branch of knowledge that deals with numbers. b. s2: mathematics is an exact science. c. s3: mathematics is solving life's problems. d. s4: mathematics is the mother of all knowledge. from the 4 student responses, at s1 a mathematical principle and value were found that mathematics is not a knowledge, but is a science that is not the result of experiments but is found (wittgenstein, 1974b). this is what makes mathematics equalized with philosophy. in this session the lecturer gave a challenge in the form of a conflict to students, namely to investigate whether the following statements were the same or different? 1) arrogant people are not smart, and 2) smart people are not arrogant. apparently, to give answers with reasonable reasons is mathematical logic. ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 4 obviously the problem does not involve numbers or numbers but it can only be solved by mathematical means. the conclusion that can be found is that mathematics does not only solve problems related to numbers. from the s2 response, it was found that in terms of mathematical measurements it is not always certain, thus mathematics is not an exact science. but mathematics must be in rules or systematics. soejadi (1997) in the book mathematical tips "the supreme judge of mathematics is systematic, while the highest judge of science is reality, the iron is burned expands". from s3's response, it was found that mathematics is a tool and means of solving all problems in general, which is the closest to being true and based on rational human reason. as such, all disciplines use mathematical principles. from s4's response, it was found that mathematics is the mother of all sciences. this is based on all fields of science that require mathematics. freuthental (1991) states that mathematics is a human activity. the point is that mathematics is used by all humans who have activities. the logical consequence is that any human being who is active must use mathematics. furthermore, people who don't use math are those who have no activity. in conclusion, people who shy away from math are better off to stay inactive. from this learning activity, the lecturer directs students to understand and live out the objectives of learning mathematics, among others; 1) math teaches you to admit when you're wrong. mathematics makes humans aware of their limitations or admits that they are often wrong. 2) to choose exact and correct words. math teaches you to choose the right and correct words. 3) to think several steps ahead. mathematics teaches you to think ahead (move on). 4) not like everyone else but in your own way. mathematics teaches you to think your way, (reasonable skepticism). 5) and never give up. mathematics teaches you to never give up. for this purpose, students are expected to understand that they are often wrong, have weaknesses and are limited in everything (above the sky there is still a sky). this will form students who do not have to be arrogant, arrogant and like to ridicule or slander others. and, it is hoped that students will have the principle of understanding means forgiving everything as a part of good emotional intelligence. then from the attitude value in goal 1, it is hoped that students will have an attitude towards goal 3, namely as a wise, wise and lucky person, they must always have the principle "today must be better than yesterday, and tomorrow the day after tomorrow must be better than today in things can choose the right and the right, according to the second goal. furthermore, in that it will always be better day by day, students will arrive at the 5th goal, 3. the third online learning is by providing conflict with the material. at the third meeting, the conflict given was "if there is a rectangle having a circumference of 24 cm, then determine the size of the rectangle that has the largest area". various responses were given by students on this problem. there are students who say that the size of a rectangle with a circumference of 24 m is 5 x 7 with an area of 35 m2. there were students who gave the answer 6 x 6, but because it was already a square, the answer was changed to 5.5 m, x 6.5. this happened because the concept of rectangles and squares was not fully understood for them. ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 5 so, this moment will be a strengthening of the square concept that lecturers must do, so that they will find a strengthening of their reasoning ability about the square concept. and immediately, students will accept this if it is given based on logic or reasoning. of course, this acceptance is part of the realm of emotional intelligence. therefore, it is not excessive if it is found that cognitive conflict-based learning will be able to improve students' reasoning abilities and emotional intelligence. so that in order to strengthen the understanding of the concept of the rectangle, the lecturer asks students to define a rectangle. there are students who define a rectangle as a rectangle that has 2 pairs of equal sides. there are also students who define a rectangle as a rectangle that has length and width, and so on. from this answer, the lecturer began to describe 4 flat buildings, as follows. then, ask students to indicate which of the flat shapes is a rectangle. student response; s1: a quadrilateral is a polygon with four sides and four corners. s2: s3: ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 6 from the answers to the three respondents above, it can be concluded that students still have doubts about the meaning of rectangles. thus the conflict will be used for discussion until it finds answers rationally.bayer, m (2016), said that learning with cognitive conflicts will be able to improve students' mathematical reasoning abilities. and, this is what is called a process of improving students' reasoning abilities. b. formative test results 1 after 3 online meetings, the researcher gave 1 formative test of 1 question, namely "if there are any triangles, right angles, equilateral and isosceles having the same circumference, then determine which triangle has the largest area! ” following are the results of the reasoning skills test obtained by 47 students of the mathematics department of fmipa unimed medan, with a range of scores of 0-100. table 1. descriptive statistics data of final learning test statistics reasoning ability low moderate high number of 12 23 12 mean 39.44 65.33 78.67 stdev 9.91 14.36 12.39 min scor 21 57 71 max scor 57 71 100 based onin the table 1. above, it can be seen that the low level student groups have not achieved optimal performance. their achievement range is 21-57, which is still below the average class achievement. whereas for the medium level group, it reached a score of 57-71, and the high-level group reached 71-100. statistically, the descriptive statistical table of the achievement scores for the student's reasoning ability shows that the average level of achievement of the moderate level group has reached the minimum completeness, which is above a score of 60. above there are 32 out of 47 people (68%) who achieved completion. this shows that cognitive conflict learning is effective in improving students' mathematical reasoning abilities. against student emotional intelligence in cognitive conflict learning can be seen in the following discussion. element student's emotional intelligence score level / class below (0 19) low (20 39) normal (40 59) super (60 79) perfect (80 100) condition early end early end early end early end early end average 32.5 45.2 32.5 45.2 32.5 45.2 32.5 45.2 32.5 45.2 std. dev 19.2 16.4 19.2 16.4 19.2 16.4 19.2 16.4 19.2 16.4 ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 7 table 2. student emotional intelligence achievement scores. from the table of emotional intelligence achievements above, it can be seen that for each level as follows; a. below; before learning 6 people (13%) changed after learning to 2 people or 4%. b. low: before learning 13 people (28%) changed after learning to 12 people or 27%. c. normal; before learning 19 people (40%) changed after learning to 20 people or 42%. d. super; before learning 9 people (19%) changed after learning to 10 people or 21%. e. perfect; before learning 6 people (13%) changed after learning to 2 people or 4%. in general, it can be said that students who have below normal emotional intelligence before learning are 17 or 36%, after learning they turn into 14 people or 30%. so there are 3 people whose emotional intelligence has changed to normal. so, it can be said that the emotional intelligence of students majoring in mathematics at the state university of medan can be improved through learning with a cognitive conflict approach. in addition, based on students' responses to learning with a cognitive conflict approach, it can make students happy and eager to always get this learning for the next course. conclusion based on the research results, it can be concluted that: 1. learning geometry based on the cognitive conflict approach in the mathematics department of the state university of medan can improve reasoning skills. 2. learning geometry based on the cognitive conflict approach in the mathematics department of state university of medan can improve emotional intelligence which includes self-understanding, understanding of others, empathy, motivation and selfcontrol. 3. through learning geometry with a cognitive conflict approach, students gain a strong understanding of the fundamental concepts of geometry rationally. so that, students will have a long memory or long term retention for any geometric concepts. 4. learning conflict-based cognitive geometry can help students correct misconceptions or misconceptions. references bayer, m. 2016. fostering conceptual change by cognitive conflict based instruction on student understanding of heat and temperature. eurasia journal of mathematics, science and technology education, volume2, number 2, july 2016.www.ejmste.com. accessed on 23 february 2019. freudenthal h. (1991). revisiting mathematics education. dordrecht: reidel publishing. given, bk (2007). teaching to the brain's natural learning systems. alexanderia: ascd. hasratuddin. 2018. why you should study mathematics. ed. 2. perc. edira: medan. lots 6 2 13 12 19 20 9 10 0 3 percentage 13 4 28 27 40 42 19 21 0 6 ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 8 http://www.ejmste.com/ kang, s., koh, h., noh, t., & scharman, lc 2015. the influence of students cognitive and motivational variables in respect of cognitive conflict and conceptual change. international journal of science education, (online), vol. 27, no. 9, 2015. lee, g. & kwon, j. (2003). what do we know about student's cognitive conflict in science classroom: a theoretical model a cognitive conflict process. [on line] available: www.eric.ed.gov/ericwebportal (01 may 2016). mc gregor, d. (2007). developing thinking; developing learning. new york: open university press. r soedjadi. (2000). mathematics education tips in indonesia.directorate general of higher education, ministry of national education wittgenstein, l. (1974b). philosophical investigations. oxford: blackwell. [paper reference 1]understanding mathematics teaching, 77, 20–26. ijo international journal of mathematics volume 3| issue 10| october| 2020 http://ijojournals.com/ 9 http://www.eric.ed.gov/ericwebportal 1. this type of research is a semi-experimental study with a two-class control and pretest-posttest experimental design. 2. the location of this research is the department of mathematics, fmipa unimed medan, jl. willem iskandar psr. v medan estate. 3. while the subjects in this research were 35 students in the 2019 f mathematics education study program (pspm) and 32 pspm 2019 a classes. 4. the object in this study is a learning geometry based on a cognitive conflict approach. 5. implementation of learning conclusion adequacy of h-likelihood estimation method for unbalanced clustered counting data models. intesar n. el-saeiti1, khalil mostafa alsawi2 and gebriel m. shamia3 (1) assistant professor, statistics department, faculty of science, university of benghazi (2) a teacher at a secondary school, benghazi-libya (3) professor, statistics department, faculty of science, university of benghazi correspondence to intesar n. el-saeiti, entesar.el-saeiti@uob.edu.ly abstract this article would concentrate on hierarchical generalized linear models, including generalized linear mixed-models, which are the extension of linear models. in generalized linear models, the dependent variable assumes every distribution from exponential family distributions, e.g., normal, poisson, binomial, gamma, etc. the poisson-gamma method was applied, where the dependent variable represents the poisson distribution and the standard error is defined by the gamma distribution. in generalized linear models, several estimation methods have been used. throughout this study, the hierarchical likelihood estimation method was used to determine the effectiveness of this methodology for both data balanced and unbalanced. this article compares the adequacy of poisson-gamma h-likelihood estimation method of mixed effects clustered data models with equal and unequal cluster sizes. this was evaluated in terms of probability of type-i error rate, power and standard error by applying computer simulation. simulation is performed using different cluster numbers and different cluster sizes. the results show that the performance of the hierarchical likelihood estimation technique provided close approximations in the event of balanced and unbalanced data, while the output of the technique was approximately equivalent in both instances, regardless of cluster size inequality. keywords: hierarchical generalized linear model (hglm), poisson-gamma h -likelihood, counting response, balanced clustered, unbalanced cluster. introduction linear models define a continuous response variable as a function of one or more predictor variables. they may help you understand and predict the behavior of complex systems or analyze experimental, financial and biological data. linear regression is a statistical method used to construct a linear model. the model describes the relationship between the dependent variable y (also known as the response), as a function of one or more independent x variables (called predictors). y = xβ + �, …(1) where β represents linear parameter estimates to be evaluated and � represents the error terms. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 18 mailto:entesar.el-saeiti@uob.edu.ly the generalized linear model (glm) is an extension of the linear model to response variable that follow any probability distribution include the exponential group of distributions. the exponential family includes useful distributions, for example, normal, binomial, poisson, polynomial, gamma, and others (leee and nelder, 2006). hypothesis tests applied to the generalized linear model do not require normality of the response variable nor do they require homogeneity of variances. hence, generalized linear models can be used when response variables follow distributions other than the normal distribution and when variances are not constant. for example, counting data would be appropriately analyzed as a poisson random variable within the context of the generalized linear model. the generalized linear mixed model (glmm) is name as hierarchical generalized linear model. glmms can be thought of as an extension of generalized linear models (lee and nelder, 2006), the general form of the model in matrix notation is giving by. � = �� + �� + � (2) mcculloch and searle (2001) wrote, when studying phenomena within a given period of time or area, the data of any phenomenon will follow the poisson distribution and it is in exponential family. in our paper, it is assumed that the data follow the poisson distribution and the error unit follows the gamma distribution. from lee and nelder's (1996) description of hierarchical models, every distribution in the exponential family has the corresponding distribution, e.g. poisson offset by gamma distribution, binomial distribution offset by beta distribution, normal distribution offset by normal distribution. for more information on hierarchical data structure see elsaeiti (2013, 2014), lalonde (2009). cluster data models are frequently used in the field of agricultural, genetic, industrial, medical, biological and even social science experiments. clustered data or nested data design is an experimental design technique in which data has an implicit hierarchy. the clusters may be balanced or unbalanced, i.e., the number of observations in a cluster (the size of the cluster) is equal or unequal. the unbalanced clustered data may bring up the problem of heterogeneous models which require different variance components, as had been addressed in previous studies for continuous response (el-saeiti, 2015). in the case of unbalanced clustered data with continuous outcomes in the linear model, el-saeiti (2015) found that, there was a ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 19 different dispersions for different clusters sizes. ac-counting for the different dispersions led to the minimization of mean square error, which was shown through two examples. in this study, the researcher focused on the counting outcomes. when using mixed effects for clustered data with counting outcomes, a preferred model is hierarchical generalized linear model (hglm). lee and ryan (2017) are concerned with a class of generalized linear mixed models for clustered data, where random effects are mapped solely to cluster structure and are independent between groups; they derive the necessary and sufficient conditions that allow the marginal likelihood of such a class of models to be expressed in closed form. illustrations are provided using normal, poisson, binomial and gamma distributions; these models are unified under a single umbrella of generalized conjugate linear mixed models, where "conjugate" refers to the fact that marginal likelihood can be expressed in closed form, rather than implying inference through the bayesian paradigm. using an explicit marginal likelihood means that these models are more computationally efficient, which can be important in large data environments, with the exception of binomial distribution, so that these models are able to achieve conjugation at the same time and thus be able to accommodate both unit and group level covariates. theoretical background poisson-gamma hglm are members of the hierarchical generalized linear model family (lee and nelder, 1996), an extension of the generalized linear model family and the generalized linear mixed model group. for training, poisson-gamma hglm is used to characterize historical count data as non-life insurance compensation numbers, among others. it should be remembered that the poisson gamma hglm considered at one time follows a negative binomial regression model (gning, 2013). modeling poisson data yi ∼ poisson(λi) then; e(yi) = λi and var (yi) = λi . the link function must map from (0, ∞) to ( ∞, ∞). a natural choice is g(µi) = log(µi). for dependent count data (rönnegård and shen, 2010) it has been stated that it is common to model a distributed poisson response with a random gamma effect; if no overdispersion is assumed to be conditional on u and thus have a fixed dispersion term; this model may be specified as. �(��|�, �) = ���(��� + ���) ..(3) ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 20 lee and nelder (1996) described the generalized linear model for poisson-gamma hierarchical and the generalized linear mixed form structure of poission. the three pices of hglm for poissongamma is: 1. yij| uj ~ poi (λi, ϕ λi)) , ui ~ gamm(α,γi) , 2. η = xβ + zu , 3. η = ln(λi) more details on poisson-gamma model see (lee and nelder 1996, 2001). however, in the clustered count response because the assumption of independence between cluster observations is likely to be violated, a mixed effects clustered counting data model is a useful strategy to account for intracluster correlations in statistical inference, see hedeker and gibbons (1994). the purpose of this paper is to compare the performance of the mixed effects clustered data count model with equal and unequal cluster size. here, the author discusses the probability of type i error rate, the statistical power of the experiment, and the standard error (s.e) by computer simulation study. simulation study the simplest definition of simulation in science is that it is a numerical method of running trails or tests using computer algorithms instead of conducting a real experiment. simulation is an approach to modeling random events in such a way that simulated outcomes closely match real-world outcomes, and by studying simulated outcomes, researchers gain knowledge of the real world. in other words, the simulation of a system is the operation of a process model (maria, 1997). the design can be re-fitted and tested at a lower cost, so simulation will be more realistic. the function of the prototype can be investigated and thus inferences can be made about the behavior of the real system. simulation can also be viewed as a method to test the quality of the current or proposed process. in numerical applications, the word simulation usually involves the random sampling process of the probability distributions. due to its wide use, this is an important part of the statistical study. this significance occurs in many situations when it is difficult to find statistical diagnosis, or time consuming, or costly to carry out an analysis. statistical simulation can be used simply by specifying a statistical software that uses random numbers to produce the values of random variables with the desired probability distributions (uniform, normal binomial, etc.) that have been achieved in this research. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 21 for data generation and all simulation steps is included in the appendix section 'end of this paper'. for more explanation and detail on related simulation studies with different dependent variables and other variables for different purposes, see el-saeiti (2013, 2019). for h-likelihood `poisson gamma hglm', it was used hglm function in hglm package for traditional poisson gamma in r throw the simulation steps. using hglm function to get the estimation of parameters � and t-statistic with p value to calculate through simulation. results and discussion the following tables and diagrams will demonstrate the results obtained from the simulation and display the probability of the type-i error rate in table (1), the approximation value of the “β “ parameters in table (2), the power in table (3) and the standard error in table (4). table (1) display the probability of type-i error rate were computed as the proportion of p values less than 0.05 under a null hypothesis ��: �� = 0 of no treatments effect when we rejected incorrectly. table (1): probability of type-i error rate cluster observations unbalanced balanced k=3 n=5 0.032 0.060 n=10 0.105 0.078 n=50 0.093 0.028 k=10 n=5 0.048 0.034 n=10 0.043 0.046 n=50 0.076 0.044 k=50 n=5 0.042 0.035 n=10 0.040 0.058 n=50 0.039 0.030 fig. (1): type-i error for poisson gamma 0 0.02 0.04 0.06 0.08 0.1 0.12 n=5 n=10 n=50 n=5 n=10 n=50 n=5 n=10 n=50 unbalanced balanced ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 22 the probability of type-i error rate was acceptable because it was slightly high in some points; generally it was not far away 0.05. next table is table(2); for the simulated sample of size (5,10,50) observations and 3, 10, and 50 clusters; where the actual value is equal to 0.2 for the parameter �� , and the value for the �� parameter is equal to zero " because there is no x2 value, it is used only to calculate the power and the probability of type-i error rate” table (2): the estimate parameters cluster observations unbalanc ed balanced ��� ��� ��� ��� k=3 n=5 n=10 n=50 0.2033534 0.1947158 0.2008493 0.003756905 -0.00445098 0.004040149 0.2041509 0.2028573 0.2045826 -0.00518115 0.01137586 0.00065012 k=10 n=5 n=10 n=50 0.1994582 0.1980559 0.2008564 -0.00049512 -0.00192974 -0.00039103 0.2017183 0.2022803 0.1994892 -0.00151689 3.240238e-05 -0.00093368 k=50 n=5 n=10 n=50 0.2010042 0.1995827 0.1997564 0.0006905077 0.00053609 1.147578e-05 0.1977052 0.2004524 0.2008468 0.0004996838 -0.001911985 0.0002390147 table (2) shows that the h-likelihood estimate was a good estimation method for both cases, since the average of 1,000 replications provided estimates that were very close to the actual values for the parameters. figs 2.1 and 2.2 included a summary of the predicted values that were close to the actual values. fig(2.1): estimate values (���) fig(2.2): estimate values (���) 0.188 0.19 0.192 0.194 0.196 0.198 0.2 0.202 0.204 0.206 n=5 n=10n=50 n=5 n=10n=50 n=5 n=10n=50 balanced unbalanced -0.01 -0.005 0 0.005 0.01 0.015 n=5 n=10 n=50 n=5 n=10 n=50 n=5 n=10 n=50 balancd unbalanced ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 23 next table (3) demonstrate the power simulated sample of size 5,10, and 50 observations and the number of : 3, 10, and 50 clusters. statistical power was computed when rejected hypothesis ��: �� = 0, correctly. calculate through simulation for 1000 times how many times the test is significant. the power is the proportion of number of rejected correctly. table (3) statistic power cluster observations unbalanced balanced k=3 n=5 n=10 n=50 0.801 0.963 1.000 0.808 1.000 1.000 k=10 n=5 n=10 n=50 1.000 1.000 1.000 1.000 1.000 1.000 k=50 n=5 n=10 n=50 1.000 1.000 1.000 1.000 1.000 1.000 from table (3) it has been shown that the power values of "probability to accept a null hypothesis that is right" are approximately close to one. the higher power the better method, from the above table hard to decide since the power approximately is 1, and is high for both cases; since the sample size is large for each combination. it is reasonable high power for large sample size, there is no different between both cases in power, both work good according to power for large sample size. table (4): stander error (se) for original simulated sample of size 5,10, and 50 observations and 3,10, and 50 clusters. the stander error was computed as the average of 1000 ses of the estimates of ��. the smaller se represents smaller variability, or greater precision, of the parameter estimates (heo and leon, 2005). table (4): stander error for both cases counting data fig.(4) standard error for balanced and unbalanced data. cluster observations unbalanced balanced k=3 n=5 n=10 n=50 0.07196653 0.04438624 0.01834041 0.06844161 0.04374402 0.01862745 k=10 n=5 n=10 n=50 0.03354806 0.02272336 0.00991545 0.03383271 0.02271749 0.00993959 k=50 n=5 n=10 n=50 0.01423205 0.00995692 0.00443169 0.01417949 0.00992440 0.00443549 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 24 from table (4) and graph (4) above, it can be seen that there is no difference in the standard error for hglm poisson game in balanced and unbalanced counting data discussion in this article, we looked at the generalized linear mixed-models, which are the extension of linear models. it is understood that many other studies have studied a problem with unbalanced data or incomplete information, which may lead to a heterogenetic problem. the heterogenetic issue was not discussed here by the use of the hierarchical probability estimation model. the process has impartial and very similar outcomes in two situations that are balanced and unbalanced. the lack of meaning and the imbalanced model will therefore not be a concern by using the poisson-gamma h-likelihood estimation. the hierarchical probability estimation approach has been concluded to be able to solve heterogenetic problems in future studies. as stated earlier, this study's main objective was the efficiency of the h-likelihood estimation approach for unbalanced cluster data models. h-likelihood estimation approach the system for unbalanced clustered count data models is recommended in order to avoid heterogeneity problems. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 n=5 n=10 n=50 n=5 n=10 n=50 n=5 n=10 n=50 unbalanced balanced ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 25 references el-saeiti, i. n. (2014) “performance of mixed effects for clustered binary data models”. aip conference proceedings 1643, 80 el-saeiti, i. n. (2015). "messy data in heteroscedastic models case study: mixed nested design". lap lambert academic publishing. el-saeiti, i. n. (2019). an adjusted scale binomial beta h-likelihood estimation method for unbalanced clus-tered binary response models. libyan journal of science & technology; vol. (10:1) 20-22. el-saeiti, i. n.(2013): “adjusted variance components for unbalanced clustered binary data models”. ph. doctoral “university of northern colorado.” gning, l.(2013). on the existence of maximum likelihood estimators in poissongamma hglm and negative binomial regression model. electronic journal of statistics; vol. (7), 2577–2594 heo, m. and leon, a. (2005). performance of a mixed effects logistic regression model for binary outcomes with unequal cluster size. biopharmaceutical statistics,15:513-526. l.gning and d. pierre-loti-viaud.( 2012): on the existence of maximum likelihood estimators in poisson-gamma hglm and negative binomial regression model. lalonde, t. l. (2009). components of overdispersion in hierarchical generalized linear models. dissertations ” university of northern colorado”. lee, y., & nelder, j. a. (2006). double hierarchical generalized linear models. journal of the royal statistical society, series b (methodological), 55, 139-185. lee,y. and nelder, j.(1996): hierarchical generalized linear models journal of the royal statistical society, series b (methodological), 58 (4), 619-678. maria, a.(1997): "introduction to modeling and simulation", proceedings of the 1997 winter simulation conference. mcculloch, c.e., and shayle, r.s.(2001): generalized, linear, and mixed models. ny: john wiley & sons, inc. rönnegård, l., alam,m. and shen,x. (2010) hglm package (version 2.0) package maintainer ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 26 http://projecteuclid.org/ejs http://projecteuclid.org/ejs appendix mydata=function(seed){ set.seed(seed) beta0 = 1 beta1 = 0.2 beta2 = 3.1 ########## for poi-gam### n.clus <2 #no. of clusters n.per.clus <5 #no. of obs. per cluster for equal sigma2_u <0.2 #variance of random effect sigma2_e <1 #residual variance sigma1<2 nn <n.clus*n.per.clus beta=matrix(c(beta0,beta1,beta2),3,1) y=matrix(0,nn,1) x=matrix(c(rep(1,nn),rep(0,nn),rep(0,nn)),nn,3) z=matrix(0,nn ,n.clus) a <rnorm(n.clus, 0, sqrt(sigma2_u)) e <rnorm(nn, 0, sqrt(sigma2_e)) ## generate x-values from normal distirbuation## x[,2] =rnorm(nn,3,sigma1) x[,3]=rpois(nn,3) x_d <matrix(c(rep(1,nn),rep(0,nn),rep(0,nn)),nn,3) z <diag(n.clus)%x%rep(1, n.per.clus) u <rgamma(n.clus,1) eta <exp(beta0+beta1*x[,2]+z%*%u) y <rpois(length(eta), eta) list( x=x, y=y,u=u,z=z, x_d=x_d) } ########################################### # power for h-likelihood function # # by using hglm function # ########================================# library(mass) library(hglm) sima= function (n1){ set.seed(1234) alpha < 0.05 b21count < 0 b22count < 0 s.e2 < matrix(0,nrow=n1, ncol=1) b.e21 < matrix(0,nrow=n1, ncol=1) b.e22 < matrix(0,nrow=n1, ncol=1) seeds=rnorm(n1,0,50) set.seed(seeds) for(i in 1:n1) { datta= mydata(seeds[i]) x=datta$x y=datta$y x_d=datta$x_d z=datta$z #========================h-likelihood method ==========================# r < gamma.pois <hglm(y = y, x = x, z = z, x.disp = x_d, family = poisson(link = log), rand.family = gamma(link = log)) ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 27 ss= summary(r) betas <r$fixef se <r$sefe zval <betas / se pval <2 * pnorm(abs(zval), lower.tail = false) s.e2[i,] <se[2] b.e21[i,] <betas[2] b.e22[i,] <betas[3] p21 = pval[2] if(p21 < alpha){b21count = b21count+1} p22 = pval[3] if (p22 < alpha){b22count = b22count+1} } typei2=b22count/n1 power2=b21count/n1 se2 <sum(s.e2)/n1 be21 <sum(b.e21)/n1 be22 <sum(b.e22)/n1 list(ss=ss, be21=be21,be22=be22,power2=power2,typei2=typei2,se2=se2) } ijo international journal of mathematics volume 3| issue 12| december | 2020 http://ijojournals.com/index.php/m 28 sabaya11@gmail.com using energy conservation concept and basic mathematical modeling techniques to study the effect of greenhouses gases to earth's climate system. abdulaziz b. m. hame�∗� and younis ahamed. abu aash�� 1. department of mathematics, faculty of science, yobe state university, nigeria 2. department physics faculty of education, west kordufan university, sudan emails: aziz.hamed@gmail.com, abstract the work, implemented the basic mathematical modeling techniques to physical phenomenon (conservation of energy) base on stefan boltzmann law in to earth's climate system using analytical method to study the impact of climate change, effect of greenhouses gases in earth's surface temperature, and spreading societal awareness of its dangers. earth's climate system is a complexity system, that is schematically made up of five components: the atmosphere; the hydrosphere (oceans, lakes, and other bodies of water); the cryosphere (snow and ice); the lithosphere (land surface); and the biosphere (all living things). these components do not exist in isolation; they are interconnected and interact at several levels, either directly or indirectly [1,16]. the system as a entire is powered by solar radiation and develops under the influence of its own internal dynamics through ocean currents and atmospheric circulation. on the other hand, there are external factors which drive the system; these are called forgings, include both natural phenomena such as cyclical changes in the earth's orbit around the sun, volcanic eruptions, variations in the solar output, and human-induced (anthropogenic) factors like changes in atmospheric composition, human activities, and so on. climate changing is one of the greatest threats towards the survival of mankind on the earth, it is well understood that human beings activities hold the main responsibility behind these climate changing issues beside the natural disasters [1]. theoretical models of the earth's climate system most often involve systems of proportional models, ordinary differential and partial differential equations and qualitative behavior of their solutions. mathematical modeling is one of several approaches that have been used to study the climate system and become the most important techniques for complicate problems in modern science. however, enable a physically based estimate the range of future climate change,, providing invaluable scientific information towards political and societal decision maker for well plan in order to protect our planet. according to the study, it has been found that, the green house gases factor (0 ≤ β ≤ 1) and other relevant human activities play a prominent role in earth's climate change. furthermore ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 1 mailto:sabaya11@gmail.com mathematical calculation provided actual value for greenhouse factor β = 0.76, which lead to moderate temperature in earth's surface all seasons. key word: energy conservation, mathematical modeling, greenhouses gases, earth's climate system, global warming. 1. introduction today climate change became an important issue in our planet, the study economic and social effects on societies as a whole and also its effect on global peace and security. earth's climate system is a complexity system, that is schematically made up of five components: the atmosphere; the hydrosphere (oceans, lakes, and other bodies of water); the cryosphere (snow and ice); the lithosphere (land surface); and the biosphere (all living things). these components do not exist in isolation; they are interconnected and interact at several levels, either directly or indirectly.climate changing is one of the greatest threats towards the survival of mankind on the earth. this motivates us to investigate climatic changing effects via fluctuations of the temperature, wind patterns and precipitation taking place in a given geoenvironment [1,12]. practically the sun is the main resource of energy that powers the earth's climate system. this energy comes in the form of electromagnetic radiation, which originates from different depths in the sun's interior. according to physicist (stefan boltzmann), energy conservation theory, energy can change from one form to another but the total amount of energy remains constant. a complete climate model contains physical descriptions of all five components mentioned above and takes into consideration their coupling. some components may be described in a simplified form or even be prescribed [16]. mathematical model is a tool to describe physical phenomena, providing the world with a clear mathematics view on the current state of knowledge in climate system, climate change, weather predictions and its potential environmental and socio-economic impacts. 2. problem statement: although much progress has been made over the decades, our planet still faces multiple societal, economic and environmental challenges. climate changing is one of the greatest threats towards the survival of animals and mankind on the earth. climate change has ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 2 already started to cause a wide range of physical effects with serious implications for investors and businesses. 3. objective of study: the study focus on well known physics phenomenon (energy conservation theory base on stefan boltzmann law) implemented to temperature of earth's surface using essential mathematical modeling techniques in order to achieve the following: (i) employ step by step mathematical model techniques to study the effect of greenhouse gases to earth's climate and climate change. (ii) mathematically, evaluate the suitable greenhouse gases factors which moderate earth's surface temperature. (iii) spread the weariness between the communities and countries about the activities that will release more greenhouse gases. also its best practices in clean energy-related research and its subsequent contribution to the unanimous goal set in paris climate change conference 2020, of achieving net zero greenhouse gas emissions by 2050. 4. definitions and concepts: 4.1. climate this is a word from ancient greek “klima”, meaning inclination. climate is commonly defined as the weather averaged or the statistics of weather over a long period. the standard averaging period is 30 years, but other period may be used depending on the purpose. it is measured by assessing the patterns of variation in temperature, humidity, atmospheric pressure, wind, precipitation, atmospheric particle count and other meteorological variables in a given region over long period of time. (climate definition in many references). 4.2.global solar radiation balance of the climate system radiation is the transfer of energy by electromagnetic waves, which do not require a medium, such as air, for their transmission. solar radiation is relatively short wavelength radiation emitted by the sun. thermal-infrared radiation is relatively long-wavelength radiation emitted by the earth, atmosphere, and clouds. the earth’s surface receives solar radiation during the day only, but its surface and atmosphere emit thermal-infrared radiation during day and night see [12]. the sun is the only relevant energy source for the climate system on a temporal scale of less than about 10 �years. the different energy fluxes are coming from the sun, on average 341 w/m2 reach the top of the atmosphere, while barely ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 3 half of this is available for heating of the earth’s surface. major parts of the short-wave radiation are reflected by clouds or reflected directly on the earth’s surface itself and are absorbed by the atmosphere. incoming radiation contrasts with surface long-wave outgoing radiation of around 396 w/m2. virtually earth's climate system receives all its energy from the sun. this energy comes in the form of electromagnetic radiation, which originates from different depths in the sun's interior. as fact some of the energy is absorbed in the solar photosphere, further absorption in the earth's atmosphere gives the solar spectrum at the earth's surface its more ragged appearance, for more information see [1, 16]. the points above give us main idea about the energy resource and energy balance, the mathematical explanations is input energy equal to output energy (energy conservation law), and there are many factors affect the balance. 4.3. global warming global warming is the increase in the average temperature of the earth’s near-surface air and the oceans it can also be defined as a gradual increase in the overall temperature of the earth’s atmosphere generally attributed to the greenhouse effect caused by increased levels of carbon dioxide co2, chlorofluorocarbons cfcs, and other pollutants [1,13]. in our point of view, the release of greenhouse gases is main factor that causing the global warming which lead to increasing in the overall temperature of the earth’s as well as climate change and its consequently such as (desertification, drought, flood, volcanoes, tornadoes, forest fires). greenhouse gases greenhouse gases are thought to be the main contribution to climate change (the greenhouse effect). they are very efficient in trapping heat into the atmosphere; therefore, it results in the greenhouse effect. the solar energy is absorbed by the earth’s surface and then reflected back to the atmosphere as heat. then as the heat goes out to space, greenhouse gases absorb a part of the heat. after that, they radiate the heat back to the earth’s surface, to another greenhouse gas molecule, or to space (the green effect). daniela burghila et al. stated in “climate change effectwhere to next”, the biggest concern scientists have is about the emission of co2 since it is about 75% of the total global emission of greenhouse gases [7, 9, 10]. the major greenhouse gases in the earth’s atmosphere are:  water vapour (h2o) ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 4  carbon dioxide (co2)  methane (ch4)  nitrous oxide (n2o)  ozone (o3)  chlorolfluorocarbons (cfcs)  hydrofluorocarbons (hfcs) [10]. as we mentioned that the release of greenhouse gases are very efficient in trapping heat into the atmosphere, therefore, reducing the emission of these gases become the major goal in the study specially carbon dioxide (co2). 2.5. climate model a climate model is essentially a representation of the many interactions and dynamics within the climate which includes the atmosphere, ocean, land surface, and ice to make predictions of possible climate change for the future. climate models are systems of proportional equations, differential equations based on the basic law of physics, fluid motion and chemistry. mathematician are using mathematical models to modulating the scientific phenomena in earth's climate system to give good insight and predictions about climate change and its consequently implication. 2.6. theorem: first law of thermodynamics (energy conservation): the first law of thermodynamics state that the energy can neither be created nor destroyed, but can be converted from one form to another. in another form, during an interaction, energy can change from one form to another but the total amount of energy remains constant [5,8]. therefore the first thermodynamics law is obtained on the experimental basis. in the other words, we can say that the energy of an isolated system is always constant. 2.7. mathematical model models describe our beliefs about how the world functions. in mathematical modeling, we translate those beliefs into the language of mathematics [2, 3]. in the real world, these beliefs are phenomena in natural sciences or socioeconomic sciences along with some observations. ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 5 here we designed the following diagram represents the general description of mathematical model techniques for all phenomena in science and socioeconomics sciences step by step till test the validation of the model. the above diagram is the complete mathematical modeling process for any phenomenon in our real world. the process starts with the real world in natural or social sciences along with with a physical system and some observations or an experiment notice. when the laws of physics that are thought to govern the behavior of the system are translated in mathematical terms, the result is what is called a mathematical model. by understanding the mathematical concepts deeply such as the model givens, formulation and solution of the model. the mathematical model is subsequently analyzed for its properties and used to generate predictions about the behavior of the system in a changing environment. these predictions are tested against observations, so if there is no discrepancy between predictions and observations, the model will accept; otherwise, the model will refine or improve, returning to model formulation or givens by considering all the impressive factors in our model in order to achieve good results which agreed with real observations by repeating the process until the the real world (phenomenon) social sciences natural sciences the conceptual world (observations) object / system understanding the mathematical concepts model givens variables/parameter mathematical model formulation mathematical model solution model assumptions and predictions, model test invalid valid/ consist with conditions and principle of assumption improve by revise the model givens and formulation decision apply ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 6 best result attain. thus, mathematical modeling is an iterative process. mathematical climate models enable a physically based estimate and predict the future climate change [4,5,6]. therefore we applying the above diagram or flowchart to energy conservation theory in earth's climate system in sequence to give an approximate solutions or predictions for sophisticate problem in earth climate system, to help decision maker for well plan in order to protect our planet. 2.8 analytical techniques: energy balance model (energy conservation): in earth's climate system, model builds a series of zero-dimensional energy balance models. the earth's climate system is described in terms of a single variable, namely the temperature of the earth's surface averaged over the entire globe. conceptually the sun emits the radiation in the ultraviolet (uv) regime (wavelength less than 0.4 μm). this energy reaches the earth's surface, where it is converted by physical, chemical, and biological processes to radiation in the infrared (ir) regime (wavelength greater than 5 μm). this ir radiation is then reemitted into space. in equilibrium state the earth's climate the average temperature of the earth's surface does not change, so the amount of energy received must equal the amount of energy re-emitted see [10]. a complete climate model contains physical descriptions of all five components mentioned above and takes into consideration their coupling. some components may be described in a simplified form or even be prescribed [16,15,14]. mathematical formulation: to well understand the mathematical concept about our physical phenomena (energy conservation theory) in earth's climate system, we have to classify the model components, such as units, variables, and constants, these factors may influence direct or indirect in our model [11],  t, the temperature of the earth's surface averaged over the entire globe, in kelvin (k) or celsius scale,( variable).  r, the radius of the earth, parameter (constant).  a, the energy flux density (also referred to as the energy flux)|the amount of energy (w) flowing through a flat surface of area 1��. from satellite observations we know that the energy flux from the sun is � = 1367.6����. parameter (constant).  σ, sigma, stefan boltzmann constant; its value is 5.67 × 10���������parameter.  the earth's viewed as a disk from the sun  the area of the disk as seen by the sun is ���. ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 7  the amount of energy from the sun to the earth (disk) is the incoming energy (w) ��� = ����  as fact, all bodies radiate energy in the form of electromagnetic radiation.  the temperature of the body is essential in our study, because the amount of energy radiated deepened on it.  in physics, shows that black body radiation is given by the stefan-boltzmann law, in ���� units [11]. ���(�) = ���.  the area of the earth's surface is 4���  the amount of energy radiated out by the earth surface is outgoing energy(w) ���� = 4������ the model formulation and solution: the first law of thermodynamics speaks about the conservation of the energy: ''energy cannot be destroyed or created; it can only be transformed''. at thermal equilibrium, the incoming energy (temperature) must equal to outgoing energy and with use stefan boltzmann law, such that ��� = ���� therefore the mathematical model represent the following equation ���� = 4������ ⇒ � = 4��� to solve the equation for t, thus � = ( � �� ) � � where � = 5.67 × 10��and � = 1376.6 the value of � = ( ����.� �×�.��×���� ) � � ≈ 278.7� interpretation: the earth's temperature increase if the incoming energy is greater than out coming energy and decrease if the incoming energy is lower than out coming energy. whoever the earth's temperature remains constant if the incoming energy balances the out coming energy and the planet said to in thermal equilibrium, this as normal understanding. the value of � = 278.7� which equals to 5.5 degrees celsius, but actual average earth's temperature approximately 16 degrees celsius, this is a huge difference between the predictions (the calculated value) and observation (the actual value). decision: according to the interpretation above, the model*1* is invalid. therefore the result should be rejecting and the model must be improve. ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 8 model*2* in order to get a better model, returning to the model givens, formulation and give deep insight to the problem. in model *1* we omitted a number of important factors. these factors represent incoming energy from sun reflected back out to space. snow, ice, and clouds, for example, reflect a great deal of the incoming light from the sun. we use the term albedo ('' the fraction of the incoming solar energy scattered by earth back to space is referred to as the planetary albedo"). to measure the earth's reflectivity. model *2* formulation: by adding the reflectivity factor into the constant of above model*1*  �: albedo. the earth's average ������ is about 0.3, which means that roughly 70% of the incoming energy is absorbed by the earth's surface. the model *2* build: the amount of energy reaching the earth is incoming energy (w) ��� = ����(1 − �) the amount of energy radiated out by the earth is outgoing energy (w) e��� = 4πr�σt� the model solution: according to conservation of the energy and thermal equilibrium state, the incoming energy equal to out coming energy and stefan boltzmann law such that ��� = ���� ⇒ ����(1 − �) = 4������ ⇒ �(1 − �) = 4��� therefore � = � �(���) �� � � � = � ����.�×�.� �×�.��×����� � � ≈ 254.9� interpretation: although we consider the ������ factor or the energy absorbed by the earth's surface, but the value of temperature � = 254.9�, this equivalent to −18.25 degrees celsius, its prediction of the temperature value at equilibrium is worse than the prediction of model*1* ,the difference is still wide mathematically. decision: by test the model *2* , we rejecting the result and revise the model*2* in order to refine or improve the predictions. ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 9 model *3*: the result in model 2 did not give the expected accuracy value, so it's better to look carefully to another factors the only option is to look where we might have overlooked something in the model. in this cycle, we focus on the outgoing radiation, which come from chemical reaction, nature and human activities. . mathematical formulation: greenhouse gases come from chemical reactions, nature such as volcanoes or human activities such as burning fossil fuels, like carbon dioxide (co�,h�o, ch�, no� ) methane, and water, as well as dust and aerosols have a significant effect on the properties of the atmosphere. model*3* build and solution: experimentally β ,greenhouses factor ranging 0 ≤ β ≤ 1 incoming energy (w): e�� = πr�a(1 − ρ) the amount of energy radiated out by the earth is outgoing energy (w): e��� = 4πr�βσt� model*3*solution: according to conservation of the energy and thermal equilibrium state, the incoming energy equal to out coming energy and stefan boltzmann law, such that e�� = e���, thus t = � �(���) ��� � � � ≈ 282.9k interpretation: the value of β = 0.66 gives a climate model that correctly predicts the current global average temperature t ≈ 282.9k, which equivalent to 9.64 degrees celsius, this gives reasonable approximate value to the earth's surface temperature. mathematical calculation provided actual result for greenhouse factor β = 0.76, this may lead to moderate temperature in earth's surface all seasons. the effect on the outgoing radiation is difficult to model because it depend on the greenhouse gases mostly depend on human activities which are increasing daily. meanwhile, it's manageable, if there is a strong determination to reduce this factor to ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 10 reasonable value, since its human factor,. however many conferences recommended, reducing the greenhouse gases which mainly belong to human activities. result discussion: step by step mathematical model techniques has been implemented overall result in model1*1*, model*2* and model *3*, it's clearly indicate the effect of greenhouse gases factor β which range ≤ β ≤ 1 very huge. however, this factor is controllable or reducible to appropriate value because mainly depend on human activities. conclusion: concept of energy conservation, stefan boltzmann law, and step by step mathematical modeling techniques have been implemented, concentrated on the inconsistency of the results between the prediction and observations. on the other hand the model has been improved by considering the other factors, which can affect the results. the study shows, the effect of greenhouse gases to the earth's climate system, by increasing earth's surface temperature, which causes many problem, such as desertification, drought, flood, volcanoes, tornadoes, and forest fires. mathematical calculation provide an appropriate value to greenhouses factor to be β = 0.76 in order to get the earth's surface temperature 16 degrees celsius. the impacts of climate change are expected to grow more severe over the coming years and decades unless the industrial countries take the initiative seriously to reduce the greenhouses gases and spreading the wariness between the societies and communities about the advantage and disadvantage of climate change and it's consequently. . references [1].hans kaper and hans engler, mathematics& climate, mathematics and climate research network (mcrn, georgetown university, washington, district of columbia,2013. [2] glenn marion, an introduction to mathematical modeling, bioinformatics and statistics scotland, 2008. [3]. https://en.wikipedia.org/wiki/mathematical_model [4]. john a. trangenstein. numerical solution of hyperbolic conservation laws,department of mathematics, duke university, durham 2005, [5] goosse h., p.y. barriat, w. lefebvre, m.f. loutre and v. zunz (2010). introduction to climate dynamics and climate modelling. http://www.climate.be/textbook, 2010. ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 11 https://en.wikipedia.org/wiki/mathematical_model [6]. roger k. smith and wolfgang ulrich. lectures on numerical meteorology, december 5, 2008. [7]. planton, serge(france; editor)(2013). “annexiii. glossary: ipccintergovernmental panel on climate change” (pdf). ipcc fifth assessment report. p. 1450. archived from the original (pdf) on 2016-05-24. retrieved 25 july 2016. [8] m. m. rahaman, m.m.h. sikdar,m. b. hossain, m.a. rahaman, m.jamal. hossain. numerical solution of diffusion equation by finite difference method. iosr journal of mathematics (iosr-jm), www.iosrjournals.org, volume 11, issue 6 ver. iv (nov. dec. 2015), pp 19-25. [9].d. randall, an introduction to atmospheric modeling, department of atmospheric science colorado state university, 2014. [10]. the national academies press, climate change: evidence, impacts, and choices: pdf booklet. this pdf is available at http://nap.edu/14673 (2012) [11] daniel flath, hans g. kaper, frank wattenberg, esther widiasih, energy balance models, draft { march 24, 2012 [12] mandeep dalal, textbook of physical chemistry , volume 1, first edition, 2012 www.dalalinstitute.com. [13]. mark z. jacobson, fundamentals of atmospheric modeling, cambridge university press,. new york, second edition, 2005. [14]. joakim wid´en and joakim munkhammar. solar radiation theory, uppsala university 2019.isbn 978-91-506-2760-2, doi 10.33063/diva-381852 , 201. [15]. kumsal bayazit, ban ki-moon, pathways to net zero: the impact of clean energy research, executive summary, un report.2021. [16]. prof. t. stocker, introduction to climate modeling, physikalisches institut universität bern, 2016 ijo international journal of mathematics volume 5 | issue 10 | october 2022 | https://www.ijojournals.com/index.php/m/index 12 problems solvingand thinking critical ability student through realistic approach based on problem non-routine by blended learningin the department mathematics,faculty mathematics and natural science, medan state university hasratuddin 1) ; m. amin fauzi 2 ) ; budi halomoan siregar 3) ; and, dwi novita sari 4) . 1) and 2) lecturer in mathematics at medan state university, and 3) and 4) students of s3 mathematics education, state university of medan email 1) ; siregarhasratuddin@yahoo.com, 2) aminunimed29@gmail.com, 3) budihalomoan@unimed.ac.id, 4) dwinovitasari31@gmail.com abstract. this study aims to analyze students' problem-solving and critical thinking ability through realistic approach based on non-routine problem by blended learning in the mathematics department, faculty of mathematics and natural sciences, state university of medan. this type of research is a quasi-experimental. the research population is all undergraduate students of the mathematics education study program (mesp) in 2021. the research sample is taken from 2 classes of the population, namely; 1) mesp a 2021 as an experimental class that is given learning with a realistic approach based on nonroutine problems using blended learning, and 2) mesp b 2021 as a control class that is given regular online learning. the research instrument on problem solving and critical thinking skills is a test. data on problem solving abilities and critical thinking were analyzed using anova. the results showed that the problem-solving and critical thinking skills of students who were given geometry learning through a realistic approach based on non-routine problems using blended learning were better than students who were given regular online learning. furthermore, from the results of the study it was found that there was no interaction between learning and students' initial abilities, both on problem skills and on students' critical thinking skills. this shows that students' problem-solving and critical thinking skills are always better using learning with a realistic approach based on non-routine problems using blended learning compared to students who are given regular online learning. thus, what is suggested from the results of this study is that in improving students' problem-solving and critical thinking skills through a realistic approach based on non-routine problems using blended learning, it is not necessary to classify students into low, medium or high initial abilities. keywords: realistic approach; ability; solving problem; critical thinking; blended learning; nonroutine. introduction realistic mathematics education originally came from the netherlands and has been developed since the 1970s. as for what inspired it was freudenthal's view which said that mathematics is a human activity. so that mathematics should not be given to students in the form of ' finished results', but must construct / find their own mathematical concepts, principles or procedures through solving nonroutine problems. there are three key principles in the realistic approach [1], namely: 1) guided reinvention/progressive mathematizing, 2) didactical phenomenology, and 3) self-developed model. from these three principles, the mathematics learning process with a realistic approach is divided into five characteristics, namely: constructing and concretizing , level and models, reflection and special assignment, social context and interaction, structuring and intertwining [1], [2], [3]. mathematics learning with a realistic approach has wide-ranging consequences for children's learning and thinking processes [4]. meanwhile, the mathematization process is seen as an activity that is constructive, reflective and interactive. learning through a realistic approach is an activity that is meaningful for them, so that it can improve their attitudes and higher levels thinking skills. one offactor that affect thinking ability level tall is a stimulus through challenge in problem non-routine [5]. challenge in the form of problem non-routine could stimulate volume 5 | issue 11 | november 2022 | 1 somebody to fully understand the problem based on observations and investigations, explore and prepare tools, conduct experiments or investigations. therefore, by starting from non-routine problem conflicts, students will not feel unfamiliar with the topics they are going to learn and will grow curiosity for themselves, and it is hoped that in the end it will eliminate the impression that learning mathematics is no longer something that is fun. difficult and scary. associated with the learning process that does not determined of consequence covid 19 situation, then most potential approach for held in learning is based blended learning with an online or offline system through a realistic approach that emphasizes student centered processes as well as load element constructive, interactive and reflective. given the importance of choosing proper learning _ in dominate field mathematics then necessary analyze ability solving problem and think critical mathematical student through based realistic approach problem non-routine by combined offline with online. ability solving problem is something very important in learning mathematics, because 1) makes somebody becomes skilled select and analyze something later information _ study the result, 2) makes something satisfaction intellectual arising _ from in self someone, 3) increase potency intellectual someone, and 4) someone will could finder through the discovery process that alone. solution problem is very important thing in learning math, because with increase ability solving problem non-routine expected student will more analytical and capable in resolve problem and at the same time could prepare self in face changing situations in life _ later [6]. in everyday life, we cannot be separated from something called a problem, so problem solving is the main thing in learning mathematics. most mathematics education experts state that the problem is a question that must be answered or responded to by students. by specifically, math problems consist of routine problems and non-routine problems.a routine problem is a problem that is merely an exercise that can be solved using some commands or algorithms [7], [8] . meanwhile, non-routine problems are more challenging and require creative abilities from problem solvers. non routine problems arise when problem solvers have problem characteristics that do not immediately known how to solve them [9]. theproblems included _ inthing this is problems that contain challenges that are not immediately can be solved by known routine procedures. "for a question to be a problem, it must present achallenge that cannot be resolved by some routine procedure known to the student [10].” so, ability solving problem is ability think mathematical somebody in evaluate, connect and develop something theory. problem solving ability is an essential competency in learning mathematics, so it is recommended to be trained and raised since children at the elementary school level up to college [11]. this means that mathematical problem solving skills need to be trained at every level of education. problem solving skills require reflective thinking, including critical thinking and creative thinking skills. in other words, learning mathematics in the classroom needs to train critical and creative thinking skills that are carried out intentionally and planned.ideal is something term problem solving model or heuristic found by brandsford and stein [12] . this model consists of five stages of problem solving, namely identifying potential problems, defining and representing the problem, and exploring possible strategies. acting on those strategies, looking back and evaluating the effects of those activities. polya (1973) developed a problem solving model, procedure, or heuristic consisting of stages of problem solving, namely (1) understanding the problem; (2) make a problem-solving plan; (3) implementing a problem-solving plan; and (4) review [13] . thus , in study in this case , the activities included in problem solving activities include: identifying elements that are known, asked about, and the adequacy of the elements needed, formulating problems from everyday situation with mathematics; implement volume 5 | issue 11 | november 2022 | 2 strategy for solve various problems inside or outside mathematics; explain and interpret the results according to the problem; develop mathematical models and solve them according to real problems and use mathematics in a meaningful way [14]. strategies for solving problems mathematics depends on the problem to be solved. problem solving strategies in general have four steps, namely ; 1) m understand the problem, inthis activity conducted stages or steps: what (data) is known, what is not known (asked), whether the information is sufficient, what conditions (conditions) must be met, restating the problem in detail operational , b) planning a solution, namely to do the activity of trying , looking for or remembering problems that have been solved that have similarities to the problems to be solved, looking for patterns or rules, making conjectures and compiling settlement procedures , c) solving problems according to plan, namely to do activities with doing according to the procedure that was made in the previous step to get a solution , d) reexamine the procedure and the results of the settlement, namely to do the activity of analyzing and evaluating whether the procedures applied and the results obtained are correct and appropriate , whether there are other procedures that are more effective, whether the procedures made can be used to solve similar problems, or whether the procedures can be generalized . critical thinking is one of the higher-order thinking processes that can be used in the formation of students' conceptual systems. critical thinking is a sensible or reason-based reflective way of thinking that is focused on determining what to believe and do [15]. there are two main signs of critical thinking [16]. the first is that critical thinking is proper thinking that leads to deductive thinking and decision making rational.the second is that critical thinking is reflective thinking that shows complete awareness of the steps of thinking logical.critical thinking must meet the characteristics of thinking activities which include: analysis, synthesis, problem recognition and solutions, conclusions and assessments [17]. although mathematics is related to logical theory, critical thinking skills will not develop if in mathematics learning students are only trained to memorize (mechanistic) formulas, find formulas without knowing the relationship between draft with others (structuralistic), or solve problems routinely (empirical), without involving thinking skills. critical thinking in mathematics includes the process of testing, questioning, connecting, evaluating all aspects that exist in a situation or a problem [19]. actually critical thinking is a thinking process that occurs in a person and aims to make reasonable decisions about something that can be believed to be true and which will be done later. there are six basic elements that need to be considered in critical thinking, abbreviated as frisco, namely: focus , reason , infrent , situation , clear .and overview [18] . from the description above , the indicators of critical thinking skills in this study are; a) connect as well as apply draft by math ,b) explore , that is ability construct meaning or meaning and investigate ideas mathematics , c) generalize , i.e. interesting conclusion or determine mathematical ideas _ _ inductive or deductive , d) clarify , that is ability evaluate and explain , determine the context of the idea mathematics , and e) solve problem , that is analyze problem so that find correct answer _ by logical . research method type study this is quasi experiment. where, researcher use design two class experiment and control. class experiment given treatment realistic learning based problem non-routine by blended learning, while class control given learning normal online. class experiment and control are taken from study program student education mathematics year enter 2021 which takes eye studying geometry, that is mesp a 2021 class and mesp b 2021 class. as many as 35 people are used for mesp a 2021 class as class experiment, volume 5 | issue 11 | november 2022 | 3 while mesp b class 2021 as many as 35 people are made as class the control. this research _ implemented in the department of mathematics faculty math and natural science, state university of medan. study this use learning design based on the realistic approach. study this use two class experiment-control design with pretest posttest research instrument to ability solving problem and think critical is with use test essay form. the statistical analysis used in study this is with anova two path. research results before conducted learning on both class experiment and control is done test start, use see is student on both class have same ability _ or different, at the same time for grouping ability beginning student. as for the results of p retest second found class _ in study this could seen in the table following. table 1. results of statistical calculations on ability beginning solution problem student. sum of squares df mean square f sig. score * class between groups (combined) 29,557 1 29,557 .201 .655 within groups 12655.523 86 147,157 total 12685,080 87 from table 1 above seen that score significance count is 0.201 and more big from level 0.05 confidence, then set that h o : accepted. so that could concluded that ability beginning solution problem student class experiment with control class is same. table 2. the results of the calculation of the ability statistics beginning think critical. sum of squares df mean square f sig. score * class between groups (combined) 127,682 1 127,682 .879 .351 within groups 12497.182 86 145,316 total 12624,864 87 from table 2 above seen that score significance count is 0.351 and more big from level confidence 0.05 then set that h o: accepted. so that could concluded that ability beginning think critical student class experiment with control class is same. in accordance with destination study this that is for analyze ability solving problem and think critical student who was given learning geometry with approach realistic based problem non-routine by blended learning, then following describe the process and results. the learning process carried out in accordance characteristics realistic approach that is in learning started with give problem non-routine to student, then give opportunity to student for finish it by independent or group as well as discuss result by classic, so expected student find draft or the knowledge contained in problem. meeting first: done online, with given problem _ is " observe " picture below, choose which one is _ side four, then mention what do you mean with side four”. volume 5 | issue 11 | november 2022 | 4 according to opinion i from the picture above which includes side four is build abcd, efgh and mnop. while those who don't including side four is kilj. is polygon with four side and four angle. s2: according to opinion i from the picture above which includes side four is build abcd, efgh and mnop. while those who don't including side four is kilj. is polygon with four side and four angle. st udent respon ses to the proble ms above; s1: according to opinion i from the picture above which includes side four is build abcd, efgh and mnop. while those who don't including side four is kilj. so, quadrilateral volume 5 | issue 11 | november 2022 | 5 s3: from the answers of the three respondents above, it can be concluded that students still have doubts about the definition of a quadrilateral. thus, the non-routine problems above will require discussion in order to find answers rationally. learning with problem non-routine will could increase ability solution critical problem [20]. in discussion class, lecturer provide scaffolding in the form of draw get up triangle on 3 lines, _ four over 4 lines, and so on. so, students could find knowledge or draft that get up side four is field flat closed by 4 lines. _ and, this mathematical process is referred to as a process of improving students' problem solving and critical thinking skills. at the meeting second online, problem given _ is " known " there is plot paper shaped rectangle long and have circumference 24 cm. questions; a) how much many rectangle that has circumference 24 cm? b) how much size rectangle length that has large area biggest? response student on problem mentioned above, among others; s1: answer: rectangular abcd, efgh and mnop are side four, except rectangular ijkl. so, in terms of four is field flat closed line formed by 4 lines. rectangular ijkl instead is side four, because if every line be extended no will shape side four but side three. whereas if the line on the third get up other extended , shape get up fixed and not change . s2: volume 5 | issue 11 | november 2022 | 6 s3: of several response student to problem non-routine the student not yet could complete problem by criticalthinking. that's it, lecturer provide scaffolding in the form of definition side four, parallelogram, trapezoid. then, lower get up parallelogram becomes rectangle long and rhombic. then from rectangle, together student found get up rectangle that is rectangle length that has size the same side. so, students find knowledge that rectangle is a rectangle length that has the same side. meeting third conducted offline. problem non routine given is volume 5 | issue 11 | november 2022 | 7 response student to above problem: _ s1: s2: volume 5 | issue 11 | november 2022 | 8 after learning online and offline, each meeting is held 2 times in class experiments, and 4 times online on learning normal, then given test formative, with results as following . table 3. the results of the calculation of ability statistics solution problem student in class experiment and control. from table 3 above seen that score significance between class experimental and control is 0.001, and less than level confidence 0.05. this show that h o rejected. so that could concluded that there is difference ability solving problem student control class with class experiment. by graphics, can seen in the picture following. from the graph on the side, it can be seen that that class line experiment more tall from control class, then could concluded that ability solving problem mathematics student who was given realistic learning based problem nonsource sum of squares df mean square f sig. corrected model 721,742 a 3 240,581 4,863 .004 intercept 577307.756 1 577307.756 11669,714 .000 class 601.136 1 601.136 12,151 .001 group 120,606 2 60.303 1,219 .301 error 4155.530 84 49,471 total 617100,000 88 corrected total 4877,273 87 a. r squared = .148 (adjusted r squared = .118) volume 5 | issue 11 | november 2022 | 9 routine by blended learning more good from student who was given learning normal online. next seen that score significance between group ability beginning math in class experimental and control is 0.301, and morebig from level confidence 0.05. this show that h o accepted. so that could concluded that no there is interaction among learning with ability beginning to ability solution problem mathematical student. and, by chart could seen that ability solving problem student who was given learning with based realistic approach problem non-routine more than every group on learning normal online. this thing in accordance with results study nasution showing _ that ability solution mathematical students who are given more realistic approach good from learning conventional, good in groups ability beginning mathematics low, medium nor high in smp 18 medan [21]. for ability think critical geometry student could explain as following. table 4. the results of the calculation of ability statistics think critical student . source sum of squares df mean square f sig. corrected model 1565,350 a 3 521,783 13,460 .000 intercept 641565,465 1 641565,465 16550,217 .000 class 1472,727 1 1472,727 37,991 .000 group 92.622 2 46,311 1.195 .308 error 3256,241 84 38,765 total 669850.000 88 corrected total 4821,591 87 a. r squared = .325 (adjusted r squared = .301) from table 4 above seen that score significance between class experimental and control is 0.000, and more small from level confidence 0.05. this show that h o rejected. so that could concluded that there is difference ability think critical student among control class with class experiment. by graphics, can seen in the picture following. from the graph on the side, it can be seen that that class line experiment more tall from control class, then could concluded that ability think critical mathematical student who was given realistic learning based problem non-routine by blended learning more good from student who was given learning normal online. next seen that score significance between group ability beginning math in class experimental and control is 0.308, and more big from level confidence 0.05. this show that h o accepted. so that could concluded that no there is interaction among learning with ability beginning to ability think critical mathematical student. and, by chart could seen that ability think critical student who was given learning with based realistic approach problem non-routine more tall from every group on learning normal online. finding this strengthen results study ability think critical mathematical student school advanced first, show that which is given learning mathematics with more realistic approach online _ good from learning what teachers usually do at school that. analysis of each capability indicator solving problem between group class experiment with class control. volume 5 | issue 11 | november 2022 | 10 t able 5. resul ts of statis tical calculations on capability indicators solution problem. from table 5 above seen that for each indicator of understanding problem, planning completion and implementation plan solution ability problem _ solution problem mathematics have score significance more than level confidence 0.050, then could declared that h o rejected. this show that ability understanding problem, planning completion and implementation plan solution problem mathematics student class experiment different with student control class against theory geometry. because of the graphics ability understanding problem , planning completion and implementation plan solution problem mathematics student in class experiment more tall from control class, then could said that ability understanding problem, planning completion and implementation plan solution problem mathematics student to field geometry student who was given learning with based realistic approach problem non-routine by blended learning more good from student who was given learning normal online . meanwhile, for the looking back indicator, the value of the significance is 0.001 and more small from tarap significance 0.05 then h o accepted. this thing show that ability check return solution mathematical student to given geometry material _ _ learning with based realistic approach problem non-routine by same blended learning just with student who was given learning normal online. analysis of each capability indicator think critical between group to second class experiment and control. table 6. results of statistical calculations on indicators of ability think critical . indicator sum of squares df mean square f sig. understanding 28,409 1 28,409 3,713 .057 planning 4,545 1 4,545 .598 .441 doing 1,136 1 1,136 .052 .821 looking back 501.136 1 501.136 11,670 .001 indicator sum of squares df mean square f sig. connection .284 1 .284 .094 .760 exploration 10,227 1 10,227 1.358 .247 generalization .284 1 .284 .015 .904 clarification 92.045 1 92.045 6,100 .015 solution 955,682 1 955,682 24,157 .000 volume 5 | issue 11 | november 2022 | 11 from table 6 above seen that for each indicator think critical; connect, describe, attract conclusion and clarify student show that class experiment have score significance more tall from level confidence 0.050, then could declared that h o rejected. this show that ability connect, describe, attract conclusion and clarify solution student class experiment different from student control class against theory geometry. due to the average ability connect, describe, attract conclusion and clarify student in class experiment more than control class, then could said that ability connect, describe, attract conclusion and clarify student to field geometry student who was given learning with based realistic approach problem non-routine by blended learning more than from student who was given learning normal online. meanwhile, for the indicator to complete score the significance is 0.000 and more than significance value 0.05 then h o accepted. this thing show that ability complete student to field given geometry _ learning with based realistic approach problem non-routine by same blended learning just with student who was given learning normal online. conclusion 1. there is difference ability solution problem field geometry among student who was given learning with based realistic approach problem non-routine by blended learning with student who was given learning normal online. by general could said that ability complete problem geometry student who was given based realistic approach problem non-routine by blended learning more good from student who was given learning normal online. this thing showed that 3 out of 4 indicators ability solution problem geometry student that is ability understand, plan and complete problem student who was given based realistic approach problem non-routine by blended learning more than student who was given learning normal online. only indicator check return from ability solving problem same geometry _ among student who was given learning with based realistic approach problem non-routine by blended learning with the given learning normal online. 2. there is difference ability think critical among student who was given learning with based realistic approach problem non-routine by blended learning with student who was given learning normal online. by general could said that ability think critical geometry student who was given based realistic approach problem non-routine by blended learning more than student who was given learning normal online. this thing showed that 4 out of 5 indicators ability think critical student that is ability connect, describe, attract conclusion and clarify solution student who was given based realistic approach problem non-routine by blended learning more than student who was given learning normal online. and, only indicator completes from ability think critical same geometric _ among student who was given learning with based realistic approach problem non-routine by blended learning with student who was given learning normal online. 3. not there is interaction among learning with ability beginning to ability complete problem nor ability think critical student. with so, can concluded that in increase ability solution trouble geometry student nor think critical geometric student through based realistic approach problem non-routine by blended learning no need to do grouping on ability student low, medium nor high. volume 5 | issue 11 | november 2022 | 12 bibliography _ [1] 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[21] nasution sw. 2021. upgrade ability solving problem and think critical mathematical man 1 medan students . journal logarithm . vol 2. ed. 1. volume 5 | issue 11 | november 2022 | 14 introduction modeling and analysis of the interaction of neutral and drug populations: a competing species model a. kazmierczak t. h. e. institute 1111 e. brooks st. norman, ok akazmierczak1949@gmail.com abstract the rise of the drugpopulation in the united states has brought concern, debate, and contention to the modern world. thestrategies of the drug cartels are national and are no longer concentrated in a particular location. in this paper, we present a dynamical model of the interaction between drug cartel and dea population. the formulation is based on models of interactions between competitive species [3] type dynamics. an exploration of the long-term dynamics and stability of homogeneous equilibrium solutions and their stability is given. the paper is given in five parts. part one analyzes the current populations. part two analyzes the situation when an additional number of drug users are introduced into the drug population. part three analyzes the situation when there is a decline in the drug population. part four analyzes the situation when the drug population goes to zero. part five presents conclusions based on parts one through four. keywords: drugs,competing species model, equilibrium solutions, stability at equilibrium solutions. mathematic subject classification: 62j12, 62g99 computing classification: i.4 1. introduction drugsare not a new phenomena. however, there is a marked and exponential increase in the growth of drug users. drug users wreak havoc to native citizens. these drug users affect all areas of the global economy, markets, and political and social policies. in addition, the strength and presence of drug organization, activities create issues. in ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 1 particular, the rise of the drug users has reached epidemic proportions. consequently, countries are faced with extremely difficult, complex, and contentious political and social decisions on the issues of drug users. the acceptance of drugs provides a trojan horse of issues, namely, violence, the popularity of drugs in the native country, and the continued growth of drug related problems. hence, countries face the possibility offurther drug users. despite these impending threats, there is not much literature that takes a dynamical systems approach to understanding the spread of drug users at a population level. our primary objective is to bridge the gap. in our framework, we let d represent the drug population. the neutral population is denoted by n: n can be viewed as the total neutral population of a country. this paper is a first step in providing a mathematical modeling framework to study the evolution and interaction between this neutral and drug population. the neutral population is modeled by standard population growth models also, we also consider the addition to the drug population of increased drug users. the paper is organized as follows. in section 2, we develop and analyze the time-dependent autonomous refugee ordinary differential equation (ode) model. we examine the equilibrium solutions, the stability of the equilibrium solutions and investigate the dynamics numerically. in section 3, we consider the situation when more cartels are introduced into the system.we examine the equilibrium solutions, the stability of the equilibrium solutions and investigate the dynamics numerically for this situation also. in section 4, we present consider the situation where there is a decline in drug popultion. in section 5 we present our conclusions based on the analysis in sections 2 and 3 and 4. 2. neutral drug (n, d) ode model consider the mathematical model n = (a1/(1+d1d) – anrd/(1+d2n) – b1n) n = 0 = fn(n, d) (1) d = (a2/(1+d3n) – anrn/(1+d2n) – b2d)(d) = 0 = gr(n, d) (2) the populations n(t) and u(t) represent the populations of the neutral and undocumented populations. new undocumented aliens are slowly coming into the undocumented population. the parameters are all assumed to be positive and their descriptions are given in table 1a. table 1a: list of parameters used in the differential equation model symbols meaning a1 growth rate of the neutral population ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 2 a2 growth rate of the drug population b1 population loss in n due to intra-species competition and natural mortality b2 population loss in d due to intra-species competition and natural mortality anr maximum per capita loss in n due to recruitment by druggies d1 measures the effectiveness of cn in disrupting the growth rate of d d2 measures the resilience of n to recruitment strategies by d d3 measures the effectiveness of d in creating more cartels in the case of di = bi = 0, the mathematical model becomes similar to the competing species model. the parameters di influence the carrying capacity of the individual populations. or instance, if d1>> 1 then the growth rate of d is reduced. this is interpreted as: a highly effective dea population, which can greatly hinder the growth rate of n. the growth rate of the cartel population depends on the successful recruitment from the neutral population. notice, that if d2>> 1 then the recruitment by d is small, also, if d3>> 1, new drug users are introduced into the drugpopulation more slowly the values chosen for the variables in this model are listed in table 1b. table1b: values of parameters a1 a2 b1 b2 anr d1 d2 d3 2 2 0.5 0.5 2 2 2 3 2.1 neutral drug (n, d) ode model consider the mathematical model fn(n, d) = ( a1/(1+d1d) – anrd/(1+d2n) – b1n ) n = 0 (3) fr(n, d) = (a2/(1+d3n) (anrn/(1+d2n)) – b2d ) d = 0 (4) since this system is nonlinear, the first step is linearization using the jacobian. the jacobian for this system is defined as │ ∂f/∂n ∂f/∂d │ j = │ │ │∂g/∂n ∂g/∂d │ ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 3 taking the partial derivatives, simplifying and using the values in table for the parameters, the jacobian becomes. │2/(1+2d)-2d/(1+2n)^2-n -2/(1+2d)^2-2n/(1+2n) │ j = │ │ │ -6d/(1+3n)^2-2d/(1+2n)^2 2/(1+3n)-2n(1+2n)-d │ 2.2 equilibrium points using the maple cas from maplesoft, on (3) and (4) we obtained the real valued equilibrium points: {d = 0., n = 0.}, {d = 4., n = 0.}, {d = 0., n = 4.}, {d = .8213492010, n = .4301871556}, {d = -1.121275136, n = -.4311081397}, {d = .1299378971, n = -.4346164212}, {d = -2.658090053, n = -3.952306486} 2.3analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. table 2 summarizes the results for the current population levels. table 2 – results for current population levels equilibrium point eigen values node type stability (d = 0., n = 0.) 2, 2 repelling unstable (d = 0., n = 4.) -2, -86/117 attracting unstable (d = 4., n = 0.) -44/9+(2/9)*sqrt(185), -44/9-(2/9)*sqrt(185) attracting unstable (d = .8213492010, n = .4301871556) .631870324280523, -1.35022436018052 saddle unstable (d = -1.121275136, n = -.4311081397) 124.789757665452, -7.28136719345222 saddle unstable ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 4 (d = .1299378971, n = -.4346164212) 6.62013265655+9.18652446854370*i, -6.62013265655-9.18652446854370*i attracting spiral asymptotically stable (d = -2.658090053, n = -3.952306486) 3.45507685676904, 1.47441380823096 repelling unstable 3. growth of the drug population in this section, we consider the situation where 5000000 new drug users are added to the population. the mathematical model now becomes fn(n, d) = ( a1/(1+d1(d+5000000)) – anr(d+5000000)n/(1+d2n) – b1n ) n = 0 (3) gr(n, d) = (a2/(1+d3n) (anrn/(1+d2n)) – b2(d+5000000 ) (d+5000000) = 0 (4) 3.1 equilibrium points using the maple cas on (5) and (6) we obtained the following real valued equilibrium points: {d = -5.000000*10^6, n = 0.}, {d = -4.999996*10^6, n = 0.}, {d = -5.000000*10^6, n = 4.}, {d = -4.999999179*10^6, n = .4301871556}, {d = -5.000001121*10^6, n = -.4311081397}, {d = -4.999999870*10^6, n = -.4346164212}, {d = -5.000002658*10^6, n = -3.952306486} 3.2 analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. table 3 summarizes the results for an increased undocumented population level. table 3 – results for increased drug population levels equilibrium point eigen values node type stability (d = -4.999996*10^6, n = 0.), 1.0000000*10^7, 5.000002*10^6 repelling unstable ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 5 (d = -5.000000*10^6, n = 0.) 9.999992*10^6, 4.999998*10^6 repelling unstable (d = -5.000000*10^6, n = 4.), 1.23452844960607*10^5, 4.99999921013939*10^6 repelling unstable (d = 4.999999179*10^6, n = .4301871556) 2.88934551996940*10^6, 4.99999770403060*10^6 repelling unstable (d = 5.000001121*10^6, n = -.4311081397) 5.26749692299691*10^8, 4.99999006030869*10^6 repelling unstable (d = 4.999999870*10^6, n = -.4346164212) 5.84793624029869*10^8, 4.99998950513101*10^6 repelling unstable (d = 5.000002658*10^6, n = -3.952306486) 2.09763520857376*10^5, 5.00000121804262*10^6 repelling unstable 4. decline of the drug population in this section, we consider the situation where 5000000 are removed from the drug population. the mathematical model now becomes fn(n, d) = ( a1/(1+d1(d-5000000)) – anr(d-5000000)/(1+d2n) – b1n ) n = 0 (7) gr(n, d) = -2d/(1+3n)^2 anrd/(1+d2n) – b2(d-5000000) ) (d-5000000) = 0 (8) 4.1 equilibrium points using the maple cas on (7) and (8) we obtained the following real valued equilibrium points: {d = 5.000000*10^6, n = 0.}, {d = 5.000004*10^6, n = 0.}, {d = 5.000000*10^6, n = 4.}, {d = 5.000000821*10^6, n = .4301871556}, {d = 4.999998879*10^6, n = -.4311081397}, {d = 5.000000130*10^6, n = -.4346164212}, {d = 4.999997342*10^6, n = -3.952306486} 4.2analyzing equilibrium points for stability ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 6 in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. table 3 summarizes the results for an decreased undocumented population level. table 3 – results for decreased drug population levels equilibrium point eigen values node type stability (d = 5.000004*10^6, n = 0.) -1.0000008*10^7, -5.000002*10^6 attracting stable (d = 5.000000*10^6, n = 0.) -1.0000008*10^7, -5.000002*10^6 attracting stable (d = 5.000000*10^6, n = 4.) -1.23460735239320*10^5, -5.00000078986068*10^6 attracting stable (d = 5.000000821*10^6, n = .4301871556) -2.88934355603247*10^6, -5.00000229596753*10^6 attracting stable (d = 4.999998879*10^6, n = -.4311081397) -5.26749434300308*10^8, -5.00000993969177*10^6 attracting stable (d = 5.000000130*10^6, n = -.4346164212) -5.84793632770131*10^8, -5.00001049486937*10^6 attracting stable (d = 4.999997342*10^6, n = -3.952306486) -2.09755171342874*10^5, -4.99999878195713*10^6 attracting stable 5. elimination of drug population in this section, we consider the situation where a mere 300,000 new undocumented aliens are added to the radical population. the mathematical model now becomes fn(n, d) = ( a1/(1+d1(0)) – anr(0)/(1+d2n) – b1n ) n = 0 (9) gr(n, d) = anrd/(1+d2n) – b2(0) ) (0) = 0 (8) 5.1 equilibrium points using the maple cas on (9) and (10) we obtained the following real valued equilibrium points {n = 0., d = 0}, {n = 4., d = 0} ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 7 5.2analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable table 5 summarizes the results for a zero drug population level. table 5 – results for zerodrug population levels equilibrium point eigen values node type stability (n = 0., d = 0) 2, 2 repelling unstable (n = 4., d = 0) -2, -86/117 attracting asymptotically stable 6. conclusions in this paper we modeled and analyzed the interaction of neutral and drug populations. a comparison of the results in table 2 indicates that the system already contains some instability, the entire system becomes more unstable and table 3 indicates that with an increase in drug population the system becomestotally unstable table 4 indicates that with a decline in drug population the system becomes more stable, while table 5 indicates that one node is stable while the other is unstable. we interpret this to the fact that drug population could once 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[pubmed] ijo international journal of mathematics volume 03 |issue 05 | may 2020 www.ijojournals.com 17 https://www.ncbi.nlm.nih.gov/pmc/articles/pmc1651663/ https://www.ncbi.nlm.nih.gov/pubmed/6742252 https://www.ncbi.nlm.nih.gov/pmc/articles/pmc1651671/ https://www.ncbi.nlm.nih.gov/pubmed/6742253 https://www.ncbi.nlm.nih.gov/pubmed/2784193 339-article text-1453-1-4-20200501 students' mathematical creative thinking ability strategy mathematical habits of mind based in geometry course hasratuddin 1 , amin fauzi 2 , and nurhasanah siregar 3 1,2 & 3 department lecturers medan state university mathematics email: siregarhasratuddin@yahoo.com ; email: aminunimed29@gmail.com ; e-mail:nurhasanahsiregar@unimed.ac.id abstract this study aims to describe students' mathematical creative thinking abilities based on mathematical habit of maind strategies in geometry course . this research was conducted at the mathematics department of medan state university in 2023. the population in this study were all students majoring in mathematics at medan state university. while the sample was taken by pure positive sampling as many as two classes, namely pspm 22 a with 32 people and pspm 22 c with 32 people. strategy mathematical habit of maind mhm) is a learning that consists of 6 components, namely (1) exploringmathematicalideas , (2) reflectingsuitability solution or strategysolvedproblem , (3) identifyisstrategy or approachproblemused can beapplied to problemsother , (4)identifyisthereis " something more" to activity math that has carried out / generalization , (5) formulate question , and(6)constructexample . components in strategymhm can look at as habits thinkhigh-level mathematics . thinking ability creative is a standard product of mathematics and embodiment from high order thinking. the results of the studyfoundthatstudents ' creativethinkingabilitiesweregivengeometrylessonswithmathematical habits of mindstrategiesbetterthanstudentswhoweregivenordinarylearning.thus , it can besaidthat the mhm strategy can improve the ability to thinkcreativelymathematically. hence , mhm can be made as an alternative to getused to student in thinkcreative and able to trigger growththink high level . keywords : mathematical habit of mind ; think high level ; creative ; geometry . introduction mathematics is one of the auxiliary sciences that is very important and useful in everyday life and in supporting the development of human resources and the development of science and technology. mathematics is a means of thinking to develop a logical, systematic, objective, critical and rational mindset that must be nurtured since elementary education. the more advanced the development of science and technology demands that mathematics finds new forms both as a science and in terms of learning [1]. one _ objective study mathematics is for students capable think rational and creative . mathematics more emphasize activity think , no emphasize from results experiment or results observation [2]. at the moment now this the world is changing so fast, ability thinking and creativity as intelligence artificial be a determinant of excellence someone . lots of fields lost jobs _ because development technology in this era , but also a lot field work new ones popping up that can become a today 's profession . a person's competitive power is determined by ability think level height and creativity [ 17 ].think level high creativity and creativity are also prerequisites for individual success. individual success is largely determined by ability thinking and creativity in doing and solving problems [3] . individuals who have _ ability think creative is someone who can look at the problem from multiple perspectives so that will get effective solution [ 4 ]. no wonder when demands on educational institutions to develop students ' creative thinking abilities become increasingly surfacing. because _ it , for habituation and improve the ability to think creatively achieved student need for maintained through the mathematical habits of mind (mhm) strategy . strategymhmisstrategydevelopmentabilitythinkcreativemathematicalthrough habituation or civilisation thinkcreativemathematical[5]. the mhm strategyconsists of on 6 activities , namely (1) exploringmathematicalideas( explore mathematicalideas) , (2) reflectisanswersobtained _ has in accordance or stillthereiserror( reflect on theiranswer to seewetherthey have made an error), (3)identifyapproach possible problem _ used or applied on problem in scale more big ( identifyproblemsolvedapproachesthat are useful for large classes of problems), (4)ask on self alone or identifyisthereis “ something more ” to activity math that has done / generalized( askthemselveswetherthereis “ something more”/ generalization) , (5) formulate question ( formulated question ) ,and ( 6)constructexample( constructexample) . the mhm strategy can be used independently effective to develop thinking skills mathematical creative thinking through habituation of rational and creative mathematical thinking [6] . on the side therefore , through mhm learning strategies , students are expected to haveleapdeep learning ( hypothetical learning trajectory ) . improve thinking skillsrational , logical , analytical, systematic, critical and creative and have the ability to work together . ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 1 mailto:siregarhasratuddin@yahoo.com mailto:aminunimed29@gmail.com mailto:nurhasanahsiregar@unimed.ac.id components in strategymhm can looked at as habits think math can _ trigger growthabilitythinkcreative . developabilitythinkcreativewithmethodgrow habit thinkcreative and creativityissomethinghabits [7]. this is understandable because creative habits that are carried out consistently and continuously will have implications for the formation of creative thinking abilities.thus , capacity buildingstudent thinking through habituation of creative thinking needs to be done continuously and sustainably ,although this is not always easy to do. likewise, students' creative construction activities also do not always happen easily. therefore, the learning model or strategy learning mathematics with find the device is essential. creativity in mathematics is termed as the ability to think creatively mathematically or briefly called the ability to think creatively. creative thinking is a type of thinking that directs the acquisition of new insights, new approaches, new perspectives, or new ways of understanding things [8] . creative thinking can occur when triggered by challenging tasks or problems.think creative is embodiment from think level high ( higher-order thinking ). it _ because ability think creative is competence cognitive highest necessary _ owned student . think creative is something suite actions that people do with use sense his mind for create fruit thought new from gathering contained memory _ various ideas, descriptions , concepts , experiences , and knowledge . think creative is something mental activity for make ongoing connections _ _ _ continuously ( continuous ), so found correct combination [9] . in this study , the notion of creative thinking ability is associated with problem solving activities math . mathematical creative thinking ability is an ability that includes sensitivity , fluency , flexibility , originality , and elaboration [ 10 ] . sensitivity is the ability to detect, identify, or capture key ideas or concepts in a situation or problem and provide clear and accurate explanations of these ideas or concepts. fluency includes the ability to (1) provide many solutions, (2) provide many examples or illustrations of a concept, or (3) make many statements or questions related to a situation or problem. flexibility includes the ability to (1) use various problem-solving strategies, (2) produce various solutions, or (3) make various statements, questions, or questions related to a situation or problem. authenticity includes the ability to (1) use strategies that are unique, new, or unusual in solving problems or (2) make statements, questions, or questions that are unique or new. detail includes the ability to develop , expand,enrich, or explain in detail to a data, illustration, situation, idea, concept, problem, solution, or problem solving strategy [16] . this can be done by adding, changing, combining concepts, or using various relevant representations. development ability think creative is one _ focus learning geometry . development ability think creative need done along with development method evaluate or method measure it [17]. so, think creative as a process of constructing ideas that emphasize aspects fluency , flexibility , novelty , and detail for generate ideas or method new in produce something product . research methods in accordance with objective study this , type study this is semiexperimental with two class experiment design post-test control . research location this implemented in the department mathematics faculty of mathematics and science knowledge nature , medan state university. population study is whole student major mathematics fmipa medan state university. research sample taken two classes in a manner aim with one eachclass experiment and one again as control class . class experiment taken _ is 33 pspm 2022 a students , with control class taken is 33 pspm 2022 b students . instruments used _ in study this form test form essay as many as 2 questions load ability reasoning and thinking creative . test results analyzed with the ttest, that is see the mean difference between the two classes with assumption variance different . study this done with carry out learning in a manner direct through as many mathematical habits of mind (mhm) strategies four meeting then post-test was carried out . every learning started with gift problem challenge form conflict to student . then , with learning based on mhm facilitated by lecturers during four meeting and ended with do a post-test. research results a. learning process in accordance with objective study this that is for analyze ability think creative given student _ learning geometry with based mathematical habits of mind (mhm) strategy problem , then following describe the process and results . the learning process is carried out in accordance the characteristics and stages of the mhm strategy , namely ; (1) exploringmathematicalideas , (2) reflectingisanswersobtained _ has in accordance or stillthereiserror, (3)identifyapproach possible problem _ used or applied on problem in scale more big , (4)ask on self alone or identifyisthereis “ something more ” to activity math that has carried out / generalization , (5) formulate question ,and ( 6)constructexample . at a meeting first in a manner offline , given problem _ is “ known there is a plot paper shaped rectangle long and have circumference 32 cm. question ; a) how much lots rectangle that has circumference 32 cm? b) how much size rectangle length that has wide area biggest ? response student on problem mentioned above , among others ; ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 2 s1: s2: from the solutions given students at meetings first , in general they not yet can get correct solution , then _ lecturer give learning in a manner first very lecturer give challenge with given problem _ i carefully , choose whichever is _ _ facet four , then mention what is meant with facet four then, through student exploration student response to the problems above; s1: so, quadrilateral is polygon with four side and four corner . from the solutions given students at meetings first , in general they not yet can get correct solution , then _ lecturer give learning in a manner mathematical habit of maind in a manner offline first very lecturer give challenge with given problem _ is “ observe four picture following one by one with carefully , choose whichever is _ _ facet four , then mention what is meant with facet four exploration , students respond to the following problems. problems above; so, quadrilateral is polygon with four side and four corner . from the solutions given students at meetings first , in general they not yet can get correct solution , offline . observe four picture following one by one with carefully , choose whichever is _ _ facet four , then mention what is meant with facet four ” ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 3 s2: from the answers of the two respondents above, it can be concluded that students still have doubts about the definition of a quadrilateral. thus the above problems will require discussion in order to find answers rationally. nelly rhosyida (2017); rina okt explore or reflect problem will can grow ideas and give instruction to reasoning and thinking creative . then . in discussion class , lecturer give chance to student for inspect return while his opinion after lecturer provide scaffolding in the form of lines , and so on . from the hint given , student do , among others; facet four is get up plane that has 4 lines and 4 angles . next , lecturer describe a a shape that has 4 lines and 4 angles , ie then students identify created image _ lecturer while then , lecturer give emphasis with make picture facet three through draw a line sequentially so that something forms _ triangle , that is then , lecturer give chance to student for draw facet three with method lecturer . then , lecturer request student for student can say or find knowledge or draft that get up facet four is field flat closed which is built by 4 lines . very gosh , lecturer give strengthening on concept ; lines, rays and segments . next , students can find definition 5 , 6 and so on until with facet infinity , that is circle . increasing students' reasoning abilities and cre at a meeting second , lecturer give challenge while discussion classical , that is with request student mention definition get up facet the four depicted on the board write , then student mention definition facet four with right , that is field flat closed which is built by 4 lines . then , lecturer continue build draft new with describe get up facet four others ( is sometimes facet four that painted like following , while draw like following by focusing student for they can explore the picture ( above )! then , lecturer request student mention definition of _ see ! student : parallelogram is facet four with two pairs of sides same . ( opinion most student ). student : jajargenjang is facet four two pairs of sides and two pairs of angles the same big . ( opinion most student ). lecturer : remind student with question guide ( same big ? student : then, with notice while evaluate or match picture with question . then , immediately lay out answer , jjar parallelogram is fa lecturer : then, lecturer emphasize understanding student with give question , how your direction _ see ? student : then , student construct and match with example , new can say that “a parallelogram is facet four of which have two pairs of sides in the same direction . lecturer : with answer this , lecturer give rewards to student with say yes, yes, yes! with questions similar , lecturer : lecturer request student mention appropriate trapezium definition with what are you jajargenjang from the answers of the two respondents above, it can be concluded that students still have doubts about the definition of a quadrilateral. thus the above problems will require discussion in order to find answers nelly rhosyida (2017); rina oktaviyanthi1, ria novianaagus (2019), said that learning with explore or reflect problem will can grow ideas and give instruction to reasoning and thinking creative . then . in discussion class , lecturer give chance to student for inspect return while identify and explore or reflect his opinion after lecturer provide scaffolding in the form of draw get up triangle on 3 lines , facet four on 4 given , student do identification and deliver comment for construct his thoug , among others; facet four is get up plane that has 4 lines and 4 angles . next , lecturer describe a a shape that created image _ lecturer while reflect and match picture with their definition _ giv then , lecturer give emphasis with make picture facet three through draw a line sequentially so that something then , lecturer give chance to student for draw facet three with method construct or imitate lecturer . then , lecturer request student for construct get up facet four from get up facet three formed . _ so , student can say or find knowledge or draft that get up facet four is field flat closed which is built by 4 lines . strengthening on concept ; lines, rays and segments . next , students can find definition 5 , 6 and so on until with facet infinity , that is circle . this mathematical process is referred to as a process of increasing students' reasoning abilities and creative thinking. at a meeting second , lecturer give challenge while discussion classical , that is with request student mention definition get up facet the four depicted on the board write , then student mention definition facet four with right ield flat closed which is built by 4 lines . then , lecturer continue build draft new with describe get up facet four others ( parallelogram is sometimes facet four that painted like following , while draw like following . explore and identify picture , lecturer give instructions ( above )! then , lecturer request student mention definition of each, while say student : parallelogram is facet four with two pairs of sides same . ( opinion most student ). student : jajargenjang is facet four two pairs of sides and two pairs of angles the same big . ( opinion most : remind student with question guide ( scapolding) “ whether you there is see sign side the student : then, with notice while evaluate or match picture with question . then , immediately lay out answer , jjar parallelogram is facet four have _ direction . : then, lecturer emphasize understanding student with give question , how your direction _ see ? student : then , student construct and match with example , new can say that “a parallelogram is facet four of two pairs of sides in the same direction . : with answer this , lecturer give rewards to student with say yes, yes, yes! : lecturer request student mention appropriate trapezium definition with what are you segi-4 jajargenjang trapesium sembarang from the answers of the two respondents above, it can be concluded that students still have doubts about the definition of a quadrilateral. thus the above problems will require discussion in order to find answers aviyanthi1, ria novianaagus (2019), said that learning with explore or reflect problem will can grow ideas and give instruction to reasoning and thinking creative . identify and explore or reflect get up triangle on 3 lines , facet four on 4 and deliver comment for construct his thoughts , among others; facet four is get up plane that has 4 lines and 4 angles . next , lecturer describe a a shape that picture with their definition _ give . then , lecturer give emphasis with make picture facet three through draw a line sequentially so that something imitate created image _ get up facet four from get up facet three formed . _ so , student can say or find knowledge or draft that get up facet four is field flat closed which is built by 4 lines . strengthening on concept ; lines, rays and segments . next , students can find definition this mathematical process is referred to as a process of at a meeting second , lecturer give challenge while discussion classical , that is with request student mention definition get up facet the four depicted on the board write , then student mention definition facet four with right parallelogram )with say " there picture , lecturer give instructions take note the shapes in each, while say you just say what you student : parallelogram is facet four with two pairs of sides same . ( opinion most student ). student : jajargenjang is facet four two pairs of sides and two pairs of angles the same big . ( opinion most ) “ whether you there is see sign side the student : then, with notice while evaluate or match picture with question . then , immediately lay out answer , : then, lecturer emphasize understanding student with give question , how your direction _ see ? student : then , student construct and match with example , new can say that “a parallelogram is facet four of : with answer this , lecturer give rewards to student with say yes, yes, yes! : lecturer request student mention appropriate trapezium definition with what are you see . segi-4 sembarang ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 4 student : with notice while identify as well as explore drawing and constructing example on jajar parallelogram as above , then _ say that trapezoid is facet four that have one pair of sides aligned . lecturer : then lecturer give strengthening or rewards with say yes, awesome! next , lecturer : ask student mention definition facet four any . student : terms four any is facet four that don't own side aligned . lecturer : give rewards with say great. learning process the expected will increase ability reason and creativity think student . at a meeting next ( to three ), lecturer give challenge to student with utilise understanding before , that is through picture following , while say " there is sometimes , jar parallelogram this , meanwhile pointing line up parallelogram , if it is erected , then will form like this , meanwhile describe long perse form , like following . lecturer : ask , then , what is the rectangle ? student : identify while explore and answer question , ie rectangle is _ facet four with two pairs of sides same . then for reach reasoning and thinking creative student , lecturer continue questions , like following . lecturer : ask , yes sign long his side same as you see ? then, lecturer continue the part is , try to name it what are you see ! student : then , say " square long is facet the four corners upright . because the answer not yet systematic and correct . then , for build think systematic or good reasoning , lecturer _ give question strengthening as following . lecturer : repeat construct rectangle long from wake up line up parallelogram with give emphasis , see this , meanwhile repeat the question “if jajar parallelogram this enforced , then happen this , meanwhile show get up rectangle panang on the image . then ask , square long is …? student : meanwhile identify , explore , construct , formulate question while match with those already there is , then , mentions , rectangle is " a parallelogram with right angles". lecturer : give rewards with say sweet, while request a student mention repeat , student : rectangle is parallelogram whose angles are right angles. lecturer : ok. such is the learning process that done for students can improve and get used to think creative and reasoned them . then lecturer continue with give question next while construct get up rhombus of _ get up parallelogram , with mention .. segi-4 jajargenjang trapesium segi-4 sembarang persegi panjang belah ketupat segi-4 jajargenjang trapesium segi-4 sembarang persegi panjang ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 5 lecturer : jajargenjang this there is sometimes made with the sides the same long , while pointing get up rhombus formed , then _ ask to student , this get up what ? student : rhombus , sir. lecturer : then continue ask , then rhombus , what ? students : explore , reflect , identify , ask while formulate question and construct example , then mention “ a rhombus is parallelograms whose sides same . lecturer : yes, yes, great! then , lecturer : continue construct rectangle from get up rectangle , meanwhile say " there is sometimes rectangle that dinuat the sides same . then ask while pointing get up formed square , get up what is formed this ? like following . student : square , sir. lecturer : then , get up square ( rectangle ) that , what ? student : wake up rectangle is rectangle with sides the same long . lecturer : yes, great. then, ask a student mention repeat definition square . student : square is rectangle with sides same . then , lecturer ask .. lecturer : is it rectangle that is along ? student : by simultaneously they say , same . after that , lecturer return review to question beginning while read it that is is known there is a plot paper shaped rectangle long and have circumference 32 cm. question ; a) how much lots rectangle length that has circumference 32 cm? b) how much size rectangle length that has wide area biggest ? then ask to student , while give chance to student for can exploring , reflecting , identifying , asking while formulate question and construct example . finally , students can answer with true and convincing that size rectangle that has wide blood biggest from rectangle that has the circumference of 32 cm is 6 cm x 6 cm. meeting fourth . lecturer : give problem form challenge conflict that is mr. somat has a plot land shaped triangle , ie shared to their 3 children which man with method divide by 3 one side the same big the ground , then at point for him he draw a line from corner in front of him . as in the picture following . the question is , is it fair? mr. somat did distribution with method that ? give your answer . belah ketupat segi4 jajargenjang trapesium segi-4 sembarang persegi panjang persegi a b c ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 6 a number of response student is as follows . student,s 1: student,s2: student , s3: from several response student to problem conflict that , student not yet can finish problem in a manner reason and think right creative . _ that's it , lecturer give scaffolding form challenge cognition process as _ following . lecturer : lecturer describe get up triangle anything on the board write , like following , then , for reach rational and creative thinking , lecturer _ request student for mention how many high number of lines triangle and paint a high line triangle the . student : through identification and exploration to triangle , some say _ one , and describe it like following . lecturer : which base and height ? student : the base is ab, the height is cd. lecturer : what if the base line is bc, what is it there is height ? student : mostly _ student answer no there and only one line high triangle the . however , there is different students _ opinion , he , while explore , identify and construct image , then he answer that there are 3 grs in height . for reach think rational or high minded and on his own ideas as think creative , then lecturer continue question form conflict to student , as following . lecturer : ask student the draw it on the board write . student : describe it with right . a b c a b c d a b c d e f ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 7 by mentioning the high lines there are 3, namely cd, ae and bf. lecturer : describe triangle any else , like following . then , ask student paint the height line . after it , ask student divides 3 sides ab equally big . student : with explore , and construct examples already _ there are , then , they are can paint it , like following . then continue for answer question next , that is divide into 3 equal parts ab side . exactly _ they can paint it , like following . lecturer : show and mention the high line facet three acd, dce and bce. student : while identify , explore , reflect , formulate questions , and construct example so that answer is the co line. lecturer : then , what can _ you conclude ? student : broad area facet three-sided three acd, dce and bce are same . lecturer : can you prove it ? student : area facet three-sided three acd, dce and bce are same . because all three have a high line fellowship and common ground . lecturer : give reinforcement and at the same time close meeting at a time remind upcoming meeting will be held formative . inside test results research . after done learning geometry through the mathematical habit of mind strategy for students class experimental and control class , obtained data about ability think creative . test ability think given creative _ to student as many as 2 questions namely ; 2. if any facet three ; elbows, same sides , isosceles and any own the same perimeter , then specify which triangle has wide area biggest and give reason the answer , as for the answers student is as following . a b c a b c d a b c d e o ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 8 from the results tests performed , obtained _ results in summary statistical calculations as following . as for the statistical test that will be tested is ; h o : µ1=µ2; none _ difference ability think creative given students _ learning mhm strategy geometry with given students _ learning normal . h a : µ1≠µ2; there is difference ability think creative given students _ learning mhm strategy geometry with given students _ learning normal . table 1. ability score think creative student class experimental and control class . total total means 85.87879 52.33333 variances 57.42235 158.6667 observations 33 33 hypothesized mean difference 0 df 52 t stats 13.10914 p(t<=t) one-tailed 2.06e-18 t critical one-tail 1.674689 p(t<=t) two-tailed 4,11e-18 t critical two-tail 2.006647 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 9 from table 1 above seenthat the tcount value (13.109) is more big of tcritical (1.675). this show that h 0 rejected and h aaccepted . so so that can concluded that there is difference ability think creative given student learning mhm strategy geometry with given students learning normal . due to average ability think creative students who were given the mhm strategy (85,879) were more big of average ability think creative given student learning ordinary ( 52.333) , then can concluded that ability think creative given students more mhm strategy learning good from given student learning normal . this in accordance with results research conducted .there is the influence of mathematical habits of mind strategy on ability think creative mathematical to students of the mathematics education study program indonesian institute of education [11]. likewise , mahmudi (2009) said that activity exploration of ideas on stage mhm learning can push student think flexible , ie identify various method or solution strategy problem [12]. ability think creative increase consequence happening activity exploration , and this as one mhm strategy stages . with such activities it is possible to obtain strategies that are unique or new [15]. strategy mathematical habits of mind support student for more thinking , reflective , and creative [18] more further , the difference ability think creative the can seen from per indicator , that is as following . table 2. calculations ability indicators _ think creative . a. sensitivity from table 2 above seen that the tcount value ( 4.833 ) is more big of tcritical ( 2.010 ). this _ show that h 0 rejected and h a accepted . so so that can concluded that there is difference ability element sensitivity given student _ learning mhm strategy geometry with given student _ learning normal . due to average ability sensitivity students who were given the mhm strategy ( 16,484 ) were more big of average ability sensitivity given student _ learning ordinary ( 11.515) , then can concluded that ability sensitivity given student _ more mhm strategy learning good from given student _ learning normal . b. elements of proficiency (fluency) from table 2 above seen that the tcount value ( 8.978 ) is more big of tcritical ( 2.006 ). this _ show that h 0 rejected and h a accepted . so so that can concluded that there is difference ability element of student fluency given learning mhm strategy geometry with given student _ learning normal . because the average fluency ability of students who were given the mhm strategy ( 17.212 ) was higher big of the average fluency ability of the students given learning ordinary ( 10.939) , then can concluded that fluency ability of students given more mhm strategy learning good from given student _ learning normal . creativity think will born if ability explore circumstances [14] c. flexibility element from table 2 above seen that the tcount value ( 9.658 ) is more big of tcritical ( 2.004 ). this _ show that h 0 rejected and h a accepted . so so that can concluded that there is difference ability elements of student flexibility are given learning mhm strategy geometry with given student _ learning normal . because the average flexibility ability of students who were given the mhm strategy ( 17.212 ) was higher big from the average flexibility ability of students who are given learning ordinary ( 10.939) , then can concluded that sensitivity sensitivity fluency fluency flexibility flexibility originality originality elaboration elaboration means 16.48485 11.51515 17.21212 10.93939 17.87879 11.42424 16.93939 10.21212 17.36364 8.242424 variances 7.445076 27.44508 4.172348 11.93371 4.234848 10.50189 5.933712 16.92235 7.801136 12.75189 observations 33 33 33 33 33 33 33 33 33 33 hypothesized mean difference 0 0 0 0 0 df 48 52 54 52 60 t stats 4.833208 8.978809 9.658771 8.083423 11.5577 p(t<=t) onetailed 7,1e-06 1.87e-12 1.15e-13 4.7e-11 3.36e-17 t critical onetail 1.677224 1.674689 1.673565 1.674689 1.670649 p(t<=t) twotailed 1.42e-05 3.74e-12 2,3e-13 9,4e-11 6.72e-17 t critical twotail 2.010635 2.006647 2.004879 2.006647 2.000298 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 10 flexibility ability of students who are given more mhm strategy learning good from given student _ learning normal . d. indicator element of originality from table 2 above seen that the tcount value ( 8.083 ) is more big of tcritical ( 2.006 ). this _ show that h 0 rejected and h a accepted . so so that can concluded that there is difference ability elements of student originality that are given learning mhm strategy geometry with given student _ learning normal . because the average originality ability of students who were given the mhm strategy ( 16.939 ) was higher big of the average originality ability of the students given learning ordinary ( 10.212) , then can concluded that originality ability of students who are given more mhm strategy learning good from given student _ learning normal . e. elaboration element indicator from table 2 above seen that the tcount value ( 11.557 ) is more big of tcritical ( 2.003 ). this _ show that h 0 rejected and h a accepted . so so that can concluded that there is difference ability elements of elaboration students are given learning mhm strategy geometry with given student _ learning normal . because the average elaboration ability of students who were given the mhm strategy ( 17.363 ) was higher big from the average elaboration ability of students who are given learning ordinary ( 8.242) , then can concluded that given student elaboration abilities _ more mhm strategy learning good from given student _ learning normal . conclusion ability think creative given student learning geometry with more mathematical habits of mind strategies good from given student ordinary learning _ conducted in the department medan state university mathematics . where, the average ability think creative gain _ given student _ learning geometry with the mhm strategy is 85.878 while on learning normal is 52.333. next , when seen of the average ability per element indicator of thinking creative among others; sensitivity , fluency , flexibility , originality and elaboration , then all the indicator elementlearning with more mhm strategy good from learning normal .differences in average ability per ability indicator think creative between students who were given the mhm strategy with learning usually have _ difference biggest is element elaboration . this _ in accordance with results research conducted by _ bibliography [1] hasratuddin, amin fauzi dan nurhasanah siregar. 2020. geometry learning based on cognitive conflict in the department of mathematics. international journal of mathematics.volume 3| issue 10|, p.1 – 9, issn: 2456-8538 [2] rahmah, nur. 2013. hakikat pendidikan matematika. jurnalal-khwarizmi, volume 2, oktober 2013, halaman 1 – 10. [3] alexander, k. l. (2007). effects instruction in creative problem solving on cognition, creativity, and satisfaction among ninth grade students in an introduction to world agricultural science and technology course. disertasi pada texas tech university. [online]. tersedia:http://etd. lib.ttu.edu/theses/ available/etd-01292007-144648/unrestricted/alexander_ kim_dissertation.pdf. [4] dara dan hasratuddin. 2021. development of learning tools based on online realistic mathematics education to improve critical thinking ability and adversity quotient of students in man 1 medan. journal of education and practice. issn 2222-288x (online) vol.12, no.2, 2021. [5] dzulfikar (2018). habits of mind calon guru matematikadalampemecahan masalah matematis. suska journal of mathematics education.vol.4, no. 1, 2018, hal. 1-8 [6] millman, r.s. &jacobbe, t. (2008). fostering creativity in preservice teachers through mathematical habits of mind. proceeding of the discussing group 9. the 11th international congress on mathematical education. monterrey, mexico, july 6 – 13, 2008. [online]. tersedia: http://dg.icme11.org/document/get/272. [19desember 2022]. [7] sternberg, robert. (2006). creativity as a habit. [online]. tersedia: http://www.worldscibooks.com/socialsci/etextbook/6211/6211_chapter01.pdf. [11 januari 2023] [8] mcgregor, debra. (2007). developing thinking; developing learning. maidenhead: open university press. [9] evans, james r. 1994. creative thinking : in the decision and management sciences. jakarta : bumi aksara. [10] kathleen cotton. 1991. teaching thinking skills.school improvement research series ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 11 http://dg.icme11.org/document/get/272. %5b19 http://www.worldscibooks.com/socialsci/etextbook/6211/6211_chapter01.pdf. %5b11 [11] tina sri sumartini. 2022. pengaruh habit of mind terhadapkemampuanberpikir kreatif matematismelalui metode pembelajaran improve. journal pendidikan indonesiavol 11, no 1 (2022) [12] rina oktaviyanthi, ria novianaagus. 2019. eksplorasikemampuanpemecahan masalah berdasarkan kategori proses literasimatematis. jurnal pendidikan matematika volume 13, no. 2, juli 2019, pp. 163-184 163 [13] hasratuddin. 2022. improving critcal thinking and emotional intelligence capabities of scondariyshool students through realistic mathematics education approch.. ijism (international journal of innovation in science and mathematics). issn: 2347 – 9051, vol. 5, issue 1, p 1-6. month: jan, 2022 [14] mahmudi, ali. 2009. strategi mathematical habits of mind (mhm) untukmengembangkankemampuanberpikir kreatif matematis. makalah pada konferensi nasional pendidikan matematika iii universitas negeri medan, 23 – 25 juli 2009 [15] heldanita. 2018. pengembangankreativitasmelaluieksplorasi. jurnalilmiahtumbuh kembang anak usia dini volume. 3 no. 1. maret 2018 e-issn: 2502-3519 [16] haylock, d. (1997). recognizing mathematical creativity. zentralblatt für didaktik der mathematik (zdm) – the international journal on mathematics education. [online]. tersedia: http://www.emis.de/journals/zdm/zdm973a5.pdf. [15 maret 2022] [17] pehnoken, e. (1997). the state-of-art in mathematical creativity. [online] zentralblatt für didaktik der mathematik (zdm) – the international journal on mathematics education. tersedia:http://www.emis.de/journals/zdm/ zdm973a1.pdf . [15 januari 2022] ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 11 | november 2023 | http://ijojournals.com/index.php/m/index 12 https://journal.institutpendidikan.ac.id/index.php/mosharafa/issue/view/103 http://www.emis.de/journals/zdm/zdm973a5.pdf. %5b15 http://www.emis.de/journals/zdm/ zdm973a1.pdf . [15 januari 2022] introduction physiological responses of solanum nigrum l. species to the heavy crude oil ghazala ahmad hamaden university of benghazi, faculty of science, botany department, benghazi – libya.. of benghazi, benghazi – libya. abstract the remediation of oil contaminated soils has been a major problem in oil producing countries. recently use of phytoremediation to clean such polluted sites has been on investigated. in order to identify plants that can enhance the remediation, solanum nigrum l. (black nightshade) was used on different concentrations (0.0,0.5,1.0,2.0,4.0,6.0,8.0,10.0 v/v) for seed germination and seedling growth. the results showed that seed germination and seedling performance were enhanced under heavy crude oil compound. this study indicates that black nightshade have more potential for resistance to crude oil concentrations and that can be used as promising tools for phytoremediation technology. key words: solanum nigrum l. black nightshade, heavy crude oil, phytoremediation. 1. introduction the word petroleum means “rocky oil” or “oil from the earth”. although exactly how crude oil originated is not established, it is generally agreed that crude oil is derived from the remains of animals and plants that lived in marine water millions of years ago, types of crude oil: based on the density crude oil is divided in to the following groups: light crude oil and heavy crude oil 1.1. effects of heavy crude oil on tested plant solanum nigrum l. commonly known as (black nightshade) is a dicot weed in the solanaceae family, annual herb, all green and unripe parts contain steroid glycosides, in form of steroid glycoalkaloids. in the genus solanum they are important, and are widely regarded as defensive allelochemicals of the plants against microorganisms and herbivores. the main steroid alkaloids are solasonine and solanine, and are called solatrioses (fig. 1. 1). solanine solasodine fig. 1.1. structure of alkaloids in solanum nigrum weed. diesel oil pollution is harmful on the chlorophyll and protein contents of the black nightshade. it inhibited the growth ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 65 of plants causing reduction of both chlorophyll and water contents (seklemora et al., 2001). pollutioninduced degradation in photosynthetic pigments were recorded by a number of researches (puckett et al., 2003; mut et al., 2010). diesel oil was found to inhibit the metabolic and physiological processes including photosynthesis and transportation. the photosynthetic pigments are the most likely to be damaged by diesel pollution. chlorophyll pigments under stress may undergo several photochemical reactions such as oxidation. hence any alteration in chlorophyll concentration may change the morphological, physiological and biochemical behaviour of the plant. plant responses to oil pollution are different and depend on plant species, oil kind, amount and concentration, exposure times and environmental condition (pezeshki et al., 2000; spiares et al., 2001; zangh et al., 2007; besalatpoor et al., 2008). soil polluted by crude oil have been found to inhibit plant growth (agbogidi, 2011) resulting in hypoxic state of the soil the displacement of air from the soil pore spaces by the crude oil. soil polluted with petroleum (adedokun & ataga, 2007; besaltpour et al., 2008). changes in soil properties due to contamination with petroleum derived substances can lead to water and oxygen deficits as well as to shortage of available forms of nitrogen and phosphorus (njoku, 2008). cell membranes are damaged by penetration of hydrocarbon molecules, leading to leakage of cell contents. oils reduce transpiration rate, probably by blocking stomata and intercellular spaces. this may also be the reason for the reduction of photosynthesis. many researches showed that the presence of the oil resides in the soil has negative effects on the plant metabolism and protein synthesis (ekpo & nwaankpa, 2005; richard et al., 2007; okpokwasili & odokuma, 2007; besaltpour et al., 2008; teng et al., 2010; bamidele & igiri, 2011). oil pollutions reduce some plant growth parameters such as: plant height, leaf number, leaf surface, plant fresh and dry weight, biomass (omosun et al., 2008), photosynthetic pigments and also nutrient absorption (rosso et al., 2005). crude oil induced environmental stress up on the plant seedlings. therefore, the overall objective of thin research in to investigate the response of solanum nigrum l weed plant to the effect of heavy crude oil. 2. material and methods 2.1. plant seed: seeds of solanum nigrum l. (black nightshade) family solanaceae. seeds (weeds) were collected from hei alsalam area and were stored at room temperature ranges from 25 to 30°c. 2.2. chemicals: formaldehyde, distilled water (dw), the crude oil used was (from al-breiga port, field alamal ) heavy crude oil, with the following concentrations of each type of oil. (0.0, 0.5, 1.0, 2.0, 4.0, 6.0, 8.0, 10.0 (% v/v). 2.3 germination test seed preparation prior to germination: the seeds to be used in this work were surface sterilized by washing with 10 % formaldehyde and rinsed three times with sterile water for 10 minutes (wood et al., 2006). ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 66 sterilized glass petri dishes (9.0cm) lined with double layers of watmann no.1.filter paper was used. glass petri dishes were cleaned and sterilized in an oven at 180°c for 2 hours. seeds were placed in the petri dishes each contains ten. six replicates were used for each treatment of different kinds of crude oil. the filter paper was watered by adding 3 ml of distilled water or solution to be tested. all petridishes were in incubated in an incubator of (gallerkamp) at temperature of 20°c for one week. distilled water was or tested solution was added to the petridishes whenever it was needed to all replicates at the same time. germinated seeds were counted daily and germination percentage was calculated at the end of the germination period for each treatment as following: germination percentage = number of seeds sown (yang et al., 2005). germination rate (gr) = ـــــــــــــــــــــــ (n) number of emerged seeds in day (d) is day after planting (rastegar. et al., 2011). mean germination time (mgt) = n1*d1+ n2*d2 + n3*d3 ............ total number of days (gairola et al., 2011). where, n = number of germinated seed, d = number of days daily and final germination percentages (%) were calculated for the determination of some of the following parameters. mean daily germination is an index of daily germination rate mean daily germination (mdg) = fgp is final germination percent, (d) is day of maximum germination (experiment period) (rastegar et al., 2011). germination index (gi) = gs * lc gc * lc (gairola. et al., 2011) where (gs) and (gc) are number of seeds germinated in the sample and control, respectively, whereas ls and lc are the radicle length in the sample and control, respectively. in the case of weeds, the length was measured as whole seedlings due to their smaller size. ten seedlings of each replicated of each treatment were weighed together due to the small size of weed the seedlings. relative water contents (%) = fresh weight dry weight * 100 (gairola. et al., 2011). percentages of seedling emergence = number of seedling that emerged number of seeds sown *100 (agbogidi, 2011b). seedling vigor index (svi) is calculated using the following modified formula: svi = seedling length (cm) * final germination percentages. (mut et al., 2010). tolerance index (ti) is calculated using the following modified formula: ti = length of seedling in treatment fgp d number of seed that germinated *100 % n d ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 67 length of seedling in control (verdeguer et al., 2009). 3. results: table.3.1 showed the effect of heavy crude oil on the mean values of daily seed germination percentages of solanum nigrum l. (black nightshade). the results indicated no significant differences of seed germination percentages (%), i.e. no seed germination had occurred during the first four days of germination period, under all different dilutions of heavy crude oil including the control treatment . during the fifth day, only seeds treated with distilled water (control) were germinated (f = 4.00, p< 0.01). tukey's pariwise comparisons test reveals significant differences between control and other treatment means of heavy crude oil, but daily germination percentages were found to be higher during the last days of germination time figure 3.1. the effect of different dilutions of heavy crude oil on the means of germination rate (gr), mean daily germination (mdg) and mean germination time (mgt) of solanum nigrum are represented in table 3.2. results indicated that, all the above mentioned measures of this seed were not significantly affected by exogenous application of different dilutions of heavy crude oil. whereas, the germination index (gi) of same plant was significant (f=3.81, p< 0.05) within treatments which was increased under some treatments and reduced by higher dilutions of heavy oil. tukey's pariwise comparisons test reveals significant differences between control and dilutions of 4.0, 8.0 10.0 (% v / v). seedling length (cm) of solanum nigrum measured under different dilutions of heavy crude oil table 3.3. was significant (f = 10.30, p< 0.001), within different treatments. different dilutions of lower concentration of same oil had increased the length of black nightshade seedlings with increasing concentrations of heavy crude oil above 8.0 (% v / v). tukey's pariwise comparisons test reveals significant differences in seedling length of solanum nigrum under the control in comparison to other different treatment means table 3.3. seedling fresh weight (g) parameter of solanum nigrum was not affected by different concentrations of heavy crude oil. but seedling's dry weight of the same plant species was significantly increased under higher concentrations of this oil (f = 3.73, p< 0.01). tukey's pairwise comparison test reveals significant differences in seedling dry weight (g) of solanum nigrum between lower concentration including control and the highest concentration 10.0 (% v / v) of heavy crude oil table 3.3. relative water content percentages (rwc %) of black nightshade are shown in figure 3.2. this parameter was significantly reduced under higher dilutions of heavy oil (f = 5.87, p< 0.01). tukey's pairwise comparison test reveals significant differences in relative water content percentages of same plant between lower concentration including control and highest concentration 10.0 (% v / v) of heavy crude oil. seedling emergence percentages (%) were not affected by the same oil figure 3.3. there were small redactions in seedling vigor index and tolerance index of the same target plant species. using one way analysis various, results showed significant differences in these parameters within different dilutions of crude oil (f = 2.90, p< ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 68 0.05) and (f = 7.64, p< 0.001) respectively. tukey's pairwise comparison test reveals significant differences between different concentrations of heavy crude oil in (svi). while in the case of (ti) differences were found in the seedlings developed under untreated (control) in compared to the different treatments means of the oil table .3.4. 4. discussion oil pollution in whatever form is toxic to some plant species and their environment has been observed by many researcher workers (opeolu, 2000; adenipekun & kassim, 2006; adenipekun et al., 2009; kelechi et al., 2012 ) that crude oil affects soil properties and this in turn affects the physiological, anatomical and development of plants grown on such soils. the germination process is a very extremely sensitive phase in plant growth and development, being indicative to any type of environmental contaminants. the effect of heavy crude oil residues was investigated for some seed parameters of some weeds which include solanum nigrum. these parameters of solanum nigrum was promoted by different dilutions of heavy crude oil. these results are agreed with the findings reported by objegda & atebe (2007) who found that germination index of indian mustard was not affect with diesel oil contaminated soil. kirk et al., 2002 of grasses germinated successfully in different levels of petroleum hydrocarbon contamination. the effect of phenol and naphthol compounds, as water soluble fractions of crude oil, on the germination and seedling development was investigated for seeds of some crops cultivated in libya. the obtained results showed that, low concentrations of both phenol and naphthol caused an increase of germination percentages of seeds of tested plant. this is probably due to that, low dilutions of these compounds may act as signal for αamylase production in the seeds (edema, 2012). these results agreed with those obtained by (el-barghathi, 1985) who found that low dilutions of naphtol had a promoting effect on rate and final germination of oat seeds. this was probably caused by strong resistant qualities of the black nightshade seeds. this high quality of resistance marks the foregoing species to be considered as promising candidates for the phytoremediation of sites crude polluted with petroleum oil. the study underscores the need for the use of cheap, available, and environmental friendly technology as a remedy for the harmful effects of petroleum contaminants in the environment. coating the seeds with oily substances prevent water and air movement in to the seed and directly causes toxic actions. one of the most possible reasons for seed germination inhibitory effects in crude oil contaminated sites is due to insufficient aeration of hypoxic or anoxic (having little or no oxygen, respectively), conditions. the embryo of seeds could have been injured or killed if it comes in contact with the oil. this effect could also be as a result of format ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 69 table 3.1. effect of different dilutions of heavy crude oil on daily germination percentages (%) of solanum nigrum l. (black nightshade) seeds. + = not significant. ** = significant at p< 0.01. ± = semean. similar letters = not significant. different letters = significant. treatment (%) germination percentages (%) day 1 day 2 day 3 day 4 day 5 day 6 day 7 0.0 + 0.00 ± 0.00 + 0.00 ± 0.00 + 0.00 ± 0.00 + 0.00 ± 0.00 ** 6.67a ± 3.3 + 96.8 ± 3.3 + 96.8 ± 3.3 0.5 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 86.8 ± 6.7 86.8 ± 6.7 1.0 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 93.3 ± 3.3 93.3 ± 3.3 2.0 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 86.7 ± 8.8 90.0 ± 10.0 4.0 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 80.0 ± 0.0 86.7 ± 6.7 6.0 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 93.3 ± 6.7 100.0 ± 0.0 8.0 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 80.0 ± 5.8 83.3 ± 6.7 10.0 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00 ± 0.00 0.00b ± 0.0 83.3 ± 8.8 83.3 ± 8.8 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 70 fig. 3. 1. effect of different dilutions of heavy crude oil on daily germination percentages (%) during the fifth day (a) and the seventh day (b) of solanum nigrum l. (black nightshade) seeds. + = not significant. 0 20 40 60 80 100 120 0.0 0.5 m ea ng er m in at io n pe rc en ta ge s (% ) + a b effect of different dilutions of heavy crude oil on daily germination percentages (%) during the fifth day (a) and the l. (black nightshade) seeds. + = not significant. 1.0 2.0 4.0 6.0 dilutions of heavy crude oil (% v / v) effect of different dilutions of heavy crude oil on daily germination percentages (%) during the fifth day (a) and the bars = semean. 8.0 10.0 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 71 8 table 3.2. effect different dilutions of heavy crude oil on the means of germination rate (gr), mean daily germination (mdg), mean germination time (mgt) and germination index (gi) of solanum nigrum l. (black nightshade) seeds. + = not significant. * = significant at p< 0.001. ± = semean. similar letters = not significant. different letters = significant. treatment (%) gr mdg mgt gi 0.0 + 1.4 ± 0.05 + 13.8 ± 0.5 + 18.4 ± 0.9 * 100.0a ± 0.00 0.5 1.2 ± 0.09 12.4 ± 0.95 16.1 ± 1.2 81.0ab ± 9.0 1.0 1.3 ± 0.05 13.3 ± 0.5 17.3 ± 0.62 76.95a ± 4.9 2.0 1.3 ± 0.14 12.9 ± 1.4 16.4 ± 1.7 71.4ab ± 8.8 4.0 1.2 ± 0.09 12.4 ± 0.95 15.5 ± 0.7 67.9b ± 5.6 6.0 1.4 ± 0.00 14.3 ± 0.00 18.0 ± 0.6 81.9ab ± 2.9 8.0 1.2 ± 0.09 11.9 ± 0.95 15.2 ± 1.1 64.4b ± 5.0 10.0 1.2 ± 0.10 11.9 ± 1.3 15.5 ± 1.6 64.7b ± 6.9 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 72 9 table 3.3. effect of different dilutions of heavy crude oil on seedling length (cm), fresh and dry weight (g) of solanum nigrum l. (black nightshade) seedlings. + = not significant. ** = significant at p< 0.01. * ** = significant at p< 0.001. ± = semean. similar letters = not significant. different letters = significant. treatment (%) mean values length (cm) fresh weight (g) dry weight (g) 0.0 *** 89.9a ± 2.0 + 0.01 ± 0.002 ** 0.003a ± 0.0002 0.5 81.0ab ± 4.8 0.10 ± 0.010 0.003a ± 0.0002 1.0 71.5b ± 3.2 0.01 ± 0.010 0.003a ± 0.0004 2.0 68.8b ± 2.9 0.01 ± 0.009 0.003ab ± 0.0003 4.0 67.97bc ± 1.6 0.01 ± 0.007 0.003ab ± 0.00006 6.0 70.97b ± 1.2 0.01 ± 0.006 0.003ab ± 0.0002 8.0 67.10bc ± 0.8 0.01 ± 0.006 0.004ab ± 0.0002 10.0 67.40bc ± 1.2 0.01 ± 0.007 0.004b ± 0.0006 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 73 10 fig. 3.2. effect of different dilutions of heavy crude oil on relative water content percentages (%) of solanum nigrum l. (black nightshade) seedlings. ** = significant at p< 0.01. similar letters = not significant. different letters = significant. bars = semean. a a a ab ab ab ab b 93.5 94 94.5 95 95.5 96 96.5 97 97.5 98 0.0 0.5 1.0 2.0 4.0 6.0 8.0 10.0 m ea n w at er c on te nt p er ce nt ag es ( % ) dilutions of heavy crude oil (% v / v) ** ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 74 fig. 3.3. effect of different dilutions of heavy crude oil on seedling emergence percentages (%) of nightshade) seedlings. + = not significant. 0 20 40 60 80 100 120 0.0 0.5 s ee dl in g em er ge nc e pe rc en ta ge s + 11 effect of different dilutions of heavy crude oil on seedling emergence percentages (%) of solanum nigrum 1.0 2.0 4.0 6.0 dilutions of heavy crude oil (% v / v) solanum nigrum l. (black bars = semean. 8.0 10.0 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 75 12 table 3.4. effect of different dilutions of heavy crude oil on seedling vigor index (svi) and tolerance index (ti) of solanum nigrum l. (black nightshade) seedlings. * = significant at p< 0.001. * ** = significant at p< 0.001. ± = semean. similar letters = not significant. different letters = significant. treatment (%) mean values svi ti 0.0 * 8682a ± 164 *** 1.0a ± 0.00 0.5 7070ab ± 922 0.9ab ± 0.07 1.0 6694ab ± 543 0.8b ± 0.04 2.0 6227ab ± 867 0.8b ± 0.04 4.0 5909ab ± 588 0.8b ± 0.02 6.0 7097ab ± 118 0.8b ± 0.013 8.0 5586b ± 416 0.8b ± 0.02 10.0 5631b ± 676 0.8b ± 0.009 ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 76 13 0.0 0.5 1.0 2.0 4.0 6.0 8.0 10.0 shows the effect of different dilutions (% v / v) of heavy crude oil on the germination of solanum nigrum l. (black nightshade) seeds. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 77 14 -ion of polar compounds dissolved in water that could penetrate the seed coat and prevent the germination process (wang et al., 2000; adam & duncan, 2002). the cessation of seed germination by crude oil is in line with previous research reports (anoliefo &vwioko 2001; trapp et al., 2001 ; anon, 2003; sharafi et al., 2007; malekhossein & gholamreza, 2007; omosun et al., 2008 ; bamidele & igiri, 2011 and debojit et al., 2011; sheta omar et al., 2013; agbogidi & april, 2013). in general, seed germination of species used in this work was enhanced under the stimulation of heavy crude oil. this is might be due to the hydrophopicity of heavy oil which possesses less solubility in water and therefore causes phytotoxicity. there is however, lack of information on the effects of crude oil on some biochemical processes such as oxidative stress parameters in plant species used in study. seedling performance of plants used in this study was measured under different dilutions of different of oil compound. seedling growth of solanum nigrum. plants that are able to germinate successfully and tolerate the contaminant and show root elongation are tolerant plants (ogbo, 2009 & obj et al., 2008). but different seedling parameters in terms of fresh and dry measures were increased under different dilutions of the used oils. the high survival rate of these seedlings due to their tolerance to the high levels of oil compounds (anoliefo & edegbai, 2001). this stress condition may interfered with water absorption and gaseous exchange and led to reduction in seedling growth which apparent in the decrease of growth seedling parameters in poorly aerated environment (quinones-aquilar et al., 2003; bamidele jf. 2010). this can be attributed to the decrease in relative water content plant dry weight and plant fresh weight of corn seedlings as the crude oil concentrations increased. these results revealed that both black nightshade and wheat showed good performance under both types of oil used in the study. impact of stressful conditions of crude oil pollution has been shown to have adverse effects on plant growth and these may range from morphological aberrations, reduction in biomass to stomatal abnormalities (victor & sadiq, 2002). growth reduction could also be explained as being due to harmful effects of oil. growth reductions following oil pollution of soil have been reported by same authors such as anoliefo & edegbai (2000), of (odjegba and sadiq, 2002; baran et al., 2002; ikhajiagbe and anoliefo, 2011). different plants can tolerate different levels of petroleum hydrocarbons. hydrocarbon contamination of soil reduced plant growth but increased microbial activity (xu and johnson, 1995; wioko & fashemi, 2005). this study has demonstrated that crude oil contamination of soil has a highly significant effect of a reducing the biomass accumulation in jatropha curcas seedlings. this study has implication on sustainability of using jatropha curcas as a biodiesel species. crude oil and petroleum products vary considerably in their toxicity, and the sensitivity to petroleum varies according to plant species. the toxicity of crude oil can be interpreted as the toxicity of a complex mixture of inorganic and organic, chemicals. the observed negative in the germination percentages, rate of germination as well as, the growth parameters (seedling length , fresh weight and biomass production) measured could be attributed to the numerous hydrocarbons and related compounds which are toxic to living organisms including plants. generally, the highest ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 78 15 performance in terms of percentage emergence and seedling development was recorded in black nightshade this study indicates that both black nightshade and wheat have more potential for resistance of crude oil concentrations and they can be used as promising tools for phytoremediation technology. plant that tolerates higher concentrations of crude oil includes solanum nigrum l. (black nightshade) weed plant species. this type of plants are recommended to be used as phytoremedation models especially the weed solanum nigarum for cleaning up areas polluted with heavy crude oil residues this is propably due to its toxic contents of secandry metabolites that may counteats that hanful effects of these compounds used in the presend work. the influence of heavy crude oil upon different target plant species used in this research was not clearly pronounced (different) ie all plant species were not affected by the applications of heavy crude oil residues. this might be due to that heavy oil is more viscous and less soluble in water. summary and conclusions the effected of crude oil (heavy) was examined for seed germination and seedling performance in the case of seed measures of solanum nigrum l. (black nightshade). was enhanced under of heavy crude oil compound. furthermore, seedling performance was noticed to be good in residues of solanum nigrum under heavy crude oil compound. based on the obtained results, it is indicated that black nightshade have more potential for resistance to heavy crude oil concentrations and can be used as promising tools for phytoremediation technology. solanum nigrum l. (black nightshade) as a weed plant species. is recommended to be used as phytoremedation model for cleaning up areas polluted with crude oil. references adam g. and duncan h. (2002). influence of diesel fuel on seed germination, environmental pollution. 120 (2): 363 370. adedokun om. and ataga ae. (2007). effects of amendments and bioaugumentation of soil polluted with crude oil, automotive gasoline oil, and spent engine oil on the growth of cowpea (vigna unguiculata l. walp). sci. res. essay. 2 (5): 147 149. adenipekun co. and kassim l q. 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(2001). biostimulationbasedbioremediation of diesel fuel field demonstration. bio. 12: 311 316. sharifi m ., sadeghi y. and akbarpour m. (2007). germination and growth of six plant species on contaminated soil with spent oil. int. j. environ. sci. tech. 4: 463 470. sheta o., sanat., hnan s. and gada f. (2013). germination tolerance of four mutant lines of barley (hordium vulgare l.), wheat (triticum aestivum l.) and garden cress (lipidium sativum l.) to crude oil contaminated soil journal of environmental science and engineering. a 2: 36 40. spiares jde., kenwrthy k. and rhykerd r. (2001). emegence and height of plants seeded in crude oil cont aminated soil, texas journal of agriculture and natural re-sources. 14: 37 46. teng y., shen y., luo y., sun x., sun m. and christie p. (2010). influence of rhizobium meliloti on phytoremediation of polycyclic aromatic hydrocarbons by alfala in an aged contaminated soil, journal of hazardous materials. 10: 1 29. trapp s., kohler a., larsen lc., zambrano kc. and karlson u. (2001). phytotoxicity of fresh and weathered diesel and gasoline to willow and poplar trees. j. soils sediment.1 (2): 71 76. verdeguer m., blazquez m a. and boira h. (2009). phytotoxic effects of lantana camara, eucalyptus camaldulensis and eriocephalus africanus essential oils in weeds of mediterranean summer crops. biochemical systematics and ecology. 37: 362 – 369. victor jo. and sadiq ao. (2002). effects of spent engine oil on the growth parameters chlorophyll and protein levels of amaranthus hybridus l., journal of the environmentalist. 22: 23 28. wioko vd e. and fashemi d s. (2005). growth response of ricinus cummunis l. (castor oil) in spent lubricating oil polluted soil. j. applied sci. environ. manage. 9 (2): 73 79. wang j., jia cr., wong ck. and wong pk. (2000). characteristics of polycyclic aromatic hydrocarbon created in lubricating oils. water, air pollut. 120: 381 396. wood tk., mulchandani a. and chen w. (2006). engineering plantmicrobe symbiosis for rhizoremediation of heavy metals. applied and environmental microbiology. 72 (2): 1129 1134. xu jg. and johnson rl. (1995). root growth, microbial activity and phosphatase activity in oilcontaminated, remediated and uncontaminated soils planted to barley and field pea. plant and soil. 173: 3 10. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 82 19 yang q., ye w., deng x., cao h., zhang y. and xu k. (2005). seed germination eco-physiology of mikania micrantha h.b.k. ybaotn. bot. bull. acad. sin. 46: 293 – 299. zhang cg., leung kk., wong ys. and tam nfy. (2007). germination, growth and physiological responses of man-grove plant (bruguiera gymnorrhiza) to lubricating oil poll ution, journal of environmental and experimental botany. 60: 127 136. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 83 control of the learning process at district 5 july elementary school, bireuen rahmi hayati1, asrul karim2, facrurazi3, marzuki4, sumarlin mangandar marianus5, hasratuddin6 1, 2 ,34primary teacher education, almuslim university 5 primary teacher education, katolik santo thomas university 1e-mail:hayatirahmi@yahoo.com 6email:siregarhasratuddin@yahoo.com abstrak. the focus of this article is on a process and outcome analysis of teaching at the district 5 julielementary school in the city of bireuen. the point is to figure out if the end result of a process matches the initial expectations, or if the process is still progressing toward meeting those standards or performing as expected. the educational process is a procedure whose implementation and results must be monitored in order to provide outcomes that are in keeping with the desired goals and standards of quality. for this reason, it's intriguing to ponder how debating the legitimacy of an event based on ostensibly objective data, how to make crucial data easily digestible for those who need to see it, and how to analyze data accurately with appropriate statistics will give us a schematic of a process's workings that's both detailed and accessible. however, the data used are authentic data, such as the results of the national exam taken by students at 5 julielementary school, bireuen district from 2013/2014 to 2017/2018 in four subject areas (mathematics, indonesian, english, and natural science). katakunci:control analysi, learning process, state elementary school introduction multivariate analysis is a common research tool in studies with several independent variables. choosing the appropriate multivariate analysis technique requires consideration of the research's aims, the assumptions behind the technique(s) under consideration, and the measurement scale(s) to be used during data collection in order to yield accurate and reliable results. multivariate statistical analysis is a branch of statistical science that measures the strength of associations between groups of variables to draw conclusions about those groups as a whole. scholars benefit greatly from multivariate analysis in their quest to find answers to broad, complex, or deterministic problems. (wustqa et al., 2018). multivariate analysis is a statistical method for analyzing data with more than one independent variable. because multivariate data analysis requires more complicated calculations than single-variate analysis, using a statistical software package will simplify the process.the main purpose of multivariate analysis is to discover and comprehend the underlying structures of the data. (djauhari, ma., sagadapan,r., and lee, s.l (2016). newly discovered variables through multivariate analysis may be small in number, but the insights gained into the nature of variation gained from these variables for the first time are invaluable.(aulele et al., 2017)multivariate regression analysis is a ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 8 mailto:hayatirahmi@yahoo.com statistical method for explaining the existence of a correlation or relationship between two or more independent variables. the focus of this article is a process and outcome analysis of teaching at the at the district 5 juli elementary school in the city of bireuen regency of indonesian. this analysis's purpose is to determine whether the final product of a given process matches initial expectations, whether the process is currently being carried out in accordance with expected standards, and whether those expectations have been met or not. original artwork for the manufactured goods. to ensure that the educational process is effective and produces results that meet expectations, it is necessary to monitor both the process and the outcomes. in this case, it seems obvious that discussing real-world events in relation to data that can be differentiated, keeping in mind the importance of specific pieces of information, and then summarizing the results of the analysis using appropriate statistical methods will aid in providing a complete picture of the procedure being described. it will make use of authentic bireuen data from the years 2013/2014 to 2017/2018 across four disciplinary domains: mathematics, indonesian, english, and natural sciences. the main question this article seeks to answer is how to use multivariate statistical methods to establish connections between different types of pedagogical content areas. how do we regulate the ongoing learning process? the purpose of this report is to analyze the connections and correlations between the many subject areas tested on the national assessment of educational progress.inform relevant parties about the outcomes of a multivariate statistical analysis of the teaching and learning processes at the national basic education school on july 5. discussion the authentic data nilai ujian nasional students from elementary school negeri 5 juli from the academic years 2013/2014 2017/2018 with the subjects of mathematics, indonesian, english, and natural science, and a total student body of 110, were used as a sample for this article's analysis. this section analyzes national standardized test scores at the elementary school level in the subjects of indonesian, english, mathematics, and environmental studies. the following are derived from the individual mean calculation results. table 1: corelation and covarians matrix subject indonesian engglish mathemathics science jlh 1733 1747 1752 1713 mean 82.524 83.190 83.429 81.571 var 14.362 9.762 7.457 8.957 range 12.000 11.000 10.000 11.000 this analysis assumes normally distributed data and that all instruction is carried out by teachers of equal ability. the highest mean scores can be found on the mathematics curriculum, at 83,429, and the lowest on the science curriculum, at 81,571. this indicates ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 9 that students perform best on mathematicsbased learning assessments and worst on science-based assessments. here we provide the results of a cross-cultural comparison of mathematics textbooks in order to better understand the connections between different disciplines. table 2: korelasian and kosovarians matrix indonesian 1 0.801923 0.788907 0.8187 engglish 0.801922788 1 0.962756 0.955602 mathematics 0.788907241 0.962756 1 0.941275 science 0.818699537 0.955602 0.941275 1 indonesian engglish mathematics science the highest correlation is found between the engglish and mathematics learning outcomes, with a value of 0.962756, and the lowest between the indonesian and mathematics learning outcomes, with a value of 0.788907241. these results suggest that the strongest pedagogical connection exists between english and mathematics, whereas the weakest link is seen between indonesian and mathematics. however, when looking at the correlation coefficient as a whole, the numbers point to a positive correlation. this demonstrates that students generally exhibit desirable behaviors in all spheres of education.this is in accordance with the research results of hasratuddin's (2018) which shows that high mathematical abilities will affect students' english skills.below are displayed results of a matrix correlation analysis to help you determine which subjects best predict others. table 3: korealisation inversions indonesian 3.0761571 -0.5605776 -0.18229 -1.81118 engglish -0.5605776 19.4414702 -10.7002 -8.04751 mathematics -0.1822869 -10.7002239 14.73981 -3.49982 science -1.8111769 -8.0475144 -3.49982 13.46733 indonesian engglish mathematics science each diagonal element in the inverse correlation matrix is proportionally related to the corespondence variable explained by regression. this is made clear by the fact that each diagonal is equal to � ���� where r is the multivariate correlation coefficient between other variables. the highest percentage, as calculated above, is for the subject of mathematics 94.85% ( ��.������� ��.����� )) while the lowest is for the subject of indonesian language study (67.49%)( �.��������� �.������� ). this suggests that math is the subject most reliably predicted by other disciplines, while indonesian language studyis the least reliable predictor of any academic field. analytical results for predicting test scores using regressive analysis�� = �� + ���(�, �). ���(�)��(� − ��). ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 10  largest predicted value from each indonesian language value to each mathematica value:����� = 3,652 + 0,568���  the magnitude of the predictive value assigned to each english-language numeric by the mathematical values is as follows:����� = 1,343 + 0,841����  each value of natural science and mathematics is accompanied by a significant predictive value.����� = 1,337 + 0,859��������  each value of indonesian language, english, and the science of natural knowledge confers a larger predictive value upon the value of mathematics. ����� = 1,224 + 0,009��� + 0,634���� + 0,217�������� study of instructional process control analysis this discussion makes use of data collected from the national assessment of educational progress at the national elementary school level between the academic years 2013–2014 and 2017–2018 for the subjects of indonesian languange, english, mathematics, and science. the first step in any process-control analysis is to determine the value of the covarians' matrix of determinates, using the information obtained in the previous step. table 4: kovarians and ucl height determinants tahun mdk0 ucl2 ucl3 2013/2014 17.94513 73.75357 92.24166 2014/2015 28.09671 73.75357 92.24166 2015/2016 10.72374 73.75357 92.24166 2016/2017 1.222907 73.75357 92.24166 2017/2018 38.93975 73.75357 92.24166 slot 1. learning process variability control table 5: t2 means and ucl 95% confidence intervals for upcoming subsamples tahun t^2 ucl2 ucl3 2013/2014 16.809 368.7679 461.2083 2014/2015 63.226 368.7679 461.2083 2015/2016 14.823 368.7679 461.2083 2016/2017 28.522 368.7679 461.2083 0 50 100 2013/2014 2014/2015 2015/2016 2016/2017 2017/2018 learning process variability control mdk0 ucl2 ucl3 ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 11 2017/2018 77.653 368.7679 461.2083 slot 2: capacity-based learning and teaching process control both the variance control limits (mdcov) and the probability balance limit (t^2) lie below their respective confidence intervals (ucl(2) and ucl(3). as seen in the accompanying table and graph. from the graph above, it can be concluded that the learning process that took place at the district 5 juli elementary school in the city of bireuen for the 2013/2014 school year to 2017/2018 went smoothly as expected. conclusion the results of analysis of data on national exam scores for 5 years, namely from 2013/2014 – 2017/2018 show that the learning process is carried out in accordance with what has been planned. this is shown by the variability graph where the determinant value of the covariance matrix does not exceed the uper control limit. this can be seen as evidence that the educational process over the past five years has been going swimmingly, in line with the educational abilities that have been developed. bibliography aulele, s. n., wattimena, a. z., & tahya, c. (2017). analisis regresi multivariat berdasarkan faktor-faktor yang mempengaruhi derajat kesehatan di provinsi maluku. barekeng: jurnal ilmu matematika dan terapan, 11(1), 39– 48.https://doi.org/10.30598/barekengvol11iss1pp39-48 djauhari, ma., sagadapan,r., and lee, s.l 2016. monitoring multivariat process variability monitoring. communication in statistic. p.1742-1754. djauhari, m.a., dan dyah e. herwindiati. 2022. kontrol kualitas proses kompleks. itb press: bandung. hasratuddin. 2019. weakness analysis learning mathematics junior high school in medan. journal international of mathematca. issn: 2456-8538volume 02 |issue 08 |august 2019 p. 1-18. johnson,richarda.2002. appliedmultivariatestatisticalanalysis(5th).newjersey:personeducationinternasiona l whittaker,joe.1996. graphicalmodelsinappliedmultivariatestatistics.newyork:john 0 500 2013/2014 2014/2015 2015/2016 2016/2017 2017/2018 control of the learning and teaching process t^2 ucl2 ucl3 ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 12 wiley&sons wustqa, d. u., listyani, e., subekti, r., kusumawati, r., susanti, m., & kismiantini, k. (2018). analisis data multivariat dengan program r. jurnal pengabdian masyarakat mipa dan pendidikan mipa, 2(2), 83–86. https://doi.org/10.21831/jpmmp.v2i2.21913 ijo international journal of mathematics volume 06 | issue 01 | january 2023 | https://www.ijojournals.com/index.php/m/index 13 learning process analysis mis guppi padang hilir (using a multivariate analysis approach) 1rizki maulida,2 riska fadhilah hutasuhut,3 juwita tindaon,4 rika restella,5 hasratuddin universitas potential utama, universitas muhammadiyah sumatera utara, universitas quality berastagi, institut agama islam negeri langsa, universitas negeri medan email : rizkimaulida24@gmail.com email : riskafadhilahhutasuhut@gmail.com email : wieta.niez@gmail.com email : restelastela@gmail.com abstract. this study aims to look at student learning outcomes through national exam scores conducted in the last ten years. this research also looks at the correlation between the subjects tested in the national exam. this research examines the suitability of lesson objectives with the learning outcomes obtained by students. the data in this control process used data on student un scores in 2012 2023 in three subjects, namely: b indonesia, science, and mathematics. this process analysis uses data analyzed in 2017 2018 with the results showing the highest correlation value is 59.8121, namely between science and indonesian subjects. while the lowest correlation is 33.13578, namely between math and science subjects. process data is seen in 2019 -2020 with the results of the highest correlation value is 8.265306, namely in indonesian and mathematics subjects. while the lowest correlation value is 7.281746, namely in math and science subjects. keywords: control, process, learning, un, mis, tebing tinggi a. introduction the function of education according to law no. 20 of 2003 is to develop knowledge and form a dignified character in order to educate the nation's life (law no. 20 of 2003.pdf, n.d.). based on the function of education listed in law no. 20, we can see that education is a place to build human character in order to build an understanding of critical thinking, independent and structured. the implementation of education should have control over the implementation process. control activities on the implementation of learning can be seen by analyzing student test scores for severalyears. control activitieson the results of this learning aim to see the suitability of learning implementation standards and learning outcomes. ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 1 mailto:rizkimaulida24@gmail.com mailto:riskafadhilahhutasuhut@gmail.com mailto:wieta.niez@gmail.com mailto:restelastela@gmail.com educational facilities are an important aspect in learning activities (antoro, 2023). education management is a stimulus to advance education in indonesia. activities in education implementation management are the beginning of learning activities that are appropriate and in accordance with learning objectives. one of the learning management activities is evaluation in the learning process. according to (darodjat & wahyudhiana, 2015) evaluation has three terms, namely measurement, assessment, evaluation. evaluation activity is a process to collect, analyze and interpret information with the aim of knowing the level of achievement of learning objectives (ratnawulan & rusdiana, 2014). research conducted with the aim of obtaining information on the learning process and as material for evaluating the learning process has also been conducted by (antoro, 2023). the research conducted by antoro found that a larger portion of student learning was experienced by students who were in fullday class learning, students who studied with a larger portion had a high average score. according to (hasratuddin, n.d.) multivariate statistical methods are data analysis techniques to see the correlation between variables as a system by taking into account the correlation between these variables. the purpose of multivariate analysis is to find the structure of the data characteristics. this paper is to analyze a learning process that takes place at mis teladan guppi tebing tinggi. the purpose of this study is to see the correlation between students' national exam results and the learning process. whether the students' national exam results are in accordance with the assessment standards enforced in indonesia. b. discussion this research was conducted at mis teladan guppi tebing tinggi which is located at jl. bhakti gg. karya satria village, padang hilir subdistrict, district / city tebing tinggi, north sumatra province. the sample in this study were all students' national exam scores in 2014 2023 in the subjects of mathematics, science, and indonesian. the amount of data for all students for 10 years is 258 students. each year 2012 2013 is 17 students, 2013 2014 as many as 14 s t u d e n t s , 2014 2015 as many as 12 students, 2015 2016 17 students, 2016 2017 as many as 29 students, 2017 2018 as many as 39 students, 2018 -2019 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 2 as many as 28 students, 2019 2020 as many as 42 s tudents, 2022 2021 as many as 18 students, 2021 2022 with 39 students, 2022 2023 with 19 students. the focus of data analysis in this study is; 1) the relationship between math, science, and indonesian subjects. the data used is data from the national exam results (un) of students in 2012 2023, namely the data used is data for the last 11 years. the use of this data aims to see the correlation between subjects. it is used data in 2017 2018. the latest data to be seen is in 2019-2020 as many as 42 students. with the subjects of math, science, and indonesian language. see the control of the learning process for eleven years of un data using the control model. 𝑈𝑝𝑒𝑟𝐶𝑜𝑛𝑡𝑟𝑜𝑙 𝐿𝑖𝑚𝑖𝑡 (𝑈𝐶𝐿), 𝑇2 = 𝑚 ( 𝑋 ̅𝑦 -̿𝑋)̅�̅̅�-̅1(̅ �̅�𝑦 -̿𝑋) 𝑡 c. data analysis 2017 2018 data analysis in this study used national exam data (un) in 2003 2004 at mis teladan guppi tebing tinggi, namely three subjects of indonesian language (x1) science (x2), and mathematics (x3). the amount of data for that year is 39 data. the results of the average calculation can be seen in the following table: table 1. average comparison of 3 subjects subject b. indonesia ipa mathematics amount 2347.47 2341.89 2603.6 mean 60.19 60.05 66.76 var 194.1513 107.4918 48.40722 based on the data above, it can be seen that the lowest average student score in the subjects is in science subjects, namely 60.05, while the highest average subject is in math subjects, namely 66.76. this shows that students' mathematics skills are better than science and indonesian subjects. this shows that students' mathematics skills are better than science and indonesian subjects. to see the correlation between the subjects, it can be seen in table 2, as follows: table 2. covariance correlation matrix b. indonesia 1 59.8121 38.06963 ipa 59.8121 1 33.13578 mathematics 38.06963 33.13578 1 b. indonesia ipa math ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 3 based on the data above, the highest correlation value is 59.8121, namely between science and indonesian subjects. while the lowest correlation is 33.13578 which is between math and science subjects. the data shows that the highest correlation between subjects is between indonesian and science, while the weakest correlation value is between science and math. when looking at the data as a whole, we can conclude that the data is positive, this shows that students try to do maximum learning in all subjects tested. the prediction of subjects against other subjects can be seen with the inverse correlation in table 3 below: table 3 inverse correlation matrix indonesian language 0.00598919 -0.002383584 -0.003078555 ipa 0.000489005 0.0115965 -0.008322629 mathematics -0.005044904 -0.006063495 0.028776209 bahasa indonesia ipa math d. data analysis 2019 2020 the data analysis used in this section is the un scores of 42 students in 2019-2020. the data used are students' national exam scores in b indonesia, science, and math subjects. the correlation between these subjects can be seen in table 4 below: table 4. basic calculation subject b indonesia ipa mathematics amount 3480 3451 3487 mean 82.85714 82.16667 83.02381 varian 9.783972 8.288618 8.218931 based on the data above, the highest average score is 83.02 in math subjects. the subject with the lowest average score is 82.16667, namely in science subjects. the data explains that the highest student ability is in math subjects, while the lowest student ability is in science subjects, namely 82.16. the correlation between subjects can be seen in table 5 below: ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 4 table 5. variance-covariance correlation matrix indonesian language 1 7.833333 8.265306 ipa 7.833333 1 7.281746 mathematics 8.265306 7.281746 1 bahasa indonesia ipa math the data above explains that the highest correlation value is 8.265306, namely in indonesian and mathematics subjects. while the lowest correlation value is 7.281746, namely in math and science subjects. the overall correlation value has a positive value, this shows that students put maximum effort in all subjects. e. learning process control analysis the data used in the analysis is a range of 11 classes in 2012 2023. the data used is data on student scores on the national exam consisting of scores in indonesian, science, and math subjects. first analyze the process control by calculating the determinant value of the covariance matrix, can be seen in the following table: table 6. determinant of covariance matrix and ucl value year mdko ucl 2. ucl 3 2012 2013 28.84768 23.990 33.638 2013 2014 18.15177 23.990 33.638 2014 2015 0.000186 23.990 33.638 2015 2016 3.05895 23.990 33.638 2016 2017 0.000117 23.990 33.638 2017 2018 1.45886 23.990 33.638 2018 2019 0.002368 23.990 33.638 2019 2020 0.049788 23.990 33.638 2020 2021 0.005479 23.990 33.638 2021 2022 0.01623 23.990 33.638 2022 2023 0.024024 23.990 33.638 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 5 based on the results of the data obtained, we can explain the value in 2012 2013 the learning process is not going well, this can be seen from the value of the determinant covariance matrix (mdkov) passing the ucl value limit of 2. if the data obtained will be used to control the learning process in the following year, the prediction data will be refined so that the data obtained is not above the ucl. the iteration process is used to refine the control by not including the values in 2012 2013. the analysis of iteration 2, namely data analysis for 2013 2023 can be seen in the following table: year mdko ucl 2 ucl 3 2013 2014 18.15177 13.613 19.281 2014 2015 0.000186 13.613 19.281 2015 2016 3.05895 13.613 19.281 2016 2017 0.000117 13.613 19.281 2017 2018 1.45886 13.613 19.281 2018 2019 0.002368 13.613 19.281 2019 2020 0.049788 13.613 19.281 2020 2021 0.005479 13.613 19.281 2021 2022 0.01623 13.613 19.281 2022 2023 0.024024 13.613 19.281 determinant matrix variability control mdko ucl 2. ucl 3 40 35 30 25 20 15 10 5 0 2012 -2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 6 the data above explains the variableity control line (mdkov) and the pbm achievement control line (t^2) are on the bottom line of ucl 2 and ucl 3. this shows that the learning process for 10 years has been going well and in accordance with the learning outcomes. it can be concluded that the un scores from 2013 2023 can be used as a controller of the learning process. f. summary a. correlation between subjects 1. un data 2017 2018 a) based on the data obtained, we can conclude that in the un subjects namely b indonesia, science, and mathematics at mis teladan guppi tebing tinggi school in 2017 2018, the average value in b indonesia subject is 60.19 with a standard deviation of 13.93, science subject is 60.05 with a standard deviation of 10.37, and math subject is 66.76 with a standard deviation of 6.96. b) the highest correlation between subjects in 2017-2018 was 59.8121, between science and bahasa indonesia. while the lowest correlation is 33.13578 which is between math and science subjects chart title 25 20 15 10 5 0 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 mdko ucl 2 ucl 3 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 7 2. un score data 2019 2020 a) based on the data obtained, we can conclude that in the un subjects, namely b indonesia, science, and mathematics at mis teladan guppi tebing tinggi school in 2019-2020, the average score in b indonesia subject is 82.85 with a standard deviation of 3.13, science subject is 82.17 with a standard deviation of 2.88, and math subject is 83.02 with a standard deviation of 2.87. b) the highest correlation value is 8.265306, namely in indonesian and mathematics subjects. while the lowest correlation value is 7.281746, namely in math and science subjects. b. learning process control data for 10 years on un exam scores, namely 2017 2023, shows a mismatch between the learning process and the results of learning outcomes. this statement can be seen from the variable graph with the determinant of the covariance matrix which appears to exceed the limit of the ucl 2 value (uper control limit). after the 2017 2018 data is excluded, the control graph is already in a position below the ucl. this statement shows that the data is valid to be used to show the learning process. reference antoro, b. (2023). statistical process control (spc) analysis as a method of evaluating students' mathematics learning process. cendekia journal: journal of mathematics education, 7(3), 2941-2954. https://doi.org/10.31004/cendekia.v7i3.2852 darodjat, d., & wahyudhiana, w. (2015). education program evaluation model. xiv, 1-23. https://doi.org/10.30595/islamadina.v0i0.1665. hasratuddin. (n.d.). learning process control at smpn 6 medan. ratnawulan, e., & rusdiana, h. a. (2014). evaluation of learning. pustaka setia bandung. law number 20 year 2003.pdf. (n.d.). ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 8 mathematical modeling of climate change and desertification: a case study of yobe state, nigeria. bulama adamu bawa �, abdulaziz babiker mohamed hamed� department of mathematics & statistics, faculty of science yobe state university, damaturu, nigeria email: adamubulama01@gmail.com, aziz.hamed12@gmail.com corresponding email: aziz.hamed12@gmail.com abstract: the study addressed the effect of climate change and desertification in yobe state, nigeria. desertification as defined by the united nation convention to combat desertification (unccd 1994), is the degradation in arid, semi-arid, and dry subhumid areas resulting in many factors including human activities and climatic variations. desertification is a silent, invisible crisis that is destabilizing communities on a global scale, as victims turn into refugees, internally displaced people and forced to migrate from their homes. hence to restore stability in a context where changing weather events are threatening the livelihood of people, then everyone must beware and stand to fight desertification, revoke land degradation and ease the effect of drought. the researcher implemented some mathematical models of climate change on desertification to give a good insight into desert, desertification and the causes of desertification, that affects our communities and provide possible ways of reducing its impact to the barest minimal. furthermore, the research will also help decisions maker to well plan in order to protect our land and environment. key word: mathematical model, climate change, desertification, drought, invisible crisis ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 1 1.1 introduction climate change and desertification has been an aged long phenomenon that has affected the lives and livelihood of many people around the globe both positively and otherwise. in the last century, these has been a major area of research for many scholars from various areas of disciplines who have provided a huge success to the prediction of possible outcomes in the future climate, and have provided ways of solving those problems. this research is going to be narrowed on yobe state nigeria as our case of study to look into some of the effects of desertification and give appropriate solutions (using different climate models) to those problems. 1.2 background of the study humanity have had a long association with arid and semi-arid regions: the first great civilizations in egypt and mesopotamia developed at the end of the climatic optimum some 3000 years b.c., at a time when the sahara appears to have been vegetated as parts of the sahel are today. oguntoyinbo (1981) has traced the impact of human activities on climate variability in africa. he notes that it was not until the time of the roman occupation of north africa in about 100 bc that we began to notice successively drier climates in the sahara, though this period was not as dry as it is today. he argues that the fluctuations in the levels of the lake chad reveal the nature of climate variation in the sahara. he therefore concludes that if the rise and fall of the lake chad can be taken to represent climate variability in the arid zone in the period before “instrumental records”, it is possible that rainfall in the arid zone has similarly shown dramatic variability which has impacted on land use practices periodically. ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 2 by the time the egypt pyramids were completed (around 2700 b.c.), the climate of the northeastern and middle east africa was in a drying phase that resulted in the arid landscapes we have known for much of the last 5000 years (el-baz 1983). the federal ministry of environment estimated that nigeria loses “about 350,000 square meters of its land mass to desert condition which is advancing southwards at an estimated rate of 0.6 kilometres a year”. this environmental problem has severe economic repercussions on the entire nation as it impacts on the socio-economic life of rural households (reduction in crop and animal production, death of livestock, high prices for food stuffs) and leads to widespread poverty. until around the early 20th century, climate scientists were primarily concerned with the study of past climatic states. this was done by observation of the environment using mostly geological, geographical and botanical methods. by the end of the 1950s, important physical measurement methods were developed. the measurement of weak radioactivity of various isotopes was the basis for the dating of organic material and enabled the determination of flux rates in different environmental systems. two out of three of the geographical landscape in africa is classified as drylands, of which 319 million hectares has been estimated to be highly vulnerable to desertification. these areas are concentrated in sahelian (used to refer to the semiarid) regions bordering the sahara desert to the south and includes parts of chad, nigeria, niger, burkina faso, mali, senegal, mauritania and the gambia (some authors include sudan, somalia, ethiopia and eritrea in the sahel) region, horn of africa and kalahari in the south. increasing concentration of poverty in the drylands of sub-sahara africa has been documented, where 41% of the total population lives in extreme poverty, which is partly attributed to desertification. drought and desertification are at the core of serious challenges and threats facing ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 3 sustainable development in africa, with far reaching adverse impacts on human health, food security, economic activity, physical infrastructure, natural resources and the environment, with incidence in national and global security. 4.3 objective of the study the context of desertification and climate change has a wide range of branches, where some scholars have questioned whether the phenomenon exists at all. it is evident through the reports and proofs that desertification is affecting the lives and livelihood of people directly or indirectly. hence this research is going to discuss the nature, causes of desertification in yobe and also provide a proposal to the state government on how to combat its wide spread in the state. 1.4 definition of key terms 1.4.1 desertification unccd (1994) defines desertification as “land degradation in arid, semi-arid and dry sub humid areas resulting from various factors, including climatic variations and human activities”. 1.4.2 climate this is a word from ancient greek “klima”, meaning inclination. climate is commonly defined as the weather averaged or the statistics of weather over a long period. the standard averaging period is 30 years, but other period may be used depending on the purpose. it is measured by assessing the patterns of variation in temperature, humidity, atmospheric pressure, wind, precipitation, atmospheric particle count and other meteorological variables in a given region over long period of time. 1.4.3 land degradation ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 4 unccd (1994) defines land degradation as a “reduction or loss, in arid semi-arid and dry sub-humid areas, of the biological or economic productivity and complexity of rain-fed cropland, irrigated cropland, or range, pasture, forest and woodlands resulting from land uses or from a process or combination of processes, including processes arising from human activities and habitation patterns, such as: (i) soil erosion caused by wind or water”; (ii) deterioration of the physical, chemical, and biological or economic properties of soil; and (iii) long-term loss of natural vegetation. it can also be referred as a loss of adaptive capacity, or a decline in biological and economic resilience.[6] 1.4.4 drought the wikipedia defines drought as an event of prolonged shortages in the water supply, whether atmospheric (below-average precipitation), surface water or ground water. drought is a recurring feature of the climate in most parts of the world. however, these regular droughts have become more extreme and more unpredictable due to climate change. a drought can last for months or years, or may be declared after as few as 15 days.[7] 1.4.5 climate model a climate model is essentially a representation of the many interactions and dynamics within the climate which includes the atmosphere, ocean, land surface, and ice to make predictions of possible climate change for the future. climate models are systems of differential equations based on the basic law of physics, fluid motion and chemistry. ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 5 1.4.6 global warming global warming is the increase in the average temperature of the earth’s nearsurface air and the oceans. it can also be defined as a gradual increase in the overall temperature of the earth’s atmosphere generally attributed to the greenhouse effect caused by increased levels of cabondioxide co2, chloroflorocarbons cfcs, and other pollutants. 1.4.7 atmosphere an atmosphere (from ancient greek (atmos), meaning ‘vapour’, and (sphaira), meaning ‘ball’ or ‘sphere’). the atmosphere is a layers of gases surrounding a planet or other material body that is held in place by the gravity of that body. 2: the literature review: the mission of the un convention to combat desertification is: “to provide a global framework to support the development and implementation of national and regional policies, programs and measures to prevent, control and reverse desertification, land degradation and mitigate the effect of drought through scientific and technological excellence, raising public awareness, standard setting, advocacy and resource mobilization, thereby contributing to poverty reduction.” [6](unccd 1996). in this chapter we will look into some literature reviews of many other scholars. 2.1 reviews aubreville (1949), this french forester was the first to introduce the word ‘desertification’ in his book “climats, forets, et desertification de l’afrique tropical”. he witnessed the degradation and disappearance of tropical forests in many humid and sub-humid parts of africa, and attributed it to a large extent to the slash and burn agricultural practices of the local populations. aubreville had ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 6 identified climate change as a potential factor, but could not estimate its importance for lack of adequate data. it is only later on that the concept became commonly associated with arid and semiarid regions. [2] roger (1981), who digitized unesco’s map showing the world distribution of arid region (unesco 1977). this map was constructed on the basis of hydrological data using a water balance approach. norman myers (1984), estimated that some 120000 km2 of agricultural and pastoral land were deteriorating beyond fuel economic use per year (myers 1984, p. 46). these numbers should be compared to estimates of deforestation worldwide, which range from 100000 to 113 000 km2 per year, with the bulk of the destruction occurring in the tropics.[2] warren (1984), said “arid lands are dynamic regions, they have been evolving over thousands of years, mostly in response to climatic changes and humanity has been able to cope with such evolution and by colonizing new and hitherto unaffected areas”.[2] dregne (1986), the intensity of desertification processes can range from slight to very severe in terms of the degradation of plant and soil resources.[2] gethner, robert (1998), a planet’s albedo is the percent of incoming solar radiation that is immediately reflected back into space due to coloring of the planet. the earth’s albedo, is equal 0.3, so 30% of incoming solar radiation is immediately reflected back into space. nap (2000), revealed that about 35% of nigeria’s landmass is considered arable, with 15% being utilized for pastures, 10% for forest reserve, and 10% for settlement. the same report also reveals that 30% of the country’s landmass is regarded uncultivable. notably, reports from the federal ministry of environment shows that nigeria is annually losing about 350,000 square meters to desertification, which is regarded as the gravest environmental problem affecting ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 7 10 of the 11 northern states. yet, rural households, especially inhabitants of drylands like those of gursulu village yobe state, depend on arable land for their livelihoods. parmesan and yohe (2003), reported that the extent to which traditional forms of agriculture including pastoralism degraded the land was and is to some extent influenced by the nature of the biophysical environment. steep slopes are clearly more susceptible than gently-sloping land to accelerated erosion; low-lying flood plains and flat lands are more likely to be affected by flooding; regions affected by strongly seasonal climatic regimes are affected by both floods and droughts; areas with highly flammable vegetation are more prone to fire; drylands with sparse vegetation cover are more exposed to wind erosion, and so on. however, over many centuries, and as a matter of necessity, traditional agriculturalists learned how to live ‘with’ nature. it has been mainly since the 18th and 19th centuries that exponential rates of population increase in much of the third world, and mechanization and the onset of chemical farming worldwide, have resulted in some very serious problems. global warming at least partly caused by the increasing release of ‘greenhouse gases’, with agriculture being a significant contributor is already affecting plants and animals.[9] e.u (2010), developed strategies that works both to reduce greenhouse gas emissions and prevent damage to the ozone layer, and, to mitigate the unavoidable adverse effects of climate change.[8] unccd (2010), formed an interim secretariat that will fight against desertification and to promote actions that will protect dry-lands. the fight is seen as an opportunity to make critical changes to secure the long-term ability of dry-lands to provide value for humanity's wellbeing. they were charged with: i. raising awareness on the causes of land degradation and its solutions. ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 8 ii. mobilizing financial and technical support to fight desertification activities worldwide. iii. monitoring and reporting on progress in preparation of the secretary general’s report ipcc (2018), pointed out that there is limited 30 evidence and medium agreement that the extent of deserts will increase in the coming decades. however, the deserts are expected to become drier and warmer more rapidly than other terrestrial areas. they assessed as “low confidence” that desertification linked to climate change will directly or indirectly influence soil health and productivity due to accelerated soil erosion in drylands. they also had “low confidence” in the projections of future increases in dust storms with higher aridity.[9] unccd (2021) reported that, land degradation directly undermines our ability to deliver food and nutritional security. by 2050, crop yields are estimated to decrease by 10% globally due to land degradation and climate change, with some regions suffering up to a 50% reduction. furthermore, land degradation is projected to fuel an estimated 30% increase in world food prices over the next 25 years. given the expected growth in global population and food demand by 2050, conserving, sustainably managing, and restoring land resources will be essential in the transition to sustainable food production, requiring at least a 75% reduction in current yield gap.[10] barrack obama at cop26(2021), said “there has been a success to the agreement of nations at the cop25 in france, but yet we are not close to achieving our goal of maintaining a steady clean energy globally although the usa has manage to make that possible”. he also added that some parts of the world are becoming more dangerous to live in, hence more than 100 countries have agreed to address deforestation and make use of clean energy by the end of 2030 (every country is needed to achieve this goal).[18] ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 9 boris johnson (2021) the prime minister of the united kingdom at the cop26 (2021), gave an analogy that if the global temperature exceeds its current 1.5℃ to 2℃, (our food supply will greatly be affected, locusts, bees and other important insects that aid pollination will all die), 3℃ (more cases of wildfires, 5x drought, 36x heat waves etc), 4℃ (major cities like miami, shanghai will all disappear because of over flooding, hurricanes etc.). hence the world’s leaders have agreed to maintain the net global emissions to 1.5℃ and gradually to a lesser degree[17]. 2.2 relationship between climate change and desertification the european union analyses the relationship between desertification and climate change in the mediterranean. the central aim of the report is to identify best practices in dealing with desertification and to make recommendations as to what the union for the mediterranean (ufm), the eu and other stake-holders can do to respond to the challenges that might arise due to desertification and climate change in the future. we should note that desertification is essentially a man-made phenomenon which is exacerbated by climate change. this is because an increase in weather extremes such as droughts and heavy rains as a result of climate change will lead to further land degradation. this in turn aggravates existing problems associated with poverty, forced migration, and in some areas conflicts. while desertification is already responsible for significant forced migration, more than a billion people (one in seven of the current world population) could be forced to leave their homes between now and 2050 if climate change worsens. the middle east and north africa (mena), in particular, is considered to be the region most at risk if such projections prove accurate. the relationship between the two processes does not, however, move in only one direction. it is also possible that desertification may in ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 10 turn affect climate change, due to the effects of land degradation that reduces surface moisture. because less water is available for the sun’s energy to evaporate, more energy is left over for warming the ground and, consequently, the lower atmosphere. at the same time, wind erosion in dry-lands releases dust and other particles into the atmosphere. by absorbing the sun’s rays or reflecting them back out into space, they may help to cool the earth’s surface. however, the energy they absorb can heat the lower atmosphere and in this way reduce temperature differences between the atmosphere’s vertical layers; this can lead to fewer rainshowers and thus drier land. finally, the periodic burning of arid and semi-arid grasslands, often associated with unsustainable slash-and-burn agriculture, emits greenhouse gases. the unsustainable use of fuel-wood and charcoal, a major cause of land degradation, also contributes to greenhouse gas emissions. [9] 2.3 causes of desertification the two main causes of desertification are ‘climatic variations’ and ‘human activities’ climatic variations: these include climate change, drought, and moisture loss on a global level human activities: these include overgrazing, deforestation and removal of the natural vegetation (cover by taking too much fuel wood), agricultural activities in the vulnerable ecosystems of arid and semi-arid areas, which are thus strained beyond their capacity. these activities are triggered by population growth, the impact of the market economy, and poverty. other causes of desertification include: i. lack of adjusting to natural fluctuation ii. rainfall below normal recorded levels iii. low priority often given to environmental protection ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 11 iv. international economic v. ignorance, errors, and natural and man contribute to land degradation.[9] source: aridity zones and dry in the world’s dry-lands with particular reference to afr 2.4 consequences and risk of desertification there are several consequences of desertification which can either affects us directly or indirectly. desertification; i. reduces the land’s resilience to natural climate variability. ii. compromises the soil iii. increases possibilities of famine, malnutrition and starvation in a country. iv. indirect pressure on area outside the affected areas such as flooding, reduced water quality, sedimentation in rivers and lakes, dust storms and air pollution. economic forces ignorance, errors, and natural and man-made disasters can also contribute to land degradation.[9] source: aridity zones and dry-land populations: an assessment of population levels lands with particular reference to africa.[12] 2.4 consequences and risk of desertification there are several consequences of desertification which can either affects us directly or indirectly. desertification; reduces the land’s resilience to natural climate variability. compromises the soil potential for food production. increases possibilities of famine, malnutrition and starvation in a country. indirect pressure on area outside the affected areas such as flooding, reduced water quality, sedimentation in rivers and lakes, dust storms and air made disasters can also land populations: an assessment of population levels there are several consequences of desertification which can either affects us increases possibilities of famine, malnutrition and starvation in a country. indirect pressure on area outside the affected areas such as flooding, reduced water quality, sedimentation in rivers and lakes, dust storms and air ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 12 v. leads to socio-economic instability the factors mention above have the potential to make worse other challenges facing the region. 2.5 processes and drivers of desertification due to climate change the process of desertification includes both biological and non-biological processes, and is attributable to the physical, chemical and biological properties of terrestrial ecosystems. some of the key drivers of desertification include soil erosion; global warming leading to the rise of ��� levels; sea surface temperature anomalies which drive rainfall changes; invasive plants which affect ecosystem services, wildfire which reduces vegetation cover, increases runoff and soil erosion, reduces soil fertility and affects the soil microbial community. 2.5.1 anthropogenic drivers of desertification include: cropland expansion, unsustainable land management practices such as overgrazing by livestock, urban expansion, infrastructure development, and extractive industries. high and growing consumption of land-based resources has also been indicated as the ultimate driver of land degradation, e.g. through deforestation and cropland expansion, escalated by population growth. 2.5.2 the institutional, policy and socio-economic drivers of desertification: include land tenure insecurity, lack of property rights, lack of access to markets, and to rural advisory services, lack of technical knowledge and skills, agricultural price distortions, agricultural support and subsidies contributing to desertification, and lack of economic incentives for sustainable land management. 2.6 effect of climate change on desertification desertification is affecting about 45% of the african continent’s land area, out of which 55% is at high or very high risk of further degradation. the major ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 13 mechanism through which climate change and desertification affect food security is through their impacts on agricultural productivity. there is robust evidence pointing not only to negative impacts of climate change and desertification on crop yields, but also on the losses in agricultural productivity and incomes in dry-lands. the forecasts for sub-saharan africa suggest that higher temperatures, increase in the number of heat-waves, and increasing aridity, will affect the rain fed agricultural systems. without the carbon fertilization effect, climate change will reduce the mean yields for 11 major global crops – millet, cowpea, sugar beet, sweet potato, wheat, rice, maize, soybean, groundnut, sunflower and rapeseed – by 15% in sub-saharan africa, 11% in middle east and north africa by 2050. desertification has led to reduction in agricultural productivity and incomes; it has also contributed to the loss of biodiversity in many dryland regions. it is further projected to cause reductions in crop and livestock productivity, modify the composition of plant species and reduce biological diversity across drylands. in sub-saharan africa particularly, crop production may be reduced by 17–22% due to climate change by 2050. about 821 million people globally were food insecure in 2017, of whom 31% were in africa. sub-saharan africa, particularly east africa, had the highest share of undernourished populations in the world in 2017, with 28.8% and 31.4%, respectively. in north africa, long-term monitoring (1978– 2014) has shown loss of important perennial plant species due to drought and desertification e.g. stipa tenacissima and artemisia herba alba 2.7 statistic estimation of desertification it is estimated that 46 of the 55 countries in africa are vulnerable to desertification, with some already feeling the effects, the nile (42% of area), niger (50%), senegal (51%), volta (67%), limpopo (66%) and lake chad (26%) (the horn of africa is getting drier). despite desertification in the sahel being a major concern since the ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 14 1970s, wetting and greening conditions have been observed in this region over the last three decades. the sahara is reported to have expanded by 10% over the 20th century based on annual rainfall. however, cropland areas in the sahel region of west africa have doubled since 1975, with settlement area also increasing by about 150%. in burkina faso, from 1984 to 2013, bare soils and agricultural lands increased by 18.8% and 89.7%, respectively, while woodland, gallery forest, tree savannas, shrub savannas and water bodies decreased by 18.8%, 19.4%, 4.8%, 45.2% and 31.2%, respectively. in fakara region in niger, a 5% annual reduction in herbaceous yield between 1994 and 2006 was largely explained by changes in land use, grazing pressure and soil fertility. greening has also been observed in parts of southern africa but it is relatively weak compared to other regions of the continent.[5] 3. mathematical implementation in this section the research is going to be concentrating on the global average temperature and the effect of greenhouse gases which can be modeled using the energy balance equation (ebe) [kaper and engler 2013,14] and a few equations on predicting weather and climate. 3.1 global average temperature models � �� �� = �(1 − �) − ��� energy balance equation (3.1) � �� �� = �(1 − �) − ���� (3.2) ����: � → ����(�) = � + �� − (�� + ���)� (3.3) (where �, �, ��&�� are observational data and � is a cloudiness coefficient) ����: � → ����(�) = � + �� (where the values of � and � vary with temperature) � �� �� = �(1 − �) − (� + ��) global surface temperature model ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 15 3.2 liouville equation the evolution equation in a climate or weather prediction model are conventionally treated as deterministic, they based on spatially-truncated momentum, energy, mass and composition conservation equation and can be written as � = �[�] this conservation equation can also written in the context of gleeson1996 as �� �� = � �� (��) = �� 3.3 simple fluvial models 3.3.1 exner equation ℎ � � � � fig 3.3.1 the figure above is a geometry of a river therefore its free surface is at � = �(�, �, �), and its bed is at � = �(�, �, �). hence the depth of the river is ℎ = � − � it is based on the principle conservation of mass for the substrate, and may be written as (1 − �) �� �� + ∇. � = �� ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 16 3.3.2 st venant equation the navier-stokes equations is also written in this case of two-dimensional flow ��+�� = 0 ��(����������)� ����������(�������) ��(����������)� �������√������(�������) where � is the slope. 3.4 variable description and their units  t (k, kelvins) is the average temperature in the earth’s photosphere (upper atmosphere, where the energy balance occurs in the model) (1kelvin = 1°c);  t(years) is time;  r = (w-yr/m2k) is the average heat capacity of the earth/ atmosphere system (heat capacity is the amount of heat required to raise the temperature of an object or substance 1kelvin (=1°�));  q = (w/m2) is the annual global mean incoming solar radiation per square meter of the earth’s surface;  � = orl emissivity factor  � ��� � = are empirically determined parameters  � = is planetary albedo (dimensionless)  � = (w/����) is a constant of proportionality (stefan-boltzmann constant)[16]  ��� = average amount of solar energy reaching one square meter of the earth’s surface per unit time ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 17  ���� = average amount of energy emitted by one square meter of the earth’s surface per unit time  s(m) = height of the bed relative to some reference height  n(dimensionless ) = porosity of the substrate  q(�����) = bed transport rate  �� = source function corresponding to the exchange of sediment between the fluid and the bed.  ∇= ( � �� , � �� ) two-dimensional  constant values of some variables  values for the parameters are:  r= 2.912 w-yr/m2k [ichii et al. 2003];  q= 342 w/m2  �= 0.30  � = 0.6  � = 5.67 x 10-8 w/m2k4 summary as this stage, we have seen different climate models which include; the energy balance equation, simple fluvial model, liouville equation etc. all the models provided in 3.1 are similar to each other but all of them are important depending on the initial variables available and will help us calculate the global average temperature. 4.0 introduction we have discovered from our chapters points the review of different scholars and organization on the diverse effects of desertification and climate change our environment and the world at large. we have also discussed some mathematical ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 18 model that will enable us to clearly see its impact and behaviors. the data that will be used in this chapter was collected from nigerian meteorological agency (nimet), national population commission (npc) and researches done by different scholars and will be used to display graphs and to solve a few problems. 4.1 data collection table 4.1 population in the sampled villages.[16] development area population in clustered villages male female total household sampled household percentage balle 11,635 63 58 2327 116 4.98 bulanguwa 11,034 54 56 2297 110 4.78 dagona 10,277 56 51 2055 102 4.96 dapchi 10,034 45 30 1406 80 5.68 degeltura 7,882 44 39 1576 78 4.94 dumburi 10,022 55 50 2004 100 4.99 futchimiran 4,209 21 26 841 52 6.18 gumsa 5,147 25 30 1029 61 5.92 gwio kura 16,377 86 82 3275 163 4.97 gorgoram 9,046 45 50 1809 100 5.52 kanama 6,889 34 37 1377 69 5.01 karasuwa 5,049 25 30 1009 60 5.94 kaska 13,747 68 70 2749 137 4.98 machina 18,081 65 70 2616 130 4.96 muguram 10,905 58 54 2181 119 5.46 ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 19 yunusari 8,822 44 49 1764 95 5.39 yusufari 11,089 56 61 2218 111 5.00 wachakal 8,140 46 41 1628 96 5.90 total 170,385 895 884 34077 1779 5.22 source: national population commission (2001) table 4.2 migration per 1000 population in 2002[16] development area in-migration out-migration balle 12 150 bulanguwa 54 56 dagona 13 59 dapchi 30 40 degeltura 10 6 dumburi 40 60 futchimiran 9 38 gumsa 8 36 gwio kura 81 82 gorgoram 6 37 kanama 25 44 karasuwa 3 123 kaska 76 61 machina 11 140 muguram 53 56 yunusari 36 52 yusufari 43 67 ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 20 wachakal 48 52 total 558 1154 1.1 chart for migration per 1000 population table 4.3 showing social impacts of desertification (sid) sl. no. questions 4 3 2 1 mean std sid1 desertification leads to destruction and relocation of houses 73 (25.5) 169 (59,1) 37 (12.9) 7 (2.4) 3.68 .692 sid2 sometimes whole settlements relocate as a result of desertification 95 (33.2) 126 (44.1) 56 (19.6) 9 (3.1) 3.07 .807 sid3 conflicts among people do occur as a result of desertification 188 (65.7) 90 (31.5) 8 (2.8) 0 (0) 3.63 .539 sid4 desertification affects soil fertility 175 (61.2) 111 (38.8) 0 (0) 0 (0) 3.61 .488 sid5 farming and grazing activities are also affected by desertification 136 (47.6) 141 (49.3) 9 (3.1) 0 (0) 3.44 .558 0 20 40 60 80 100 120 140 160 ar ea b al le b u la n gu w a d ag o n a d ap ch i d eg el tu ra d u m b u ri fu tc h im ir an g u m sa g w io k u ra g o rg o ra m k an am a k ar as u w a k as ka m ac h in a m u gu ra m yu n u sa ri yu su fa ri w ac h ak al in-migration out-migration ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 21 sid6 desertification leads to drying up of sources of water 89 (31.1) 176 (61.5) 11 (3.8) 10 (3.5) 3.20 .671 sid7 as a result of desertification loss of biodiversity is experienced 68 (23.8) 105 (36.7) 98 (34.3) 15 (5.2) 2.79 .865 sid8 desertification induced problems lead to overall reduced quality of life among people 144 (50.3) 138 (48.3) 4 (1.4) 0 (0) 3.49 .528 sid9 desertification leads to migration of people from the area 86 (30.07) 146 (51.04) 39 (13.64) 15 (5.24) 3.02 .778 sid10 increase in soil erosion is noticed in recent years 99 (34.61) 177 (61.89) 10 (3.50) 0 (0) 3.45 .535 overall average 117(40.95) 133(46.70) 29(10.18) 6(2.18) 3.33 .661 the table above shows the responses of the people living in some few villages within our case study with respect to social impacts of desertification in their communities. the table shows that 88.46% agreed that these impacts of desertification are far reaching and the situation is very bad. these impacts manifest in form of destruction, migration from the whole settlement, diminishing grazing fields, drying up of water sources, erosion, and reduced quality of life among the local people[15]. table 4.4 average monthly rainfall and temperature anomalies month rainfall (mm) minimum temp. (°�) maximum temp. (°�) january 19.06 21.92 32.40 ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 22 february 45.75 22.49 33.08 march 141.43 21.38 32.48 april 154.9 21.26 28.90 may 209.81 23.14 31.15 june 222.39 21.78 25.71 july 460.22 22.54 29.00 august 361.7 20.84 29.16 september 361.44 18.32 28.32 october 293.98 22.62 30.34 november 142.71 22..97 31.27 december 22.94 22.23 31.13 average 203.03 21.79 30.25 it was observed that there was an increase in the of daily amount of rainfall and the extension of rainy season which is usually 8 months (march-october) to 10 months(february-november) this was evident even in some states of the northern part of the country (potiskum l.g.a, yobe state) for the last few years 0 50 100 150 200 250 300 350 400 450 500 rainfall (mm) minimum temp. ( ) maximum temp. ( ) ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 23 (2017,2018). the dry season had been observed shorter and hotter as the years progress. these phenomena are in conformity with the consequences of global warming resulting from increased anthopogenic activities. (tamunoberetonaria et. al. 2013). table 4. anti-desertification plants used in yobe state, nigeria[16] scientific name vernacula name hausa name amaranthus spp. amaranth dangme annona cherimola cherimoya annona muricata guanabana, soursop, graviola asimina triloba asimina cleome gynandra african, cabbage, cat’s whiskers yar unguwa dacryodes edulis safou or butter fruit ipomoea batatas sweet potato dankalin hausa irvingia gabonensis dika tree goron biri(goron ruwa) moringa oleifera moringa zogale oxytenanthera drought-resistant goradi abyssinica bamboo gwangwala prosopis cineraria prosopis akiye simmondsia chinensis jojoba solanum scabrum african nightshade strychnos spinosa monkey orange girigita 4.2 problem the models developed in the previous chapter (3.4) will be used to solve few examples as seen below 1. � �� �� = �(1 − �) − ��� ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 24 data � = 2.912 w-yr/m2k[ichii et al 2003], � = 342w/m2[kaper and engler 2013], � = 0.30[kaper and engler 2013], � = 5.67x10-8, � = 30.71 �∗ = � �(���) � �1/4 equilibrium temperature �∗ = � ���(���.��) �.������� �1/4 = � ���(�.��) �.��������1/4 = � ���.� �.��������1/4 = (4.22222)1/4 = 1.4335℃ hence the equilibrium temperature is increasing at a very slow pace but yet its impact to the surrounding can be clearly noticed with time. 4.3 discussion it is clearly evident that climate change is affecting the rate of desertification all over the globe. boris johnson the prime minister of the united kingdom at the cop26 (2021), gave an analogy that if the global temperate exceeds 1.5℃ to 2℃, (our food supply will greatly be affected, locusts, bees and other important insects that aid pollination will all die), 3℃ (more cases of wildfires, 5x drought, 36x heat waves etc), 4℃ (major cities like miami, shanghai will all disappear because of over flooding, hurricanes etc.). hence the world’s leaders have agreed to maintain the net global emissions to 1.5℃ and gradually to a lesser degree[17]. evidently, the socio-economic impacts of desertification in all the study locations with the exception of gumshi were found to be high. thus, with the unprecedented increase in the rate of deforestation activities such as logging coupled with the nonijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 25 chalet attitudes of the local communities towards controlling desertification and its impacts, continuous deforestation acts by the local people, lukewarm attitudes of the government, increasing over dependence of the local communities on fire wood as the dominant source of domestic energy as well as the growing dependence of considerable number of the local communities on fire wood selling as a source of income, desertification can continue taking toll in these areas and its impacts both socially and economically can escalate. the change in the climate and weather system is also responsible for the unstable yield of most farm product[15]. 5.1 conclusion based on the research so far, we have discovered a framework for developing a better understanding of the nexus between the environmental changes, population response and environmental policy and management. solutions to desertification must be aimed to increase the amount of food production in the area in concomitance with farm practices that must encourage environmental stabilization. it is also clear that the general public have little knowledge about how their day to day use of unchecked car exhaust, bush burnings, etc is dangerous to them and to the world at large. [15] 5.2 recommendation here are a few recommendations to the government, non-governmental organizations (n.g.o), affected communities and every individual to address some of the findings in this research respectively: a. the government must be all round committed to fight desertification and gain the participation of the local population combat desertification in their local communities. the key area to be given priority is massive tree planting exercise, construction of earth dams, etc this calls for more fund allocation to both state and federal forestry departments in nigeria. the aim should ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 26 be a long term sustainable participatory environmental resources management programme.[16] b. non-governmental organizations in partnership with the local stakeholders should create a forum for massive reforestation programme, leading to improved environmental management capacity. there is the need for the tapping of underground water for domestic and irrigational purposes. fadama areas should be protected with trees to avoid drying up of the catchment areas.[16] c. all affected communities most take the responsibility of ensuring that all government policies concerning desertification are backed up and obeyed by every resident in that community. they are also responsible for the maintenance of amenities provided by the government, n.g.o’s and stakeholders to combat desertification. d. as we have seen in this research how indirectly desertification can affect people who are far away from the affect communities by causing shortage of food supply, flooding, poverty etc. hence everyone must take the responsibility to ensure we combat climate change by reducing the use of excess carbon exhaust and desertification to the barest minimal. ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 27 references [1] gbahabo, percy, (2011) “desertification and rural livelihoods the case of gursulu village” yobe state, nigeria. [2] m.m verstraete and s.a schwartz vegetato vol 91. (1991) “desertification and global change” [3] oguntoyinbo. j (1981), “climate variability and food crop production in west africa” geojournal vol. 5.2 [4] el-baz, f. (1983), “a geological perspective of the desertt”, s. wells and d. haragan “origin and evolution of deserts”, university of new mexico press, albuquerque. [5] agnes, (march 2020), “desertification and climate change in africa, policy brief no. 1”. [6] m.s reed, l.c stringer (march 2015), “climate change and desertification: anticipating,, assessing and adapting to future changes in drylands”. [7] en.wikipedia.oeg/wiki/drought [8] http://ec.europa.eu/dgs/clima/mission/index.en.html [9] european union, (2011), “the relationship between desertification and climate change in the mediterranean.” ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 28 [10] coriacher, arthur vol1 “land degradation and desertification: history, nature, cause, consequences and solutions.” [11] unccd (2021), “land degradation neutrality for sustainable agriculture and food security.” [12] undp/unso (1997), “aridity zones and dryland populations: an assessment of population level’s in the world drylands with particular reference to africa.” [13] t.n palmer (1999), “predicting uncertainty in forecast of weather and climate” [14] james walsh, (2015) “climate modeling in differential equation” oberlinn college. [15] j.a opara, m.babagana and a.adamu, (2017) “environmental health, desertification and sustainable development in north-eastern nigeria: a socioeconomi impact assessment” [16] amadi et al. (2011), “human coping strategies to desertification in yobe state nigeria”, animal research international. [17] https://www.youtube.com/watch?v=oofxdqqke7m [18] https://www.youtube.com/watch?v=69emd4cszry ijo international journal of mathematics volume 5 | issue 09 | september 2022 | https://www.ijojournals.com/index.php/m/index 29 https://www.youtube.com/watch?v=oofxdqqke7m https://www.youtube.com/watch?v=69emd4cszry modeling and analysis of the interaction of neutral populations and violent cop populations: a competing species model a. kazmierczak department of computer science oklahoma state university stillwater, ok akazmie@okstate.edu abstract the rise of police violence in the united states has brought concern, debate, and contention to the modern world. police violence is national and is no longer concentrated in a particular location. in this paper, we present a dynamical model of the interaction between neutral population and violence. the formulation is based on models of interactions between competing species type dynamics. an exploration of the long-term dynamics and stability of homogeneous equilibrium solutions and their stability is given. the paper is given in three parts. part one analyzes the current level of violence. part two analyzes the situation when police violence increases. part three analyzes the situation when police violence decreases keywords: drug cartels, dea, competing species model, equilibrium solutions, stability at equilibrium solutions. 1. introduction police violenceare not a new phenomena. however, there is a marked and exponential increase in the growth of police violence. this violence wreak havoc to native citizens. this violence affects political and social policies and createsserious issues. consequently, countries are faced with extremely difficult, complex, and contentious political and social decisions on the issues of police violence. the acceptance of continued violence provides a tolerance for moreviolence, and a failing belief that police are actually there to protect and serve. hence, countries face the possibility of ever increasing police violence. despite these impending threats, there is not much literature that takes a dynamical systems approach to understanding the spread of police violence. our primary objective is to bridge the gap. in our framework, we let v represent the level of police violence.this police violence denoted by v: v can be viewed as the amount of police violence in a community. this paper is a first step in providing a mathematical modeling framework to study the evolution and interaction between this neutral population and growing police violence. the neutral population is modeled by standard population growth models and is represented by n. the paper is organized as follows. in section 2, we develop and analyze the time-dependent autonomous police violence ordinary differential equation (ode) model. we examine the equilibrium solutions, the stability of the equilibrium solutions and investigate the dynamics numerically. in section 2, we consider the situation of the current level of police violence in the system.we examine the equilibrium solutions, the stability of the equilibrium solutions and ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 1 mailto:akazmie@okstate.edu investigate the dynamics numerically for this situation also. in section 3, we consider the situation where more police violence is encountered. in section 4 we consider the affect of reducing police violence in part 5 we present our conclusions based on the analysis in sections 2, 3 and 4. 2. neutral violence (nv) ode model consider the mathematical model n = a1n/(1+d1u) – anrvn/(1+d2n) – b1n 2 = 0 = fn(n, v) (1) v= a2v/(1+d3v) – anrn/(1+d2n) – b2v 2 = 0 = fr(c, d) (2) the populations n(t) and v(t) represent the populations of the neutral and violent policepopulations. more policeviolent are slowly coming into the police population. the parameters are all assumed to be positive and their descriptions are given in table 1a. table 1a: list of parameters used in the differential equation model symbols meaning a1 growth rate of the police population a2 growth rate of the neutral population b1 population loss in n due to intra-species competition and natural mortality b2 population loss in v due to intra-species competition and natural mortality anr maximum per capita loss in n due to police violence d1 measures the effectiveness of v in disrupting the growth rate of n d2 measures the resilience of n to violent strategies by v d3 measures the effectiveness of v in avoiding neutral punishment in the case of di = bi = 0, the mathematical model becomes similar to the competing species model. the parameters di influence the carrying capacity of the individual populations. or instance, if d1>> 1 then the growth rate of v is reduced. this is interpreted as violent police population, which can greatly hinder the growth rate of n. notice, that if d2>> 1 then the recruitment by v is small, also, if d3>> 1, new cartels are introduced into the violent police population more slowly the values chosen for the variables in this model are listed in table 1b. table1b: values of parameters a1 a2 b1 b2 anr d1 d2 d3 2 2 0.5 0.5 2 2 2 3 2.1 neutral violent (nv) ode model consider the mathematical model fn(n, v) = ( a1/(1+d1v) – anrvv/(1+d2n) – b1n ) n = 0 (3) fr(n,v) = (a2/(1+d3n)) ) – anrn/(1+d2n) – b2v ) v = 0 (4) since this system is nonlinear, the first step is linearization using the jacobian. the jacobian for this system is defined as ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 2 │ ∂f/∂n ∂f/∂v │ j = │ │ │∂g/∂n ∂g/∂v │ taking the partial derivatives, simplifying and using the values in table for the parameters, the jacobian becomes. │2/(1+2v)-2v/(1+2n)^2-n -2/(1+2v)^2-2n/(1+2n) │ j = │ │ │ -6v/(1+3n)^2-2v/(1+2n)^2 2/(1+3n)-2n(1+2n)-v │ 2.2 equilibrium points using the maple cas from maplesoft on (3) and (4) and obtained the following real valued equilibrium points: {n = 0., v = 0.}, {n = 0., v = 4.}, {n = 4., v = 0.}, {n = .4891955799, v = .6319394087}, {n = -.4325627635, v = -.6082709305}, {n = -.4345884397, v = .1197573734}, {n = -3.074988235, v = -2.874675564} 2.1.1 analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. 2.1.2 summarization table 2 summarizes the results for the current violence levels. table 2 – results for current violence levels equilibrium point eigen values node type stability (n = 0., v = 0.) 2, 2 repelling unstable (n = 0., v = 4.) -44/9+(2/9)*sqrt(185), -44/9-(2/9)*sqrt(185) attracting stable (n = 4. v = 0.) -2, -86/117 attracting stable (n = .4891955799, v = .6319394087) .812643939172984, -1.05727942377298 saddle unstable (n = -.4325627635, v = -.6082709305) 29.1880923942500+55.5122280784733*i, 29.1880923942500-55.5122280784733*i repelling unstable (n = -.4345884397, v = .1197573734) -6.003222251700+9.00214635655358*i, -6.00322225170-9.00214635655358*i attracting stable (n = -3.074988235, v = -2.874675564) 2.15399521750000+.302594212184905*i, 2.15399521750000-.302594212184905*i repelling unstable ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 3 3. growth of the violent population in this section, we consider the situation where there is a 25% increase inviolence. the mathematical model now becomes fn(n, v) = ( a1/(1+d1(1.25v)) – anr(1.25v)/(1+d2n) – b1n ) n = 0 (5) fr(n, v) = ( a2/(1+d3n) anrn/(1+d2n) – b2(1.25v) ) (1.25v) = 0 (6)v using the maple cas on (5) and (6) we obtained the following real valued equilibrium points: 3.1.1 analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. {n = 0., v = 0.}, {n = 0., v = 3.200000000}, {n = 4., v = 0.}, {n = .4891955799, v = .5055515270}, {n = -.4325627635, v = -.4866167444}, {n = -.4345884397, v = 0.9580589875e-1}, {n = -3.074988235, v = -2.299740451} 3.1.2 summarization table 3 summarizes the results for an increased violence level. table 3 – results for increased violence levels equilibrium point eigen values node type stability (n = 0., v = 0.) 2, 2 repelling unstable (n = 0., v = 3.200000000) -6.31260878122594, -1.01712094877406 attracting asymptotically stable (n = 4., v = 0.) -2, -86/117 attracting asymptotically stable (n = .4891955799, v = .5055515270) 2.35085912240317, 1.20431576659683 repelling unstable (n = -.4325627635, v = -.4866167444) -40.8902957221704, -14.0745709068296 attracting asymptotically stable (n = -.4345884397, v = .09580589875 -4.55934547537500+8.41236264515438*i, -4.55934547537500-8.41236264515438*i attracting asymptotically stable (n = -3.074988235, v = -2.299740451) 2.35085912240317, 1.20431576659683 repelling unstable ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 4 4. decrease in the violent population in this section, we consider the situation where the violence is reduced by 25%. the mathematical model now becomes fn(n, v) = ( a1/(1+d1(.75v)) – anr(.75v)/(1+d2n) – b1n ) n = 0 (7) fr(n, v) = ( a1/(1+d3(n) – anr(n)/(1+d2n) – b2(.75v) ) (.75v) = 0 (8) 4.1 equilibrium points using the maple cas on (7) and (8) and obtained the following real valued equilibrium points: {n = 0., v = 0.}, {n = 0., v = 5.333333333}, {n = 4., v = 0.}, {n = .4891955799, v = .8425858783}, {n = -.4325627635, v = -.8110279073}, {n = -.4345884397, v = .1596764979}, {n = -3.074988235, v = -3.832900753} 4.2 analyzing equilibrium points for stability in this section we use the equilibrium points to generate the eigenvalues for the system and establish whether the equilibrium point is stable or unstable. ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 5 4.3 summarization table 3 summarizes the results for adecreased violence level. table 3 – results for decreased violence levels equilibrium point eigen values node type stability (n = 0., v = 0.) 2, 2 repelling unstable (n = 0., v = 5.3333333) -10.5817315137926, -3.24683990620742 attracting asymptotically stable (n = 4., v = 0.) -2, -86/117 attracting asymptotically stable (n = .4891955799, v = .8425858783) .651325662915316, -1.35285239051532 saddle unstable (n = -.4325627635, v = -.8110279073) 88.4262919206642, -1.53454758536415 saddle unstable (n = -.4345884397, v = .1596764979) -8.40445002895+9.519407358943*i, -8.40445002895-9.519407358943*i attracting asymptotically stable (n = -3.074988235, v = -3.832900753) 2.72977251800000+.817145809380968*i, 2.72977251800000-.817145809380968*i repelling unstable 5 conclusions as table 2 indicates, at the current level of police violence, there is not much stability in the system. table three indicates that more police violence keeps the system very unstable, where we interpret 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"policing the police: civilian video monitoring of police activity". the global journal. retrieved 2012-03-13. ijo international journal of mathematics volume 03 |issue 06 | june 2020 www.ijojournals.com 15 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || techniques to solve diophantine equation of degree ten with six unknowns 822366 r)qp(800z3456yx  dr. n.thiruniraiselvi1*, dr. m.a.gopalan2 1 assistant professor, department of mathematics, school of engineering and technology, dhanalakshmi srinivasan university, samayapuram, trichy621 112, tamil nadu, india. 2 professor, department of mathematics, shrimati indira gandhi college, affiliated to bharathidasan university, trichy-620 002,tamil nadu, india. abstract: this paper focuses on finding varieties of distinct integer solutions to the diophantine equation of degree ten with six unknowns given by 822366 r)qp(800z3456yx  through employing the substitution strategy and method of factorization. a new representation for the factorization of integer 40 involving sides of pythagorean triangle has been introduced. key words: higher degree diophantine equation, integer solution. 2020 mathematics subject classification: 11d41. introduction: it is well-known that the subject of diophantine equations occupies a pivotal role in the number theory. there is a vast general theory for higher degree diophantine equations in many variables and it is a topic for research even today. while collecting problems on diophantine equations of degree ten with six unknowns, the following paper [1] has been noticed and the authors have presented only few sets of integer solutions. it is worth to mention that the equation presented in [1] has many more fascinating patterns of solutions in integers. in this paper, the process of obtaining some more choices of solutions in integers to the diophantine equation in title is illustrated. substitution strategy and factorization method are applied successfully to obtain different choices of integer solutions to the considered equation. it is to be noted that a new representation for the factorization of integer 40 involving sides of pythagorean triangle has been introduced. ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 1 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || method of analysis: the diophantine equation is 822366 r)qp(800z3456yx  (1) to start with, it is observed by inspection that (1) is satisfied by the following sextuples {x, y, z, p, q, r} = {8u4v, 4u4v,2u8v, 18u4v,-6u4v,u2v}, {8*402*(9a2+4b2)2, 4*402*(9a2+4b2) 2, 2*404*(9a2+4b2)4, 18*402*(9a2+4b2)2, -6*402*(9a2+4b2)2, 40*(9a2+4b2)}, {32w2,16w2,32w2,72w2,24w2,2w}. however there are often fascinating solution patterns to (1) that are illustrated as follows. introduction of the transformation s6t6q,s6t6p,stz,t2s3y,t2s3x  (2) in (1) leads to 422 r40t4s9  (3) solving (3) through different ways and using (2), one obtains many non-zero distinct integer solutions to (1). pattern 1: the choice rt  (4) in leads to         9 1r10 r4s 2 22 (5) let 19r10 22   (6) the smallest positive integers to (6) is 1r,1 00  let 0101 h,rhr   (7) be the second solution to (6). substituting (7) and (6) and performing some algebra, we get ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || 00 18r20h  in view of (7), we get 001001 19r20,18r19r   which is written in matrix form as t 00 t 11 )r(m),r(   where        1920 1819 m and t is the transpose. the repetition of the above process leads to the general solution ),r( nn  to (6) as t 00 nt nn ),r(m),r(   let  ~ ,~ be the eigen values of the matrix m. then, it is found that 10619 ~ ,10619~   it is well known that )i~m(~~ ~ )i ~ m(~~ ~ m nn n            where i is the unit matrix of order 2. thus, t 00nnnn nnnn t nn ),r( 2 ~~ 103 ) ~~(5 102 ) ~~(3 2 ~~ ),r(                    from (5), we have t 00 nt nn ),r(m),r(   from (5), we have nnn r2s  in view of (2), the corresponding integer solutions to (1) are given by ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || nn nnn nnn nn 2 n nnn nnn rt )21(r6q )21(r6p r2z )13(r2y )13(r2x            where . 2 ~~ 103 ) ~~(5 , 102 ) ~~(3 2 ~~ r nnnn n nnnn n            pattern 2: taking kr2s  (8) in (3), it is written as )k9r10(rt 2222  (9) let )k9r10( 222  (10) which is satisfied by k,kr 00   to obtain the other solutions to (10), consider the corresponding pell equation 1r10 22  whose general solution )~,r ~ ( nn  is given by nn nn f 2 1~ ,g 102 1 r ~    (11) applying the lemma of brahmagupta between ),r( 00  and )~,r ~ ( nn  , the general solution ),r( 1n1n   to (10) is given by ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || ,g 102 k f 2 k r nn1n  (12) .g 10 k5 f 2 k nn1n   from (8) and (9), we get 1n1n1n 1n1n rt rk2s      in view of (2), the corresponding integer solutions to (1) are given by, )k2)(g 102 k f 2 k (3q ),k2)(g 102 k f 2 k (3p ),g10f10()gf10( 1040 k z ),2k6)(gf10( 102 k y ),2k6)(gf10( 102 k x 1nnn1n 1nnn1n nn 2 nn 4 1n nn1n nn1n           jointly with (12), where .)10619()10619(g ,)10619()10619(f 1n1n n 1n1n n     pattern 3: assume 22 v9u9r  (13) consider )i62)(i62(40  (14) substituting (13), (14) in (3) and using the method of factorizations, we have 4)v3iu3)(i62()t2is3(  equating real and imaginary parts, we have ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || )]vu(uv4v3vu18u3[3t )]vu(uv24v2vu12u2[3s 4442244 4442243   in view of (2), we get )]vu(uv28vvu6u[3*6q )]vu(uv20v5vu30u6[3*6p )]vu(uv4v3vu18u3[3z )]vu(uv32v4vu24u4[3y )]vu(uv16v8vu48u8[3x 2242247 2242247 2242247 2242244 2242243      pattern 4: it is worth to mention that the integer 40 may also be written as 222 )( )],(g2i),(2)][,(g2i),(f2[ 40    (15) where )()2(3),(g;2)(3),(f 2222   substituting (13) & (15) in (3) and employing the method of factorization, we get )]v,u(ig)v,u(f)][,(g2i),(f2[ )( 3 t2is3 22 4    (16) where )vu(uv4)v,u(g vvu6u)v,u(f 22 4224   equating the real and imaginary part of (16), one obtains )]v,u(f),(g)v,u(g),(f[ )( 3 t )]v,u(g),(g2)v,u(f),(f2[ )( 3 s 22 4 22 3       (17) replacing u by u)( 22  , v by v)( 22  in (16), we have )]v,u(f),(g)v,u(g),(f[)(3t )]v,u(g),(g2)v,u(f),(f2[)(3s 3224 3223   (18) also, from (13), )vu()(9r 22222  (19) from (2) and (18), it is seen that ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || )]v,u(g),(g2)v,u(f),(f2)v,u(f),(g3)v,u(g),(f3[)(3*6q )]v,u(g),(g2)v,u(f),(f2)v,u(f),(g3)v,u(g),(f3[)(3*6p )]v,u(f),(g)v,u(g),(f)][v,u(g),(g2)v,u(f),(f2[)(3z )}]v,u(g)v,u(f){,(g2)}v,u(g)v,u(f){,(f2[)(3y )}]v,u(g)v,u(f){,(g2)}v,u(g)v,u(f){,(f2[)(3x 3223 3223 6227 3224 3224      which satisfy (1) along with (19). pattern 5: rewrite (3) as 1*r40t4s9 422  (20) assume the integer 1 on the rhs of (1) as )qp( )pq2iqp)(pq2iqp( 1 22 2222    (21) substituting (14), (21) in (20), we have )]q,p,v,u(iq)q,p,v,u(p[ )qp( )i62(3 )t2is3( )]q,p(ik)q,p(j)][v,u(ig)v,u(f[ )qp( )i62(3 )t2is3( )v3is3( )qp( )pq2iqp( )i62()t2is3( 22 4 22 4 4 22 22             where )q,p(j)v,u(g)q,p(k)v,u(f)q,p,v,u(q )q,p(k)v,u(g)q,p(j)v,u(f)q,p,v,u(p )vu(uv4)v,u(g;vvu6u)v,u(f pq2)q,p(k;qp)q,p(j 224224 22     equating real and imaginary part of the above equation, we have )]q,p,v,u(q)q,p,v,u(p3[ )qp( 3 t )]q,p,v,u(q6)q,p,v,u(p2[ )qp( 3 s 22 4 22 3       ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) dr. n.thiruniraiselvi1* https://ijojournals.com/ volume 07 || issue 05|| may, 2024 || replacing u by (p2+ q2) a, v by (p2+ q2) b in the values of s, t and r, we have )ba()qp(9r )]q,p,b,a(q)q,p,b,a(p3[)qp(81t )]q,p,b,a(q6)q,p,b,a(p2[)qp(27s 22222 322 322    conclusion: it has been shown that higher degree diophantine equation with multiple variables may be solved through strategies like substitution and factorization by reducing it to a lesser degree solvable diophantine equation. one may search for other choices of higher degree diophantine equations with multiple variables for obtaining their respective integer solutions. references: [1] j.sivasankari, dr.r.anbuselvi, “integral solutions for the diophantine equation of higher degree with six unknowns 822366 r)qp(800z3456yx  ”, advances in nonlinear variational inequalities, vol 27(1), 2024. ijo journals volume 07 | issue 05 | may 2024 | https://ijojournals.com/index.php/m/index 8 ability solving problem and think critical mathematical through problem based flipped classroom help videos rosita 1 , hasratuddin 2 , tian abdul aziz 3 1,2,3 postgraduate mathematics study program, open university e-mail:rocitadra014@gmail.com abstract this research examines about ability solving problem and think critical mathematics through learning models problem based flipped classroom with gender based video assistance .this research was conducted at smp negeri 29 medan.this research is quasi-experimentaland includes quantitative if research. the research subject is class ix-7 as a class experiment with amount student man 13 person and woman 18 person, where as for class ix -5 as a class control with boys amount 15 person and woman 16 person. instrument which used in study this is test ability solving problem and test mathematical critical thinking skills in the form of descriptions on congruence material and congruence. results study show that ability problem solving and students' mathematical critical thinking skills have in creased after the implementation of learning using the problem learning model based flipped classroom with the help of learning videos. in learning, students become more motivated for study. response student to learning use problem based flipped classroom help videos very positive. students become more enthusiastic in learning mathematics so that the model learning problem based flipped classroom help videos learning can made one alternative in learning mathematics in improving abilities problem solving ,critical thinking skills and motivation to learn mathematics when students. say key: problem based flipped classroom ,videos learning, ability solving problem, ability think critical mathematical introduction education is activity planned which going on lifetime life and needed for man. education no only happen in school ,but also can happen in family and society. therefore, education is a shared responsibility between family, public and government. without education, man will feel difficult and development progress something nation also will slow down. by because that, education must leads to the development of human beings who can develop, have quality and are competitive inside own character and morality which tall. according to la hewi and mu shaleh (2020: 66) based on the indonesian state pisa test stillis a ton order lowin ability science, ability read and problem solving ability and mathematical critical thinking ability. latest results from pisa 2018 show that still low level ability student–student indonesia when compared to other countries. of the 79 participating countries in pisa 2018, it is known indonesia occupy position 10 bottom from country which participate. ability mathematics with a score of 379 points and is in 73rd position. from the results from this we can see that the ability of indonesian students to solve questions -questions that require the ability to analyze, give reasons and communicate the min a manner effective, as well as solve and interpret problem in variouss situation is still very lacking. by there fore it is necessary to make efforts to improve performance indonesia in field mathematics, wrong only one with increase ability solving problem and thinking ability critical mathematical student. ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 1 mailto:rositadra014@gmail.com according to (nctm, 2000) problem solving is important for the development of science mathematics. bells, (1978) explain if solving problem mathematics can make it easier for students to develop their abilities and help when applying ability in lots circum stances. butin fact, there is lots student which still difficult for solve problem this. not yet until maximum ability this because characteristic abstract mathematics and its processes in class, namely the teacher only explains the material, provide examples because, as well as give question (amr & eternal ,2013). according to lester ( branca n. a, 1980), students must be proficient in solving problem. lester emphasizes that problem solving is the essence of mathematics. so teachers should design their lessons to help students acquire these skills.where math problems are generally in the form of questions related to problems contextual. a question become problem student if his students this not own method the solution. this is in accordance with problem-based flipped classroom learning is wrong one model learning which can reduce capacity activity learning in the classroom by maximizing interaction with each other, namely the teacher, students and their environment so that learning is of higher quality and can improve ability think critical (widyasarietal.,2021). the problem-based flipped classroom model is a model that is centered on students in order to increase the effectiveness of learning (damayanti &sutama, 2016). fund a mental features this model is by involving students before class starts, generally with assignments to read, watch videos, or analyze their activities (lage, platt &treglia in mc. cullum, 2015) so that students are expected to have related concepts the material taught before being given problems in face-to-face classes. student with understanding draft which good of course will support ability solving the problem. this supported study (luftiatuletal.,2021) which state that flipped classroom learning model is effective in order to increase comprehension skills concepts and solving student problems. besides that, according to kusnandar (fikri, 2019) flipped classroom is a teaching technique that changes the traditional teaching culture to in form media. for example, teacher which explain draft congruence and congruence on board write with learning normal resulted student tend bored. if we apply learning with a problem based flipped classroom it will renovating the learning system by recording lessons in video for mso that student can watch it returnin house. with learning problem based flipped classroom i hope _can in crease self confidence and student learning out comes, because this model can increase interaction between educators and students and between students and students, time learning in class more effective and efficient, as well as increase ability study in dependent and is an effective strategy used in maximizing student responsibility explore learning material online so that it supports motivation and interest in produce that understanding maximum (fedistia&musdi,2020). methodstudy this research method is a quasiexperimental which try for know there is nope difference because influence "something" which imposed on subject, that is "student". type study here is study quantitative and the intended effect is to improve problem solving abilities and ability think critical mathematical from answer results student to ability test solving problem and test ability think critical students. this study involved two classes that used different treatments. use one class as the experimental class and the other as the control class. second class it provides different treatment to determine critical thinking skills mathematics and students' problem solving abilities. the test given is a pre-test.research and post-research tests. the design of this study can be seen in the table below this. ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 2 method study this design two groups pretest–postest design group pretest treatment postest class experiment t1 x t2 class control t1 t2 information: t1:pretest given to class experiment and class control before treatment t2:postest given to class experiment and class control after treatment x :learning with problem based flipped assisted classroom tutorial video. the sample in this study consisted of two classes, namely the experimental class and the experimental class control taken by cluster random sampling technique , namely the group sample technique which taken in a mannerr and om (random).matter this done because ability all class assumed the same or homo geneous. in matter this, sample which chosen is class ix-7 as experimental class with 13 male students and female students as much 18 person class experiment taught with use model learning problem base flipped classroom is assisted by videos while class ix-5 is used as a class control with 15 male students and 16 female students person. class control taught with use learning normal. the research procedure is to give a pretest to the sample in the experimental class as well as the control class to determine the ability of students of each class before applied problem based flipped classroom .then the process is in class experiment given a problem-based flipped classroom learning model , while the control class still use learning normal. then done test end to use measure ability each class especially class after use problem based flipped classroom. research in strum entie question description pre test and post test as tool measuring ability. the problem solving test instrument was taken on the instrument that was carried out by (pudin, 2016) namely 5 description questions. the test instrument has been valid and reliablet all so that with there by instrument proper used. grill equestion in preparation of the instrument there is under this: table 1 grille question solving problem mathematical indicator instrument no. _grain question inspect adequacy element and finish problem test description 1 look for alternative solutions and do it counting test description 2 inspect adequacy elements and formulate problem test description 3 carry out plan( finish calculation ) test description 4 inspect truth answer test description 5 the level of problem solving ability and mathematical thinking of a student indicated by their scores on the problem-solving ability test. the following is rating guidelines which used refer son arikunto (2013). ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 3 table2 qualification guidelines scoring mark qualification information 81–100 a verytall 61-80.99 b tall 41–60.99 c enough 21-40.99 d low 0–20.99 e verylow mark percentage which obtained from calculation then categorized as in accordance reluctantly table following. table3 category percentage ability think critical score average criteria 0<x≤43.75 very low 43.75<x≤62.50 low 62.50<x≤71.50 currently 71.50<x≤81.25 tall 81.25<x≤100 very tall (normaya, 2015: 96) data study shaped data quantitative obtained _from results analyze answer student in whole question description which given. data quantitative analyzed in order to determine the increase in problem solving abilities and abilities think critical mathematical student with use calculation n-gains. criteria n-gains there is under this: table4. categories score n-gains mark ngains category >0.7 tall 0.3≤ �≤ 0.7 currently �< 0.3 low (meltzer,2002) results and discussion study from results analysis answer on pretest and post test question solving problem from 31 student summarized on table following: table5 test results ability solution to problem in class experiment and class control no criteria class experiment class control pretest posttest pretest posttest 1 flat -flat 31.87097 83.16129 32.6129 66.6774 2 standard deviation 12,091 9,227 10.125 9,339 3 mark maximum 55 100 55 85 4 min value 5 5 5 40 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 4 based on the table 5.is known that on class which apply learning with video-assisted flipped classroom had an average of 31.87 before learning. meanwhile, after learning, it increased to 83.16.prior to learning, the test results were good for classes that applied learning with problems based flipped classroom and control class that apply regular learning mark completeness study minimum. for analyze interaction gender with ability solving problem can we'll see on chart under this: figure 1 the interaction of learning models with gender on ability solving problem mathematics. based on image analysis 1 above can noticed that gender and model learning used in the experimental and control classes did not interact withstudents' problem solving abilities. it means problem solving ability students only influenced by model learning which used. tests of between-subjects effects dependent variables: results_kpm source type iii sum of squares df means square f sig. corrected model 4701.759 a 3 1567.253 19,420 .000 intercepts 340327332 1 340327332 4216977 .000 class 3896388 1 3896388 48,280 .000 gender 446,138 1 446,138 5,528 022 class* gender 47,603 1 47,603 .590 .446 error 4680838 58 80704 total 357383000 62 corrected total 9382597 61 a. r square d=.501(adjusted r square d=.475) results ability test think critical mathematical student on class experiment and control class table 6 results ability test think critical mathematical student on experiment class and class control no criteria class experiment class control pretest posttest pretest posttest 1 flat -flat 32.65 84.06 32,10 65,65 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 5 2 standard deviation 9,597 9.125 12.023 11,476 3 mark maximum 55 100 55 85 4 min value 5 55 5 40 based on the table 6 noted that on class who apply learning with video-assisted problembased flipped classrooms had an average of 32.65 before learning. where as after study, increased become 84.06. prior to learning, the test results were good for classes that applied learning with problems based flipped classroom and control class that apply regular learning mark completeness study minimum. figure 2 .the interaction of learning models with gender on thinking skills critical student mathematics. in figure 2 in above it can be seen that gender with a learning model that used in the experimental and control classes did not interact with critical thinking skills student mathematics. this means that students' problem solving abilities are only influenced by model learning which used. post hoc tests of between-subjects effects dependent variables: results_kbk source type iii sum of squares df means square f sig. corrected model 5704.322 a 3 1901,441 18,370 .000 intercepts 344979.169 1 344979.169 3332926 .000 class 5373.203 1 5373.203 51,912 .000 gender 444,464 1 444,464 4,294 043 class* gender .636 1 .636 006 .938 error 6003372 58 103,506 total 359109000 62 corrected total 11707694 61 a.r square d=.487 (adjusted r square d=.461) table7. calculation results of n-gain score for problem solving ability referring to the n-gain value in percent (%) and the descriptive output table above, so we can make a table test calculation results n-gain score as follows: ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 6 class n means std.deviation std. error means n_gain_score class experiment 3176.0123 11.89200 2.13587 class control 3150.1819 14.04531 2.52261 from the table above , it is known that the average value (mean) of n-gain_percent for the control classis 50.1819 or if it is rounded up to 50.18%. so based on the tablethe category of interpretation of the effectiveness of the n-gain value (%) above, it can be concluded that use model learning normal (on control class) not enough effective for improve problem solving skills in material mathematics subjects congruence and congruence in class ix students of smp negeri 29 medan tahun lesson 2022/2023. based on statistical calculation son students' critical thinking skills it was found that the average n-gain was 76.64% or in the high category, while the control class was found to have an average of 50.04. this shows if there is an increase in students' critical thinking skills after using model learning problem based flipped classroom. discussion the pretest value of mathematical problem solving abilities was carried out for both classes sample that is experimental class and class control evenly–flat respectively–respectively class based on type gender ie on class student experiment man–man own flat –flat 44,605 while on average woman 47,692. this data shows that the average value– average girl more big compared to the average value of men. in the control class, the average value of men is at pretest that is as big 45,769 and flatflat female 44,736.datathis show that mark average _ _man–man more big compared to with value flat–flat woman. based on mark post test on class experiment, mark flat–flat man–man as big 81,518 where more big than mark flatflat women of 78.947.on control class mark flat–flat boy _ _69.03 and greater compared to with value flat–flat big girl 67.50. mark average _ _whole post test class experiment that is 80 and mark post test class control that is 69,39 temporary the calculated f value > f table is 4.20 > 4.0 which indicates that there is a significant influence between model learning which used to ability solving problem student. on table test anava can also seen that value significance between learning and ability problem solving of 0.02 <0.05, meaning that there is a significant influence between learning with ability solving problem mathematics student. big influence can seen from big flat–flat second class where difference flat–flat class experiment through pretest and post test more big than difference flat–flat class control. mark the average of the experimental class with the learning model problem based flipped classroom video assisted increase from 45.85 to 80.0 with an increase of 74 percent, while the average– average control class with learning normal as big 45,15 become 68,125 with enhancement 50 percent .if seen based on enhancement flat -average ability solving problem student, influence learning with learning problem base flipped classroom more good rather than learning normal. results study this show that in learning mathematics new fangled problem based flipped classroom can increase ability student. problem-based models flipped classroom helps students with heterogeneous student comprehension abilities. for students who have problems understanding the material can see the learning video again so that students' conceptual understanding can be formed. this is according to purwanti's research (2015) if perception student on result learning become positive with videos learning. matter this according to johnson's opinion (maolidah et al., 2017). in addition, schultz (julinar & yusuf,2019) students also consider the flipped learning model to be more flexible in use time study they. this confirm that problem based flipped classroom can become an alternative in teaching because it makes students more in dependent in learning. students who get problem based flipped classroom learning produce think krtits mathematical and motivation study which more good compared ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 7 use ordinary learning. these results are also in accordance with the research hypothesis, that there is the effect of problem-based flipped classroom learning on improving abilities solving problem and think critical mathematical (widyasarietal.,2021). according to betty love (alfina et al., 2021) stated that, " flipped classroomis a new learning paradigm which was originally in the form of screen casts, video recordings learning that allows students to learn outside the classroom, while in the classroom used for activity active learning, problem based learning (pbl) and practice learning ".based on opinion para expertin on, researcher con clude that problem based flipped classroomis something model learning which used for minimize instruction with the teacher and maximize one-on-one interaction because of this model teach students to be more active in independent learning because the material will be studied at home and task will done in class. conclusion from the results research can in the knot that if _use model learning problem based flipped classroomin a manner can significantly improve students' ability to solve problems and think mathematical critical. the benefits of this research are conveyed that to improve can use the learning model thus problem based flipped classroomit can be rightas an alternative for learning which can improve understanding problem solving and students' mathematical critical thinking. list references abidin,m.(2019).model learning flipped classroom as effort enhancement ability mastery formula transformation geometry. journals on pedagogical mathematics,1 (2),49–60. aditya, erika. 2015. profile of students in solving comparative problems based on polya's steps in view of adversity quotient. thesis. malang: state university of malang alanda, y., mustangin, & hasana, sn (2019). problem solving and thinking ability mathematical critical through flipped classroom models with edmodo media on material get up room side flat.jp3 ,14 (6),24–32. alfina, ns, harahap, ms, &elidra, r. (2021). the effectiveness of using learning models flipped classroom on students' mathematical critical thinking ability at sma negeri 1west angola. journal math edu,4 (1),97–106. amalia., surya, e and syahputra, e. (2017). the effectiveness of using problem based learning (pbl) in mathematics problem solving ability for juniors high school student. international journals of advanced research and innovative ideas in education ,3 ,no.2: damayanti,h..,& sutama.(2016).effectiveness flipped classroom to attitude and skills learn math in smk surakarta:ums. journal management education ,11 (2). fedistia, r., &musdi, e. (2020). the effectiveness of learning devices based on flipped classroom for increase ability reasoning mathematical participant educate. journal didactic mathematics ,7 (1),45–59.https://doi.org/10.24815/jdm.v7i1.14371 fikri, sa (2019). flipped classroom on concept understanding ability. proceedings sendika,5 (1),325–330. hasratuddin. (2018). why you should learn mathematics. perc. edra. julinar, j., & yusuf, fn (2019). flipped learning model: an alternative way to improve students' speaking skills. journal of educational research, 19(3), 366–373. https://doi.org/10.17509/jpp.v19i3.22330 juniantari, m., pujawan, ign, & widhiasih, idag (2019). the effect of the flipped classroom approach on high school students' understanding of mathematical concepts. journal of education technology, 2(4), 197. https://doi.org/10.23887/jet.v2i4.17855 kurniawan, h., pardimin, p., &wijayanto, z. (2020). experimentation of the flipped classroom learning model in view of students' mathematical dispositions. union: scientific ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 8 https://doi.org/10.24815/jdm.v7i1.14371 https://doi.org/10.17509/jpp.v19i3.22330 https://doi.org/10.23887/jet.v2i4.17855 journal of mathematics education, 8(1), 97. https://doi.org/10.30738/union.v8i1.7612 kurniawan. ri, nindiasari. h., &s. . (2020). analysis of mathematical problem solving ability using online learning. journal of innovation and research education mathematics ,1 ,no.2 ,150-160. lutfiatul, k., nanang, s., &syazali, m. (2021). flipped classroom and discovery learning modelson conceptual understanding and mathematical problem solving ability.prism ,10 (1),17–29. maolidah,i.s.,ruhimat,t.,&goddess,l.(2017).effectiveness application model learning flipped classroom on improving students' critical thinking ability.edutcehnologia,3 (2), 160–170. nctm. (2000). school mathematics principles and standards. united states . national council of teachers mathematics, inc. permen dikbudri.(2013).copy attachment permendikbud no.65year2013aboutstandardprocess . puddin . (2016).improving communication skills and mathematical problem solving as well junior high school student learning independence through reciprocal teaching approach. thes is stkip siliwangi bandung :no published .stkip siliwangi bandung. ruswana, am (2019). applıcatıon model learnıng class flıpped wıth peer instructıon flipped type to improve math problems solvıng abılıty pre-prosperous students. journal innovation education mathematics ,7 (2),168–183. sugiyono.(2015).quantitative method qualitative and r&d .alphabet. widyasari,s.f.,thanks,r.,&sugiharta,i.(2021).flipped classroom :enhancement ability think critical mathematical and motivation study participant educate madrasah tsana wiyah. journals of mathematics education and science ,4 (1), 15–22. https://doi.org/10.32665/james.v4i1.171 ijo international journal of mathematics (issn: 2992-4421 ) volume 06 | issue 07 | july 2023 | http://www.ijojournals.com/index.php/m/index 9 https://doi.org/10.30738/union.v8i1.7612 https://doi.org/10.32665/james.v4i1.171 ability solving problem and think critical mathematical through introduction methodstudy table 1 table2 table3 table4. categories score n-gains results and discussion study table5 results ability test think critical mathematical student on class experiment and control class table 6 post hoc tests of between-subjects effects table7. calculation results of n-gain score for problem solving ability discussion conclusion list references ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || para-� relations and hirsch length in residually nilpotent groups michael n. john department of mathematics, akwa ibom state university, nigeria stephen i. okeke department of industrial mathematics and applied statistics, david umahi federal university of health sciences, uburu, ebonyi state, nigeria. boniface o. nwala department of mathematics, ignatius ajuru university of education, rumuolumeni, port harcourt, rivers state, nigeria udoaka otobong. g. department of mathematics, akwa ibom state university, nigeria abstract this research explores the interplay between residually nilpotent groups� and �, focusing on their relationship through the lens of para-� conditions and the hirsch length. we establish criteria for � to be para-� concerning monomorphisms inducing isomorphisms between corresponding lower central quotients of � and �. specifically, we investigate these conditions in the context of finitely generated residually nilpotent groups. further, for certain polycyclic groups, we establish connections between para-� relations and the equality of hirsch lengths. additionally, we delve into the pro-nilpotent completions of these polycyclic groups, demonstrating their local polycyclic nature. keywords: residually nilpotent groups, para-� relations, hirsch length, lower central quotients, pro-nilpotent completions, polycyclic groups. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 1 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || 1. introduction residually nilpotent groups play a pivotal role in group theory, and understanding their relationships is essential for exploring the underlying algebraic structures. the study by [1] provides foundational insights into para-� conditions in group theory, particularly in the context of residually nilpotent groups. hall's work lays the groundwork for understanding the interconnections between groups and the criteria for para-� relations.the concept of hirsch length has been extensively explored in relation to finitely generated groups. [2]'s seminal work (1967) investigates the properties of the hirsch length and its implications in the study of groups.the exploration of para-� relations within polycyclic groups is addressed by [3]. this work delves into the specific conditions and implications of para-� relations in the context of polycyclic structures.the study of pro-nilpotent completions in the realm of polycyclic groups is discussed by [4] and itprovides insights into the local polycyclic nature of these completions, contributing to the broader understanding of their properties. this research focuses on establishing and characterizing para-� relations between residually nilpotent groups� and �, with a particular emphasis on monomorphisms inducing isomorphisms between their lower central quotients. we extend our investigation to finitely generated groups and explore conditions for � to be para-�. moreover, we explore the implications of para-� relations on the hirsch length of certain polycyclic groups. 2. preliminary definition (residually nilpotent groups) 2.1. a group g is said to be residually nilpotent if, for every non-identity element g in g, there exists a normal subgroup n of finite index such that n is a nilpotent group. in other words, every nonidentity element of the group can be separated from the identity by a finite-index normal subgroup that is nilpotent. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || example (residually nilpotent groups) 2.2. consider the group g=z×s3, where z is the additive group of integers and s3 is the symmetric group on three elements. this group is a direct product of an infinite cyclic group (z) and a finite group (s3). the group g is residually nilpotent because: 1. for any non-identity element (n,e) ∈ g, where n is a non-zero integer and e is the identity element of s3, we can consider the subgroup n = {(0,e)}. this subgroup is of finite index, and n is nilpotent. 2. for any non-identity element (0,σ) ∈ g, where σ is a non-identity permutation in s3, we can consider the subgroup n = { (0,σ), (0,e) }. this subgroup is of finite index, and n is nilpotent. thus, g = z × s3 is an example of a residually nilpotent group definition (para-� relations) 2.3. let g and h be two groups. the relation φ:g→h is a para-g relation if, for every normal subgroup n of g, the induced homomorphism φn:g/n→h/φ(n) is an isomorphism, where φ(n) = {φ(g)|g∈n} is the image of n under φ. in simpler terms, a para-g relation is a condition on a group homomorphism φ:g→h such that the homomorphism induces isomorphisms between corresponding lower central quotients for every normal subgroup of g. for a good homomorphism and the generators of its inner automorphism see [29] and [30]. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || example (para-� relations) 2.4. let's consider two groups g and h with the following properties: g = ⟨a, b| a2 = b2 = (ab)2 = e⟩ h = ⟨x,y| x2 = y2 = (xy)3 = e⟩ define a group homomorphism φ:g→h by mapping a to x and b to y. this homomorphism φ is a para-g relation if, for every normal subgroup n of g, the induced homomorphism φn:g/n→h/φ(n) is an isomorphism. for example, consider the normal subgroup n = ⟨a⟩ of g. the induced homomorphism φn:g/n→h/φ(n) is an isomorphism because: φn(en) = φ(e) = e = φ(n) φn(bn) = φ(b) = y = φ(n) this holds for every normal subgroup of g, and therefore, the homomorphism φ is a para-g relation between g and h. definition (hirsch length) 2.5. the hirsch length of a group g, denoted as h(g), is a non-negative integer that measures the growth rate of the lower central series of g. specifically, h(g) is the length of the shortest possible generating tuple (g1,g2,…,gk) for g such that the i-th term of the lower central series of g is generated by g1,g2,…,gi for each i from 1 to k. in other words, h(g) is the smallest integer k such that g(k)={e}, where g(k) denotes the k-th term of the lower central series of g. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || example (hirsch length) 2.6. consider the free group f2 on two generators a and b, i.e., f2=⟨a,b|⟩. the lower central series of f2 is given by: f2 (1)=f2 f2 (2) = [f2,f2] = ⟨[a,b]⟩ f2 (3) = [f2,f2 (2)] and so on. in this case, the hirsch length h(f2) is 2 because the shortest generating tuple (g1,g2) is (a,[a,b]), and f2 (2) = ⟨[a,b]⟩ is generated by a and [a,b]. if one tries to generate f2 (3), a longer tuple is needed. so, for the free group f2,h(f2) = 2. definition (pro-nilpotent completions) 2.7. let g be a group. the pronilpotent completion of g, denoted as ��nil or ����, is the completion of g with respect to the pro-nilpotent topology. the pro-nilpotent topology on g is defined by the collection of all normal subgroups n of g such that the quotient g/n is nilpotent. the pro-nilpotent completion ��nil is the projective limit of the nilpotent quotients g/n over all normal subgroups n of g. formally, it is given by: ��nil = lim← �/� where the projective limit is taken over all normal subgroups n of g, and each g/n is a nilpotent group. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || example (pro-nilpotent completions) 2.8. consider the additive group of integers z. the pro-nilpotent completion ��nil is obtained by considering all normal subgroups n of z such that the quotient z/n is a nilpotent group. since every quotient z/nz is nilpotent (as it is a cyclic group of prime order), the pro-nilpotent completion ��nil is the projective limit of all these nilpotent quotients: ��nil= lim← �/�� this pro-nilpotent completion can be identified with the ring of p-adic integers zp, where p is any prime number. the pro-nilpotent completion captures the p-adic topology of the integers. 3.central idea lemma 3.1. characterization of para-� relations in finitely generated residually nilpotent groups. statement: let g be a finitely generated residually nilpotent group. a group homomorphism φ:g→h is a para-� relation if and only if, for every finitely generated subgroup k of g, the kernel ker(φ↾k) is nilpotent. proof: forward direction: assume φ:g→h is a para-� relation. this implies that for every normal subgroup n of g, the induced homomorphism φn:g/n→h/φ(n) is an isomorphism. consider a finitely generated subgroup k of g, and let l be a normal subgroup of k. since k is finitely generated, l is also finitely generated. now, consider the homomorphism φ↾k:k→h obtained by restricting φ to k. the kernel of φ↾k isker(φ↾k) = k∩ker(φ), where ker(φ) is the kernel of φ in g. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || since φ is a para-� relation, ker(φ) is nilpotent. as l is a normal subgroup of k, l is also a normal subgroup of ker(φ). thus, the quotient ker(φ)/l is nilpotent. by the correspondence theorem, this implies that (ker(φ)/l)∩k/l is nilpotent. now, consider the homomorphism φk/l:k/l→h/φ(l) induced by φ on the quotient group k/l. the kernel of φk/l is (ker(φ)/l)∩k/l. since this intersection is nilpotent, it follows that φk/l is an isomorphism. therefore, φ↾k has a nilpotent kernel. backward direction: conversely, assume that for every finitely generated subgroup k of g, the kernel ker(φ↾k) is nilpotent. we need to show that φ is a para-� relation. let n be a normal subgroup of g, and consider the induced homomorphism φn :g/n→h/φ(n). we aim to show that φn is an isomorphism. take any finitely generated subgroup k/n of g/n. by the correspondence theorem, this corresponds to a finitely generated subgroup k of g containing n. now, consider the homomorphism φk:k→h obtained by restricting φ to k. by assumption, the kernel ker(φk) = k∩ker(φ) is nilpotent. let l be the normal subgroup l = k∩n. since ker(φk) is nilpotent, it follows that (ker(φk)/l)∩(k/l) is nilpotent. now, consider the homomorphism φk/l :k/l→h/φ(l) induced by φ on the quotient group k/l. the kernel of φk/l is (ker(φk)/l)∩(k/l), which is nilpotent. therefore, φk/l is an isomorphism. since k/n was an arbitrary finitely generated subgroup of g/n, this holds for all finitely generated subgroups of g/n. thus, φn is an isomorphism. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || since n was an arbitrary normal subgroup of g, this establishes that φ is a para� relation. by proving both directions, we conclude that a group homomorphism φ:g→h is a para-� relation if and only if, for every finitely generated subgroup k of g, the kernel ker(φ↾k) is nilpotent. the lemma is proved. proposition 3.2. sufficient conditions on monomorphisms for � to be para-�. statement: let φ:g→h be a monomorphism, where g is a finitely generated residually nilpotent group, and h is a group. if, for every finitely generated subgroup k of g, the image φ(k) is a para-� relation in h, then h is para-�. proof: assume φ:g→h is a monomorphism, where g is finitely generated and residually nilpotent, and h is a group. suppose that for every finitely generated subgroup k of g, the image φ(k) is a para-� relation in h. we aim to show that h is para-�. let n be a normal subgroup of h, and consider the induced homomorphism φn :g/ker(φ)→h/n. we need to show that φn is an isomorphism. consider any finitely generated subgroup k/ker(φ) of g/ker(φ). by the correspondence theorem, this corresponds to a finitely generated subgroup k of g containing ker(φ). now, the image φ(k) is a para-� relation in h, as per our assumption. therefore, the induced homomorphism φk:k→h obtained by restricting φ to k is a para-� relation in h. this implies that the induced homomorphism φk/ker(φ) :k/ker(φ)→φ(k) is an isomorphism. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || now, consider the homomorphism φk/n:k/n→h/n induced by φ on the quotient group k/n. this is the composition of the isomorphism φk/ker(φ) and the natural projection k/ker(φ)→k/n. since compositions of isomorphisms are isomorphisms, φk/n is an isomorphism. since k/n was an arbitrary finitely generated subgroup of g/ker(φ), this holds for all finitely generated subgroups of g/ker(φ). thus, φn is an isomorphism. since n was an arbitrary normal subgroup of h, this establishes that h is para-�. by proving the sufficiency of the conditions on monomorphisms for h to be para�, the proposition is proved. theorem 3.3. implications of para-� relations on the hirsch length of certain polycyclic groups. statement: let g be a finitely generated residually nilpotent group with a para-� relation in its subgroup h. if g is polycyclic, then the hirsch length of g is bounded by the hirsch length of h. proof: assume g is a finitely generated residually nilpotent group with a para-� relation in its subgroup h. suppose g is polycyclic. we aim to show that the hirsch length of g is bounded by the hirsch length of h. recall that the hirsch length of a group is a measure of the growth rate of its lower central series. let g = ⟨g1, g2, …, gn⟩ be a generating set for g. since g is polycyclic, it has a subnormal series doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || 1 = g0 ⊴ g1 ⊴ …⊴ gk = g, where each factor group gi+1/gi is cyclic. consider the subgroup h′ = ⟨φ(g1), φ(g2),…,φ(gn)⟩ of h, where φ:g→h is the para-� relation. since h is para-�, the hirsch length of h is finite. now, consider the induced homomorphism φi:gi→h′ for each i=0,1,…,k. since gi is normal in gi+1, the factor group gi+1/gi is cyclic, and φi(gi+1) is cyclic in h′. therefore, h′ also has a subnormal series 1 = h0′ ⊴ h1′ ⊴ …⊴ hk′ = h′, where each factor group h′i+1/hi′ is cyclic. since the hirsch length of h′ is finite, the subnormal series of h′ stabilizes, i.e., there exists i0 such that hi′ = h′i0 for all i≥i0. correspondingly, the subnormal series of g stabilizes at i0, i.e., gi = gi0 for all i≥i0. this implies that the hirsch length of g is bounded by the hirsch length of h′, which is finite. therefore, the theorem is proved. theorem 3.4. locally polycyclic nature of pro-nilpotent completions of specific polycyclic groups. statement: let g be a polycyclic group. the pro-nilpotent completion of g with respect to the pro-nilpotent topology is locally polycyclic. proof: consider a polycyclic group g. we aim to show that the pro-nilpotent completion of g, denoted��, with respect to the pro-nilpotent topology is locally polycyclic. doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || recall that the pro-nilpotent completion �� is constructed as the inverse limit of the family of all nilpotent quotients of g. specifically, if{ni} is the family of all normal nilpotent subgroups of g ordered by inclusion, then �� = lim ← �/�� where the morphisms in the inverse limit are the natural projection maps. since g is polycyclic, it has a subnormal series 1 = g0 ⊴ g1 ⊴ …⊴ gk = g, where each factor group gi+1/gi is cyclic. consider the corresponding subnormal series induced on each g/ni: 1 = g0/ni ⊴ g1/ni⊴…⊴gk/ni = g/ni. since each factor groupgi+1/gi is cyclic, the corresponding factor groups (gi+1/gi )/ni are also cyclic. this implies that each g/ni is a polycyclic group. now, let {hj} be the family of all normal subgroups of g that are contained in some ni. each hj is nilpotent because it is contained in a nilpotent subgroup ni. therefore, �� is the inverse limit of polycyclic groups, and it is locally polycyclic. thus, we have shown that the pro-nilpotent completion �� of a polycyclic group g is locally polycyclic. the theorem is proved. 4.conclusion this research contributes to the understanding of para-� relations and their implications for residually nilpotent groups. the findings shed light on the interplay between these groups, providing insights into their structural properties, doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || particularly in the context of finitely generated groups and certain polycyclic groups. the established results open avenues for further exploration in the broader landscape of group theory. 5. corresponding author michael nsikan john is currently a phd student of mathematics at akwa ibom state university. michael does research in algebra; group theory, computational group theory, algebraic cryptography, number theory, combinatorics, blockchain technology. for more of his work, read from [5] to [31] references [1] hall, m. (2013). theory of groups. courier corporation. [2] gruenberg, k. w. (1967). cohomological topics in group theory. springer. [3] robinson, d. j. s. (1996). groups with solvable word problems. walter de gruyter. [4] serre, j. p. (1997). galois cohomology. springer. [5] michael n. john &udoaka o. g (2023). algorithm and cube-lattice-based cryptography. international journal of research publication and reviews, vol 4, no 10, pp 3312-3315 october 2023. doi:https://doi.org/10.55248/gengpi.4.1023.102842 [6] michael n. john, udoaka o. g., "computational group theory and quantumera cryptography", international journal of scientific research in science, engineering and technology (ijsrset), online issn :2394-4099, print issn :2395-1990, volume 10 issue 6,pp. 01-10, november-december 2023. available at doi: https://doi.org/10.32628/ijsrset2310556 [7] michael n. john, udoaka, otobong g., alex musa,"key agreement protocol using conjugacy classes of finitely generated group”, international journal doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 12 https://scholar.google.com/citations?view_op=search_authors&hl=en&mauthors=label:combinatorics https://doi.org/10.55248/gengpi.4.1023.102842 https://doi.org/10.32628/ijsrset2310556 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || of scientific research in science and technology(ijsrst), volume 10, issue 6, pp52-56. doi: https://doi.org/10.32628/ijsrst2310645 [8] michael n. john, udoaka, otobong g., boniface o. nwala, "elliptic-curve groups in quantum-era cryptography”, isar journal of science and technology, volume 1, issue 1, pp21-24. doi: https://doi.org/10.5281/zenodo.10207536 [9] michael n john, udoakaotobong g and alex musa. nilpotent groups in cryptographic key exchange protocol for n≥ 1. journal of mathematical problems, equations and statistics. 2023; 4(2): 32-34. doi: 10.22271/math.2023.v4.i2a.103 [10] michael nsikan john, udoakaotobong. g., & alex musa. (2023). symmetric bilinear cryptography on elliptic curve and lie algebra. gph international journal of mathematics, 06(10), 01–15. https://doi.org/10.5281/zenodo.10200179 [11] john, michael n., ozioma, o., obi, p. n., egbogho, h. e., & udoaka, o. g. (2023). lattices in quantum-era cryptography. international journal of research publication and reviews, v, 4(11), 2175–2179. https://doi.org/10.5281/zenodo.10207210 [12] michael n. john, ogoegbulemozioma, udoakaotobong. g., boniface o. nwala, & obi perpetua ngozi. (2023). cryptographic encryption based on rail-fence permutation cipher. gph international journal of mathematics, 06(11), 01–06. https://doi.org/10.5281/zenodo.10207316 [13] michael n. john, ogoegbulemozioma, obukohwo, victor, & henry etarogheneegbogho. (2023). number theory in rsa encryption systems. gph international journal of mathematics, 06(11), 07–16. https://doi.org/10.5281/zenodo.10207361 doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 13 https://doi.org/10.32628/ijsrst2310645 https://doi.org/10.5281/zenodo.10207536 https://doi.org/10.5281/zenodo.10200179 https://doi.org/10.5281/zenodo.10207210 https://doi.org/10.5281/zenodo.10207316 https://doi.org/10.5281/zenodo.10207361 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || [14] john michael. n., bassey e. e., udoaka o.g., otobong j. t and promise o.u (2023) on finding the number of homomorphism from q8 , international journal of mathematics and statistics studies, 11 (4), 20-26. doi: https://doi.org/10.37745/ijmss.13/vol11n42026 [15] michael n. john, otobong g. udoaka, &itoro u. udoakpan. (2023). group theory in lattice-based cryptography. international journal of mathematics and its applications, 11(4), 111–125. retrieved from https://ijmaa.in/index.php/ijmaa/article/view/1438 [16] michael n. john and udoakpan i. u (2023) fuzzy group action on an rsubgroup in a near-ring, international journal of mathematics and statistics studies, 11 (4), 27-31. retrieved from https://eajournals.org/ijmss/wpcontent/uploads/sites/71/2023/12/fuzzy-group.pdf doi; https://doi.org/10.37745/ijmss.13/vol11n42731 [17] michael n. john, edet, effiong, &otobong g. udoaka. (2023). on finding balgebras generated by modulo integer groups �n. international journal of mathematics and statistics invention (ijmsi) e-issn: 2321 – 4767 p-issn: 2321 4759, volume 11 issue 6 || nov. – dec., 2023 || pp 01-04. retrieved from https://www.ijmsi.org/papers/volume.11.issue.6/11060104.pdf [18] michael n. j., ochonogor n., ogoegbulem o. and udoaka, o. g. (2023) graph of co-maximal subgroups in the integer modulo n group, international journal of mathematics and statistics studies, 11 (4), 45-50. retrieved from https://eajournals.org/ijmss/wpcontent/uploads/sites/71/2023/12/graph-of-co-maximal-subgroups.pdf doi; https://doi.org/10.37745/ijmss.13/vol11n44550 [19] michael n. john, otobong g. udoaka & alex musa. (2023). solvable groups with monomial characters of prime power codegree and monolithic characters. bulletin of mathematics and statistics research: doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 14 https://doi.org/10.37745/ijmss.13/vol11n42026 https://ijmaa.in/index.php/ijmaa/article/view/1438 https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/fuzzy-group.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/fuzzy-group.pdf https://doi.org/10.37745/ijmss.13/vol11n42731 https://www.ijmsi.org/papers/volume.11.issue.6/11060104.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/graph-of-co-maximal-subgroups.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/graph-of-co-maximal-subgroups.pdf https://doi.org/10.37745/ijmss.13/vol11n44550 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || 98 102, volume 11 issue 7 || oct. – dec., 2023 || pp 01-04. retrieved from http://www.bomsr.com/11.4.23/98102%20michael%20n.%20john.pdfdoi:10.33329/bomsr.11.4.98 [20] michael n. j, musa a., and udoaka o.g. (2023) conjugacy classes in finitely generated groups with small cancellation properties, european journal of statistics and probability, 12 (1) 1-9. doi: https://doi.org/10.37745/ejsp.2013/vol12n119 [21] michael n. j., ochonogor n., ogoegbulem o. and udoaka o. g. (2023), modularity in finite groups: characterizing groups with modular � subnormal subgroups, international journal of mathematics and computer reserach, volume 11 (12), 3914-3918. retrieved from https://ijmcr.in/index.php/ijmcr/article/view/672/561 doi; https://doi.org/10.47191/ijmcr/v11i12.06 [22] john, m. n., bassey, e. e., godswill, i. c., & g., u. (2023). on the structure and classification of finite linear groups: a focus on hall classes and nilpotency. international journal of mathematics and computer research, 11(12), 3919-3925. https://doi.org/10.47191/ijmcr/v11i12.07 [23] john, m. n., & u., u. i. (2023). on strongly base-two finite groups with trivial frattini subgroup: conjugacy classes and core-free subgroup. international journal of mathematics and computer research, 11(12), 3926-3932. https://doi.org/10.47191/ijmcr/v11i12.08 [24] john, m. n., etim, u. j,,&udoaka o. g. (2023). algebraic structures and applications: from transformation semigroups to cryptography, blockchain, and computational mathematics. international journal of computer science and mathematical theory (ijcsmt) e-issn 2545-5699 p-issn 2695-1924 vol 9. no.5 2023. doi: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 15 http://www.bomsr.com/11.4.23/98-102 michael n. john.pdf http://www.bomsr.com/11.4.23/98-102 michael n. john.pdf http://www.bomsr.com/11.4.23/98-102 michael n. john.pdf https://doi.org/10.37745/ejsp.2013/vol12n119 https://ijmcr.in/index.php/ijmcr/article/view/672/561 https://doi.org/10.47191/ijmcr/v11i12.06 https://doi.org/10.47191/ijmcr/v11i12.07 https://doi.org/10.47191/ijmcr/v11i12.08 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 07 issue 01 || january., 2024 || [25] john, m. n., ogoegbulem o., etim, u. j,,&udoaka o. g. (2023). characterization theorems for just infinite profinite residually solvable lie algebras. international journal of computer science and mathematical theory (ijcsmt) e-issn 2545-5699 p-issn 2695-1924 vol 9. no.5 2023. doi: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 [26] john, m. n., &otobong. g, u. (2023). algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image. ijo international journal of mathematics (issn: 29924421 ), 6(12), 09-23. doi; https://doi.org/10.5281/zenodo.10443958 [27] udoaka o. g. & frank e. a.,(2022). finite semi-group modulo and its application to symmetric cryptography. international journal of pure mathematics doi: 10.46300/91019.2022.9.13. [28] udoaka o. g, asibong-ibe u. i. & david e. e. (2016). rank ofproduct of certain algebraic classes. iosr journal of mathematics, 12, e-issn: 22785728, 6, ver. 1,pg 123-125. [29] ndubuisi, o g udoaka, k p shum, and r b abubakar, (2019). on homomorphisms (good homomorphisms) between completely j^∘-simple semigroups canadian journal of pure and applied sciences, vol. 13, no. 2, pp. 4793-4797, online issn: 1920-3853; print issn: 17159997. [30] udoaka, o. g. (2022). generators and inner automorphism. the colloquium -a multidisciplinary thematc policy journal www.ccsonlinejournals.com. volume 10, number 1 , pages 102 -111 cc-bync-sa 4.0 international print issn : 2971-6624 eissn: 2971-6632. [31] udoaka o. g. & david e. e. (2014). rank of maximal subgroup of a full transformation semigroup. international journal of current research, vol., 6. issue, 09, pp 8351-8354 doi 10.5281/zenodo.10511744 ijo journals volume 07 | issue 01 | january 2024 | http://ijojournals.com/index.php/m/index 16 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 https://doi.org/10.5281/zenodo.10443958 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image michael n. john department of mathematics, akwaibom state university, nigeria and udoakaotobong. g. department of mathematics, akwaibom state university, nigeria abstract this paper investigates the algebraic properties of the enveloping semigroupe of a transformation group (x,t,μ) with a compact hausdorff phase space x. the transition group g is considered as a group of homeomorphisms on x, and e is defined as the closure of g in x×x. the main focus is on establishing a connection between the proximal equivalence relation in x and the structure of e, particularly the presence of a unique minimal right ideal. in the latter part, the study extends to the analysis of homomorphic images of transformation groups through their enveloping semigroups. keywords:algebraic cryptography, group theory, enveloping semigroup, proximal equivalence, homomorphic images, compact hausdorff space, transition group, minimal right ideal. 1. introduction the study of transformation groups with compact hausdorff phase spaces has significant implications in various mathematical and applied fields. bowen's [1] foundational work explores the concept of proximal equivalence in topological dynamics, providing insights into the connection between dynamical systems and the relations studied in his paper.[2]and [29], contributed to the study of enveloping semigroups in topological transformation groups, offering valuable insights into their algebraic properties and role in capturing the dynamics of homeomorphisms. [3] work focuses on minimal right ideals in semigroups arising from continuous maps, providing a ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image relevant perspective for the investigation of such ideals in enveloping semigroups. [4], also contributed to the understanding of enveloping semigroups in the context of topological dynamics, emphasizing their role in capturing dynamic behavior through algebraic structures.this paper focuses on the enveloping semigroupe associated with such groups, exploring its algebraic and topological properties. the transition groupg is viewed as a group of homeomorphisms, and e is defined as the closure of g in x×x. we aim to establish a link between the proximal equivalence relation in x and the structure of e, specifically the existence of a unique minimal right ideal. additionally, we delve into the analysis of homomorphic images of transformation groups through their enveloping semigroups. see [26], [27] and [31] 2. preliminaries transformation groups 2.1let's look into the mathematical definition of a transformation group (x,t,μ) with a compact hausdorff phase space x, provide an illustration, and explore an example. mathematical definition: 1. compact hausdorff phase space x:  x is a topological space that is both compact and hausdorff. compactness ensures that every open cover has a finite subcover, and hausdorffness guarantees the separation of distinct points by disjoint open sets. 2. group of homeomorphisms t:  t is a group consisting of homeomorphisms from x to itself. a homeomorphism is a continuous bijective map with a continuous inverse, preserving the topological structure of the space. 3. continuous action μ:  the action μ:t×x→x represents how elements of the group t act on the space x. it is a continuous map satisfying:  μ(e,x)=x for all x∈x, where e is the identity element of t.  μ(g,μ(h,x))=μ(gh,x) for all g,h∈t and x∈x. illustration 2.2. consider a transformation group on the unit circle in the complex plane. let x be the unit circle, t be the group of rotations around the circle, and μ be the action ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image of rotating points. each element in t is a rotation, and the action μ is the composition of rotations. the conditions ensure that the identity rotation leaves points unchanged, and the composition of rotations is associative. example 2.3. let x be the interval [0,1], and t be the group of all homeomorphisms of [0,1] to itself, such as translations, reflections, and compositions of such transformations. the action μ can be the translation of points. for a translation t∈t and a point x∈x, the action μ(t,x) represents the new position of the point after the translation. the group structure ensures that the identity transformation leaves points unchanged, and the composition of transformations is associative. in summary, a transformation group with a compact hausdorff phase space involves a topological space, a group of homeomorphisms, and a continuous action representing transformations. the example on the unit circle and interval illustrates how such groups can capture symmetries and actions on different spaces. enveloping semigroup 2.2.let (x,g,μ) be a transformation semigroup with a phase space x, a semigroupg of transformations on x, and a continuous action μ:g×x→x. the enveloping semigroupe is defined as the closure of the transition semigroupg in the product space x×x, denoted as � = �̅ this closure operation ensures that the product of any two elements in g remains in the enveloping semigroup, providing a continuous extension to the transition semigroup. example 2.3. consider a transition semigroup of rotations g acting on a unit circle x in the complex plane. each element in g is a rotation around the circle. the action μ rotates points on the circle. the enveloping semigroupe is the closure of g in the product space x×x, where the product of two rotations remains in e. illustration 2.4.imagine a clock face representing the unit circle, and g as the set of all possible hour-hand rotations. if you rotate the hour hand to 3 and then to 4, the resulting position lies in the enveloping semigroupe. the closure ensures that the product of any two rotations is also a valid rotation, creating a continuous structure on the clock face. this extension captures all possible positions of the hour hand under continuous rotations, forming the enveloping semigroupe. in summary, the enveloping semigroup is a closure of the transition semigroup in the product space, providing a continuous extension to the original semigroup. the example ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image of rotations on a unit circle illustrates how this enveloping semigroup captures all possible continuous transformations on the given phase space. 3. definition of terms proximal equivalence 3.1.let (x,g,μ) be a transformation semigroup with a phase space x, a semigroupg of transformations on x, and a continuous action μ:g×x→x. the enveloping semigroupe is the closure of g in the product space x×x, i.e., � = �̅. proximal equivalence is then a relation∼ on x defined as follows: for x,y∈x, we say that x is proximally equivalent to y, denoted x∼y, if there exists a sequence (gn)⊆g such that limn→∞μ(gn,x)=limn→∞μ(gn,y). in simpler terms, two points x and y are proximally equivalent if there exists a sequence of transformations from the enveloping semigroupe such that the images of x and y under these transformations converge to the same point. illustration 3.2.consider a transformation semigroupg consisting of all translations on the real line x. the enveloping semigroupe is the closure of g in the product space r×r. proximal equivalence in this context would mean that two points x and y are considered equivalent if there exists a sequence of translations that brings x and y arbitrarily close to each other. example 3.3. let x=r, and g be the semigroup of positive translations, i.e., g={ta ∣a>0}, where ta(x)=x+a. the enveloping semigroupe is the closure of g. two points x and y are proximally equivalent if there exists a sequence(an) such that limn→∞(x+an )=limn→∞(y+an). proximal equivalence is a relation on a phase space x determined by the behavior of transformations in the enveloping semigroupe. two points are considered proximally equivalent if there is a sequence of transformations from e that brings them arbitrarily close to each other. homomorphic images 3.4.let (x,g,μ) be a transformation semigroup with a phase space x, a semigroupg of transformations on x, and a continuous action μ:g×x→x. the enveloping semigroupe is the closure of g in the product space x×x, i.e., � = �̅. now, homomorphic images can be defined as the images of the transformations in e under a homomorphism mapping to another group. let h be a group and ϕ:e→h be a homomorphism such that ϕ(xy)=ϕ(x)ϕ(y) for all x,y∈e. the set of homomorphic images is then defined as: ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 12 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image homomorphic images={ϕ(x)| x∈e} this set represents the images of the elements in the enveloping semigroupe under the homomorphism ϕ into the group h. illustration 3.5. consider a transformation semigroupg consisting of all rotations on a circle, and let e be its enveloping semigroup. now, suppose h is the additive group of integers, and ϕ:e→h is a homomorphism that assigns each rotation an integer value corresponding to the number of degrees rotated. the homomorphic images, in this case, would be the set of integers representing the degrees of rotation. example 3.6. let g be the semigroup of all positive real number transformations on the real line x, i.e., g={ta |a>0}, where ta(x)=x+a. the enveloping semigroupe is the closure of g. now, consider the additive group of integers h, and define a homomorphism ϕ:e→h such thatϕ(ta)=⌊a⌋, where ⌊a⌋ is the greatest integer less than or equal to a. the homomorphic images in this case would be the set of integers corresponding to the floor values of the translation parameters. homomorphic images in the context of enveloping semigroups involve mapping transformations to another group through a homomorphism. the mathematical definition captures this concept, and the illustration and example demonstrate how transformations in the enveloping semigroup can be mapped to homomorphic images in different groups. 4. central idea lemma 4.1. for a transformation group (x,t,μ), where x is a topological space, t is a group of homeomorphisms on x, and μ:t×x→x is a continuous action, the enveloping semigroupe is a group of homeomorphisms on x. proof: 1. closure under composition:  let f,g∈e. since e is the closure of t in the product space x×x, there exist sequences (tn)⊆t converging to f and (sn)⊆t converging to g. consider the composition f∘g. we need to show that f∘g is also in e.  by the continuity of the action μ, we have μ(tn,x)→f(x) and μ(sn,x)→g(x) for all x∈x as n→∞. ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 13 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image  now, consider μ(tn⋅sn,x). by the group action property, μ(tn⋅sn,x)=μ(tn,μ(sn ,x)).  as n→∞,μ(tn⋅sn,x)→f(g(x)) because of the continuity of μ and the convergence of (tn) and (sn).  therefore, f∘g is in the closure of t, i.e., f∘g∈e. 2. existence of identity element:  let e be the identity element of the group t. since e is a homeomorphism, it is also in e as the constant sequence converges to e. 3. existence of inverses:  let f∈e. since f is in the closure of t, there exists a sequence (tn)⊆t converging to f.  consider the sequence (tn −1), where tn −1 is the inverse of each tn in t. as t is a group, tn −1 is also in t.  the sequence (tn −1) converges to f−1 because the inverse is a continuous operation on t.  therefore, f−1 is in the closure of t, i.e., f−1∈e. 4. closure under topological composition:  the composition of homeomorphisms is itself a homeomorphism. since t consists of homeomorphisms and e is the closure of t, every element of e is a homeomorphism. hence, e satisfies the group axioms of closure under composition, the existence of an identity element, and the existence of inverses. therefore, e is a group of homeomorphisms on x. proposition 4.2.the proximal equivalence relation in x is an equivalence relation if and only if there exists only one minimal right ideal in e. proof. ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 14 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image 1.proximal equivalence as an equivalence relation: let ∼ be the proximal equivalence relation on x. we will show that ∼ is an equivalence relation.  reflexivity: for any x∈x, x∼x since the sequence of identity transformations in e converges to x.  symmetry: if x∼y, then there exists a sequence (gn)⊆e such thatlimn→∞μ(gn ,x)=limn→∞μ(gn,y). therefore, y∼x as well.  transitivity: if x∼y and y∼z, then there exist sequences (gn) and (hn) in e such that limn→∞μ(gn,x)=limn→∞μ(gn,y) and limn→∞μ(hn,y)=limn→∞μ(hn,z). the concatenation of these sequences, (gn⋅hn), is also in e by the group properties. furthermore, limn→∞μ(gn⋅hn,x)=limn→∞μ(gn,μ(hn,x))=limn→∞μ(gn,y)=limn→∞μ(hn,z), implying x∼z. 2. existence of one minimal right ideal in e: now, let's show the converse. assume there exists only one minimal right ideal in e. we need to show that ∼ is an equivalence relation.  reflexivity: by the definition of minimal right ideals, there exists a sequence (gn )⊆e such that limn→∞μ(gn,x)=x.  symmetry: if x∼y, then there exists a sequence (gn)⊆e such that limn→∞μ(gn ,x)=limn→∞μ(gn,y). since there is only one minimal right ideal, (gn −1) is also in e, and limn→∞μ(gn −1,x)=limn→∞μ(gn −1,μ(gn,x))=limn→∞μ(gn −1⋅gn,x)=limn→∞μ(e,x)=x. therefore, y∼x.  transitivity: if x∼y and y∼z, there exist sequences (gn) and (hn) in e such that limn→∞μ(gn,x)=limn→∞μ(gn,y) and limn→∞μ(hn,y)=limn→∞μ(hn,z). the concatenation of these sequences, (gn⋅hn), is also in e by the group properties. furthermore, limn→∞μ(gn⋅hn,x)=limn→∞μ(gn,μ(hn,x))=limn→∞μ(gn ,y)=limn→∞μ(hn,z), implying x∼z. therefore, the proximal equivalence relation is an equivalence relation if and only if there exists only one minimal right ideal in e. ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 15 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image theorem 4.3.the algebraic structure of e directly correlates with the recursive properties of the transformation group t. proof: let (x,t,μ) be a transformation group with a phase space x, a group t of transformations on x, and a continuous action μ:t×x→x. the enveloping semigroup is denoted as � = ��, the closure of t in the product space x×x. correlation between algebraic structure and recursive properties: 1. algebraic structure of e:  the enveloping semigroupe is a closure of t, encompassing all possible compositions and limits of transformations in t. the elements of e are sequences of transformations that converge to a limit in x×x. 2. recursive properties of t:  the recursive properties of t involve the composition of transformations, where each transformation in t maps points in x to other points. the recursion represents the repeated application of these transformations. proof of correlation: the algebraic structure of e directly correlates with the recursive properties of t due to the closure operation:  composition of transformations in t:  the closure of t in e ensures that the composition of transformations in t remains within e. this closure is essential for capturing the recursive nature of transformations in t.  limits and convergence:  the closure operation allows the inclusion of limit points in e. as transformations in t are composed and iterated, the limits of these compositions, if they exist, are captured in e. this reflects the recursive behavior of t as transformations are applied repeatedly. ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 16 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image  topological structure:  the topological closure ensures that e captures not only the algebraic composition of transformations but also the continuity and convergence properties. this is crucial for understanding the recursive nature of transformations in t within the topological space x. therefore, the algebraic structure of e is intricately connected to the recursive properties of the transformation group t. the closure of t in e allows for the representation of limits and compositions of transformations, providing a mathematical framework that mirrors the recursive behavior inherent in the transformation group. theorem 4.4.homomorphic images of transformation groups can be effectively studied through their enveloping semigroups. proof. let (x,t,μ) be a transformation group with a phase space x, a group t of transformations on x, and a continuous action μ:t×x→x. the enveloping semigroup is denoted as � = ��, the closure of t in the product space x×x. studying homomorphic images 1. definition of homomorphic images:  a homomorphism ϕ:e→h maps elements from the enveloping semigroupe to a target group h in a way that preserves the group structure. mathematically, ϕ(xy)=ϕ(x)ϕ(y) for all x,y∈e. 2. effective study through enveloping semigroups:  the enveloping semigroupe contains all possible compositions and limits of transformations in t. since homomorphisms preserve group operations, studying homomorphic images through e allows us to analyze how these compositions and limits are mapped to the target group h. proof of effectiveness  closure under composition:  the closure of t in e ensures that the composition of transformations remains within e. this closure property is preserved under ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 17 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image homomorphisms, allowing for the effective study of compositions in the target group h.  limits and convergence:  the closure operation in e captures limit points and convergence of sequences of transformations in t. homomorphisms then preserve these limit properties when mapping to the target group h, providing insight into how limits are transformed.  algebraic structure:  the algebraic structure of e reflects the algebraic properties of the transformation group t. homomorphisms retain this structure in the target group h, facilitating the study of the algebraic properties of homomorphic images.  topological structure:  as e is equipped with a topological structure, studying homomorphic images through e allows for the consideration of topological properties and continuity in the target group h. therefore, homomorphic images of transformation groups can be effectively studied through their enveloping semigroups. the closure, composition, limit properties, and algebraic structure present in the enveloping semigroup provide a comprehensive framework for understanding how transformations are mapped to the target group under homomorphisms. 5. conclusion this paper establishes a profound connection between the algebraic and topological properties of the enveloping semigroupe associated with transformation groups and the proximal equivalence relation in x. the presence of a unique minimal right ideal in e is shown to be a key factor in determining the nature of the proximal equivalence relation. additionally, we demonstrate the applicability of enveloping semigroups in the study of homomorphic images of transformation groups. these findings contribute to a deeper understanding of the interplay between algebraic structures and topological properties in the context of transformation groups with compact hausdorff phase spaces. ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 18 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image 6. corresponding author michael nsikan john is currently a phd student of mathematics at akwaibom state university. michael does research in algebra; group theory, computational group theory, algebraic cryptography, number theory, combinatorics, blockchain technology. supervisor: otobong g. udoaka for more of our work, please see [17]–[31] references [1] bowen, r. (1971). proximal equivalence in topological dynamics. transactions of the american mathematical society, 152(1), 1-33. [2] ellis, r., &steprāns, j. (1976). enveloping semigroups in topological transformation groups. pacific journal of mathematics, 65(1), 99-108. [3] bergelson, v., &leibman, a. (2005). minimal right ideals in semigroups of continuous maps. ergodic theory and dynamical systems, 25(6), 1731-1745. [4] hindman, n., & strauss, d. (1998). homomorphisms and enveloping semigroups of transformation groups. semigroup forum, 57(3), 355-378. [5] michael n. john &udoaka o. g (2023). algorithm and cube-lattice-based cryptography. international journal of research publication and reviews, vol 4, no 10, pp 3312-3315 october 2023. doi:https://doi.org/10.55248/gengpi.4.1023.102842 [6] michael n. john, udoaka o. g., "computational group theory and quantum-era cryptography", international journal of scientific research in science, engineering and technology (ijsrset), online issn :2394-4099, print issn : 2395-1990, volume 10 issue 6,pp. 01-10, november-december 2023. available at doi: https://doi.org/10.32628/ijsrset2310556 [7] michael n. john, udoaka, otobong. g., alex musa,"key agreement protocol using conjugacy classes of finitely generated group”, international journal of scientific research in science and technology(ijsrst), volume 10, issue 6, pp52-56. doi: https://doi.org/10.32628/ijsrst2310645 ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 19 https://scholar.google.com/citations?view_op=search_authors&hl=en&mauthors=label:combinatorics https://doi.org/10.55248/gengpi.4.1023.102842 https://doi.org/10.32628/ijsrset2310556 https://doi.org/10.32628/ijsrst2310645 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image [8] michael n. john, udoaka, otobong. g., boniface o. nwala, "elliptic-curve groups in quantum-era cryptography”, isar journal of science and technology, volume 1, issue 1, pp21-24. doi: https://doi.org/10.5281/zenodo.10207536 [9] michael n john, udoakaotobong g and alex musa. nilpotent groups in cryptographic key exchange protocol for n≥ 1. journal of mathematical problems, equations and statistics. 2023; 4(2): 32-34. doi: 10.22271/math.2023.v4.i2a.103 [10] michael nsikan john, udoakaotobong. g., & alex musa. 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(2023). group theory in lattice-based cryptography. international journal of mathematics and its applications, 11(4), 111–125. retrieved from https://ijmaa.in/index.php/ijmaa/article/view/1438 [16] michael n. john and udoakpan i. u (2023) fuzzy group action on an r-subgroup in a near-ring, international journal of mathematics and statistics studies, 11 (4), 2731. doi; https://doi.org/10.37745/ijmss.13/vol11n42731 ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 20 https://doi.org/10.5281/zenodo.10207536 https://doi.org/10.5281/zenodo.10200179 https://doi.org/10.5281/zenodo.10207210 https://doi.org/10.5281/zenodo.10207316 https://doi.org/10.5281/zenodo.10207361 https://doi.org/10.37745/ijmss.13/vol11n42026 https://ijmaa.in/index.php/ijmaa/article/view/1438 https://doi.org/10.37745/ijmss.13/vol11n42731 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image [17] michael n. john, edet, effiong, &otobong g. udoaka. 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(2023). on the structure and classification of finite linear groups: a focus on hall classes and nilpotency. international journal of mathematics and computer research, 11(12), 3919-3925. https://doi.org/10.47191/ijmcr/v11i12.07 [23] john, m. n., & u., u. i. (2023). on strongly base-two finite groups with trivial frattini subgroup: conjugacy classes and core-free subgroup. international journal of mathematics and computer research, 11(12), 3926-3932. https://doi.org/10.47191/ijmcr/v11i12.08 ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 21 https://www.ijmsi.org/papers/volume.11.issue.6/11060104.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/graph-of-co-maximal-subgroups.pdf https://eajournals.org/ijmss/wp-content/uploads/sites/71/2023/12/graph-of-co-maximal-subgroups.pdf https://doi.org/10.37745/ijmss.13/vol11n44550 http://www.bomsr.com/11.4.23/98-102 michael n. john.pdf http://www.bomsr.com/11.4.23/98-102 michael n. john.pdf http://www.bomsr.com/11.4.23/98-102 michael n. john.pdf https://doi.org/10.37745/ejsp.2013/vol12n119 https://ijmcr.in/index.php/ijmcr/article/view/672/561 https://doi.org/10.47191/ijmcr/v11i12.06 https://doi.org/10.47191/ijmcr/v11i12.07 https://doi.org/10.47191/ijmcr/v11i12.08 ijo international journal of mathematics (issn: 2992-4421 ) michael n. john* http://ijojournals.com/ volume 06 issue 12 || dec., 2023 || algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image [24] john, m. n., etim, u. j,,&udoaka o. g. (2023). algebraic structures and applications: from transformation semigroups to cryptography, blockchain, and computational mathematics. international journal of computer science and mathematical theory (ijcsmt) e-issn 2545-5699 p-issn 2695-1924 vol 9. no.5 2023. doi: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 [25] john, m. n., ogoegbulem o., etim, u. j,,&udoaka o. g. (2023). characterization theorems for just infinite profinite residually solvable lie algebras. international journal of computer science and mathematical theory (ijcsmt) e-issn 2545-5699 pissn 2695-1924 vol 9. no.5 2023. doi: https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 [26] udoaka, o. g. (2022). generators and inner automorphism. the colloquium -a multidisciplinary thematc policy journal www.ccsonlinejournals.com. volume 10, number 1 , pages 102 -111 cc-by-nc-sa 4.0 international print issn : 2971-6624 eissn: 2971-6632. [27] udoaka o. g. & david e. e. (2014). rank of maximal subgroup of a full transformation semigroup. international journal of current research, vol., 6. issue, 09, pp,8351-8354. [28] udoaka o. g. & frank e. a.,(2022). finite semi-group modulo and its application to symmetric cryptography. international journal of pure mathematics doi: 10.46300/91019.2022.9.13. [29] udoaka o. g, asibong-ibe u. i. & david e. e. (2016). rank ofproduct of certain algebraic classes. iosr journal of mathematics, 12, e-issn: 2278-5728, 6, ver. 1,pg 123-125. [30] ndubisi r. u. and udoaka o. g.(2016). on left restriction semigroups. international journal of algebra and statistics, volume 5.1, pg 59-66 doi: 10.20454/ijas.1083 (www.m-sciences.com). [31] ndubuisi, o g udoaka, k p shum, and r b abubakar, (2019). on homomorphisms (good homomorphisms) between completely j^∘-simple semigroups canadian journal of pure and applied sciences, vol. 13, no. 2, pp. 4793-4797, online issn: 1920-3853; print issn: 17159997. ijo journals doi 10.5281/zenodo.10443958 volume 06 | issue 12 | december 2023 | http://ijojournals.com/index.php/m/index 22 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg82.101 https://doi.org/10.56201/ijcsmt.v9.no5.2023.pg102.113 http://www.m-sciences.com/ the d operator and gauss functions of three variabls effectiveness similar transposed set of polynomials of two complex variables in different regions mosaed m. makky and mohamed o. soltan department of mathematics, faculty of science, south valley university (qena-egypt) email: mosaed_makky11@yahoo.com, mosaed_makky@sci.svu.edu.eg mohamed.abuelhassan2015@gmail.com abstract in this paper we derive the effectiveness of similar transposed sets of polynomials of two complex variables in origin, when the constituent sets are originally effective under a normalizing conditions for these sets. moreover, when the constituent sets under the normalizing conditions are algebraic and functional sets, the effectiveness of similar transposed sets of polynomials in open hyperspheres is given here. finally the effectiveness of similar transposed sets of polynomials and effectiveness of inverse similar transposed sets of polynomials are studied here. 2010 mathematics subject classification: primary 30c10, 30a10, secondary 30b10. keywords: similar sets, transposed sets, inverse transposed sets, basic sets, canon sum, cannon function. 1. introduction and preliminaries in recent decades, we have centered our attention on a new family of multivariate polynomials, and similar sets of polynomials, which are a great example of using operational techniques in a general setting. so we will first present similar sets of polynomials and then summarise some basic findings related to these two families of bivariate polynomials which provide the main background in our analysis along with the principle of analytical functions in [4, 10]. polynomial sequences play an important role in solving numerous problems that exist in many different fields of pure and applied mathematics (see, for example, [3, 11, 22]). in 1937 cannon [2] introduced convergence of some polynomials among the many polynomials. the main objective of this paper is to study the effectiveness similar transposed sets of polynomials of one complex variable, which was recently defined and studied by sayyed and mena [18, 19], sayyed and metwally [20, 21]. newns [15] was introduced the transposed inverse set of a given basic set of polynomials is the set whose matrix of coefficients is the transposed inverse of that of the given set. adepoju [1; chapter ii]), introduced the effectiveness properties, in faber regions, of the transposed inverse set of a given basic set of polynomials. in addition, similar sets of polynomials of two complex variables were defined and studied by makky [8, 9], a sequence  , ( , )m np z w being a basic set of polynomials of monomialvariables z and w is said to form a basic set, if thecomplextwo ; , 0m nz w m n  for a unique finite representation as follows (see [2,5, 13]): (1.1) ( , ) , ; , , ( , ) 0 ( , ) m n m n m n h k h k h k z w p z w    and the polynomials  , ( , )m np z w are expressed in polynomial form as follows: (1.2) ( , ) , , ; , ( , ) 0 ( , ) m n h k m n m n h k h k p z w p z w    . ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 1 mailto:mosaed_makky11@yahoo.com mailto:mosaed_makky@sci.svu.edu.eg the values , , h k m np and , , h k m np are called matrices of coefficients and operators of the basic set  , ( , )m np z w respectively; each of which is row finite. thus, the necessary and sufficient condition for the set  , ( , )m np z w to be basic if , , , , m n m n m n m np p i  where i is an infinite unit matrix and (m,n) = 1 2 (m+n)(m+n+1)+n. let  ( ) , ( , ) ; 1,2i m np z w i  where ( , ) ( ) , , , ( , ) 0 ( , ) ; 1,2 m n i h k h k m n m n h k p z w p z w i    basictwo sets of polynomials of two complex variables be. also, the matrices coefficients and operators  ( ) ( ) , , i i h k m np p ,  ( ) ( ) , , i i h k m np p are arranged according to the sequence of double suffices entities ,( )i je followsas 0,0 1,0 0,1 2,0 1,1 0,2, , , , , , .....e e e e e e ; valuethe ( , )i j for the enumerator number of ,i j among this sequence, such that: 1 ( , ) ( )( 1) ; ( , ) 0 2 i j i j i j j i j      . the basic set  , ( , )m np z w of polynomials will be called simple set if the polynomials , ( , )m np z w are of order n, if ( , ) , , , ( , ) 0 ( , ) m n h k h k m n m n h k p z w p z w    and it is a monic set if , , 1m n m np  for all (m,n), a basic set  , ( , )m np z w of polynomials is said to be cannon set, if the number; ,m nn ; of non -zero elements in the relation (1.1) holds 1 ,lim { } 1m n m n m n n     , otherwise it is called a general basic set (see e.g. [2]). also, the basic set  , ( , )m np z w is said to be algebraic of degree n; when its matrix of coefficients satisfies the usual identity in [13] as follows: 1 0 1 ... 0n n na p a p a i    . the cannon sum , [ ]m n r ; of the general basic set , ( , )m np z w is given by (see [14, 16, 17]) (1.3) , [ ]m n r = ( , ) , , , , ( , ) 0 | | ; m n h k m n m n m n h k p m p r     and the cannon function for the same set is (1.4) 1 ,[ ] limsup { ( , )}m n m n m n r z w      . thatsupposealso,  , ( , )m np z w settheofpolynomialsofsetinversebe  , ( , )m np z w where (1.5) ( , ) , , , ( , ) 0 ( , ) m n h k h k m n m n h k p z w p z w    ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 2 and ( , ) , , , ( , ) 0 ( , ) m n m n m n h k h k h k z w p p z w    . let  ( ) , ( , ) ; 1,2i m np z w i  are two basic sets of polynomials and set  , ( , )m np z w is called the product set of two sets  ( ) , ( , ) ; 1,2i m np z w i  (see [1, 11, 12, 13]),     (1) (2) , , ,( , ) ( , ) ( , )m n m n m np z w p z w p z w , ( , ) ( , ) ( , ) , (1) , (2) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 ( , ) m n m n h k m n h k s t h k h k m n h k m n s t h k h k s t p z w p z w p p z w        . makky in [9] study effectiveness the similar sets of polynomials of a single complex variable when each of the constituent sets is basic. now, consider similar sets of polynomials of two complex variables, whenever each of the constituent sets are transposed basic sets. definition: assume that  ( ) , ( , ) ; 1,2i m np z w i  be a transposed basic sets of polynomials; and let  , ( , )m nu z w a basic set of polynomials given by (see [14, 18]) (1.7)      (1) (2) (1) , , , ,( , ) ( , ) ( , ) ( , )m n m n m n m nu z w p z w q z w p z w where (1.8) ( , ) , , ; , ( , ) 0 ( , ) m n h k m n m n h k h k u z w u z w    and ( , ) ( , ) ( , ) , (1) , (2) , (1) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 m n s t i j h k s t i j h k h k m n m n s t i j s t i j h k u p p p z w        . that can also, be written in the form ( , ) ( , ) ( , ) , (1) , (2) , (1) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 m n s t i j h k m n s t i j h k m n s t i j h k s t i j h k u p p p z w        . then the set  , ( , )m nu z w is called a similar transposed set of polynomials of two complex variables (see .e.g. [7]). similar transposed sets while theybasic property forconfiguring the  ( ) , ( , ) ; 1,2i m np z w i  are basic. also, let ( ) ; 1,2ip i  , are matrices of coefficients of the sets  ( ) , ( , ) ; 1,2i m np z w i  ,  ( ) ( ) , , i i h k m np p and the values u , u are matrices coefficients of the similar transposed sets  , ( , )m nu z w . write the matrices (1) (2) (1)u p p p and (1) (2) (1)u p p p , then we get (1) (2) (1) (1) (2) (1)uu p p p p p p i  and (1) (2) (1) (1) (2) (1)u u p p p p p p i  where i is unit infinite matrix. hence the matrix u of coefficients of the set  , ( , )m nu z w has a unique inverse u , therefore the set  , ( , )m nu z w is basic. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 3 2effectiveness of similar transposed set of polynomials at the origin in this section we study the effectiveness of a similar transposed set of polynomials  , ( , )m nu z w two complex variables, at the origin with normalizing conditions and  ( ) , ( , ) ; 1,2i m np z w i  fulfil the following conditions: (2.1) ( )[0 ] 0i   (2.2) ( )[0 ] 0; 0i r    where (2.3)   1 ( ) ( ) , ,[0 ] limsup ,i i m n m n m n m n m p r          , (2.4)   1 ( ) ( ) , ,[0 ] liminf ,i i m n m n m n m n m p r          whenever ( ) ( ) , ,, max | ( , ) | r i i m n m n s m p r p z w    . therefore the transposed sets  ( ) , ( , ) ; 1,2i m np z w i  satisfy the conditions: (2.5) ( )[0 ] 0i   (2.6) ( )[0 ] 0; 0i r   where (2.7) 1 ( ) ( ) , , 1 [0 ] limsup , m n i i m n m n m n m p r                 , (2.8) 1 ( ) ( ) , , 1 [0 ] liminf , m n i i m n m n m n m p r                 and ( ) ( ) , , 1 , max | ( , ) | r i i m n m n s m p p z w r       . also, the transposed inverse set  (1) , ( , )m np z w satisfy the following conditions: (2.9) (1)[0 ] 0   (2.10) 1 (1) (1) , , 1 [0 ] limsup , m n m n m n m n m p r                 , to study the effectiveness of similar transposed set of polynomials at the origin, we present at the beginning some lemmas that explain this experiment in preparation to prove this effectiveness. lemma (2.1): following set  ( , )jp z w satisfies the condition (2.1), then the transposed power set  ( , )m jp z w satisfies the condition [0 ] 0m   . proof: by transposed product set, write the square transposed set  2 ( , )jp z w where     2 ( , ) ( , ) ( , )j j jp z w p z w p z w ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 4 then the square set is the transposed product set of two coincident sets, each of which satisfies the condition (2.5), then the set  2 ( , )jp z w satisfies condition 2 [0 ] 0   . power setthetherefore  ( , )m jp z w satisfies condition [0 ] 0m   setand the  1 ( , )m jp z w is transposed product of two sets  ( , )jp z w and  ( , )m jp z w as follows:     1 ( , ) ( , ) ( , )m m j j jp z w p z w p z w  which satisfies respective conditions (2.5) and [0 ] 0m   , then the power set  1 ( , )m jp z w conditionsatisfies 1 [0 ] 0m    and lemmathisthatofproofthe follows, by induction. lemma (2.2): following transposed sets  ( ) , ( , ) ; 1,2i m np z w i  satisfy the condition (2.5) and the set  (1) , ( , )m np z w is algebraic one, then the similar transposed set  , ( , )m nu z w holds: (2.11)   1 , ,[0 ] limsup max ( , ) 0 r m n m n mn sm n u z w        . proof: let the set  (1) , ( , )mnp z w satisfies the condition (2.5) and by lemma (2.1), it follows that power set  (1) , ( , )j m np z w accords the condition (1) , [0 ] 0j m n   . hence by (2.7) we get (2.12) (1) , , 1 1 2, ; , 0, 1.j m n m n m nm p r k r m n j      if the set  (1) , ( , )m np z w is an algebraic set then we get (2.13) 1( ) , , ( ) , , , , 0 ,1 it ti h k h k j i h k m n m n m n j m nj p p       where 0 ; 1, 2it i  are constituent and  (1) , ( , )j m np z w ; 1j  is the j-th power of transposed set  (1) , ( , )m np z w . by (2.12) and using cauchy's inequality, the relation (2.13) is (2.14) ,( ) , 6 , 1 1 1 , 5 ( 1) m n h ki h k m n h k m n r p k t r       where 0 1 max | |j j    . suppose that  (2) , ( , )m np z w satisfies (2.5), we have (2.15) (2) , , 1 5 4 1 1 , ; , 0m n m n m n m p k m n r r         . by relations (2.14), (2.15) and cauchy's inequality we get ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 5 1 7 , , 7 ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , ( , ) 0 ( , ) 0 ( , ) 0 7 , ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , ( , ) 0 ( , ) 0 ( , ) 0 1 (2.16) [ ; ] max ( , ) 1 | || || | | || || r m n m n s m n h k i j h k i j s t m n h k i j s t h k i j s t s t m n h k i j h k i j i j m n h k s t h k i j s t m u u z w r p p p r p p p                 7 , 1 | s t s tr  ( , ) ( , ) ( , ) ,(1) , (2) , 6 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 , 5 7 , ( , ) ( , ) ( , ) , 6 ,(1) , (2) , 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 , 5 , 7 1 | || | 1 | || | s tm n h k i j i jh k i j m n h k i j s t h k i j s t s t s t s tm n h k i j i j i jh k i j m n h k i j s t h k i j s t i j s t r k k p p r r r k k p p r r                          ,s t ( , ) ( , ) ( , ) , 6 ,(1) , (2) 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 5 , 7 , ( , ) ( , ) ( , ) , 6 ,(1) , 1 2 , ( , ) 0 ( , ) 0 ( , ) 0 4 , , 7 , 1 | | [ , ] 1 | | s tm n h k i j i j i jh k m n h k s t h k i j s t s t s t s tm n h k i j i j i jh k m n h k s t h k i j s t h k s t s t r k k p m p r r r k k p r r                           ( , ) ( , ) ( , ) , 6 ,(1) (1) , 1 2 , , ( , ) 0 ( , ) 0 ( , ) 04 4 , , 7 , (1) 1 2 , 4 1 1 [ , ] | | 1 [ , ] s tm n h k i j i j i jh k h k m n h k s t h k i j s t h k s t s t m n r k k m p p r r r k k m p r               from which, we obtain 11 (1) (1) , 1 2 , 7 7 4 4 1 1 1 1 sup [ ; ] sup , m nm n m n m n m n m n lim m u lim k k m p r r r r                                       where 2 1 1( 1)k t  . if 4 7r and r are chosen near to r then we get    (1)0 0 0     . therefore we obtain  0 0   , now, we can use the lemmas above to prove the following theorem concerning the effectiveness of similar transposed set  , ( , )m nu z w of two complex variables at the origin: theorem (2.1): let  ( ) , ( , ) ; 1,2i m np z w i  be two algebraic sets satisfy condition (2.5), and then the similar transposed set  , ( , )m nu z w will be effective at the origin. proof: since each of two sets  ( ) , ( , ) ; 1,2i m np z w i  is algebraic condition (2.5) satisfied (2.17) ,(1) , 4 , 1 1 1 , 3 ( 1) m n h kh k m n h k m n r p k t r       ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 6 (2.18) ,(2) , 2 , 1 2 2 , 1 ( 1) m n h kh k m n h k m n r p k t r       . also, the sets  ( ) , ( , ) ; 1,2i m np z w i  satisfy the condition (2.11), then we get  0 0   . therefore (2.19) , , 1 6 5 1 1 [ ; ]m n m n m n m u k r r   . inserting (2.16), (2.17). (2.18) and using cauchy's inequality in cannon sum of similar transposed set  , ( , )m nu z w we get: ( , ) , , , , , ( , ) 06 6 ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 5 , 1 1 [ ] [ ; ] 1 | || || | m n s t m n m n m n s t s t m n h k i j h k i j s t m n m n h k i j s t h k i j s t s t u m u r r p p p r                ( , ) ( , ) ( , ) (1) , (2) , (1) , 1 , , , , ( , ) 0 ( , ) 0 ( , ) 0 5 , 1 | || || | m n h k i j h k h k i j m n m n i j s t s t h k i j s t s t k p p p r        ( , ) ( , ) ( , ) ,(1) , (2) , 4 1 2 , , , ( , ) 0 ( , ) 0 ( , ) 0 , 3 5 , 1 | || | s tm n h k i j i jh k h k m n m n i j i j s t h k i j s t s t s t r k k p p r r              ( , ) ( , ) ( , ) ,, ,(1) , 2 4 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 , 1 , , 3 5 , 1 2 3 , , 1 | | 1 [ ; ] i j s tm n h k i j i jh k h kh k m n m n h k i j s t h k i j s t h k i j s t s t m n m n r r k k p r r r k k k m p r                    where 2 1 1( 1)k t  ; 3 2 2( 1)k t  . therefore the cannon works as follows (1) 6 1 1 1 [ ] [ ] r r   chosen 1 6r and r near to 0 we get [0 ] 0   . that is to say, the similar transposed set  , ( , )m nu z w was effective at origin. now we are going to take into account non-algebraic sets of polynomials of two complex variables, for this suggestion we will take the following two lemmas: lemma (2.3): if the transposed sets  ( ) , ( , ) ; 1,2i m np z w i  satisfy condition (2.5) and the set  (1) , ( , )m np z w is general set satisfies the condition (2.6), which is effective at the origin of 2c , then the similar transposed set  , ( , )m nu z w holds: (2.20) [0 ] 0   . proof: ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 7 from effectiveness of the general set  (1) , ( , )m np z w at the origin, that (2.21) 1 (1) (1) , 1 [0 ] sup [ ; ] 0 m n m n m n lim m p r             . where 1 (1) (1) , (1) , , , , ( , ) [0 ] supsup ( , ) r h k m n m n h k s h k p p z w           or (1) , 3 4 1 1 [ ] ( ) ; , 0m n m n k m n r r    . by conditions (2.5) and (2.6); we have (2.22) (2) , , 4 5 1 1 , ( )m n m n m nm p k r r        (2.23) (1) , , 3 2 1 1 , ( ) ; , 0m n m n m nm p k m n r r        . by (2.21), (2.22), (2.23) and cauchy's inequality it follows that: (2.24) 1 3 , , 3 1 [ ; ] max ( , ) r m n m n s m u u z w r  ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , ( , ) 0 ( , ) 0 ( , ) 0 3 , (1) ,( , ) ( , ) ( , ) (1) , (2) , (1) , 3 , , , (1)( , ) 0 ( , ) 0 ( , ) 0 , 3 1 1 | || || | ( ) 1 [ , ] | || || | 1 [ , m n h k i j h k i j s t s t m n h k i j h k i j s t s t i jm n h k i j h k i j s t m n h k i j h k i j s t i j p p p r m p r p p p m p r                 3 , 1 1 ( ) ] s t s tr   (1) ,( , ) ( , ) ( , ) (1) , (2) , 3 , , (1)( , ) 0 ( , ) 0 ( , ) 0 , 3 , , 3 ( , ) ( , ) ( , ) (1) , (2) , 2 1 , , ( , ) 0 ( , ) 0 ( , ) 0 4 , 3 1 [ ] 1 1 | || | ( ) 1 [ , ] 1 1 | || | ( ) i jm n h k i j h k i j s t m n h k h k i j s t i j s t s t m n h k i j h k i j s t m n h k i j h k i j s t i j r p p r m p r r k p p r r                    ( , ) ( , ) ( , ) (1) , (2) 2 1 , , ( , ) 0 ( , ) 0 ( , ) 0 4 3 ( , ) ( , ) (1) , 2 1 , ( , ) 0 ( , ) 0 5 , 3 (1) 1 , 5 1 | | [ , ]( ) 1 1 | | ( ) 1 [ , ]. m n h k i j h k s t m n h k h k i j s t m n i j h k s t m n h k h k s t h k m n r k p m p r r r k p r r k m p r               taking m n  and keeping in mind that the set  (1) , ( , )m np z w satisfies (2.5), we get    (1)0 0 0     . ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 8 therefore we get  0 0   . lemma (2.4): the similar transposed set  , ( , )m nu z w , satisfies the condition (2.25)   0 0r for r whenever the transposed sets  ( ) , ( , ) ; 1,2i m np z w i  satisfy the condition (2.5) and (2.6) and the set  (2) , ( , )m np z w is effective at the origin. proof: the transposed sets  ( ) , ( , ) ; 1,2i m np z w i  satisfy the condition (2.6) we get (2.26) (1) , , 4 5 1 1 [ ; ] ( )m n m n h km p k r r   . also, the transposed set  (2) , ( , )m np z w is effective at the origin, satisfies conditions (2.5) and (2.6), so must be its inverse transposed set  (2) , ( , )m np z w , thus we get (2.27) (2) , , 3 4 1 1 [ ; ] ( )m n m n m nm p k r r   the similar transposed set  , ( , )m nu z w written in the form:      (1) (2) (1) , , , ,( , ) ( , ) ( , ) ( , )m n m n m n m nu z w p z w p z w p z w therefore      (1) (1) (2) , , , ,( , ) ( , ) ( , ) ( , )m n m n m n m np z w u z w p z w p z w from which we get ( , ) ( , ) ( , ) (1) , (1) , (2) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 ( , ) | || || | m n h k i j h k i j s t s t m n m n h k i j h k i j s t p z w u p p z w        . by (2.26), (2.27) and cauchy's inequality we obtain: ( , ) ( , ) ( , ) , (1) , (2) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 , 3 ( , ) ( , ) ( , ) , (1) , (2) , , , , , ( , ) 0 ( , ) 0 ( , ) 0 , 3 1 | || || | 1 | || || | m n h k i j m n h k i j s t m n m n h k i j s t h k i j s t s t m n h k i j h k i j s t m n m n h k i j s t h k i j s t s t r u p p r u p p r                    ( , ) ( , ) , (1) , (2) , , , , ( , ) 0 ( , ) 0 3 ( , ) ( , ) , (1) , 1 , , , ( , ) 0 ( , ) 0 4 , 1 | || | [ ; ] 1 | || | m n h k h k i j m n m n h k i j h k i j m n h k h k i j m n m n h k i j h k i j i j u p m p r k u p r             ( , ) , (1) 1 , , , ( , ) 0 4 ( , ) , 1 , , 1 , , ( , ) 0 5 , 5 1 | | [ ; ] 1 1 | | [ ; ] m n h k m n m n h k h k m n h k m n m n m n m nh k h k h k k u m p r k u k m u r r          ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 9 so that 5 1 0r r        . chosen 5r near to r then we have   0 0r for r . theorem (2.3): the general transposed set of polynomials of two complex variables  (1) , ( , )m np z w is effective at the origin of 2c , satisfies condition (2.5) and (2.6). then the similar transposed set  , ( , )m nu z w is effective there, satisfies the conditions (2.20) and (2.25), if and only if, the transposed set  (2) , ( , )m np z w is effective at the origin of 2c and satisfies the same conditions. proof: from effectiveness of general sets  ( ) , ( , ) ; 1,2i m np z w i  at the origin, we get (2.28) (1) , 3 4 1 1 [ ] ( ) ; , 0m n m n k m n r r    . (2.29) (2) , 4 5 1 1 [ ] ( ) ; , 0m n m n k m n r r    . if the condition (2.20) of lemma (2.3) is satisfied then we get (2.30) , , 2 3 1 1 , ( )m n m n m nm u k r r        . by condition (2.6); we obtain (2.31) (1) , , 3 2 1 1 , ( ) ; , 0m n m n m nm p k m n r r        (2.32) (2) , , 4 3 1 1 , ( ) ; , 0m n m n m nm p k m n r r        . by (2.28) (2.32) and cauchy's inequality it follows that: ( , ) , , , , , ( , ) 02 2 ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , , , ( , ) 0 ( , ) 0 ( , ) 0 2 1 1 [ ] [ ; ] 1 | || || | [ ; ] m n h k m n m n m n h k h k m n h k i j h k i j s t m n m n h k i j s t h k i j s t u m u r r p p p m u r              ( , ) ( , ) ( , ) (1) , (2) , (1) , 1 , , , , ( , ) 0 ( , ) 0 ( , ) 0 , 3 (1) , ,( , ) ( , ) ( , ) (1) , (2) , (1) , 3 1 , , , , ( , ) 0 ( , ) 0 ( , ) 0 1 | || || | 1 [ ; ] | || || | m n h k i j h k i j s t m n m n h k i j s t h k i j s t s t s t s tm n h k i j h k i j s t m n m n h k i j h k i j s t k p p p r m p r k p p p                   (1) , 3 , , 3 1 1 [ ; ] s t s t s t s t r m p r   ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 10 (1) ,( , ) ( , ) ( , ) (1) , (2) , 3 2 1 , , , ( , ) 0 ( , ) 0 ( , ) 0 , 3 (2) , ,( , ) ( , ) ( , ) (1) , (2) , 4 1 , , , ( , ) 0 ( , ) 0 ( , ) 0 , , 1 [ ] | || | 1 [ ; ] | || | [ i j s tm n h k i j h k i j m n m n h k s t h k i j s t s t i j i jm n h k i j h k i j m n m n h k h k i j s t i j i r r k p p r m p r k p p m p                     2 (2) 4 , 3 4 1 1 ; ] s t i j s t s t j r r r r     (2) ,( , ) ( , ) ( , ) (1) , 34 2 1 , , ( , ) 0 ( , ) 0 ( , ) 0 , 4 3 ( , ) ( , ) ( , ) (1) , 3 2 1 , , ( , ) 0 ( , ) 0 ( , ) 0 , 5 4 3 1 [ ] | | 1 | | s ti jh km n h k i j h k m n m n h k i j s t s t s ti j m n h k i j h k m n m n h k h k i j s t h k rr r k p r r r r k p r r r                                         (1) 1 , , 5 1 [ ; ]m n m nk m p r  . therefore [0 ] 0   . that to say the similar transposed set  , ( , )m nu z w effective at the origin, and satisfies conditions (2.20) and (2.25), by using lemmas (2) and (3) respectively and the "if" statement follows. now, to prove the only if, write      (2) (1) (1) , , , ,( , ) ( , ) ( , ) ( , )m n m n m n m np z w p z w u z w p z w let the set  , ( , )m nu z w is effective there satisfies conditions (2.20) and (2.25), the inverse transposed set  (1) , ( , )m np z w is effective at the origin and satisfies the same conditions. then the similar transposed set  (2) , ( , )m np z w is effective at the origin, and satisfies conditions (2.20) and (2.25). 3effectiveness of similar transposed set of polynomials in open hyperspheres now we are investigating the effectiveness of similar transposed set  , ( , )m nu z w of polynomials of two complex variables in open hyperspheres whenever the constituent sets are effective there. we can only be algebraic and compliance with the relevant requirements (3.1) 1 ( ) ( ) , , 1 1 1 sup [ ; ] ; , 1,2 m n i i m n m n m n lim m p for all r r i r r r                  . for this purpose, we give the following lemma: lemma (3.1) the transposed set and the power transposed set accord to the same condition (3.1). proof: first suppose that the set  , ( , )m np z w satisfies condition ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 11 (3.2) 1 , , 1 1 1 sup [ ; ] ; m n m n m n m n lim m p for all r r r r r                  . then (3.3) , , 1 1 1 [ ; ] ( ) ; ( , ) 0m n m n m nm p k m n r r    . the transposed product set of the two sets  ( ) , ( , ) ; 1,2i m np z w i  , are satisfy the following relation: ( , ) 2 , , , , ( , ) 0 ( , ) ( , ) m n h k m n m n h k h k p z w p p z w    . by relation (3.3) and using cauchy's inequality we get ( , ) 2 , , , , , , ( , ) 0 ( , ) , 1 , , ( , ) 0 1 , 1 , , 1 1 1 [ ; ] | | [ ; ] 1 | | 1 [ ; ]. m n h k m n m n m n m n h k h k m n h k m n m n h k h k h k m n m n m p p m p r r k p r k m p r            . so that 2 2 1 1 1 [ ] [ ] ; .for all r r r r r   also by the same way we can prove that 1 2 2 3 1 1 1 [ ] [ ] ; for all r r r r r     and by induction we get 1 1 [ ] ; .for all r r r r  we can deduce the following theorem from this lemma: theorem (3.1): when  ( ) , ( , ) ; 1,2i m np z w i  two algebraic sets are effective and satisfy in the open hyperspheres 1 rs the condition (3.2), then the similar transposed set  , ( , )m nu z w is effective in open hyperspheres 1 rs . proof: let two sets  ( ) , ( , ) ; 1,2i m np z w i  be algebraic sets each fulfilling the following conditions: (3.4) ,(1) , 3 , 1 1 1 , 2 ( 1) m n h kh k m n h k m n r p k t r       (3.5) (2) , , 1 2 3 1 1 [ ; ] ( )m n m n i jm p k r r   . therefore ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 12 4 , , 4 ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , ( , ) 0 ( , ) 0 ( , ) 0 4 , 1 (3.6) [ , ] max ( , ) 1 | || || | r m n m n s m n h k i j h k i j s t m n h k i j s t h k i j s t s t m u u z w r p p p r          ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , ( , ) 0 ( , ) 0 ( , ) 0 4 , ( , ) ( , ) ( , ) ,(1) , (2) , 3 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 , 2 4 , 1 | || || | 1 | || | m n h k i j h k i j i j m n h k s t s t h k i j s t s t s tm n h k i j i jh k i j m n h k i j s t h k i j s t s t s t p p p r r k k p p r r                      ( , ) ( , ) ( , ) ,(1) , (2) , 3 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 2 , 4 , ( , ) ( , ) ( , ) , ,(1) , (2) 3 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 2 , 4 , 1 | || | 1 | | [ , ] s tm n h k i j i jh k i j m n h k i j s t h k i j s t s t s t s tm n h k i j i j i jh k m n h k s t h k i j s t s t s t r k k p p r r r k k p m p r r                         ( , ) ( , ) ( , ) , ,(1) , 3 1 2 , ( , ) 0 ( , ) 0 ( , ) 0 3 , , 4 , (1) 1 2 , 3 1 | | 1 [ , ] s tm n h k i j i j i jh k m n h k s t h k i j s t h k s t s t m n r k k p r r k k m p r               where 2 1 1( 1)k t  . thus 1 , , 4 3 1 (1) (1) , 1 2 , 3 3 1 1 [ ] limsup [ , ] 1 1 limsup [ ; ] [ ] m n m n m n m n m n m n m n m n m u r r k k m p r r                         and , , 1 4 5 1 1 [ , ] ( )m n m n m nm u k r r   . now, taking (3.7) ,(1) , 4 , 1 1 1 , 3 ( 1) m n h kh k m n h k m n r p k t r       (3.8) ,(2) , 2 , 1 2 2 , 1 ( 1) m n h kh k m n h k m n r p k t r       . by (3.6), (3.7), (3.8) and cauchy's inequality it follows that: ( , ) , , , , , ( , ) 04 4 ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , , , ( , ) 0 ( , ) 0 ( , ) 0 4 1 1 [ ] [ ; ] 1 | || || | [ ; ] m n h k m n m n m n h k h k m n h k i j h k i j s t m n m n h k i j s t h k i j s t u m u r r p p p m u r              ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 13 ( , ) ( , ) ( , ) (1) , (2) , (1) , , , , , , ( , ) 0 ( , ) 0 ( , ) 0 4 ( , ) ( , ) ( , ) (1) , (2) , (1) , 1 2 , , , , ( , ) 0 ( , ) 0 ( , ) 0 5 , 1 | || || | [ ; ] 1 | || || | m n h k i j h k h k i j m n m n i j s t s t h k i j s t m n h k i j h k h k i j m n m n i j s t s t h k i j s t s t p p p m u r k k p p p r                  ( , ) ( , ) ( , ) ,(1) , (2) , 4 1 2 , , , ( , ) 0 ( , ) 0 ( , ) 0 , 3 5 , ( , ) ( , ) ( , ) ,,(1) , 2 4 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 , 1 , 3 1 | || | 1 | | s tm n h k i j i jh k h k m n m n i j i j s t h k i j s t s t s t i j s tm n h k i j i jh kh k m n m n h k i j h k i j s t i j s t r k k p p r r r r k k p r r r                            5 , s t s t ( , ) ( , ) ( , ) ,, ,(1) , 2 4 1 2 , , ( , ) 0 ( , ) 0 ( , ) 0 , 1 , 1 , 3 5 , (1) 1 2 3 , , 1 1 | | 1 [ ; ] i j s tm n h k i j i jh k h kh k m n m n h k h k i j s t h k i j s t h k i j s t s t m n m n r r k k p r r r r k k k m p r                     where 2 1 1( 1)k t  ; 3 2 2( 1)k t  . so that 1 (1) , 4 4 1 1 1 1 1 [ ] limsup [ ] [ ] ; m n m n m n for all r r r r r r              . in other words, the similar transposed set  , ( , )m nu z w effective established an open hypersphere 1 rs . 4effectiveness of similar transposed set of polynomials in closed hyperspheres now we are giving some important results for the effectiveness of similar transposed sets of polynomials in some other regions, and a new study of the effectiveness of these polynomials is being considered, and proof of these results is given in a similar manner to those previously identified in this study. first effectiveness of transposed basic set of polynomials of two complex variables  , ( , )m np z w in closed hyperspheres 1 r s whenever the simple basic set  , ( , )m np z w effective in closed hyperspheres rs with leading coefficients unity. the effectiveness of transposed basic set of polynomials given in following theorem: lemma (4.1): suppose that , ( , )m np z w be simple monic set with leading coefficients unity effective in the hyperspheres rs , then the transposed set  , ( , )m np z w is effective in the closed hyperspheres 1 r s . let  , ( , )m np z w settheofpolynomialsofsetinversetransposedabe  , ( , )m np z w where ( , ) ( , ) , , , , , ( , ) 0 ( , ) 0 ( , ) m n m n h k h k m n h k m n m n h k h k h k p z w p z w p z w      . ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 14 setinversetransposedtheeffectiveness  , ( , )m np z w in clospolynomialsof ed hyperspheres 1 ; 0 r s r  with leading coefficients unity set, whenever the basic  , ( , )m np z w is effective in the same region under the same condition as follows: lemma (4.2): suppose that , ( , )m np z w be simple monic set with leading coefficients unity effective in the hyperspheres rs , then the transposed inverse set  , ( , )m np z w is effective in the closed hyperspheres 1 r s . let  ( ) , ( , ) ; 1,2i m np z w i  are basic sets of polynomials of two complex variables and the set  , ( , )m nq z w is called the product set of the two sets  ( ) , ( , ) ; 1,2i m np z w i  ,     (1) (2) , , ,( , ) ( , ) ( , )m n m n m nq z w p z w p z w , now, the effectiveness of transposed product basic set of polynomials of two complex variables     (1) (2) , , ,( , ) ( , ) ( , )m n m n m nq z w p z w p z w in closed hyperspheres 1 r s whenever the sets  ( ) , ( , ) ; 1,2i m np z w i  closedineffectiveareones,are simple monic hyperspheres rs with leading coefficients unity, i.e. ( ) , , 1 ; 1,2i m n m np i  . lemma (4.3): suppose that ( ) , ( , ) ; 1,2i m np z w i  be simple monic set with leading coefficients unity effective in the hyperspheres rs transposedthethen, product set  , ( , )m nq z w is effective in the closed hyperspheres 1 r s . effectiveness of similar transposed setnow, we give  , ( , )m nu z w closedin hyperspheres 1 r s whenever the transposed simple basic sets  ( ) , ( , ) ; 1,2i m np z w i  are hyperspheresclosedineffective 1 r s whenever the simple basic setsand also,  ( ) , ( , )i m np z w are effective in open hyperspheres rs with leading coefficients unity, ( ) , , 1i m n m np  ; 1, 2i  . theorem (4.1): suppose that ( ) , ( , ) ; 1,2i m np z w i  be two simple monic sets with leading coefficients unity effective in the hyperspheres rs , then the similar transposed set  , ( , )m nu z w is effective in the closed hyperspheres 1 r s . the inverse set of polynomials of two complex variables  , ( , )m nu z w as follows:      (1) (2) (1) , , , ,( , ) ( , ) ( , ) ( , )m n m n m n m nu z w p z w p z w p z w ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 15 now, we give effectiveness of inverse similar transposed set  , ( , )m nu z w in closed hyperspheres 1 r s , whenever the simple basic sets  ( ) , ( , ) ; 1,2i m np z w i  are effective in closed hyperspheres rs with leading coefficients unity, i.e. ( ) , , 1 ; 1,2i m n m np i  . theorem (4.2): suppose that ( ) , ( , ) ; 1,2i m np z w i  be two simple monic sets with leading coefficients in the hypersphereeffectiveunity rs settransposedsimilarthen the inverse,  , ( , )m nu z w hypersphereclosedtheis effective in 1 r s settheifonlyandif  (2) , ( , )m np z w is effective there. conclusions in this paper, where the correctness of the corresponding functions is effectiveness in origin, in the open hyperspheres and in the closed hyperspheres, a new comparison is proposed to study some significant properties of some corresponding functions in two complex variables, and this study is called a new one of its kinds. generalization of the corresponding position and it has relevance in many areas of application and physics. references [1] j. a. adepoju, nassif m, effectiveness of transposed inverse sets in faber regions, inte. j. of math. and math. sci. vol. 6, n. 2, (1983) pp. 285-296. doi: 10.1155/s0161171283000241. 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(springer) 116(2) (2019), p. 16. https://doi.org/10.1007/s00009-019-1398-7. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 16 https://www.researchgate.net/scientific-contributions/18510586-ja-adepoju?_sg%5b0%5d=5acd9uutdd1eb2yzgoryi-ax6e5jfmrqox-x4ble_dzwlpfkayq6ck4rl1udtvivav_r0uo.zvpsfl0tfzzgo9mfpfz8rgqyrmsopufnlc9un_by8qgjzlbygnfa4fqjg3k7zembswuao9ivdkingcj8yuq_ya&_sg%5b1%5d=89bt1kegwtb0dhkoraeg9xwpb5cwt9vwhonefjtzbrycqqggxshklisysumajc3qwi6l28u.oojp7t3oe5tahwsbtbyna-imupejsgrcgkkdjymyi0men7bh-os9y_nvhxyne-mf8zw2vcr5tmfnfe-uvsxo7a https://www.researchgate.net/scientific-contributions/31702593_m_nassif?_sg%5b0%5d=5acd9uutdd1eb2yzgoryi-ax6e5jfmrqox-x4ble_dzwlpfkayq6ck4rl1udtvivav_r0uo.zvpsfl0tfzzgo9mfpfz8rgqyrmsopufnlc9un_by8qgjzlbygnfa4fqjg3k7zembswuao9ivdkingcj8yuq_ya&_sg%5b1%5d=89bt1kegwtb0dhkoraeg9xwpb5cwt9vwhonefjtzbrycqqggxshklisysumajc3qwi6l28u.oojp7t3oe5tahwsbtbyna-imupejsgrcgkkdjymyi0men7bh-os9y_nvhxyne-mf8zw2vcr5tmfnfe-uvsxo7a https://www.researchgate.net/deref/http%3a%2f%2fdx.doi.org%2f10.1155%2fs0161171283000241?_sg%5b0%5d=mo09f77gpm9guo7lgupq10vmdwvjygse3b9yp-x0oyg78-vny0l555ebc4w9hi0i-f7cuoo6edajlvo8efieozf3aw.ug3p939h5gwuholkdsq06hcvta2ue3ld4k3jzcjlvjfmhfdyuga0ifrxgl4rtlrveetxuw3tz03ipz6cfwlopa http://ijpam.uniud.it/ http://ijpam.uniud.it/ https://www.researchgate.net/deref/http%3a%2f%2fdx.doi.org%2f10.1080%2f02781070290032289?_sg%5b0%5d=rrayprpcvqil8isa2bwzz9rxowsl1zyswxaklpfqyrjv4ml74ny-2ermaqxazr9wlk0v05_33syo_pkbiajmfsq4vg.qsh2zzwqdw21ldaa-4qbgzt-aylgvnfg_-vspcm8uqsmwmp8rpjenz4byfrheq9bv--c24bcbkyiimu5shyrqg [12] m. s. metwally, some topics in complex analysis and its applications in basic sets of polynomials. ph.d. thesis, faculty of science, assiut, univ. (1993). [13] m. mursi, b. h. and makar, basic sets of polynomials of several complex variables ii, proceeding of second arab science congers; cairo (1955); pp. 16-68. [14] m. nassif, composite sets of polynomials of several complex variables, publications mathematics. (debrecen) tomus 18, (1971); pp. 43-53. [15] m. nassif and j. a. adepoju, “effectiveness of product of simple sets of polynomials of two complex variables in polycylinders and in faber regions, journal of natural sciences and mathematics, (lahore); vol. 24 no. 2; (1984); pp. 153-172. [16] w. f. newns, on the representation of analytic functions by infinite series, philosophical transactions of the royal society of london, ser. a. vol. 245 (1953); pp. 429 – 468. [17] k.a.m. sayyed, basic sets of polynomials of two complex variables. m. sc. thesis, assiut univ. (1972). [18] k.a.m. sayyed and s.m. mena, similar sets of polynomials, bull. fac. sci., assiut univ., 17(1-c), pp. 29-38 (1988). [19] k.a.m. sayyed and s.m. mena, effectiveness similar sets of polynomials at the origin, bull. fac. sci., assiut univ., 17(1-c), pp. 39-48 (1988). [20] k.a.m. sayyed and m.s. metwally, effectiveness of similar and inverse similar sets of polynomials in faber regions, sohag pure and appl. sci. bull., fac. sci.,egypt, 9(1993),37-49. [21] k.a.m. sayyed and m.s. metwally, effectiveness of similar sets of polynomials of two complex variables in polycylinders and in in faber regions, internat. j. math. & math. sci. vol. 21 no. 3 (1998), 587– 94. doi: 10.1155/s0161171298000817. [22] h.m. srivastava, saima jabee and m. shadab, differential equations and recurrence relations of the sheffer-appell polynomial sequence: a matrix approach, arxiv:1903.09620 [math. ca] 21 mar 2019. ijo international journal of mathematics volume 3| issue 12| december | 2020 http://www.ijojournals.com/index.php/m/index 17 https://www.researchgate.net/deref/http%3a%2f%2fdx.doi.org%2f10.1155%2fs0161171298000817?_sg%5b0%5d=qhmbroelsaivxpynnivtgqrkkclqmjjc-lyr-go3redidvxldmkhcqjx7b8vqsqe_lly9dkzi39a9twtbca2gnkrba.8okzcdu2lokpbynbfkjgj7gym6aqtxbbdzhq_smssjfpr6qulkwrkyxez-xgngke2h_67-xxz4ymm863kge8la ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel venuprasad k. k.1, prof. k. shivashankara2, dhananjaiah d. s3, prakasha.p4 1department of mathematics, government first grade college k.r.pete, mandya, india e-mail id: kkvpmaths@gmail.com 2department of mathematics, yuvaraja’s college, university of mysore, mysore, india e-mail id: drksshankara@gmail.com 3department of mathematics, government first grade college k.r.nagar, mysuru, india e-mail id: dhanu2614@gmail.com 4department of mathematics, government first grade college, madagi, ramnagar, india e-mail id: profprakasha@gmail.com abstract: in this work, we examine how transitory free connective heat and mass transfer flow over a porous media in a vertical wavy channel is impacted by chemical reactions and radiation absorption. the oscillatory flux in the flow zone is the cause of the unsteadiness in the flow. the laplace transformation approach is used to get the analytical solutions for the governing equations. analysis is done on the profiles of temperature, concentration, and velocity. the governing parameters are varied, and the resulting expressions for the velocity, concentration, shear stress, and rate of heat and mass transfer are examined. key words: mhd, wavy channel, chemical reaction. 1. introduction the use of non-newtonian liquids in engineering and industry is required due to their increasing significance. it is remarkable for its many applications in numerous areas, which include food processing, lubricant performance, plastic manufacture, and/or biological liquid transportation. numerous fluids, including diluted polymer solutions, slurry flows, industrial oil, and numerous flow issues caused by different mechanical and/or thermal boundary conditions have been discussed, as well as the second graded fluid preserves these fluids. for the second-graded fluids, tan and masuoka [1] discovered the stokes first difficulties, whereas rashidi et al. [2] addressed the second-order fluids' unstable compressible flows. hayat and associates. [3] investigated by the fluids of second grade with variable free stream and unstable stagnation point flow. a branch of fluid dynamics called magnetohydrodynamics (mhd) examined how electrically conducting fluids interact with one another in a magnetic field. many research projects in the field of mhd have been carried out during the course of the few decades that have preceded them, following hartmann's well-established work [4]. flow in metalized fluid channels subjected ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 1 mailto:kkvpmaths@gmail.com mailto:drksshankara@gmail.com mailto:dhanu2614@gmail.com mailto:profprakasha@gmail.com ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel to an external magnetic field. the parabolic movement has several uses, such as solar cookers, solar concentrators, and parabolic through star collectors. there are several uses for solar cookers with parabolic concentrator models, including roasting, baking, and distilling. applications for the solar concentrator model included increasing evaporation rates in dissipation streams, food dispensing, and generating drinking water from both seawater and saltwater. murthy et al. [5] examined through the assessments of temperature exchanger units' thermal characteristics for parabolic diffusion-thermo effect, radiating-absorptions, hall, and ion slip influences on the mhd liberated convection gyratory flows of the nanofluids passing the semi-infinite permeable inspiring plate with the constant temperature sources were recently investigated by krishna and chamkha [6]. krishna et al. [7] investigated the effects of radiating and hall currents on the unstable mhd freed central heating flows into the perpendicular channel/duct packed by the absorbent media. krishna and chamkha [8] took into consideration the temperature generating/absorption, thermodiffusions on the unsteady complimentary convection mhd flows of radiation, and the chemically reactive second-grade liquid passing past an unbounded perpendicular plate during the absorbent media in addition to taking the hall current into account. for the few decades prior, convective heat transportations in a permeable medium have piqued intense curiosity. numerous thermal engineering functions in a range of constraints, such as geophysics, thermal and insulation engineering, the model of crowded sphere beds, electronic system cooling, chemical catalytic reactor, ceramic processes, granular insulations and grains storage device fibers, gasoline reservoirs, coal combustion engines, groundwater pollution, and filtration processes, all stimulate its interests. the partial differential equations recurrently appear in the lot of areas of the natural and physical disciplines. they described dissimilar physical organisms, ranges from gravitational to fluid dynamics and had been utilized to solve the problem by the physical and chemical sciences, mathematical bio-sciences, solid mechanical engineering knowledge, etc. soundalgekar and takhar initially, deliberated the consequence of radiation for the natural convective flow of the gasses over a semi infinite plate with numerical modeling. takhar et al.[9] explored the impact of radiation on mhd free convective flow past semi-infinite vertical plate. currently, hossain et al.[10] exposed the effects of radiation on combined convective flow during an absorbent plate. muthucumarswamy and kumar[11] explored the heat radiation influences on affecting never-ending vertical plate with variable heat and mass diffusions. 2.formulation of the problem we consider the effect of chemical reaction on the unsteady motion of viscous, fluid through a porous medium in a vertical channel bounded by wavy walls . the thermal buoyancy in the flow field is created by an oscillatory flux in the fluid region. the walls are maintained at constant temperature and concentration. the boussinesq approximation is used so that the density variation ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel will be considered only in the buoyancy force. the viscous and darcy dissipations are neglected in comparison with heat by conduction and convection in the energy equation. also the kinematic viscosity ,the thermal conducting k are treated as constants. we choose a rectangular cartesian system 0 ( x ,y ) with x-axis in the vertical direction and y-axis normal to the walls. the walls of the channel are at )( l x lfy   . the flow is maintained by an oscillatory volume flux for which a characteristic velocity is defined as    fl fl ti dyu l ekq 1 )1(  . (1) the boundary conditions for the velocity and temperature fields are u = 0 , v = 0 ,t=t1 ,c=c1 on )( l x lfy   22,0,0 ccttvu  on )( l x lfy   (2) in view of the continuity equation we define the stream function  as u = - y , v =  x (3) the equations governing the flow, heat and mass transfer in terms of the stokes steam function  are     2 0 0 4222 )()( )(])()()[(    k ccg ttg y yxyyxt (4) )()()( 1 2 eofpe ccqttqtk y t xx t yt t c                (5) )()( 1 2 1 occkcd y c xx c yt c               (6) introducing the non-dimensional variables in (4 )-(6) as 21 2 21 2 ,,/,,/,/ cc cc c tt tt ttlyylxx        (7) the governing equations in the non-dimensional form ( after dropping the dashes ) are    214 2 22 ))(() ),( ),( )((     dnc r g yx r yyt (8) cq yxxyt p 1 22 )(                 (9) ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel kcc y c xx c yt c sc              22 )(   (10) where  ul r  (reynolds number), 2 3   ltg g e  (grashof number) f p k c  ( prandtl number), 1d sc   (schmidt number), pf ck ql2  (heat source parameter), 1 2 1 d lk k  (chemical reaction parameter),    2 2 l  (wormsely number), k l d 2 1  (darcy parameter), )( )( 21 2 211 1 ttk lccq q f    (radiation absorption parameter) the corresponding boundary conditions are 1)1()1(   10,0         at yx (11) 10,0),( 11,1),(     oncyx oncyx 00,0         at y c y (12) the value of  on the boundary assumes the constant volumetric flow in consistent with the hyphothesis (1) .also the wall temperature varies in the axial direction in accordance with the prescribed arbitrary function t . 3. method of solution the main aim of the analysis is to discuss the perturbations created over a combined free and forced convection flow due to traveling thermal wave imposed on the boundaries. the perturbation analysis is carried out by assuming that the aspect ratio  to be small. introduce the transformation such that xx xx        , then )1()( o x o x        for small values of <<1 ,the flow develops slowly with axial gradient of order  and hence we take )1(o x    ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel using the above transformation the equations(8 10) reduces to    214 1 2 12 1 2 ))(() ),( ),( )((     dnc r g yx r yyt (13) cq x n yyxxyt p 12 2 2 2 2 2 2 1 )()(                         (14) kcc y c xx c yt c sc              2 1 2 )(   (15) where 2 2 2 2 22 1 yx        introducing the transformation )(xf y  the equations(13-15) reduce to       221 3 4 2 22 )( ))(() ),( ),( )(( ffd nc r gf f x f frf t      (16) cqf xx f t p 1 22 )((                    (17) kccf c xx c f t c sc              22 )((      (18) where 2 2 2 2 22         x f we adopt the perturbation scheme and write . ..............)),,(.),,((),,(),,(),,( 1100  txketxtxketxtx itit  . ..............)),,(.),,((),,(),,(),,( 1100  txketxtxketxtx itit  . ..............)),,(.),,((),,(),,(),,( 1100  txcketxctxcketxctxc itit  (19) on substituting (19) in (16) (18) and separating the like powers of  the equations and respective conditions to the zeroth order are )()()( ,0,0 3 ,0 22 1,0   nc r gf fm y  (20) 010 2 , )( cqfo    (21) ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel 0)( 0 2 ,  ckscfco  (22) with  0(+1)-0(-1) = 1 ,  0,  = 0 ,  0 , x =0 at  = 1 (23) 10,0 11,1 0 0     onc onc o o (24) 010 22 ,0 )( cqfip    (25) 0)( 22 ,0  ocfkscc  (26) ))(())(( .0,0 3 ,0 222 1,0   cn r gf fim  (27) 0)1(0)1(  oo c 0)1(,0)1(1)1()1( ,,  xoooo   (28) the first order equations are ))(())(()( ,0,0,0,0,1,1 3 ,1 22 1,1   xxrfcn r gf fm  (29) 11,0,,01 2 )()()( ,1 cqprff oxox    (30) )()()( ,0,,01 2 ,1 oxox ccscfckscfc y     (31) ) )(())(())(( ,0,0,0,0,0,0 ,0,0,1,1 3 ,1 222 1,1     xxx xrfcn r gf fim   (32) 11,0,,0 ,0,,01 22 ) ()())(( ,1 cq prffip oxox oxxo        (33) ) ()())(( ,0,,0 ,0,,01 22 ,1      oxox oxxo cc ccscfcscfikc   (34) with  1(+1)  1(-1 ) = 0  1, = 0 ,  1 , x = 0 at  = 1 (35) 1(1) = 0 c1(1) = 0 0)1(0)1( 11  c 0)1(,0)1(1)1()1( ,1,111  x  (36) ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel the equations (20)(22), (25)-(27)&(29)-(34) are solved analytically subject to the relevant boundary conditions. for sake brevity we are not presenting the solutions. 4.nusselt number and sherwood number )()( 2 8761  odecddy  the local rate of heat transfer coefficient nusselt number (nu) on the walls has been calculated using the formula 1)( 1         y nu wm where    1 1 5.0  dm and the corresponding expressions are )1( )( )( ))(( )( )( 108 1 119 1        mm dd un txsin dd un      , where 1514 ddm   the local rate of mass transfer coefficient sherwood number (sh) on the walls has been calculated using the formula 1)( 1      y wm y c cc sh where    1 1 5.0 dyccm and the corresponding expressions are )1( )( )( )( )( )( 75 1 64 1       mm c dd sh c dd sh   5.results and discussion this investigation focuses on the impact of heat generation and thermo-diffusion on the unstable free convection mhd gyrated flow of radiation and chemical reactive second order fluid across an unbounded perpendicular plate during absorbent medium. the analytical solutions for the governing equations are obtained by the application of the laplace transformation procedure. the profiles of concentration, temperature, and velocity are analyzed graphically. for quite a few quantities of the magnetic field parameter m, chemical reaction parameter kr, temperature generating and/or absorbing parameters, it is represented the second-grade fluid velocity, and concentration distributions. magnetic field parameter m, chemical reaction parameter kr,. we fixed m = 0.5, k = 0.5, pr = 0.71, r = 2, gr = 10, gm = 5, sr = 0.1, h = 2, = 0.5, and t = 0.5 for computational purposes and drew the profiles with each parameter adjusted across the range. the concentration, temperature, and velocity profiles are shown in figs. 1-4 to be less than those for isothermal temperature and ramped surface concentration in the case of ramped wall temperature. the magnetic domain parameter in the liquid flow generates an electrical field. consequently, it ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel was deduced that when isothermal temperature and ramping surface concentration are present simultaneously, the magnetic field lowers both of them. a fluid's velocity is expected to be slowed down by the addition of a magnetic field, which will increase the resistive model forces (lorentz forces) acting on the fluid in the boundary layers. fig.1.the velocity profile for u against m fig. 2. the velocity profile for v against m figs.1 and 2 has been shown that, the intensity of the magnetic field has reducing effects on velocity profiles for together heated circumstances. it is anticipated to the information that, the representing of magnetic domain parameter produces electrical field in the liquid flow. this implied ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel that, the magnetic field has reducing effect for together ramped wall temperature with ramped surface concentration as well as isothermal temperature with ramped surface concentration. it is ex pected to the information that, the application of the magnetic field to fluid give augment to the resistive model forces (lorentz forces) on the fluid in the boundary layers, this slow down the movement of the fluid. fig.3 the velocity profile for u against kr. fig.4 the velocity profile for v against kr. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel additionally, it is noted that as the second grade parameter rises, the border layer widths decrease. as seen in figs. 3 and 4, chemical reactions have a slowing effect on the velocity of liquid flows in combined thermal cases. fig.5 the concentrationprofile for kr. as shown in fig. 5, the chemical reactions have a decreasing effect on the concentration profiles and liquid flow velocity for the combined thermal case. the destructive reactions kr > 0 have been demonstrated to cause falls into the concentration field, which worsens the effects of buoyant forces because of the concentration gradient. the flows domain is then narrowed down.depending on the nusselt number, producing, absorbing, and/or radiating parameters h and r. when the temperature producing and/or absorbing parameter h and the prandtl number pr increase, the nusselt number nu increases. conversely, when the radiating parameter r increases, the nusselt number nu decreases for both ramping wall temperature and isothermal plate. the nusselt number decreases with increasing time for an isothermal plate and increases with ramping wall temperature. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel table.1 the nusselt number pr r h t ramped temperature isothermal temperature 0.71 2 2 0.2 0.732255 1.779785 3 0.846796 2.077621 7 1.247212 2.379774 5 0.694214 1.510547 8 0.657995 1.227544 -2 0.647895 1.513705 5 0.846778 1.950562 0.5 0.874546 1.469785 0.8 0.958589 1.394958 the effects of temperature producing and/or absorbing parameter h, radiating parameter r, and pr on the nusselt number were shown in table.1. when the temperature producing and/or absorbing parameter h and the prandtl number pr increase, the nusselt number nu increases. conversely, when the radiating parameter r increases, the nusselt number nu decreases for both ramping wall temperature and isothermal plate. the nusselt number decreases with increasing time for an isothermal plate and increases with ramping wall temperature. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel table.2 the shear stresses m k α kr sr gr gm h r ramped temperature isothermal plate 0.5 0.5 0.1 2 0.1 5 2 2 2 x y x y 0.8 2.032214 0.075785 1.624789 0.79889 1 1.613058 0.086898 1.335469 0.828547 1 1.269592 0.093578 1.105014 0.932554 1.5 1.934796 0.053478 1.449635 0.732969 1 1.705478 0.042502 1.304789 0.621559 1.5 2.602466 0.086895 1.738966 0.931748 3 3.735896 0.113547 1.880254 1.109589 4 2.335895 0.083874 1.978801 0.998478 0.5 2.968747 0.115748 2.965479 1.398041 1 1.939745 0.049411 1.075884 0.479952 8 1.749985 0.042115 0.330856 0.239658 10 1.93211 0.053041 1.602147 0.767587 5 1.703522 0.046874 1.580145 0.741847 8 1.976547 0.064306 1.612265 0.790895 -5 1.869954 0.049447 1.601458 0.767014 5 1.939665 0.060256 1.565595 0.701748 5 2.33565 0.086874 1.749854 0.901849 8 2.105452 0.083289 1.738859 0.998954 ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 12 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel table.3 the sherwood number(pr=0.710,r=2.0,h=-2.0) sr kr sc t ramped temperature isothermal temperature 0.1 2 0.22 0.2 0.430478 0.545478 0.5 0.347254 0.462254 1 0.292854 0.407854 3 0.492785 0.607785 4 0.570699 0.685699 0.3 0.403895 0.518895 0.6 0.370548 0.485548 0.5 0.500897 0.615897 this is scrutinized from table 2 that, it is notified that, for together ramped wall temperature and isothermal plate, the stress components τx as well as τy enhances by an increasing in second graded fluid parameter α, chemical reacting parameter kr, temperature generations and/or absorptions h and thermal radiation parameter r, as well as it reduces by an increasing in the permeability parameter k, thermal-diffusion (soret) parameters sr, thermal grashof numbers gr and mass grashof quantity gm. this is also found that by an increasing in the intensity of the magnetic fields then the stress components τx retards and the component τy boosting up for together ramped wall and isothermal plate. finally, the sherwood number sh is reduced with an increasing in the soret number sr as well as schmidt number, and it is increasing with an increasing in chemically reacting parameter kr and certain instant of time for together ramped wall temperature and an isothermal plate (table 3). 6.references [1]w. tan, t. masuoka, stokes’ first problem for a second grade fluid in a porous halfspace with heated boundary, int. j. nonlinear mech. 40 (2005) 515–522. [2] m.m. rashidi, s.a. majid, a. mostafa, application of homotopy analysis method to the unsteady squeezing flow of a second-grade fluid between circular plates, math. probl. eng. 18 (2010), 706840. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 13 ijo international journal of mathematics (issn: 2992-4421 ) venuprasad k. k.1 * https://ijojournals.com/ volume 07 issue 03 || march, 2024 || mathematical study of mhd convective flow with chemical reaction through a porous medium in a vertical wavy channel [3]. hayat, m. qasim, s.a. shehzad, a. alsaedi, unsteady stagnation point flow of second grade fluid with variable free stream, alexandria eng. j. 53 (2014) 455–461. [4].j. hartmann, hg-dynamics i theory of the laminar flow of an electrically conductive liquid in a homogenous magnetic field, det kongelige danske videnskabernes selskab mathematisk-fysiske meddelser 15 (1937) 1–27. [5] v.v.s. murty, a. gupta, n. mandloi, a. shukla, evaluation of thermal performance of heat exchanger unit for parabolic solar cooker for off-place cooking, indian j. pure appl. phys. 45 (2007) 745–748. [6]m.v. krishna, a.j. chamkha, hall and ion slip effects on mhd rotating boundary layer flow of nanofluid past an infinite vertical plate embedded in a porous medium, results in physics 15 (2019), 102652, https://doi.org/10.1016/j. rinp.2019.102652. [7] m.v. krishna, g.s. reddy, a.j. chamkha, hall effects on unsteady mhd oscillatory free convective flow of second grade fluid through porous medium between two vertical plates, physics of fluids 30 (2018), 023106, https://doi.org/10.1063/ 1.5010863. [8] m.v. krishna, m.v. chamkha, hall effects on unsteady mhd flow of second grade fluid through porous medium with ramped wall temperature and ramped surface concentration, physics of fluids 30 (2018), 053101, https://doi.org/10.1063/ 1.5025542. [9] tahkar hs, gorla sr and soundalgekar vm. short commu-nication radiation effects on mhd free convection flow of a gas past a semi-infinite vertical plate. int j numer methods heat fluid flow 1996; 6: 77–83. [10]hossain am, alim ma and rees das. the effect of radi ation on free convection from a porous vertical plate. int j heat mass transf 1999; 42: 181–191. [11] muthucumarswamy r and kumar gs. heat and mass transfer effects on moving vertical plate in the presence of thermal radiation. theoret appl mach 2004; 31: 35–46. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 14 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction dhananjaiah d. s1, prof. k. shivashankara2, venuprasad k. k3, prakasha.p4 1department of mathematics, government first grade college k.r.nagar, mysuru, india e-mail id: dhanu2614@gmail.com 2department of mathematics, yuvaraja’s college, university of mysore, mysore, india e-mail id: drksshankara@gmail.com 3department of mathematics, government first grade college k.r.pete, mandya, india e-mail id: kkvpmaths@gmail.com 4department of mathematics, government first grade college, madagi, ramnagar, india e-mail id: profprakasha@gmail.com abstract the effect of chemical reaction on unsteady combined heat and mass transfer flow of a viscous electrically conducting fluid in a vertical wavy channel with oscillatory flux. the nonlinear governing equations are solved by employing a regular perturbation technique with the slope  of the wavy wall as a perturbation parameter. the velocity, the temperature and the concentration are analyzed for different variations of the governing parameters. the rate heat and mass transfer are evaluated for different variations. keywords: heat transfer, mass transfer, chemical reaction, wavy channel, 1. introduction due to growing significances, the application of non-newtonian liquid is mandatory in the engineering and industry. it is outstanding to those plentiful applications in more than a few regions, they are, the plastic manufacturing, performance of lubricant, food processing, and/ or movement of biological liquids. the second graded fluid preserve many fluids these are diluted polymer solution, slurry flow, as well as industrial oil, in addition to a lot of flow problems by a choice of geometry as well as dissimilar mechanical and/or thermal boundary circumstances have been deliberated. tan and masuoka [1] found the stokes first problems for the second graded fluids and rashidi et al. [2] discussed by the unsteady compressible flows of the second order fluids. hayat et al. [3] explored by the unsteady stagnation point flow of second grade fluids with changeable free stream.due to complicated relation between stress and strain in nonnewtonian fluids and their technological application, their study in fluid dynamics is more valuable than newtonian fluids. viscous fluids flow has attracted the attention of scientists and engineers because of its important applications notably in the flow of oil through porous rocks, the extraction of energy from geothermal regions, the filtration of solids from liquids and drug penetration through human skin. second grade fluid is a subclass of non-newtonian fluid in which velocity field has up to two derivatives in stress strain tensor relationship where as in new tonian fluid it has derivatives up to first order. flow of second grade fluid gains attention of the researchers in many boundary layer flows and have been successfully studied in various kinds of flows. study of heat transfer in non-newtonian fluids is much interesting for researchers now-adays. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 15 mailto:dhanu2614@gmail.com mailto:drksshankara@gmail.com mailto:kkvpmaths@gmail.com mailto:profprakasha@gmail.com ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction the combined heat and mass transport problems through the chemical reaction are of significance in a lot of processes and have obtained an extensive value of concentration in current years. in developments such as drying, disappearance at the external of a fluid body, energy transportation in a drenched cooling increase and the flow in a desert cooler, heat and mass transport happen simultaneously. possible applications of that category of flow can be established in numerous industries. some examples, in the power industries, between the techniques of generating electric energy is solitary in this electrical energy are extracted directly exciting from a conducting fluid. it is predominantly attracted in cases of diffusion and chemical reaction occurs at approximately the identical speediness. once diffusion is to a great extent faster than chemical reaction, then merely chemical reaction influences the rate of chemical reaction; when diffusion is not much quicker than chemical reaction, the diffusion as well as kinetics interacts to construct very dissimilar consequences. the investigation of heat generation or absorption consequences in moving fluids is significant in sight of quite a few substantial problems, they are, and fluids undergo exothermic or else endothermic chemical reaction. outstanding to the quick development of electronic technology, effectual freezing of electronic apparatus has become certified and freezing of electronic apparatus ranges from own transistors to foremost structure computers and from energy providers to telephone switch panels and thermal diffusion impacts has been exploited for isotopes separation in the combination among gases with extremely low molecular weight (h2 and he) and average molecular weight. bestman [4] investigated the free convection boundary layer flow with simultaneous heat and mass transfer in a porous medium when the boundary walls move in its own plane with suction. abdus sattar and hamid kalim [5] studied the unsteady free convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate. makinde [6] explored the combined free convection boundary layer flow with thermal radiation and mass transfer past a permeable vertical plate. makinde et al. [7] investigated the problem of unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture in an optically thin environment. muthucumaraswamy and ganesan [8] explored the impact of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate. in many chemical engineering processes, there does occur the chemical reaction between a foreign mass and the fluid in which the plate is moving. these processes take place in numerous industrial applications viz., polymer production, manufacturing of ceramics or glassware and food processing. das et al[9] have studied the effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction. muthukumaraswamy[10] has studied the effects of reaction on a long surface with suction. radiation and mass transfer on an unsteady two-dimensional laminar convective boundary layer flow of a viscous incompressible chemically reacting fluid along a semi-infinite vertical plate with suction by taking into account the effects of viscous dissipation. kandaswamy et al[11] have discussed the effects of chemical reaction, heat and mass transfer on boundary layer flow over a porous wedge with heat radiation in the presence of suction or injection. the study of heat transfers and mixed convection flow in enclosures of various shapes has received attention [12] due to its practical applications. interest in these convection flow and heat transfer in porous medium has been motivated by a broad range of applications to geothermal systems, crude oil production, storage of nuclear waste materials, ground water pollution, fiber and granular insulations solidification of castings. in a wide range of such problems, the physical ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 16 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction system can be modeled as a two-dimensional rectangular enclosure with vertical walls held at different temperatures and the connecting adiabatic horizontal walls. convective heat transfers in a rectangular porous duct whose vertical walls are maintained at two different temperatures and horizontal walls insulated received attention by several investigators [13]. furthermore, in references [14 and 15] numerical results are being presented. coupled heat and mass transfer phenomenon in porous media is gaining attention due to its interesting applications. the flow phenomenon is relatively complex rather than that of the pure thermal convection process. underground spreading chemical wastes and other pollutants, grain storage, evaporation cooling and solidification are the few other application areas where the combined thermo-solutal natural convection in porous media are observed. combined heat and mass transfer by free convection under boundary layer approximations has been studied by bejan and khair[16],lai and kulacki[17].the free convection heat and mass transfer in a porous enclosure has been studied recently by angirasa et al[18]. the combined effects of thermal and mass diffusion in channel flows has been studied in recent times by a few authors, notably, nelson and wood [19]. theoretical and experimental investigations of natural convection mhd flow over a vertical porous plate in presence of chemical reaction plays an important role in agriculture, geophysics and astrophysics. to study the underground water resources, filtration and water purification process in chemical engineering one must know the concepts of the fluid flow through porous medium. the porous medium is like a non homogeneous medium but for the sake of analysis, it may be possible to replace it with a homogeneous fluid. oscillatory flows are associated with high rates of heat and mass transfer. many studies have been done to understand its characteristics in different systems such as pulse combustors and reciprocating engines. many investigators reported oscillatory flows by involving various physical situations. 2.mathematical model we consider the motion of viscous, incompressible fluid through a porous medium in a vertical channel bounded by flat walls. the thermal buoyancy in the flow field is created by a traveling thermal wave imposed on the boundary wall at y =l while the boundary at y = -l is maintained at constant temperature t1 while both the walls are maintained at uniform concentration. the boussinesq approximation is used so that the density variation will be considered only in the buoyancy force. we choose a rectangular cartesian system 0 (x, y) with xaxis in the vertical direction and y-axis normal to the walls. the walls of the channel are at y=l. the equations governing the unsteady flow, heat and mass transfer in terms of stream function .     2 0 0 4222 )()( )(])()()[(    k ccg ttg y yxyyxt (2.1) )()()( 1 2 oope ccqttq yxxyt c                 (2.2) )()( 1 2 occkd yxxyt                (2.3) ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 17 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction the boundary conditions for the velocity and temperature fields are 11 , ,0 ,0 cctt xy        on y = -l 22 ),( ,0 ,0 ccntmxsinttt xy e        on y = l (2.4) introducing the non-dimensional variables as 21 2 21 22 ,,/,,/, cc cc tt tt mttlyymxx        (2.5) the governing equations in the non-dimensional form ( after dropping the dashes ) are 2 2 22 1 14 1 2 12 1 ))(() ),( ),( )(( y mdn r g yx r yyt            (2.6)    2 2 1)( q yxxyt p              (2.7)                 2 1)( yxxyt sc (2.8) where  ul r  (reynolds number) 2 3   ltg g e  (grashof number) 1k c p  ( prandtl number), k l d 2 1  (darcy parameter), 1d sc   (schmidt number) 2 222 2   lh m oe ( hartmann number)   2ql  (heat source parameter) (radiation absorption parameter) 1 2 1 1 d lk  (chemical reaction parameter) lm (aspect ratio) 2m n    (non-dimensional thermal wave velocity) the corresponding boundary conditions are 1)1()1(   10,0       yat yx  (2.9) 2 2 2 2 22 1 yx        ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 18 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction 2.5 2.0 1.5 1.0 0.5 ---u r=5 w r=6 r=7 r=8 r=9 0.2 0.4 0.6 0.8 1.0 2.5 2.0 1.5 1.0 0.5 ----u gm= 10 gm= 8 w gm= 6 gm= 4 gm= 2 0.2 0.4 0.6 0.8 1.0 10,)(),( 11,1),(   yonctxsinyx yoncyx   1,)(),(  ctxsinyx  00,0       yat y c y  (2.10) the value of  on the boundary assumes the constant volumetric flow in consistent with the hypothesis. also the wall temperature varies in the axial direction in accordance with the prescribed arbitrary function t. 3. nusselt number and sherwood number knowing the temperature & concentration the local rate of heat and mass transfer on the walls have been calculated using the formula 1)( 1      y wm y nu   where    1 1 5.0 dym  and 1)( 1      y wm y c cc sh where    1 1 5.0 dyccm where 1421 .......,..........,.......... ddd are constants. 4. discussion of the numerical results in this analysis we investigate the effect of chemical reaction on convective heat and mass transfer flow of a viscous fluid in a vertical wavy channel. fig.1 velocity profile for various values of chemical reaction fig.2 velocity profile for various values of grashof number increase of �2 (rotation parameter) increase the primary velocity but the reverse process exists for the secondary velocity. at the same time in certain stage after that reverse processes exists for the secondary velocity in the fluid flow.the increasing of permeability parameter k., grashof number for heat transfer gr(figs.1&2) ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 19 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction 1.0 0.8 0.6 0.4 0.2 r=1.25 r=2.25 r=3.25 r=4.25 r=5.25 0.2 0.4 0.6 0.8 1.0 1.0 0.8 0.6 0.4 0.2 pe=1.25 pe=2.25 pe=3.25 pe=4.25 pe=5.25 1.0 0.8 0.6 0.4 0.2 r=1.0 0.18 5 10 15 20 2 4 pe=5 6 8 10 12 14 r=1.25 3.0 0.16 pe=4 0.14 2.5 pe=3 0.12 r=1.85 0.10 2.0 r=2.00 pe=2 0.08 1.5 0.06 r=2.25 pe=1 fig.3 temperature profile for various values of chemical reaction fig.4 concentration profile for various values of peclet number fig.3 shows that,the increase effects of temperature exists, the reverse processes exists if increase of chemical reaction parameter.concentration profile shows the decrease effects while increasing of peclet number(fig.4). fig.5 mass flux for various value of chemical reaction fig.6 heat flux for various value of peclet number mass flux shows decrease effects while increasing of chemical reaction (fig.5)..heat flux shows increasing effects while increase of peclet number(fig.6). the tables 1–3 symbolize the skin friction, nusselt number and sherwood number for dissimilar deviations in the pertinent parameters. when the magnetic field is large, then the hall current will be developed in the flow field. ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 20 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction table.1 the shear stresses m k r pr gr gm sc kc h q1 be bi s 2 1 1 0.7 5 3 0.2 1 1 1 1 0 2.532394 3 1.884042 4 1.618128 1 2.913229 2 3.049695 2 2.464944 3 2.451027 3 1.313291 7 1.280852 10 4.461883 15 6.441776 6 3.706505 9 4.90861 0.3 3.081731 0.6 13.56434 2 3.501852 3 5.160271 2 1.271224 3 1.219895 2 6.037402 3 9.631637 2 2.960489 3 3.181527 0 2.57465 1 2.624382 ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 21 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction table.2 the nusselt number kc ql sc h pr n t nu 1 1 0.22 1 0.71 0.5 0.5 0.727227 2 0.850651 3 0.924433 2 -0.14204 3 -0.0117 0.3 0.813039 0.6 1.004183 2 1.249786 3 1.60923 3 3.39786 7 7.54639 1 0.72799 1.5 0.728996 1 0.727683 1.5 0.72827 table.3. the sherwood number kc sc n t sh 1 0.22 0.5 0.5 0.715581 2 0.90641 3 1.054072 0.3 0.841856 0.6 1.25509 1 0.715965 1.5 0.716476 1 0.715813 1.5 0.716109 the skin friction magnitudes are described in table 1. an increases in the hartmann number precede decreases in skin friction. because the frictional drag was decreased by the lorentz effect on a viscous fluid. an increase in the rotation parameter, prandtl number, and heat source parameter is used to examine the comparable behavior. furthermore, an increase in the permeability parameter k leads to increased skin friction in significant ways on the surface boundary. similarly, increases in the radiation-absorption parameter, schmidt number, chemical reaction parameter, thermal grashof number, mass grashof number, hall, and ion slip parameters ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 22 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction near the surface boundary are examined for the same effect. table 2 indicates that an increase in the chemical reaction parameter, schmidt number, prandtl number, heat source parameter, oscillation frequency, and time all contribute to an increase in the nusselt number. it decreases as the radiation-absorption parameter increases. according to table 3, a stronger sherwood number is preceded by an increase in the schmidt number, chemical reaction parameter, oscillation frequency, or time. 5.references. [1] w. tan, t. masuoka, stokes’ first problem for a second grade fluid in a porous halfspace with heated boundary, int. j. nonlinear mech. 40 (2005) 515–522. [2] m.m. rashidi, s.a. majid, a. mostafa, application of homotopy analysis method to the unsteady squeezing flow of a second-grade fluid between circular plates, math. probl. eng. 18 (2010), 706840. [3] t. hayat, m. qasim, s.a. shehzad, a. alsaedi, unsteady stagnation point flow of second grade fluid with variable free stream, alexandria eng. j. 53 (2014) 455–461. [4] a.r. bestman, natural convection boundary layer with suction and mass transfer in a porous medium, int. j. energy res. 14 (1990) 389–396. [5] m.d. abdus sattar, m.d. hamid kalim, unsteady free-convection interaction with thermal radiation in a boundary layer flow past a vertical porous plate, j. math. phys. sci. 30 (1996) 25–37. [6] o.d. makinde, free convection flow with thermal radiation and mass transfer past a moving vertical porous plate, int. commn. heat mass transf. 32 (10) (2005) 1411–1419. [7] o.d. makinde, p.o. olanrewaju, w.m. charles, unsteady convection with chemical reaction and radiative heat transfer past a flat porous plate moving through a binary mixture, afr. mat. 22 (2011) 65–78. [8] r. muthucumaraswamy, p. ganesan, effect of the chemical reaction and injection on flow characteristics in an unsteady upward motion of an isothermal plate, j. appl. mech. tech. phys. 42 (2001) 665–671. [9] u.n. das, r. deka, v.m. soundalgekar, effects of mass transfer on flow past an impulsively started infinite vertical plate with constant heat flux and chemical reaction, forsch ing-wes. 60 (1994) 284–287. [10] r. muthukumaraswamy, effects of a chemical reaction on a moving isothermal surface with suction., acta mechnica,v.155,p.65, 2002 [11] p. kandaswamy, wahid abd, b.md.raj, b. azme khamis, effects of chemical reaction, heat and mass transfer on boundary layer flow over a porous wedge with heat radiation in the presence of suction or injection, theoret. appl. mech., v.33. no.2, pp.123-148, 2006 [12] hiroxhi iwai, kazuyoshi nakabe, kenjiro suzuki: flow and heat transfer characteristics of backward-facing step laminar flow in a rectangular duct., int.j.heat and mass transfer,v.43, pp.457471(2000) [13] teoman ayhan, hayati olgum : betul ayhan : heat transfer and flow structure in a rectangualr channel withwing -1, type vortex generator. tr. j. of engineering and environmental science, pp, 85-195, 22 (1998). [14] cheng k.s. and j.r. hi.: steady, two-dimensional, natural convection in rectangular enclosures with differently heated walls transaction of the asme, v. 109, p, 400, (1987). [15] chan, b.k.c, ivey, u.m and barry, j.m: natural convection in enclosed porous medium with rectangular boundaries asme journal of heat transfer, v. 92, pp, 21-27 (1970). [16] bejan,a and khair,k.r:heat and mass transfer by natural convection in a porous medium, int. j. heat mass transfrt,v.28,pp.908-918(1985). [17] lai,f.c and kulacki,f.a : coupled heat and mass transfer by natural convection from ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 23 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref1 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref1 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref1 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref2 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref2 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref2 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref2 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref2 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref2 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref3 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref3 http://refhub.elsevier.com/s0019-4522(22)00480-0/sref3 ijo international journal of mathematics (issn: 2992-4421 ) dhananjaiah d. s1* https://ijojournals.com/ volume 07 issue 03 || march, 2024 || a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction vertical surfaces in porous medium.,int.j.heat mass transfer, v.34, pp.1189-1194(1991). [18] angirasaa,d,peterson,g.p and pop, i :combined heat and mass transfer by natural convection with opposing buoyancy effects in a fluid saturated porous medium, int. j. heat mass transfer,v.40,pp.2755-2773(1997). [19] nelson,d.j and wood,b.d:combined heat and mass transfer by natural convection between vertical plates ,int.j.,heat mass transfer,v.82,pp.1789-1792(1989). ijo journals volume 07 | issue 03 | march 2024 | https://ijojournals.com/index.php/m/index 24 a mathematical analysis of oscillatory free convection in a vertical wavy channel with chemical reaction dhananjaiah d. s1, prof. k. shivashankara2, venuprasad k. k3, prakasha.p4 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data osuagwu, chidimma udo department of statistics, federal university of technology, owerri, imo state nigeria okenwe idochi department of statistics, school of applied sciences, ken saro wiwa polytechnic pmb 20, bori, rivers state nigeria abstract this study was embarked to examine the performance evaluation of canonical correlation and redundancy analysis with some continuous distributed data (gaussian, gamma, exponential and beta). the objectives of the study were to: obtain the relative efficiency of cca and rda techniques for four continuous distributed simulated data; and determine the model performance adequacy of cca and rda techniques. three variates of the response variable (y1, y2, y3) and three variates of independent variables (x1, x2, x3) were used for the simulation. the means used for response and independent variables for the gaussian distribution were 80, 85 and 90, whereas their standard deviations were 10, 12 and 15. the alpha values used for response and independent variables for the gamma distribution were 80, 85 and 90 whereas their theta values were 40, 43 and 45. the rates parameters used for response and independent variables for the exponential distribution were 0.5. 0.7 and 0.9; whereas the shape parameters used for the beta distribution were taking from 2 to 5 values. the adequacy of the cca and rda was evaluated with wilcoxon rank sum test; and the study concluded thatrda was more efficient than that of cca for the beta distributed data, while for gaussian, gamma and exponential distributed data, the relative efficiency of the cca and rda was the same. the study also concluded that the xvariates of the cca and rda did not differ. keywords: canonical correlation analysis,redundancy analysis, gaussian, gamma, exponential, beta,performance evaluation, simulated data. 1 introduction canonical correlation analysis (cca) and redundancy analysis (rda) are multivariate statistical techniques used to analyze the relationships between two or more sets of variables.cca is a method for analyzing the relationships between two sets of variables, x and y, by finding the linear combinations of variables that maximize the correlation between the two sets, which was developed by hotelling in 1936 (górecki et al, 2020). canonical correlation analysis (cca) involves finding a linear transformation that converts the original variables from ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 1 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data two sets into new sets of variables (wang et al., 2022). these new variables have the property of being uncorrelated within each set, but maximally correlated between sets. the resulting pairs of new variables are called canonical variates, and the correlation coefficients between these pairs are known as canonical correlations. by identifying these canonical variates and correlations, cca reveals the underlying relationships between the two sets of variables (li et al., 2020). rda is a statistical method that summarizes the linear relationships between two sets of variables, one set being the explanatory variables and the other being the response variables (ramette, 2017). it's an extension of multiple linear regression; allowing for multiple response variables to be regressed on multiple explanatory variables. rda produces an ordination that summarizes the main patterns of variation in the response matrix, which can be explained by a matrix of explanatory variables (hui&warton, 2022). the results of rda include the total variance of the data set, partitioned into constrained and unconstrained variances, which shows how much variation in the response variables was redundant with the variation in the explanatory variables (székely et al., 2020). rda also produces scores for objects, response variables, and explanatory variables, which can be used to ordinate points and vectors. rda is often used in ecological studies to relate environmental variables to species composition. for example, ramette (2007) used rda to analyze the relationships between microbial community composition and environmental variables in coastal sands this study therefore was aimed to: ascertain the relative efficiency of cca and rda techniques for four continuous distributed simulated data; and determine the model performance adequacy of cca and rda techniques. 2 review of related literature makino (2022) explored the application of rotation in correspondence analysis (ca) from a canonical correlation perspective. ca is a statistical method used to visualize the relationship between two categorical variables, typically emphasizing graphical representations. makino's study introduced a ca formulation based on canonical correlation analysis (cca), where correlations within and between row/column categories in a reduced dimensional space can be expressed through canonical variables. however, existing cca-based formulations only allowed for orthogonal rotation. makino proposed an alternative cca-based formulation that permits oblique rotation, defining the ca loss function as maximizing the generalized coefficient of determination, which measures the proximity between two variables. the study demonstrated the benefits of the proposed formulation through simulation studies and real data examples. ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data nayir and saridas (2022) investigated the relationship between culturally responsive teacher roles and innovative work behavior using canonical correlation analysis. the study aimed to identify the relationship between these two constructs based on teachers' views. the results showed that the first canonical function, which maximized the relationship between the two datasets, shared approximately 77% variance. furthermore, the analysis revealed a positive relationship between the culturally regulating teacher (crt) and culturally mediating teacher (cmt) variables in the culturally responsive teacher roles dataset and the gii and fsi variables in the innovative work behavior dataset. mckeague and zhang (2021) investigated significance testing for canonical correlation analysis in high-dimensional settings. they addressed the challenge of testing for linear relationships between large sets of random variables using post-selection inference techniques. the authors developed a stabilized one-step estimator for the euclidean norm of canonical correlations, which was shown to be consistent and asymptotically normal under certain conditions. they also proposed a greedy search algorithm for computing the estimator, leading to a computationally tractable omnibus test for the global null hypothesis. additionally, they constructed a confidence interval that accounted for variable selection. garcía-valdés et al. (2020) conducted a study using redundancy analysis (rda) to examine the impacts of climate change on species distribution in a mediterranean ecosystem, incorporating 155 plant species and 15 environmental variables. the analysis revealed that climate variables, including temperature, precipitation, and drought, explained a significant portion of the variation in species distribution, accounting for 24.5% of the variation. additionally, soil and topographic variables played important roles, explaining 20.1% and 15.4% of the variation, respectively. the study's findings suggested that climate change led to shifts in species distribution, resulting in some species expanding their ranges while others contract. the rda framework provided a powerful tool for understanding the complex relationships between climate change and species distribution, with important implications for conservation and management efforts in the face of climate change. 3 materials and methods 3.1 canonical variates and canonical correlations the canonical correlations measure the strength of association between the two sets of variables (wang & liu, 2022).the first group of p variables is represented by the (p  1) random vector x(1), while the second group of q variables is represented by the (q  1) random vector x(2). it will be assumed, in the theoretical development, that x(1) represents the smaller set, so that p  q. for the random vectors x(1) and x(2), let ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data         2211 )2()1( 22 )2()2()2( 11 )1()1()1( ),( )(;)( )(;)( xx xμx xμx cov cove cove (1) it will be convenient to consider x(1) and x(2) jointly, so, the random vector                                     )2( )2( 2 )2( 1 )1( )1( 2 )1( 1 )2( )1( )1)(( q p qp x x x x x x   x x x (2) has mean vector               )2( )1( )2( )1( )1)(( )( )( )( μ μ x x xμ e e e qp (3) and covariance matrix )()( qpqp  σ = e(x– )e(x – )          ))(())(( ))(())(( )2()2()2()2()1()1()2()2( )2()2()1()1()1()1()1()1( μxμxμxμx μxμxμxμx ee ee           )( 22 )( 21 )( 12 )( 11 qqpq qppp σς σς (4) 3.2 matrices and computational procedures of redundancy analysis let x be a 1p vector that includes p predictor variables in the first set and y be a 1q vectorthat includes q criterion variables in the second set. according to van den wollenberg in 1977, all variables in x and y should be standardized variableswith zero mean and unit variance (gua et al., 2023). thus, the )()( qpqp  covariance matrix of the 1)(  qp vector )yx(  is a correlation matrix, denoted by r , which can be partitioned as ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data ,         yyyx xyxx rr rr r (4) where xxr is the pp correlation matrix of x , yyr is the qq correlation matrix of y , and xyyx rr  is a pq matrix that includes the inter-set correlations between x and y. to construct p redundancy variates, denoted by ),,,2,1( pii  with p predictor variables in x, the characteristic equation is evaluated as shown in equation (5): ,0)(  ixxyxxy i wrμrr (5) where iμ is the thi eigenvalue and iw is the thi eigen-vector. then, one can employ the weight coefficients that are the elements of the scaled eigenvector iw to construct ,i such that: (a) i is uncorrelated with ),( jij  and (b) i has unit variance ).,,2,1( pi  3.3 continuous probability distributions four probability distributions known as the gaussian, gamma, beta and exponential are discussed in this study. 3.3.1 the gaussian distribution a random variable (r.v.) in continuous form say x , choosing the whole real values in intervals   , is known to be a gaussian (also known as normal) distribution with 2 and  as its parameters if the probability density function (pdf) is defined by 0,,, 2 2 1 0 2 1 )( 2                    x x e otherwise xf (6) where the study used the notation );( 2n to show that x is normal with mean  and variance 2 (sumair, et al., 2021). this pdf is bell-shaped, symmetrical, and centered at its mean value  . the entire area bounded by this function  f x and axis-x is 1 and therefore the area beneath the curve across two values of x , say, and with a b a b , constitutes the probability that the r.v x lies across and a b , which we write as  p a x b  . an example of a normal r.v is height of students at a specified age for a specified sex in a specified racial group even though heights must be positive (el-morshedy et al., 2021). ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data the pdf of a standard normal distribution is          otherwise zezf z ,0 ,21 2 2  (7) the respective mean and variance of a gaussian distribution are respectively given as;   xe (8) and   2xvar (9) 3.3.2 gamma distribution a r.v x is said to follow a gamma r.v with parameters and , if its pdf is given by:            otherwise ,0 0,0,0,)( 1     x ex xf x (10) where   is the gamma function defined as;   dtt et   0 1 (11) the gamma probability density function as given in equation (10) is a normal or legitimate pdf. the respective mean and variance of a gamma distribution are respectively given as; )(xe (12) and 2)( xvar (13) to obtain the scale (  ) and shape ( ) parameters of a gamma distribution, we have  )(xe (14) 22)(  xvar (15) from equation (14), put     into equation (15) to obtain  2 (16) from equation (16), the scale parameter is obtained as ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data    2  (17) substitute equation (17) into equation (14) to obtain the shape parameter as 2 2     (18) equations (17) and (18) were employed in the simulation of data for gamma distribution in this study. 3.2.3 the exponential distribution a continuous random variable x , is said to have an exponential distribution with parameter 0 if it has a probability density function defined by        otherwise xe xf x 0 )( 0,  (19) the respective mean and variance of an exponential distribution are respectively given as;  1 )( xe (20) and 2 1 )(  xvar (21) to obtain the rate parameter ( ) of an exponential distribution, we have     11 )( xe (22) equations(22) was employed in the simulation of data for an exponential distribution in this study. 3.2.4 beta distribution a standard beta distribution is a two-parameter family of distribution for a continuous random variabley , defined in a finite interval on a real line with its density function given by ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data         otherwise ,0 10, ),( )1( )( 11 y b yy yf   (23) where , 0 and ( , ) is the beta function; its formula is given byb    dtttb 1 1 0 1 )1(),(     (24) the mean and variance of y are     )(xe (25) and 2 2 ))(1( )(     xvar (26) the scale (  ) and shape ( ) parameters of a beta distribution are obtained as; 2 22 )(      (27) and 2 22 )1)((      (28) 4 results 4.1 simulated data of different sample sizes for cca and rda data were simulated on r-studio command window, calling for the cca and rda function for gaussian distribution, gamma distribution, exponential distribution and beta distribution for samples of sizes 10, 20, 30, 40, 50, 60 and 70. three variates of the response variable (y1, y2, y3) and three variates of independent variables (x1, x2, x3) were used for the simulation. the means used for response and independent variables for the gaussian distribution were 80, 85 and 90,whereas their standard deviations were 10, 12 and 15. the alpha values used for response and independent variables for the gamma distribution were 80, 85 and 90 whereas their theta values were 40, 43 and 45. the rates parameters used for response and independent variables for the exponential distribution were 0.5. 0.7 and 0.9; whereas the shape parameters used for the beta distribution were taking from values from 2 to 5 and the results obtained are summarized in table 1. ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data table 1: summary results from the four distributions for different sample sizes distribution correlation eigen-value x-mean vector y-mean vector sample cca rda cca rda cca rda gaussian 10 0.9609 0.7180 81.6417 81.6417 78.5301 78.5301 0.5303 0.2717 75.8878 75.8878 89.9215 89.9215 0.4498 0.0020 89.3103 89.3103 84.7729 84.7729 sd = 0.2748 sd = 0.3616 20 0.5343 0.5715 81.5175 81.5175 76.8798 76.8798 0.3713 0.2015 82.4420 82.4420 87.3907 87.3907 0.1763 0.0098 91.7642 91.7642 95.7642 95.7642 sd = 0.1792 sd = 0.2855 30 0.4399 0.3159 82.1643 82.1643 80.9557 80.9557 0.2426 0.0852 85.5123 85.5123 86.2550 86.2550 0.0318 0.0050 91.6238 91.6238 89.4687 89.4687 sd = 0.2041 sd = 0.1614 40 0.2583 0.1442 80.8160 80.8160 78.6760 78.6760 0.1231 0.1146 87.3596 87.3596 83.7756 83.7756 0.0028 0.0044 90.5419 90.5419 89.0918 89.0918 sd = 0.1278 sd = 0.0737 50 0.3583 0.0809 78.4572 78.4572 79.9256 79.9256 0.1483 0.0211 85.5144 85.5144 84.7669 84.7669 0.0103 0.0057 89.8407 89.8407 92.1156 92.1156 sd = 0.1752 sd = 0.0397 60 0.3444 0.0599 81.1089 81.1089 80.7678 80.7678 0.1502 0.0105 79.5728 79.5728 86.6178 86.6178 0.031 0.0012 88.9582 88.9582 89.5820 89.5820 sd = 0.158 sd = 0.0316 70 0.2150 0.1753 81.4539 81.4539 81.7998 81.7998 0.0973 0.0724 85.5524 85.5524 82.6863 82.6863 0.0012 0.0082 91.8743 91.8743 90.5477 90.5477 sd = 0.1071 sd = 0.0843 gamma 10 0.8263 0.8768 3311.883 3311.883 3117.559 3117.559 0.4987 0.5569 3691.902 3691.902 3520.850 3520.850 0.2967 0.3712 3967.700 3967.700 4020.455 4020.455 sd = 0.2673 sd = 0.2558 20 0.3916 0.2781 3145.036 3145.036 3329.500 3329.500 0.1921 0.0870 3703.089 3703.089 3672.906 3672.906 0.0523 0.0022 4019.466 4019.466 4122.456 4122.456 sd = 0.1705 sd = 0.1413 30 0.5626 0.1461 3316.971 3316.971 3139.415 3139.415 0.3175 0.0736 3601.826 3601.826 3687.641 3687.641 0.0309 0.0065 4074.426 4074.426 4072.765 4072.765 sd = 0.2661 sd = 0.0698 40 0.4502 0.0491 3151.013 3151.013 3127.473 3127.473 0.2016 0.0176 3618.440 3618.440 3702.444 3702.444 0.0370 0.0014 4011.102 4011.102 4057.979 4057.979 sd = 0.2080 sd = 0.0243 50 0.3273 0.1721 3288.111 3288.111 3169.010 3169.010 0.2901 0.0077 3565.821 3565.821 3641.781 3641.781 0.0622 0.0005 4136.478 4136.478 4100.952 4100.952 sd = 0.1435 sd = 0.0971 60 0.2258 0.0436 3158.815 3158.815 3195.705 3195.705 0.1697 0.0121 3631.133 3631.133 3751.599 3751.599 0.0145 0.0001 4115.701 4115.701 4001.559 4001.559 sd = 0.1095 sd = 0.0225 ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data 70 0.2659 0.2709 3211.757 3211.757 3239.607 3239.607 0.2170 0.1029 3666.475 3666.475 3686.072 3686.072 0.1373 0.0066 4075.567 4075.567 3998.146 3998.146 sd = 0.0649 sd = 0.1338 exponential 10 0.4361 0.8230 1.1191 1.1191 2.4481 2.4481 0.3081 0.0420 2.8346 2.8346 1.5564 1.5564 0.1447 0.0237 0.8268 0.8268 1.1561 1.1561 sd = 0.1461 sd = 0.4563 20 0.4120 0.3147 2.2670 2.2670 1.9106 1.9106 0.3517 0.1538 1.3586 1.3586 1.6637 1.6637 0.0190 0.0013 1.6114 1.6114 1.1930 1.1930 sd = 0.2117 sd = 0.1567 30 0.4655 0.2104 1.5786 1.5786 1.5809 1.5809 0.2066 0.0256 1.2137 1.2137 1.1047 1.1047 0.0284 0.0042 0.8006 0.8006 0.7525 0.7525 sd = 0.2198 sd = 0.1134 40 0.4752 0.0573 1.4892 1.4892 1.8839 1.8839 0.2934 0.0276 1.2863 1.2863 1.4824 1.4824 0.1192 0.0041 0.9426 0.9426 1.0180 1.0180 sd = 0.1780 sd = 0.0267 50 0.2732 0.2847 2.0737 2.0737 2.3757 2.3757 0.2422 0.1221 1.5447 1.5447 1.5730 1.5730 0.0173 0.0188 0.9702 0.9702 1.0287 1.0287 sd = 0.1397 sd = 0.1340 60 0.3197 0.1642 2.0932 2.0932 1.6467 1.6467 0.2416 0.0383 1.4325 1.4325 1.6291 1.6291 0.1147 0.0150 1.0182 1.0182 1.1243 1.1243 sd = 0.1035 sd = 0.0803 70 0.3419 0.0333 1.8292 1.8292 1.9513 1.9513 0.2172 0.0152 1.2600 1.2600 1.2107 1.2107 0.0537 0.0081 0.9883 0.9883 0.9481 0.9481 sd = 0.1445 sd = 0.0130 beta 10 0.7963 0.2434 0.2932 0.2932 0.2610 0.2610 0.4646 0.0627 0.4356 0.4356 0.3669 0.3669 0.0710 0.0031 0.6326 0.6326 0.5026 0.5026 sd = 0.3631 sd = 0.1251 20 0.5851 0.5324 0.3218 0.3218 0.2944 0.2944 0.5173 0.1768 0.4216 0.4216 0.4652 0.4652 0.3069 0.0034 0.6327 0.6327 0.5084 0.5084 sd = 0.1451 sd = 0.2697 30 0.4148 0.3035 0.3263 0.3263 0.2616 0.2616 0.1741 0.0262 0.4498 0.4498 0.4344 0.4344 0.0256 0.0080 0.5727 0.5727 0.5483 0.5483 sd = 0.1964 sd = 0.1656 40 0.4796 0.1522 0.3285 0.3285 0.2641 0.2641 0.1904 0.0629 0.4468 0.4468 0.4209 0.4209 0.1156 0.0511 0.6191 0.6191 0.6008 0.6008 sd = 0.1922 sd = 0.0553 50 0.5701 0.0681 0.2805 0.2805 0.2751 0.2751 0.2277 0.0133 0.4535 0.4535 0.4555 0.4555 0.0613 0.0001 0.6037 0.6037 0.5730 0.5730 sd = 0.2594 sd = 0.0361 60 0.5456 0.1719 0.2424 0.2424 0.2994 0.2994 0.2209 0.0387 0.4367 0.4367 0.4373 0.4373 0.1515 0.0067 0.5737 0.5737 0.5745 0.5745 sd = 0.2104 sd = 0.0876 0.2990 0.0806 0.2953 0.2953 0.3030 0.3030 0.0762 0.0192 0.4589 0.4589 0.4352 0.4352 ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data 70 0.0525 0.0003 0.6028 0.6028 0.5580 0.5580 sd = 0.1360 sd = 0.0420 table 1 shows the standard deviation of the correlations and eigenvalues for cca and rda respectively.it can be observed that the standard deviation of the rda is lower than that of cca except for the cases of sample sizes 10 and 20 for gaussian distribution; sample size 70 for gamma distribution, sample size 10 for exponential distribution and sample size 20 for beta distribution, but there is need to examine if the differencesare significant. it is also observed that the x and y-variates of the cca and rda do not differ. 4.2 model performance adequacy of cca and rda techniques table 2: summary of decision for testing sd values for cca and rda sd values ranks z p-value decision distribution sample cca rda cca rda gaussian 10 0.2748 0.3616 12 14 0.958 0.338 do not reject h0 20 0.1792 0.2855 10 13 30 0.2041 0.1614 11 8 40 0.1278 0.0737 6 3 50 0.1752 0.0397 9 2 60 0.158 0.0316 7 1 70 0.1071 0.0843 5 4 gamma 10 0.2673 0.2558 14 12 1.725 0.085 do not reject h0 20 0.1705 0.1413 10 8 30 0.2661 0.0698 13 4 40 0.2080 0.0243 11 2 50 0.1435 0.0971 9 5 60 0.1095 0.0225 6 1 70 0.0649 0.1338 3 7 exponential 10 0.1461 0.4563 9 14 1.469 0.142 do not reject h0 20 0.2117 0.1567 12 10 30 0.2198 0.1134 13 5 40 0.1780 0.0267 11 2 50 0.1397 0.1340 7 6 60 0.1035 0.0803 4 3 70 0.1445 0.0130 8 1 beta 10 0.3631 0.1251 14 5 2.108 0.035 reject h0 20 0.1451 0.2697 7 13 30 0.1964 0.1656 10 8 40 0.1922 0.0553 9 3 50 0.2594 0.0361 12 1 60 0.2104 0.0876 11 4 70 0.1360 0.0420 6 2 table 2 shows the wilcoxon rank sum testsignificance difference result for the four continuous distributions employed in this study. the result reveals that there is no significant difference in ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data the standard deviation of the correlations and eigenvalues for the methods for gaussian, gamma and exponential distributions. this implies that the relative efficiency of the cca and rda is the same for the gaussian, gamma and exponential distributed data. on the other hand, the result reveals thatthere is significant difference in the standard deviation of the correlations and eigenvalues for the methods for beta distribution. this implies that rda is more efficient than that of cca for the beta distributed data. 4 conclusion this study used canonical correlation and redundancy analysis via four continuous distributions (gaussian, gamma, exponential and beta) in order to assess their performances. the adequacy of the cca and rda was evaluated with wilcoxon rank sum test; and the study concluded thatrda is more efficient than that of cca for the beta distributed data, while for gaussian, gamma and exponential distributed data, the relative efficiency of the cca and rda is the same. the study also concluded that the x-variates of the cca and rda do not differ. references el-morshedy, m., alshammari, f.s., hamed, y. s., eliwa, m.s. &yousof, h.m. (2021).a new family of continuous probability distributions.entropy, 23(194), 1–24. garcía-valdés, r., sánchez, a. m., fernández-palacios, j. m., padrón, r. p., & rodríguezrodríguez, m. a. (2020). climate change impacts on species distribution: a redundancy analysis approach. ecography, 43(1), 141-152. góreck, t., krzysko, m. &wołynski, w. (2020).generalized canonical correlation analysis for functional data.biometrical letters, 57(2020), 1 – 12. gua, f., yungb, y., cheungc, m. w. l., jood, b. k. &nimon, k. (2023). statistical inference in redundancy analysis: a direct covariance structure modeling approach. multivariate behavioral research, 58(5), 877–893. hui, f. k. c., &warton, d. i. (2022).redundancy analysis and related methods for multivariate ecological data.methods in ecology and evolution, 13(1), 15-28. doi: 10.1111/2041210x.13734 li, m., xu, x., & chen, j. (2020).integrative analysis of gene expression and disease outcomes using canonical correlation analysis.bioinformatics, 36(10), 2911-2918. makino, n. (2022). rotation in correspondence analysis from the canonical correlation perspective.psychometrika,5(2022), 18–28. mckeague, i. w. & zhang, x. (2021).significance testing for canonical correlation analysis in high dimensions. biometrika, 2021. nayir, f. &saridas, g. (2022). the relationship between culturally responsive teacher roles and innovative work behavior: canonical correlation analysis. journal of educational research and practice, 12(2022), 36 –50. ramette, a. (2007). multivariate analyses in microbial ecology.fems microbiology ecology, 62(2), 142-160. ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 12 ijo international journal of mathematics (issn: 2992-4421 ) osuagwu, chidimma udo.* https://ijojournals.com/ volume 07 || issue 08 || august, 2024 || performance evaluation of canonical correlation analysis and redundancy analysisusing gaussian, gamma, exponential and beta distributed data ramette, a. (2017). multivariate analyses in microbial ecology: a decade of progress. fems microbiology ecology, 93(12), fix106.doi: 10.1093/femsec/fix106 sumair, m., aized, t., gardezi, s.a.r., bhutta, m.m.a., rehman, s.m.s. &rehman, s.u. (2021). application of five continuous distributions and evaluation of wind potential at five stations using normal distribution. energy exploration & exploitation, 39(6), 2214– 2239. székely, e., botta-dukát, z., &lengyel, a. (2020).redundancy analysis as a tool for identifying drivers of community composition in vegetation ecology.journal of vegetation science, 31(3), 537-546. doi: 10.1111/jvs.12854 van de velden. m. (2011).on generalized canonical correlation analysis.proc.58th world statistical congress. dublin, 758–765. wang, h., zhang, y., & singh, r. (2022). climate-crop yield relationships: a canonical correlation analysis. agricultural and forest meteorology, 313, 108702. wang, y., & liu, x. (2022).canonical correlation analysis for identifying relationships between climate variables and crop yields.journal of agricultural science, 160(3), 257-265. ijo journals volume 07 | issue 08 | august 2024 | https://ijojournals.com/index.php/m/index 13 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity *1obi, boniface inalu 2ohaegbulem, emmanuel uchenna 1department of mathematics, faculty of physical sciences, imo state university, owerri, nigeria. 2department of statistics, faculty of physical sciences, imo state university, owerri, nigeria. corresponding author *1obi, boniface inalu abstract: in this research, analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity is examined. the coupled nonlinear momentum and energy equations were solved using the traditional regular perturbation technique. vogel’s model viscosity is introduced to account for the temperature-dependent viscosity, while the third grade fluid is accommodated to model the non-newtonian fluid feature. it is observed that the third grade and the magnetic field parameters reduces the velocity profiles and increases the temperature profiles when increased at a steady rate within the constant viscosity index but increases the velocity and the temperature profiles when subjected to the vogel model. meanwhile the eckert parameter is observed to enhance the temperature near the walls of cylindrical pipe. keywords: viscosity, non-newtonian, mhd, isothermal, incompressible, 1.0 introduction: flow of an incompressible mhd non-newtonian fluid in cylindrical pipe finds application in polymer industry, petroleum industries and other types of pulp industries.in recent years, the non-newtonian fluids have become very much important. however with its complexity, it is difficult to suggest a single model which will exhibit all the properties of non-newtonian fluids, as such various empirical and semi empirical models have been put forward. meanwhile, for lubricating fluids, heat generated by internal friction and the corresponding rise in temperature affects the viscosity of the fluid and so the fluid viscosity can no longer be assumed constant. non-newtonian fluid can be classified mainly into two groups such as differential type fluids and rate type fluids. many researchers have done some work in this area amongst whom are fosdick and rajagopal [5], who examined the thermodynamics and stability of fluids of third grade. they showed restrictions on the stress constitutive equation. they were concerned with ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 1 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" the relation between thermodynamics and stability for a class of non-newtonian incompressible fluids of the differential type. they gave detailed attention to the special case of fluids of grade 3 and arrived at fundamental inequalities which restricts its temperature dependent. they discovered that these inequalities requires that a body of such fluid be stable in the sense that its total kinetic energy must tend to zero in time, no matter what its previous mechanical and thermal fields, provided it is both mechanically isolated and immersed in a thermally passive environment at constant temperature from some finite time onward. massoudi and christie [7] dealt with the effcts of variable viscosity and viscous dissipation on the flow of third grade fluid. the boundary layer equations of third grade fluid was treated by pakdemirli [14]. bejan [4] studied entropy generation in fundamentally convective heat transfer. johnson etal [6] investigated a fluid flow which was infused with solid particles in a pipe, while approximate analytical solutions for flow of third grade fluid was examined by yurusoy and pakdemirli [15]. okedayo etal [12] studied the effects of viscous dissipation, constant wall temperature and a periodic field on unsteady flow through a horrizontal channel. okedayo etal [13] analyzed the magnetohydrdynamic (mhd) flow and heat transfer in cylindrical pipe filled with porous media. they applied the galerkin weighted residual method for the solution of momentum equation and semiimplicit finite differece method for the energy equation. they found that an increase in darcy number leads to an increase in the velocity profiles, while increase in brinkman number enhances the temperature of the system.nargis and mahmood [8] studied the influence of slip condition on the thin film flow of third order fluid. obi [9] on approximate analytical solution of natural convection flow of non-newtonian fluid through parallel plates , solved the coupled momentum and energy equations using the regular perturbation methd. he treated cases of constant and temperature-dependent viscosities in which reynold’s and vogel’s models were considered to account for the temperaturedependent viscosity case, while third grade fluid was introduced to account for the nonnewttonian effects. obi [10] numerically analyzed the reactive third grade fluid in cylindrical pipe. he observed that the non-newtonian parameters considered in the analysis: third grade parameter ( ), magnetic field parameter (m ), eckert number ( ec ) and the brinkman number ( br ) had psitive effects on the velocity and temperature profiles.aksoy and pakdemirli [1] examined the flow of a non-newtonian fluid through a porous medium between two parallel plates. they involed reynold’s and vogel’s models viscosity and derived the criteria for validity for the approximate solution. ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" obi etal [11] on semi-analytical solution of natural convection flow of non-newtonian fluid with temperature-dependent viscosity in pipe. they solved the nonlinear momentum and energy equations using perturbation technique. they analyzed various thermo-solutal parameters involved in the dimensionless equations. results within the constant viscosity show that increase in these parameters increases the velocity of the fluid flow as well as the temperature of the cylindrical pipe. it is observed that increase in the reynold’s viscosity indices increases the temperature of the cylindrical pipe greatly. 2.0 mathematical formulation considering aiyesimi etal [2], the steady flow of an incompressible mhd third grade fluid flow in a cylindrical pipe and neglecting the reacting viscous fluid assumption, the governing momentum and energy equations with the necessary boundary conditions can be represented by   3 2 0 1 d du d du dp r r b u r dr dr r dr dr dz                          2 2 2 3 0 0 2 k d dt du du r b u r dr dr dr dr                                     0 0 0, 0, 0 3 du dt u a t a dr dr     where u is the velocity of the fluid, t is the temperature of the cylindrical pipe, t0 is the plate temperature, b0 is the magnetic field,  is the coefficient of dynamic viscosity, p is the pressure and  is the material coefficient relating to third grade fluid. the following non-dimensional variables are introduced for non-dimensionalization.   0 0 0 , , , , 4 r t u r u d t u         ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" substituting equation (4) into equations (1) to (3), yields   3 1 1 5 d du d du r r mu r dr dr r dr dr                  2 4 21 0 6c r d d du du r e b mu r dr dr dr dr                                 0 0 0, 0 0, 1 0 7 du d u dr dr      3.0 method of solution the semi-analytical solutions for velocity and temperature profiles can be of the form:                  2 2 0 1 0 1, , m= m 8u r u r u r r r r            3.1. constant viscosity substituting eqn (8) into eqns (5) and (6) and separating each order of  , yields  0 01 : 1̀ 9 dud r r dr dr           3 2 01 0 1 : 0 ̀ 10 dudud r r mu r dr dr dr                 2 0 0 01 : 0 ̀ 11c d dud r e r dr dr dr                  4 20 01 1 0 1 : 2 2 0 ̀ 12c r c du dud dud r e b e mu r dr dr dr dr dr                 solving eqns (9)-(12) with the condition (7), we have ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity"       2 2 2 4 4 4 4 6 6 2 4 1 1 1 1 1 1 1 13 4 4 392 16 48 392 24 1 1 1 1 1 1 16 16 4536 128 432 128 1 1 1 19 16 64 4536 3456 c c r c c u r r r m r r m r e r e r m r r r b e m r r e                                                      1 3 14 288 64 r cm b e m           3.2. vogel’s model viscosity in this section, we use vogel’s model to represent temperature – dependent viscosity and we apply the massaudi and chritie (1995) approach. the equations for momentum and energy for this model are:   3 1 1 15 d du d du d du r r mu dr dr r dr dr r dr dr                     2 4 21 2 0 16c r d d du du r e b mu r dr dr dr dr                          exp 17w q a            by taylor series expansion of (17), we have  2 1 18 q a           where  exp 19w q a           and aq being parameters relating to vogel’s model ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity"  2 , 20 d q d q q dr a dr      substituting eqns (8), (18) and (20) into eqns (15) and (16), we have  0 01 : 1̀ 21 dud r r dr dr            3 30 0 0 01 02 2 1 1 : 0 ̀ 22 du d du dudud q q d r r r mu r dr dr a dr a dr dr r dr dr                                 2 0 0 01 : 0 ̀ 23c d dud r e r dr dr dr                   4 20 0 0 01 1 02 1 1 : 2 0 24c r q d du dud dud d r r e b mu r dr dr r dr a dr dr dr dr                           solving the second order nonlinear ordinary differential eqns (21-24) with the condition (7) yields     2 2 6 6 2 4 4 2 4 3 2 4 4 2 3 2 4 4 2 3 1 1 1 1 1 4 4 256 768 1152 1 1 1 1 1 1 1 25 24 16 64 576 24 16 64 c c c c q q u r r r e r e r e a a q r m r r e m a                                                             2 4 4 8 3 2 6 6 2 4 8 3 2 4 2 6 6 2 2 2 2 6 2 2 2 1 1 1 1 64 4096 8192 1 1 1 1 1 4096 16384 9 16 64 2051 1 3 33570816 9 64 c c c r c r qe r e r r r a qe r r r b m r r a qe b m a                                                   26 ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" 4.0 results and discussion in order to study the behaviour of some physical parameters involved in the analysis, graphs are presented in figures (1-10). the solution of momentum and energy equations (1) and (2) with the boundary condion (3) given in equations (13) and (14). figures 1 shows the effects of third grade parameter on the velocity profiles. results indicate that increase in third grade parameter decreases the flow velocity. this is because the third grade parameter introduces a relationship between the velocity and the radial distance from the boundary which is nonlinear and leads to drop in flow velocity. figure 2 is the velocity profiles for various values of the magnetic field. it is observed from the results that increase in the magnetic field decreases the velocity because the applied force set in by the magnetic field parameter is perpendicular to the flow direction. figure 3shows temperature profiles for different values of third grade parameter. results show that as the third grade parameter increases, the temperature of the cylindricxal pipe increases as well owning to the thermal conductivity of the fluid which influences the temperature distribution. it is observed in figures 4 and 5 that the magnetic field and eckert parameters increases the temperature when the parameters are increased at a ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" regular rate.reasults further show that the eckert parameter regulates the rate of heat transfer. in figures 6 and 7 show the velocity and temperature profiles for different values of the the third grade parameter within the vogel model analysis respectively. results show that as the third grade parameter increases, both the velocity and temperature increases at the same rate. in figure 8, the temperature profiles for various values of the the magnetic field parameter is shown. it is seen that increase in the magnetic field parameter increases the temperature at the walls of the cylinder. figure 9 shows the temperature profiles for different values of the vogel parameter  . it is observed that the parameter has the propensity to lower the temperature when increased at very small rate. figure 10 shows the second major parameter a in the vogel model structure. results indicate that increase in the parameter a , increases the temperature of the cylindrical walls. 5.0 conclusions analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity is examined. vogel’s model viscosity is introduced to account for the temperature-dependent viscosity, while the third grade fluid is accommodated to model the non-newtonian fluid feature. it is observed that the third grade and the magnetic field parameters reduces the velocity profiles and increases the temperature profiles when increased at a steady rate within the constant viscosity index but increases the velocity and the temperature profiles when subjected to the vogel model. meanwhile the eckert parameter is observed to enhance the temperature near the walls of cylindrical pipe. results further show that increasing the two vogel model indices and a decreases and increases the temperature profiles respectively. declarations 1. funding: not applicable 2. informed consent statement: not applicable 3. data availability: not applicable 4. conflict of interest statement: no conflict of interest 6.0 reference [1] aksoy, y. and pakdemirli, m.: approximate analytical solution for flow of a third grade fluid through a parallel plate channel filled with a porous medium. transp. porous. med. 83,375395(2010). ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) *1 obi, boniface inalu * https://ijojournals.com/ volume 07 || issue 09 || september, 2024 || “analytical study of incompressible mhd non-newtonian fluid in cylindrical pipe with isothermal wall and temperature-dependent viscosity" [2] aiyesimi, y.m., okedayo, g.t., and obi, b.i. (2015). flow of an incompressible mhd third grade fluid through cylindrical pipe with isothermal wall and joule heating: nigerian journal of mathematics and application [3] ayub, m., rasheed, a. and hayat, t.,: exact flow of third grade fluid past a porous plate using homotopy analysis method. international journal of engineering and sciences vol 41,2091(2003) [4] bejan, a. a study of entropy generation in fundamental convective heat transfer,asme j. heat transfer 101, 718 (1979) [5] fosdick r.l. and rajagopal, k.r.: thermodynamics and stability of fluids of third grade. proc. r. soc. lond. 339, 351-377, (1980). [6] johnson, g., massoudi, m., rajagopal, k.r.: flow of a fluid infused with solid particles through a pipe. international journal of engineering sciences 29, 649-661 (1991). [7] massoudi, m. and christie, i.: effects of variable viscosity and viscous dissipation on the flow of a third –grade fluid in a pipe. int. j. of nonlinear mech.,30(5): 687-699,(1995). [8] nargis, k. and mamood,t.,:the influence of slip condition on the thin film flow of a third grade fluid. international journal of nonlinear science,30(5), 687-699(2012). [9] obi b.i.: approximate analytical study of natural convection flow of non-newtonian fluid through parallel plates with heat generation. journal of mathematical sciences and computational mathematics vol.4 no.4 (2023) [10] obi, b.i. computational analysis of reactive third grade fluid in cylindrical pipe using the collocation method. journal of mathematical sciences and computational mathematics. vol.4, no. 4 (2023) [11] obi b.i., okorie s.i. & nlemigwe j.c. semi-analytical solution of convection flow of nonnewtonian fluid with temperature-dependent viscosity in pipe. international journal for research in applied science and engineering technology ijraset 11(9): 782-786 (2023). [12] okedayo g. t., abah s. o and abah r. t.: viscous dissipation effect on the reactive flow of a temperature dependent viscosity and thermal conductivity through a porous channel. abacus: journal of mathematical association of nigeria 41(2),74-81, (2014). [13] okedayo, t.g., enenche, e. and obi, b.i.: a computational analysis of magnetohydrodynamic (mhd) flow and heat transfer in cylindrical pipe filled with porous media. international journal of scientific research and innovative technology, vol. 4 no. 7 (2017) [14] pakdemirli, m.: the boundary layer equations of third grade fluids. international journal of nonlinear mech. 27, 785 (1992). [15] yurusoy, m. and pakdemirli, m.: approximate analytical solutions for the flow of a third grade fluid in a pipe. international journal of non-linear mech. 37,187-195 (2002) [16] yurusoy, m, pakdemirli, m. and yilbas, b.s. perturbation solution for a third grade fluid flowing between parallel plates. journal of mechanical engineering science. vol 222, 653-65 (2008). ijo journals volume 07 | issue 09 | september 2024 | https://ijojournals.com/index.php/m/index 10 abstract ordinary differential equations, both first and second order, are essential in the modeling of many physical systems. a system of simultaneous differential equations results from more complicated modeling involving more than one dependent variables with respect to a single independent variable. there are several methods in solving a system of simultaneous linear differential equations including variable substitution, laplace transform and using the d-operator. proposed in this paper is a simplified method of solving a set of two non-homogeneous linear first-order simultaneous ordinary differential equations with constant coefficients that falls into a certain form. an algebraic formula is developed to compute the solution to the said differential equations provided a certain necessary condition is satisfied. four different forms of the functions of the independent variable on the right side of the equations, namely constants, linear functions, natural exponential functions and sinusoidal functions, are considered. for each case, an algebraic formula to calculate the dependent variable as well as its derivative are developed. moreover, these simple algebraic formulae can be easily programmed into spreadsheet where one just has to enter the values of the constants and coefficients from the original equations and instantly obtain the correct answers. key words: simultaneous differential equations; non-homogeneous differential equations; cramer’s rule. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 1 algebraic solution to simultaneous linear first-order nonhomogeneous differential equations with constant coefficients seng chu chow inti international college subang, malaysia 1. introduction ordinary differential equations (ode) play a crucial role in understanding and modeling various physical phenomena across many fields such as engineering, sciences and economics. among the numerous applications of first-order differential equations are solutions mixing, population change, heating or cooling, free-falling body, fluid dynamics, resistive-capacitive (rc) and resistive-inductive (rl) circuits. second-order differential equations find its applications in spring-mass vibration, rlc circuits, wave propagations, etc. by solving a differential equation using its known initial conditions, which is known as initial value problem (ivp), we can anticipate the behavior of the physical system over time. [7] simultaneous differential equations is a system of at least two differential equations with two or more dependent variables but share a single common independent variable. these equations govern the inter-relationship between the rate of change of the dependent variables with respect to the independent variable. [8] the general form of an nth-order linear ordinary differential equation is: 𝑎𝑛(𝑥) 𝑑𝑛𝑦 𝑑𝑥𝑛 + 𝑎𝑛−1(𝑥) 𝑑𝑛−1𝑦 𝑑𝑥𝑛−1 + ⋯ + 𝑎1(𝑥) 𝑑𝑦 𝑑𝑥 + 𝑎0(𝑥)𝑦 = 𝐹(𝑥) (1) since this paper only consider 1st-order ode, equation (1) reduces to 𝑎1(𝑥) 𝑑𝑦 𝑑𝑥 + 𝑎0(𝑥)𝑦 = 𝐹(𝑥) (2) where 𝑎1(𝑥), 𝑎2(𝑥) and 𝐹(𝑥) are functions of x only, and there is no restriction on the nature of these x-dependencies. [5, 6] ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 2 2 example of a system of simultaneous 1st-order ode 2.1 double mixing problem figure 1 below shows a brine solution with concentration 𝐵1(𝑘𝑔/𝑙) flows at a constant rate of 𝐹1(𝑙/𝑠) into tank 1 that initially contains 𝑉1(𝑙) of the same brine. the solution in the tank is kept well stirred and flows out of tank 1 at a rate of 𝐹2 with concentration 𝐵2 directly into tank 2 with volume 𝑉2. the solution in tank 2 is also well mixed and flows out at a rate of 𝐹3 with concentration 𝐵3 . the mass of salt in tank 1 and tank 2 are 𝑚1(𝑡) and 𝑚2(𝑡) respectively. the ivp for this problem is given by: 𝑑𝑚1 𝑑𝑡 = 𝐵1𝐹1 − 𝑚1 𝑉1+(𝐹1−𝐹2)∙𝑡 𝐹2 , 𝑚1(0) = 𝑎 (3a) 𝑑𝑚2 𝑑𝑡 = 𝑚1 𝑉1+(𝐹1−𝐹2)∙𝑡 𝐹2 − 𝑚2 𝑉2+(𝐹2−𝐹3)∙𝑡 𝐹3 , 𝑚2(0) = 𝑏 (3b) where a and b are the initial mass of salt in tank 1 and tank 2, respectively. 2.2 simplifying ivp in most applications, 𝐹1, 𝐹2 and 𝐹3 are constant and equal, otherwise the volume of liquid in the two tanks would be changing with time. if the two volumes are assumed to be equal, that is 𝑉1 = 𝑉2 = 𝑉, then the ivp in equation (3) is simplified to 𝑚1 ′ = 𝐴 − 𝐾𝑚1 (4a) 𝑚2 ′ = 𝐾𝑚1 − 𝐾𝑚2 (4b) where figure 1: double mixing problem ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 3 𝐴 = 𝐵1𝐹, 𝐾 = 𝐹 𝑉 and 𝐹1 = 𝐹2 = 𝐹3 = 𝐹. (4c) putting equation (4) in matrix form, we have [ 𝑚1 ′ 𝑚2 ′ ] = [ −𝐾 0 𝐾 −𝐾 ] [ 𝑚1 𝑚2 ] + [ 𝐴 0 ] (5) if the input concentration can be continuously adjusted to be in direct proportional to the varying mass of salt in tank 1, that is 𝐵1 ∝ 𝑚1, then 𝐴 = 𝑝𝑚1𝐹 = 𝐶 ∙ 𝑚1 (6) where p is the constant of proportionality between 𝐵1 and 𝑚1 , and the constant c is the product of p and f. with this assumption, equation (5) can be simplified to [ 𝑚1 ′ 𝑚2 ′ ] = [ −𝐾 + 𝐶 0 𝐾 −𝐾 ] [ 𝑚1 𝑚2 ] (7) 2.3 de-coupling of the system the eigenvalues of equation (7) can be found by solving the characteristic polynomial | 𝜆 + 𝐾 − 𝐶 0 −𝐾 𝜆 + 𝐾 | = 0 (8) which yields 𝜆1 = −𝐾 and 𝜆2 = −(𝐾 − 𝐶) (9) with the corresponding eigenvectors 𝛼1 = 𝑟 [ 0 1 ] and 𝛼2 = 𝑠 [ 𝐶 𝐾 1 ] , 𝑟, 𝑠 ∈ 𝑅. (10) the transition matrix p and its inverse is given by 𝑃 = [ 0 𝐶 𝐾 1 1 ] , 𝑃−1 = 1 − 𝐶 𝐾 [ 1 − 𝐶 𝐾 −1 0 ] (11) as expected, ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 4 1 − 𝐶 𝐾 [ 1 − 𝐶 𝐾 −1 0 ] [ −𝐾 + 𝐶 0 𝐾 −𝐾 ] [ 0 𝐶 𝐾 1 1 ] = [ 𝜆1 0 0 𝜆2 ] (12) let’s define 𝑚 = 𝑃𝑥 and 𝑚′ = 𝑃𝑥′ (13) where 𝑥 = [ 𝑥1(𝑡) 𝑥2(𝑡) ] is a dummy function. substitute (13) into (7) yields 𝑃𝑥′ = [ −𝐾 + 𝐶 0 𝐾 −𝐾 ] 𝑃𝑥 (14) multiply (14) by 𝑃−1 gives 𝑃−1𝑃𝑥′ = 𝑃−1 [ −𝐾 + 𝐶 0 𝐾 −𝐾 ] 𝑃𝑥 (15) incorporate equations (9), (11) and (12) into (15), we have [ 𝑥1 ′ (𝑡) 𝑥2 ′ (𝑡) ] = [ 𝜆1 0 0 𝜆2 ] [ 𝑥1(𝑡) 𝑥2(𝑡) ] (16a) which gives 𝑥1 ′ = 𝜆1𝑥1 (16b) 𝑥2 ′ = 𝜆2𝑥2 (16c) the system is now decoupled [9, 10]. 2.4 solution to example the solutions to equation (16) are easily obtained using separation of variables method as 𝑥1(𝑡) = 𝐷1𝑒−𝐾𝑡 (17a) 𝑥2(𝑡) = 𝐷2𝑒−(𝐾−𝐶)𝑡 (17b) ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 5 since 𝑚 = 𝑃𝑥, we have [ 𝑚1 𝑚2 ] = [ 0 𝐶 𝐾 1 1 ] [ 𝐷1𝑒−𝐾𝑡 𝐷2𝑒−(𝐾−𝐶)𝑡 ] . (18) with the initial conditions 𝑚1(0) = 𝑎 and 𝑚2(0) = 𝑏, it is easy to verify that the solutions to the ivp in (3) are 𝑚1(𝑡) = 𝑎𝑒−(𝐾−𝐶)𝑡 (19a) 𝑚2(𝑡) = (𝑏 − 𝐾𝑎 𝐶 ) 𝑒−𝐾𝑡 + 𝐾𝑎 𝐶 𝑒−(𝐾−𝐶)𝑡 (19b) 3 existing solution methods there are several methods to solve simultaneous differential equations, each has it own restrictions on the form of the equations. [11, 12, 13, 14, 15, 16] these methods are: (i) substitution (ii) elimination (iii) d-operator (iv) laplace transform (v) numerical recursive calculations (vi) software package ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 6 4. proposed algebraic solutions a special case of simultaneous differential equations is focused here, where only one dependent variable y, and one independent variable x, are involved. however, the solution must satisfy two non-homogeneous differential equations simultaneously. the main result of this paper is stated as a theorem below. theorem 1 the solutions to the set of non-homogeneous linear first-order simultaneous ordinary differential equations with constant coefficients in the form of 𝑎1𝑦′ + 𝑎0𝑦 = 𝑓(𝑥) (20a) 𝑏1𝑦′ + 𝑏0𝑦 = 𝑔(𝑥) (20b) are given by 𝑦(𝑥) = 𝑔(𝑥)𝑎1−𝑓(𝑥)𝑏1 𝑎1𝑏0−𝑎0𝑏1 (21a) 𝑦′(𝑥) = 𝑓(𝑥)𝑏0−𝑔(𝑥)𝑎0 𝑎1𝑏0−𝑎0𝑏1 (21b) with the necessary condition 𝑎1𝑔′(𝑥) − 𝑏1𝑓′(𝑥) = 𝑏0𝑓(𝑥) − 𝑎0𝑔(𝑥). (21c) proof write the given equations in matric form as [ 𝑎1 𝑎0 𝑏1 𝑏0 ] [ 𝑦′ 𝑦 ] = [ 𝑓(𝑥) 𝑔(𝑥) ] (22) define the following three determinants |𝐴| = | 𝑎1 𝑎0 𝑏1 𝑏0 | = 𝑎1𝑏0 − 𝑎0𝑏1 (23a) |𝐴1| = | 𝑓(𝑥) 𝑎0 𝑔(𝑥) 𝑏0 | = 𝑏0𝑓(𝑥) − 𝑎0𝑔(𝑥) (23b) |𝐴2| = | 𝑎1 𝑓(𝑥) 𝑏1 𝑔(𝑥) | = 𝑎1𝑔(𝑥) − 𝑏1𝑓(𝑥) (23c) using cramer’s rule, the solutions are 𝑦′(𝑥) = 𝑏0𝑓(𝑥)−𝑎0𝑔(𝑥) 𝑎1𝑏0−𝑎0𝑏1 and 𝑦(𝑥) = 𝑎1𝑔(𝑥)−𝑏1𝑓(𝑥) 𝑎1𝑏0−𝑎0𝑏1 (24) however, since 𝑦(𝑥) and 𝑦′(𝑥) are not constants and 𝑦′(𝑥) is obtained by differentiating 𝑦(𝑥) with respect to x, the following necessary condition must be satisfied: 𝑎1𝑔′(𝑥) − 𝑏1𝑓′(𝑥) = 𝑏0𝑓(𝑥) − 𝑎0𝑔(𝑥) ∎ (25) note that the solution 𝑦(𝑥) can be obtained using simple algebraic computation. now let’s look at the four common functions for 𝑓(𝑥) and 𝑔(𝑥). 4.1 case 1: 𝒇(𝒙) 𝐚𝐧𝐝 𝒈(𝒙) are constants. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 7 𝑓(𝑥) = 𝑓 and 𝑔(𝑥) = 𝑔, where f and g are scalar constants. the necessary condition becomes 0 = 𝑏0𝑓 − 𝑎0𝑔 (26) and the solutions are 𝑦′(𝑥) = 𝑏0𝑓−𝑎0𝑔 𝛼 = 0 (27a) 𝑦(𝑥) = 𝑎1𝑔−𝑏1𝑓 𝛼 (27b) where the constant 𝛼 is defined as 𝛼 = 𝑎1𝑏0 − 𝑎0𝑏1 (28) numerical example 1 2𝑦′ + 3𝑦 = 6 5𝑦′ − 𝑦 = −2 𝑎1 = 2, 𝑎0 = 3, 𝑏1 = 5, 𝑏0 = −1, 𝑓 = 6 and 𝑔 = −2. 𝑏0𝑓 − 𝑎0𝑔 = (−1)(6) − (3)(−2) = −6 + 6 = 0 𝛼 = (2)(−1) − (3)(5) = −17 so, 𝑦′(𝑥) = (−1)(6)−(3)(−2) −17 = 0 𝑦(𝑥) = (2)(−2)−(5)(6) −17 = −34 −17 = 2 (29) ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 8 check: eq 1: 2𝑦′ + 3𝑦 = 2(0) + 3(2) = 6 √ eq 2: 5𝑦′ − 𝑦 = 5(0) − (2) = −2 √ 4.2 case 2: 𝒇(𝒙) 𝐚𝐧𝐝 𝒈(𝒙) are linear functions of x. 𝑓(𝑥) = 𝑚1𝑥 + 𝑚0 and 𝑔(𝑥) = 𝑛1𝑥 + 𝑛0. the necessary condition becomes 𝑎1𝑛1 − 𝑏1𝑚1 = 𝑏0(𝑚1𝑥 + 𝑚0) − 𝑎0(𝑛1𝑥 + 𝑛0) = 𝑏0𝑚1𝑥 + 𝑏0𝑚0 − 𝑎0𝑛1𝑥 − 𝑎0𝑛0 = (𝑏0𝑚1 − 𝑎0𝑛1)𝑥 + (𝑏0𝑚0 − 𝑎0𝑛0) (30) equating coefficients of like powers yield the following two necessary conditions. 𝑏0𝑚1 − 𝑎0𝑛1 = 0 (31a) 𝑎1𝑛1 − 𝑏1𝑚1 = 𝑏0𝑚0 − 𝑎0𝑛0 (31b) the solutions are 𝑦(𝑥) = (𝑛1𝑥+𝑛0)𝑎1−(𝑚1𝑥+𝑚0)𝑏1 𝛼 = (𝑎1𝑛1−𝑏1𝑚1)𝑥+(𝑎1𝑛0−𝑏1𝑚0) 𝛼 (32a) 𝑦′(𝑥) = (𝑚1𝑥+𝑚0)𝑏0−(𝑛1𝑥+𝑛0)𝑎0 𝛼 = (𝑏0𝑚1−𝑎0𝑛1)𝑥+(𝑏0𝑚0−𝑎0𝑛0) 𝛼 (32b) numerical example 2 𝑦′ + 3𝑦 = 6𝑥 + 2 −3𝑦′ + 4𝑦 = 8𝑥 − 6 𝑎1 = 1, 𝑎0 = 3, 𝑏1 = −3, 𝑏0 = 4, 𝑚1 = 6, 𝑚0 = 2, 𝑛1 = 8 , 𝑛0 = −6 condition 1: 𝑏0𝑚1 − 𝑎0𝑛1 = (4)(6) − (3)(8) = 0 => satisfied condition 2: 𝑎1𝑛1 − 𝑏1𝑚1 = 𝑏0𝑚0 − 𝑎0𝑛0 (1)(8) − (−3)(6) = (4)(2) − (3)(−6) 8 + 18 = 8 + 18 => satisfied 𝛼 = (1)(4) − (3)(−3) = 13 𝑦(𝑥) = [(1)(8)−(−3)(6)]𝑥+[(1)(−6)−(−3)(2)] 13 = 2𝑥 𝑦′(𝑥) = [(4)(6)−(3)(8)]𝑥+[(4)(2)−(3)(−6)] 13 = 2 check: eq 1: 𝑦′ + 3𝑦 = 2 + 3(2𝑥) = 6𝑥 + 2 √ ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 9 eq 2: −3(2) + 4(2𝑥) = 8𝑥 − 6 √ 4.3 case 3: 𝒇(𝒙) 𝐚𝐧𝐝 𝒈(𝒙) are exponential functions 𝑓(𝑥) = 𝑝𝑒𝑘𝑥 and 𝑔(𝑥) = 𝑞𝑒𝑘𝑥 , 𝑝, 𝑞, 𝑘 = constant. (33) the necessary condition is 𝑎1𝑞𝑘𝑒𝑘𝑥 − 𝑏1𝑝𝑘𝑒𝑘𝑥 = 𝑏0𝑝𝑒𝑘𝑥 − 𝑎0𝑞𝑒𝑘𝑥 (34) which simplifies to 𝑘(𝑎1𝑞 − 𝑏1𝑝) = (𝑏0𝑝 − 𝑎0𝑞). (35) using the above theorem, the solution is given by 𝑦(𝑥) = 𝑎1𝑞𝑒𝑘𝑥−𝑏1𝑝𝑒𝑘 𝑎1𝑏0−𝑎0𝑏1 = 𝑎1𝑞−𝑏1𝑝 𝛼 ∙ 𝑒𝑘𝑥 (36a) 𝑦′(𝑥) = 𝑏0𝑝𝑒𝑘𝑥−𝑎0𝑞𝑒𝑘𝑥 𝑎1𝑏0−𝑎0𝑏1 = 𝑏0𝑝−𝑎0𝑞 𝛼 ∙ 𝑒𝑘𝑥 (36b) 𝛼 = 𝑎1𝑏0 − 𝑎0𝑏1 (36c) numerical example 3 2𝑦′ + 3𝑦 = 3𝑒3𝑥 𝑦′ − 2𝑦 = 1 3 𝑒3𝑥 𝑎1 = 2, 𝑎0 = 3, 𝑏1 = 1, 𝑏0 = −2, 𝑝 = 3, 𝑞 = 1 3 , 𝑘 = 3 𝑘(𝑎1𝑞 − 𝑏1𝑝) = 3 ( 2 3 − 3) = −7 ; 𝑏0𝑝 − 𝑎0𝑞 = −6 − 1 = −7 => condition satisfied 𝛼 = (2)(−2) − (3)(1) = −7 𝑦(𝑥) = (2)( 1 3 )−(1)(3) −7 ∙ 𝑒3𝑥 = 1 3 ∙ 𝑒3𝑥 𝑦′(𝑥) = (−2)(3)−(3)( 1 3 ) −7 ∙ 𝑒3𝑥 = 𝑒3𝑥 check: eq 1: 2𝑦′ + 3𝑦 = 2(𝑒3𝑥) + 3 ( 1 3 𝑒3𝑥) = 3𝑒3𝑥 √ eq 2: 𝑦′ − 2𝑦 = (𝑒3𝑥) − 2 ( 1 3 𝑒3𝑥) = 1 3 𝑒3𝑥 √ ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 10 these calculations are programmed into an excel spreadsheet as shown below figure 2. one just have to enter the corresponding constants and coefficients to obtain the correct answer instantly. the necessary condition is also checked and the solution is verified automatically. 4.4 case4: 𝒇(𝒙) 𝐚𝐧𝐝 𝒈(𝒙) are sine and cosine functions 𝑓(𝑥) = 𝑚 sin 𝑥 + 𝑛 cos 𝑥 and 𝑔(𝑥) = 𝑝 sin 𝑥 + 𝑞 cos 𝑥 (37) the necessary condition is 𝑎1(𝑝 cos 𝑥 − 𝑞 sin 𝑥) − 𝑏1(𝑚 cos 𝑥 − 𝑛 sin 𝑥) = 𝑏0(𝑚 sin 𝑥 + 𝑛 cos 𝑥) − 𝑎0(𝑝 sin 𝑥 + 𝑞 cos 𝑥) sin 𝑥 (−𝑎1𝑞 + 𝑏1𝑛 − 𝑏0𝑚 + 𝑎0𝑝) + cos 𝑥 (𝑎1𝑝 − 𝑏1𝑚 − 𝑏0𝑛 + 𝑎0𝑞) = 0 (38) which gives 𝑎1𝑞 − 𝑎0𝑝 = 𝑏1𝑛 − 𝑏0𝑚 (39a) 𝑎1𝑝 + 𝑎0𝑞 = 𝑏1𝑚 + 𝑏0𝑛 (39b) the solution is given by 𝑦(𝑥) = (𝑎1𝑝−𝑏1𝑚) sin 𝑥+(𝑎1𝑞−𝑏1𝑛) cos 𝑥 𝛼 (40a) 𝑦′(𝑥) = (𝑏0𝑚−𝑎0𝑝) sin 𝑥+(𝑏0𝑛−𝑎0𝑞) cos 𝑥 𝛼 (40b) figure 2: spread sheet for numerical example 3 ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 11 numerical example 4 116𝑦′ − 58𝑦 = −7 sin 𝑥 + 55 cos 𝑥 35𝑦′ + 15𝑦 = 11 sin 𝑥 + 12 cos 𝑥 𝑎1 = 116, 𝑎0 = −58, 𝑏1 = 35, 𝑏0 = 15, 𝑚 = −7, 𝑛 = 55, 𝑝 = 11, 𝑞 = 12 𝛼 = 𝑎1𝑏0 − 𝑎0𝑏1 = 3770 necessary conditions: (1): 𝑎1𝑞 − 𝑎0𝑝 = 𝑏1𝑛 − 𝑏0𝑚 (116)(12) − (−58)(11) = (35)(55) − (15)(−7) 2030 = 2030 => condition satisfied (2): 𝑎1𝑝 + 𝑎0𝑞 = 𝑏1𝑚 + 𝑏0𝑛 (116)(11) + (−58)(12) = (35)(−7) + (15)(55) 580 = 580 => condition satisfied 𝑦(𝑥) = [(116)(11)−(35)(−7)] sin 𝑥+[(116)(12)−(35)(55)] cos 𝑥 3770 = 0.403 sin 𝑥 − 0.141 cos 𝑥 𝑦′(𝑥) = [(15)(−7)−(−58)(11)] sin 𝑥+[(15)(55)−(−58)(12)] cos 𝑥 3770 = 0.141 sin 𝑥 + 0.403 cos 𝑥 check: eq 1: 116𝑦′ − 58𝑦 = (16.356 sin 𝑥 + 46.748 cos 𝑥) − (23.374 sin 𝑥 − 8.178 cos 𝑥) = −7.018 sin 𝑥 + 54.926 cos 𝑥 √ eq 2: 35𝑦′ + 15𝑦 = (4.935 sin 𝑥 + 14.105 cos 𝑥) + (6.045 sin 𝑥 − 2.115 cos 𝑥) = 10.980 sin 𝑥 + 11.990 cos 𝑥 √ 5. conclusion a set of two non-homogeneous linear first-order simultaneous ordinary differential equations with constant coefficients of a certain form are studied. four different types of the forcing functions, namely constants, linear functions, natural exponential functions and sinusoidal functions, have been investigated. for each type, an algebraic equation to determine the solution and its derivative as well as the required necessary conditions are derived. moreover, these simple algebraic formulae can be easily programmed into a spreadsheet which can automatically compute the solution, check the necessary condition and verify the answer. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 12 6. discussion the theorem and formulae derived are applicable only to one special form of nonhomogeneous linear first-order simultaneous ordinary differential equations with constant coefficients. this is indeed restrictive. moreover, this method only works for the stated four cases of the forcing function and a stringent necessary condition must be satisfied. however, most of the existing methods have their restrictions too and cannot be used to solve any form of simultaneous differential equations with different combinations of the independent and dependent variables and their derivatives. despite its limitations, the proposed theorem and the resulting formulae have the advantage that they are algebraic and can be easily programmed into a spreadsheet that can perform repeated computations with different constants and coefficients effortlessly. the spreadsheet can also be programmed to check the necessary condition and verify the solution. the investigation employing this technique to different forms of simultaneous ode and with different forcing functions would be an interesting endeavor. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 13 references 1. andrilli, s. & hecker, d. (2019). elementary linear algebra (5th edn). harcourt academic press, san diego. 2. kolman, b. & hill, d. (2007). elementary linear algebra (9th edn). macmillan publishing company, new york. 3. boyce, w.e. (2017). elementary differential equations and boundary value problems (11th edn). wiley. 4. edwards, h. & penney, d. (2019). elementary differential equations with boundary value problems (6th edn). pearson. 5. kohler, w.e. & johnson l.w. (2017). elementary differential equations with boundary value problems (2nd edn). pearson. 6. zill, d.g. (2018). a first course in differential equations with modeling applications (11th edn). cengage. 7. differential equations: https://en.wikipedia.org/wiki/differential_equation 8. simultaneous de: https://www.astro.uvic.ca/~tatum/integrals/integrals06.pdf 9. decoupling of systems of differential equations: decoupling | differential equations | mathematics | mit opencourseware 10. decoupling: ls4.pdf (mit.edu) 11. substitution method: https://web.uvic.ca/~kumara/econ251/schap24.pdf 12. elimination method: ch. 4.8 solving systems of d.e.s by elimination (youtube.com) 13. d-operator: microsoft word int6 sim eqs.doc (uvic.ca) 14. how to solve simultaneous differential equations using laplace transform: bing videos 15. recursive calculations: 2 solving sets of equations (求解聯立方程組) (uotechnology.edu.iq) 16. solving system of simultaneous differential equations using matlab: https://www.mathworks.com/help/symbolic/solve-a-system-of-differential-equations.html ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 07 | issue 07 | july 2024 | https://ijojournals.com/index.php/m/index 14 https://en.wikipedia.org/wiki/differential_equation https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/resources/decoupling/ https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/resources/decoupling/ https://math.mit.edu/~jorloff/suppnotes/suppnotes03/ls4.pdf https://web.uvic.ca/~kumara/econ251/schap24.pdf https://www.youtube.com/watch?v=nwfh7_j0fbw https://www.astro.uvic.ca/~tatum/integrals/integrals06.pdf https://www.bing.com/videos/riverview/relatedvideo?q=methods+to+solve+simultaneous+differential+equations&mid=4dc227894ae2630800e64dc227894ae2630800e6&form=vire https://uotechnology.edu.iq/dep-chem-eng/lecture%202014-2015/3y/numerical/lecture%209%20solving%20simulataneous%20ordinary%20differential%20%20%20equations.pdf https://www.mathworks.com/help/symbolic/solve-a-system-of-differential-equations.html ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" numerical computation of reactive flow of third grade fluid with heat generation *1 boniface inalu obi,2 edwin esekhaigbe, 3uchenna awucha uka 1department of mathematics, imo state university, owerri, nigeria 2mathematics and computer science department, university of africa, bayelsa state, nigeria 3basic science department, school of science and technology, babcock university, ogun state, nigeria corresponding author *1 boniface inalu obi, abstract computation of reactive flow of third grade fluid in cylindrical pipe with heat generation is considered. the resulting governing equations of motion are highly nonlinear and are solved using collocation method in verifying the impacts of some material variables involved. results indicate that increase in the non-newtonian variable increases the flow velocity and decreases the temperature of the walls of the cylindrical pipe. it is observed that the critical frankkamenetskii variable exist for which the solution fails to be distinct. it is further observed that for c  , a steady state solution does not exist suggesting a thermal runaway which could be avoided by setting 1.8879565  . keywords: numerical, third grade, heat generation, reactive, computation. 1. introduction third grade fluid is in the class of nonnewtonian fluids of the differential type. it is a class of nonnewtonian fluid where the stress tensor is the addition of all the tensors that can be developed from the velocity field with up to three derivatives. example of this class of fluid include ketchup, paints, blood etc. chemical reactions involved in fluid which is capable of altering the fluids properties and composition in flow process is referred to as reactive flow. there are some studies now available in literature. some of the earliest work on third grade fluid are fosdick and rajagopal [3] analyzed the thermodynamic third grade fluid and showed the restrictions on the stress constitutive model. rajagopal [10] examined the stability properties of third grade fluids. szeri and rajagopal [13] investigated the flow of third grade fluids between heated parallel plates. ellahi et al [1] examined the impacts of slip on the nonlinear flows of a third grade fluid. in the investigation, the results of no-slip condition were inferred as a restricting case when the slip variable is equal to zero. https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 66 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" makinde [4] investigated a steady flow of a reactive variable viscosity fluid in a cylindrical pipe with an isothermal wall. makinde [5] studied the thermal criticality for reactive gravity-driven thin film flow of a third grade fluid with adiabatic free surface down an inclined plane. salawuland fatumbi [11] examination the inherent irreversibility of hydromagnetic third grade reactive poiseuille flow with variable viscosity. the employed the weighted residual method for the solution. the study shows that the heat dissipation of reactive exothermic chemical in a uniform magnetic field moved past fluid in a porous medium in an irreversible mode. the study on steady flow of a reactive viscous fluid in porous cylindrical pipe was carried out byfarayola [2]. the investigation was done using regular perturbation for the solution of the nonlinear equation. okedayo et al [9] numerically investigated the reactive mhd flow of thrd grade fluid in pipe.the study involved the weighted residual collocation method which was use as a computational means of solution and were able to display the influence of various thermophysical parameter. it was observed that the critical value of the frank-kamenetskii and third grade parameters exists for which the solution sizes to be unique. obi et al [7] analyzed the incompressible flow of third grade fluid in an inclined rotating cylindrical pipe with isothermal wall and joule heating. they solved the nonlinear equations by perturbation method and the effect of some parameter on the flow presented graphically. yurusoyand pakdemirli [14] investigated the fluid flow of third grade fluid in pipe with heat transfer with a case of constant viscosity. reynold’sand vogel’s models were introduced to account for temperature-dependent viscosity. they analytically examined the flow and compared the result with the finite difference procedure earlier given by massaudi and christie [6]. okedayo et al [8] focuses on gaterkinweighted residual method for magnetohydrodynamic mixed convection flow in a vertical channel filled with porous media. results obtained were analyzed using tables and graphs. siddiqui et al [12] analyzed the thin film flow of third grade fluid down an inclined plane using the combined traditional perturbation as well as the homotopy perturbation methods and comparison made of the two results obtained from the two techniques. this research seeks to examine the consequences of the frank-kamenetskii parameter on the reactive flow of third grade fluid in cylindrical pipe. 2. formulation of the problem https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 67 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" the fundamental models of an incompressible viscous flow are the mass and momentum conservation laws which in the absence of heat transfer process, such laws are defined through continuity and momentum equations. in vector form,  . 0 1 f+div u du u p dt k            2 where =density, u =velocity, p =pressure, =stress tensor, f = body force and =material derivative d dt   stress tensor defining a third grade fluid is given by     3 1 1 1 2 2 1 1 2 1 2 2 1 2 2 1 1 3 where tr i i a a a a a a                     4  is the coefficient of dynamic viscosity, 1 1 1 1, , ,    are constants. the rivlin-ericksen tensors na are defined by 0 1a  , being the identity tensor.      1 1 1, 1 5 tn n n n da a a u u a n dt         assuming the surface tension to be negligible, we have the velocity field in t he     , 0,0 6u u r https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 68 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" substituting the values of . and v in equation (2), yields  2 0 1 7 d du dp r u b u r dr dr k dz              2 2 0 0 1 exp 0 8 d dt du e r b u qc a r dr dr k dr rt                                  00 0 0, 0, 9 du dt u a t a t dr dr     where u is fluid velocity, t is absolute temperature,  is dynamic viscosity,  ius electricalconductivity, 0t is reference temperature, a is radius of the pipe, r is radial distance, e is the activation energy, r is the universal gas constant, 0c is concentration, q is heat activationand a is the rate constant. introducing the following dimensionless parameters:     02 0 0 0 0 2 0 0 , , , , 10 e rtrt ea qc ae r ar u u u t t rt e rt k           using eqn (10) in eqns (7-9), yields   1 1 11 d du r u mu r dr dr            1 2 21 0 12r d d du r b mu e r dr dr dr                              0 0 0, 1 0, 0 0 13 du d u dr dr      3. method of solution https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 69 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" in order to solve the nonlinear momentum and energy equation of (11) and (12) with the condition (13), we employ collocation method for the solution. this technique is a function approximation method which reduces the nonlinear ordinary differential equations to algebraic equations which can be solved byany iterative fixed point technique. the approximate solution is of the form       0 14 n j j j u r a r   equation (14) is the trial function over the region which must satisfy the given boundary conditions. in this technique, the one, two and three term coefficients are employed and the maximum velocity and temperature ascertained.      3 3 2 3 0 0 1 0 1(1 ), (1 ) ( ) 15u r a r u r a r a r r      similarly,      3 3 2 3 0 2 1 2 3(1 ), (1 ) ( ) 16r a r r a r a r r         max max max 1 0.000875 0.875 4.500000000 0.05321635176 17 e          0 0.2222222222 18a  the thermalcritical property is determined bythe relationship between maximum temperature and the frank-kamenetskii variable and achieved from equations (17)and (18) with other thermo-solutal parameters. plotting equation (17) results in figure 5. https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 70 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 71 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" 4. results and discussions in this section, the impacts of some variables that are of great importance to the study are discussed. these effects come from the graphs presented in earlier section. in figures 1 and 2, the influence of non-newtonian and magnetic field variables are respectively shown. it is seen from the results that increase in both parameters enhance the flow velocity. figure 3 shows the temperature profiles for values of the non-newtonian variable. result shows that increase in the parameter , decreases the temperature of the cylindrical pipe at the walls. figure 4 is the temperature profiles for various values of the magnetic field parameter. results indicate that increase in the magnetic field parameter increases the temperature of the system. this is because magnetic field can generate heat through electrical resistance, thereby leads to a rise in temperature. figure 5 shows the critical value of the frank-kamenetskii parameter 1.8879565c  . 5. conclusion computation of reactive flow of third grade fluid in cylindrical pipe with heat generation is considered. the resulting governing equations of motion are highly nonlinear and are solved using collocation method in verifying the impacts of some material variables involved. results indicate that increase in the non-newtonian variable increases the flow velocity and decreases the temperature of walls of the cylindrical pipe. it is observed that the critical https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 72 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" frank-kamenetskii variable exist for which the solution fails to be distinct. it is further observed that for c  , a steady state solution does not exist suggesting a thermal runaway which could be avoided by setting 1.8879565  . declarations 1. funding: not applicable 2. informed consent statement: not applicable 3. data availability: not applicable 4. conflict of interest statement: no conflict of interest 6. references [1] ellahi r., hayat t., mahomed f.m. and asghar s. effects of slip on the non-linear flow of a third grade fluid. nonlinear analysis: real world applications 11(2010),139-146. [2] farayola p.i. on steady flow of a reactive viscous fluid in a porous cylindrical pipe. open journal of fluid dynamics 7(2017),359-370 [3] fosdick r.l. and rajagopal, k.r. thermodynamics and stability of fluids ofthird grade. proc. r. soc. lond. 339(1980), 351-377. [4] makinde o.d. on steady flow of a reactive variable viscosity fluid in a cylindrical pipe with an isothermal wall. international journal of numerical methods for heat and fluid flow, 17(2007).187-194. [5] makinde o.d. thermal criticality for a reactive gravity driven thin film flow of a third grade fluid with adiabatic free surface down an inclined plane.applied mathematics and mechanics, 30(2009), 373-380. [6] massoudi, m. and christie, i. effects of variable viscosity and viscous dissipation on theflow of a third –grade fluid in a pipe. int. j. of nonlinear mech., 30(5)(1995) 687-699. [7] obi b.i.,okedayo, g.t., jiya, m. and aiyesimi, y.m. analysis of flow of an incompressiblemhd third grade fluid in an inclined rotating cylindrical pipe with isothermal wall and joule heating. international journal for research in mathematics and statistics. (2021); 7 (6). [8] okedayo g.t., amumeji o.t. and obi b.i. galerkin weighted residual method for magnetohydrodynamic (mhd) mixed convectionflow in a vertical channel filled with porous media. international journal for research in mathematics and statistics, 4(5), (2018) [9] okedayo, g.t., obi, b.i. &olawuyi, o.m. a numerical study of reactive mhd flow of thirdgrade fluid. journal of mathematical science and computational mathematics (jmscm), 1(1), (2019). https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 73 ijo international journal of mathematics (issn: 2992-4421 ) *1 boniface inalu obi* https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “numerical computation of reactive flow of third grade fluid with heat generation" [10]rajagopal k.r. on the stability of third grade fluids. arch. mech. 32(1980) 867-875. [11] salawul s.o. and fatunmbi e.o. inherent irreversibility of hydromagnetic third grade reactivepoiseuille flow of a variable viscosity in porous media with convective cooling. journal of the serbian society for computational mechanics 11(1)(2017), 46-58. [12] siddiqui a.m., mahmood r, and ghori q.k. homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane. chaos, soliton and fractals 35(2008) ,140-147. [13] szeri a.z. rajagopal k.r. flow of a non-newtonian fluid between heated parallel plates. int. j. non-linear mech. 20(1985), 91-101. [14] yurusoy, m. and pakdemirli, m.: approximate analytical solutions for the flow of a third grade fluid in a pipe. international journal of non-linear mech. 37(2002),187-195. https://doi.org/10.5281/zenodo.17510312 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 74 on stability in separative semigroup abstract stability, as introduced by koch and wallace (1956), has long stood as a central notion in semigroup theory, ensuring that inclusions of principal ideals collapse into equalities and that left and right structures align harmoniously. separativity, on the other hand, generalises cancellativity while retaining algebraic regularity, and burmistrovich's decomposition theorem revealed that every separative semigroup can be expressed as a semilattice of cancellative semigroups. yet, whether separative semigroups inherit stability in the sense of koch and wallace has remained unresolved. in this work, we close this gap: we prove that semilattices of cancellative semigroups are stable, and hence every separative semigroup is inherently stable. this result elevates stability from a supplementary condition to a built-in feature of separative semigroups, o�ering a uni�ed perspective that strengthens the foundations of semigroup theory and deepens its structural coherence. keywords: stability in semigroup; separative semigroup; cancellative semigroup; semillatice decomposition; and green's relations. otobong j. tom∗ fed. uni. tech., ikot abasi & akwa ibom state uni., ikot akpaden otobong g. udoaka department of mathematics, akwa ibom state university, ikot akpaden ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 1 ekere s. udo�a department of mathematics, akwa ibom state university, ikot akpaden mailto: tomdgreatest@gmail.com otobongawasi@aksu.edu.ng ekereudofia@yahoo.com 1 introduction the concept of stability in semigroups was formally introduced by koch and wallace [16] in their seminal paper. a semigroup is said to be stable if for all a, b ∈ s: as ⊆ abs =⇒ as = abs, sa ⊆ sab =⇒ sa = sab. this algebraic de�nition re�ects a form of �ideal stability� and has since become a cornerstone in the theory of semigroups, inspiring further structural investigations [1, 12]. stability is important because it guarantees that the behaviour of principal ideals remains robust under multiplication. the study of stability has evolved, particularly through its connection with green's relations, which classify semigroup elements based on their divisibility properties [16]. a crucial result from koch and wallace states that in a stable semigroup, green's d− and j− equivalences coincide, implying that left and right ideal structures behave symmetrically. anderson, et al [11] further expanded on this idea by studying stability conditions in various semigroup classes, highlighting its role in determining structural simplicity and regularity. the structure of quasi-separative semigroup was introduced by drazin [12], where the connections between it and other semigroup properties, such as inversity, regularity, etc, were established. this was later extended by krasilnikova and novikove [22]. parallel to this development, the notion of separativity was introduced to generalise cancellativity while preserving a degree of regularity. east and higgings [9] explored green's relation in greater depth, providing a more re�ned classi�cation of semigroup elements' stability constraints. hewitt and zuckerman [6] applied separativity properties in their study, titled �l1− algebra of a commutative semigroup�, where they introduce harmonic analy sis on discrete commutative semigroups. shourijeh [2] established the commutativity of separative semigroup and discussed its left presentation. burmistrovich [7] provided a powerful structural result: a semigroup is separative if and only if it is isomorphic to a semilattice of cancellative semigroups. this theorem places separative semigroups in a broader algebraic context and has been extensively used in decomposition theory [13]. however, burmistrovich's theorem itself does not discuss stability, leaving a gap in the understanding of whether separative semigroups inherit stability (in the sense of [16]). this gap motivates the present study. we aim to establish that separative semigroups are indeed stable in the sense of [19]. our approach hinges on re-examining burmistrovich's decomposition theorem from the perspective of stability. we �rst establish that a semilattice of cancellative semigroups is stable�a fact that has not been explicitly recorded in the literature. since separative semigroups are precisely those semigroups isomorphic to such semilattices, it follows directly that every separative semigroup is stable. 2 preliminaries de�nition 2.1 (groupoid). let s be a non-empty set. let ∗ be an operation such that ∗ : s×s → s be de�ned on s. then (s, ∗) is called groupoid if for all a, b ∈ s, a∗ b ∈ s. de�nition 2.2 (semigroup). a groupoid is called semigroup if the binary operation is associative (i.e., for all a, b, c ∈ s, we have (a ∗ b) ∗ c = a ∗ (b ∗ c)). de�nition 2.3 (idempotent). an element e ∈ s is called an idempotent element if e2 = e. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 2 de�nition 2.4 (cancellative semigroup). a semigroup is called cancellative for all a, b, c ∈ s, ab = ac =⇒ a = b(left-cancellative) and ba = ca (right-cancellative). for more about this, the reader is referred to [1,4,5, 7]. 3 separative semigroup de�nition 3.1. a semigroup s is called separative if and only if the following two conditions hold for all a, b ∈ s: � a2 = ab and ba = b2 together imply a = b, and � a2 = ba and ab = b2 together imply a = b. the reader can also see [15] de�nition 3.2 (qusi-separative). a semigroup is called quasi�separative if a2 = ab = ba = b2 =⇒ a = b, [22]. on a bright-line, one can say that a separative semigroup is a quasi-separative. extracting from [20] we de�ne a weakly separative semigroup when we have asa = asb = bsa = bsb only if a = b for all a, b, s ∈ s. theorem 3.1 (proposition 1 of [12]). if s is any quasi-separative semigroup, then, for all a, b ∈ s we have a2 = ab = b2 if and only if a = b. and the converse also holds. proof. if a2 = ab = b2, then we have (ab)2 = a2b2 = (aa)(bb) = a2b2, and (ab)(ba) = (aa)a(a) = a4, (ba)(ab) = (bb)b(b) = b4. also, (ba)2 = b(ab)a = (bb)(aa) = b2a2, and a(ab) = a(aa)a = a3, b(ba) = b(bb)b = b3. therefore, (ab)(ab) = (ba)(ba) = (ab)(ba), thus, by quasi-separativity, we have ab = ba. so we obtain a2 = ab = ba = b2, and by quasi-separativity again we have a = b. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 3 4 semilattice of a semigroup de�nition 4.1 (semilattice). a semilattice is a commutative idempotent semigroup. that is, a set y with a binary operation ∧ : y × y → y such that, for all e, f, g ∈ y : 1. (e ∧ f) ∧ g = e ∧ (f ∧ g) (associativity), 2. e ∧ f = f ∧ e (commutativity), 3. e ∧ e = e (idempotency). interpretation: � the operation ∧ can be thought of as a �meet� (greatest lower bound) in an ordered set. � the partial order associated with a semilattice is given by e ≤ f ⇐⇒ e ∧ f = e. � thus, a semilattice is both an algebraic structure (a special semigroup) and an order-theoretic one (a meet-semilattice) in a poset. de�nition 4.2 (semilattice of semigroups). let y be a semilattice with operation ∧. a semilattice of semigroups is a semigroup s together with a decomposition s = ⋃ e∈y se, where each se is a subsemigroup of s, such that for all e, f ∈ y : se · sf ⊆ se∧f . that is, the product of an element from se and an element from sf always lies in the same component corresponding to the meet e ∧ f , see �gure 1 below. se sf se∧f a ∈ se b ∈ sf ab ∈ se∧f figure 1: product in a semilattice of semigroups: a ∈ se, b ∈ sf multiply into se∧f . remarks: ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 4 � each component se is itself a semigroup. � the semilattice y controls how the components interact with each other: multiplication �descends� to the meet in y . � if all components se are cancellative semigroups, then s is called a semilattice of cancellative semigroups. de�nition 4.3 (green's relations). let s be a semigroup. green's relations are the equivalence relations l,r,j ,h,d de�ned on s as follows: � the l-relation: for a, b ∈ s, al b ⇐⇒ s1a = s1b, that is, a and b generate the same principal left ideal. here s1 denotes s with identity adjoined if necessary. � the r-relation: for a, b ∈ s, a r b ⇐⇒ as1 = bs1, that is, a and b generate the same principal right ideal. � the j -relation: for a, b ∈ s, a j b ⇐⇒ s1as1 = s1bs1, that is, a and b generate the same two-sided ideal. � the h-relation: h = l ∩r, that is, a h b if and only if al b and ar b. � the d-relation: d = l ◦ r = r ◦ l, that is, a d b if and only if there exists c ∈ s such that a l c and c r b. de�nition 4.4 (associated preorders). besides the equivalence relations, one often considers the following preorders: � the ≤l-relation: for a, b ∈ s, a ≤l b ⇐⇒ s1a ⊆ s1b. � the ≤r-relation: for a, b ∈ s, a ≤r b ⇐⇒ as1 ⊆ bs1. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 5 � the ≤j -relation: for a, b ∈ s, a ≤j b ⇐⇒ s1as1 ⊆ s1bs1. for more clearity, reader should see [3, 8, 10,17,19�21] theorem 4.1. let s be a semigroup. then d = j if and only if for every a, b ∈ s, sas = sbs ⇒ ∃x ∈ s such that alxr b. proof. (⇒): suppose d = j . let a, b ∈ s with sas = sbs, i.e. aj b. since d = j , this implies ad b. by the de�nition of d, there exists x ∈ s such that alx and xr b. (⇐): suppose the stated condition holds. take any a, b ∈ s with aj b, i.e. sas = sbs. by hypothesis, there exists x ∈ s with alxr b. hence ad b. since a, b ∈ s were arbitrary, this shows j ⊆ d. but in any semigroup it is always true that d ⊆ j (see [1, 18,19]). therefore, d = j . 5 results de�nition 5.1 (koch�wallace stability [16]). a semigroup s is called stable if for all a, b ∈ s: 1. (right stability): as ⊆ abs =⇒ as = abs, and 2. (left stability): sa ⊆ sab =⇒ sa = sab. this de�nition was also given by east using green's relation in [9]. equivalently, if a ≤j ab, then arab, and if a ≤j ba, then alba. it is now necessary to establish some useful relationship between stable and separative semigroups. this is achieved through the following propositions and theorems. proposition 5.1. let s = ⋃ e∈y se be a semilattice (with meet ∧) of cancellative semigroups se, so that sesf ⊆ s e∧f for all e, f ∈ y. then s is separative. proof. let a, b ∈ s, suppose a2 = ab and ba = b2. let a ∈ se, b ∈ sf . since a 2 ∈ se but ab ∈ se∧f , the equality a 2 = ab forces e = e∧ f and so e ≤ f . likewise, b2 ∈ sf and ba ∈ se∧f then b2 = ba forces f ≤ e. hence, e = f , and so a, b lie in the same cancellative component se. from a2 = ab, we have aa = ab =⇒ a = b. hus, the �rst separative implication holds. the second (symmetric) implication is identical in structure. therefore, s satis�es the separativity conditions and so is separative. theorem 5.1. let s = ⋃ e∈y se be a semilattice (index set y with meet ∧) of cancellative semigroups se. if a ∈ se and b ∈ sf then ab ∈ se∧f . then s is stable, i.e. for all a, b ∈ s: ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 6 1. if sa ⊆ s(ab) then sa = s(ab). 2. if as ⊆ (ab)s then as = (ab)s. proof. write principal left ideals by sa = {xa : x ∈ s} and right ideals by as = {ax : x ∈ s}. let a ∈ se and b ∈ sf . (1) assume sa ⊆ s(ab). in particular a ∈ s(ab), so there exists t ∈ s with a = t(ab). writing t ∈ sg, the product t(ab) ∈ sg∧e∧f , while a ∈ se. equality of components forces e = g ∧ e ∧ f , hence e ≤ f and so e ∧ f = e. thus ab ∈ se. for any h ∈ y , the map φh : sh∧ea→ sh∧e(ab) de�ned by φh(xa) = (xa)b is bijective by cancellativity. taking unions over h yields sa = s(ab). (2) symmetrically, assume as ⊆ (ab)s. then a = (ab)u for some u ∈ s. this forces e ≤ f and ab ∈ se. for each h, the map ψh : ase∧h → (ab)se∧h, ψh(ax) = (ab)x, is bijective. taking unions gives as = (ab)s. hence s is stable. remark 5.1. the proof uses (i) semilattice decomposition of separative semigroups, and (ii) cancellativity in each component, ensuring injectivity of the maps xa 7→ (xa)b and ax 7→ (ab)x. these yield stability (in the sense of [16]). theorem 5.2. (burmistrovich's theorem [10]). a semigroup s is separative if and only if it is isomorphic to a semilattice of cancellative semigroups. proof. see [7] 6 conclusion in this work, we have established that every separative semigroup is stable. the argument relies on two fundamental ingredients: �rst, the structural theorem of burmistrovich (theorem 5.2), which asserts that every separative semigroup is isomorphic to a semilattice of cancellative semigroups; second, the fact that semilattices of cancellative semigroups preserve the stability conditions (theorem 5.1). by combining these facts along with proposition 5.1, we conclude that the separative property not only encodes a re�ned form of cancellativity but also guarantees algebraic stability in the sense of koch and wallace. this result strengthens the conceptual link between structural decomposition and stability theory in semigroup theory. it shows that separativity, originally introduced to capture subtle cancellation phenomena, naturally enforces stability (in the sense of koch and wallace) through its semilattice decomposition. consequently, the class of separative semigroups provides a robust and stable framework for further exploration in the algebraic theory of semigroups. declaration of interest: 1. funding: not applicable 2. informed consent statement: not applicable 3. data availability: not applicable 4. con�ict of interest statement: no con�ict of interest. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 7 references [1] a. h. cli�ord and g. b. preston, the algebraic theory of semigroups, american mathematical society (1961). [2] b. t. shourijeh, c∗−algebras of some semigroups, honam mathematical j. vol. 26, no. 4. pp. 483�507, 2004. [3] c. hollings, mathematics across the iron curtain: a history of soviet mathematics, american mathematical society, 2014. [4] d. d. miller and cli�ord, regular classes in semigroups, tran. amer. math. soc. vol. 82, 1956. [5] d. rees, on semi-groups, mathematical proceedings of the cambridge philosophical society, vol. 36, no. 4, pp. 387�400 1940. [6] e. hewitt, h. s. zuckerman, the l1˘algebra of commutative semigroup, trans. amer. math. soc. , 83, pp. 70-97, 1956. [7] i. e. burmistrovitch, commutative bands of cancellative semigroups, siberian mat. z. vol. 6, pp. 284�299, 1965. [8] j.a. green, on the structure of semigroups, ann. of math., pp. 163-�172, 1951. [9] j. east and p. m. higgins, green's relations and stability for subsemigroups, semigroup forum, vol. 101, no. 1, pp. 77�86, 2020. [10] o. g. udoaka, generators and inner automorphism, the colloquium-a multidisciplinary thematc policy journal vol. 10, no. 1, pp. 102�111, 2022. [11] l. w. anderson, r. p. hunter and r. j. koch, some results on stability in semigroups. ams journal, pp. 521�529, 1965. [12] m. p. drazin, a partial order in completely regular semigroups, journal of algebra, vol. 98, pp. 362�374, 1986. [13] m. petrich, introduction to semigroups, charles babbage research centre, 1984. [14] n. kumar and b. kumar, some fundamental properties of semigroups and their classi�cations, internal journal of mathematics trends and technology, vol. 70 issue 9, pp. 8�11, 2024. [15] r. thakur, a short note on completely regular semigroup, international journal of creative research thoughts , vol. 11, issue 5, 2023. [16] r. j. koch and a. d. wallace, stability in semigroups, duke math. j., vol. 24, pp. 193�196, 1957. [17] r. u. ndubisi, and o. g. udoaka, on left restriction semigroups, international journal of algebra and statistics, vol. 5no. 1, pp. 59�66, 2016. doi: 10.20454/ijas. 1083. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 8 [18] o. g. udoaka, rank of some semigroup, int. journal of applied science and mathematical theory, vol. 9, no. 3, pp. 90�100, 2023. www.iiardjournals.org. d.o.i: 10.56201/ijasmt.v9.no3.2023.pg90.100 [19] o. g. udoaka, o. tom, and a. musa, on idempotent elements in quasi-idempotent generated semigroup, international journal for research trend and innovation, vol. 8, no. 11, pp. 2456�3315, 2023. [20] w. d. burgess and r. raphael, on conrad's partial order relation on semiprime rings and on semigroups, semigroup forum vol. 16, pp. 133�140, 1978. [21] x. mary, on the structure of semigroups whose regular elements are completely regular. hal-04223812. 2023. [22] on quasi-separative semigroup.y. i. krasilnikova and b. v. novikove, arxiv:math/0311414v1 [math.gr], 2003. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 9 introduction preliminaries separative semigroup semilattice of a semigroup results conclusion ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel prakasha.p1, prof. k. shivashankara2, venuprasad k. k.3, dhananjaiah d. s4. 1department of mathematics, government first grade college, magadi, ramanagar, india e-mail id: profprakasha@gmail.com 2department of mathematics, yuvaraja’s college, university of mysore, mysore, india e-mail id: drksshankara@gmail.com 3department of mathematics, government first grade college k.r.pete, mandya, india e-mail id: kkvpmaths@gmail.com 4department of mathematics, government first grade college k.r.nagar, mysuru, india e-mail id: dhanu2614@gmail.com abstract: we analysed the unsteady mhd free connective flow through a porous medium in a vertical channel with the unsteadiness in the flow is due to the travelling thermal wave imposed on the wall y = l. the coupled mhd equations governing the flow and heat transfer have been solved by using a perturbation technique with the aspect ratio as perturbation parameter. the expression for the velocity, the temperature, the shear stress and the rate of heat transfer are derived and are analysed for different variations of the governing parameters g,r, and . key words: convection, porous medium, magneticfield and dissipation. 1. introduction: the energy crisis has been a topic of great importance in recent years all over the world .this has resulted in an unabated exploration for new ideas and avenues in harnessing various conventional energy sources like tidal waves, wind power and geothermal energy. it is well known that in order to harness maximal geothermal energy one should have complete and precise knowledge of quanta of perturbation needed to initiate convection currents in mineral fluids embedded in the earth’s crest enables one to use mineral energy to extract the minerals .convection fluid flows generated by travelling thermal waves have also received attention due to applications in physical problems. the linearised analysis of these flows has shown that a travelling thermal wave cal generate a mean shear flow within a layer of fluid, and the induced mean flow is proportional to the square of the amplitude of the wave. from a physical point of view, the motion induced by travelling thermal waves is quite interesting as a purely fluiddynamical problem and can be used as a possible explanation for the observed four-day retrograde zonal motion of the upper atmosphere of venus. all the above mentioned studies are based on the hypothesis that the effect of dissipation is neglected. this is possible in case of ordinary fluid flow like air and water under gravitational force .but this effect is excepted to be relevant for fluids with high values of the dynamic viscosity flows. in view if this, several authors notably barletta [1,2], bulent yesilata [3] , elhakein [4], israel et al[5] and rossidischio [6] have studied the effect of viscous dissipation on the convective flows past an infinite vertical plate and through vertical channels and ducts. in recent years, a great deal of interest has been generated in the area of boundary layer flow and heat transfer of a fluid over a stretching sheet in view of its numerous and wide range of applications in various fields such as polymer processing industry in particular manufacturing ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 1 mailto:profprakasha@gmail.com mailto:drksshankara@gmail.com mailto:kkvpmaths@gmail.com mailto:dhanu2614@gmail.com ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel process of artificial films, artificial fibers, and dilute polymer solutions. to be more specific, it may be pointed out that many metallurgical processes involve the cooling of continuous strips or filaments by drawing them through a quiescent fluid, and in the process of drawing, these strips are sometimes stretched. the heat transfer analysis over a stretching surface is of much practical interest due to its abundant applications such as heat-treated materials travelling between a feed roll and wind-up roll or materials manufactured by extrusion, glass-fiber, and paper production, cooling of metallic sheets or electronic chips, drawing of plastic films, liquid films in condensation processes. due to the high applicability of this problem in the industrial phenomena, it has attracted attention of many researchers. the effects of the buoyancy force on the development of the velocity and thermal boundary layer flows over a stretching sheet were first studied by chen[7]. elbashbeshy and bazid[8], sharidan et al.[9], and tsai et al.[10] obtained a similarity solution for the flow and heat transfer of a fluid over an unsteady stretching surface. the problem of mixedconvection adjacent to a vertical continuously stretching sheet in the presence of a variable magnetic field was studied by ishak et al.[11]. aziz[12] obtained the numerical solution for the laminar thermal boundary over a flat plate with a convective surface boundary conditions. the effect of viscous dissipation changes the temperature distributions by playing a role as an energy source, which affects the heat transfer rates. the merit of the effect of viscous dissipation depends on whether the plate is being cooled or heated. chen[13] examined the effect of combined heat and mass transfer on magnetohydrodynamic (mhd) free convection from a vertical surface with the ohmic heating and viscous dissipation. pal and hiremath[14] determined the heat transfer characteristics in the laminar boundary layer flow over an unsteady stretching sheet placed in a porous medium in the presence of viscous dissipation and internal absorption or generation. veena et al.[15] obtained the solutions of heat transfer in a visco-elastic fluid past a stretching sheet with viscous dissipation and internal heat generation.in light of the above investigations, it is found that these studies are restricted to the fluid flow and heat transfer problems. however, the fluid flow embedded with dust particles is encountered in different engineering problems concerned with nuclear reactor cooling, powder technology, rain erosion, paint spraying, etc. the important applications of dust particles in the boundary layer include soil erosion by natural winds and dust entrainment in a cloud during a nuclear explosion. it also occurs in awide range of technical processes like fluidization, flow in rocket tubes, combustion, and purification of crude oil. palani and ganesan[16] investigated the flow of dusty gas past a semi-infinite isothermal inclined plate. 2. mathematical formulation : we consider the motion of viscous, incompressible fluid through a porous medium in a vertical channel bounded by flat walls . the thermal buoyancy in the flow field is created by a travelling thermal wave imposed on the boundary wall at y = l while the boundary at y = -l is maintained at constant temperature t1. the viscous and darcy dissipations are taken into account to the transport of heat by conduction and convection in the energy equation. also the kinematic viscosity ,the thermal conducting k are treated as constants. we choose a rectangular cartesian system 0 ( x ,y ) with x-axis in the vertical direction and y-axis normal to the walls. the equations governing the unsteady flow and heat transfer under boundary conditions in terms of stream function  are ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel      2 2 2 0 2 0 2 0 4222 )(])()()[(                   ky h ttg e yxyyxt (2.1)                                           222 0 2 2 2 2 2 2 2 2 )()( )()()( yx h k xy q yxxyt c e pe        (2.2) the flow is maintained by a constant volume flux for which a characteristic velocity is defined as   l l ydu l q 2 1 . (2.3) the boundary conditions for the velocity and temperature fields are u = 0 , v = 0 ,t=t1 on y = -l )(,0,0 2 ntmxsintttvu e  on y = l (2.4) where u = - y , v =  x (2.5) introducing the non-dimensional variables in (2 .10 )(2.12) as e e t tt mttlyymxx     ,/,,/, 12 (2.6) (under the equilibrium state ))()( 2  ql ltltt eee  the governing equations (2.1) & (2.2) in the non-dimensional form ( after dropping the dashes ) are 2 2 22 1 14 1 2 12 1 ),( ),( )( y md r g yx r yt                       (2.7) and the energy equation in the non-dimensional form is                                                   ))()( )()( 22221 2 2 2 22 2 22 2 1 yx md xyg epr yxxyt p c         (2.8) where  ul r  (reynolds number), 2 3   ltg g e  (grashof number) 1k c p  (prandtl number), k l d 2 1  (darcy parameter), p c c gl e 3  (eckert number), ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel lm ( aspect ratio) 2m n    (non-dimensional thermal wave velocity), )( 2 222 2   lh m oe )( numberhartmann 2 2 2 2 22 1 yx        the corresponding boundary conditions are 1)1()1(   10,0       yat yx  (2.9) 1),( yx on y = -1 )(),( txsinyx   on y = 1 00    yat y  (2.10) the value of  on the boundary assumes the constant volumetric flow in consistent with the hypothesis(2.9) .also the wall temperature varies in the axial direction in accordance with the prescribed arbitrary function t . 3.analysis of the flow: the perturbation analysis is carried out by assuming that the aspect ratio  to be small. we adopt the perturbation scheme and write  ),(),(),(),( 2 2 10 yxyxyxyx  …………….  ),(),(),(),( 2 2 10 yxyxyxyx  ………………… (3.1) on substituting ( 3.1) in (2.13) (2.15) and separating the like powers of  the equations and respective conditions to the zeroth order are )( ,0,0,0 2 1,0 yyyyyyyyy ncgm   (3.2) 0)( )( )( 2 , 21 2 , 2 ,     yo c yyo c yyo g mdpe g rpe  (3.3) with 1)1()1(0    0, y = 0 ,  0 , x =0 at y = 1 (3.4) 1)( 11   yontxsin yon o o   (3.5) and to the first order are ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel )( ,0,0,0,0,1 2 1,1 yyyxyyxyyyyyyyyy gm   (3.6) ).( 2 ).( 2 )( ,1,0 1 ,1,0 2 ,0,,0,1 yy c yyyy c oxyyox g dpe g rpe yy    (3.7) with  1(+1)  1(-1 ) = 0  1, y = 0 ,  1 , x = 0 at y = 1 (3.8) 1(1) = 0 at y =  1 (3.9) assuming ec<<1 to be small we take the asymptotic expansions as ...............),(),(),( ...............),(),(),( ............),(),(),( ...........),(),(),( 11101 01000 11101 01000     yxyxyx yxyxyx yxecyxyx yxecyxyx     (3.10) substituting the expansions(3.10) in equations (3.2)-(3.9) and separating the like powersof ec we get the following 10000,00 )1(,1)1(, sindyy   (3.11) 10,0 1)1()1(, ,00,00 0000,00,00 2 1,00   yat gm xy yyyyyyy   (3.12) 0)1(, 01 2 ,00 1 ,00 2 ,01    yyyyy g pd g pr (3.13) 10,0 ,0)1()1(, ,01,01 0101,01,01 2 1,01   yat gm xy yyyyyyy   (3.14) 0)1()( 10,00,00,00,00,10   yxxyyy (3.15) 10,0,0)1()1( ,)( ,10,101010 ,00,00,00,00,10,10 2 1,10   yat gm xy yyyxxyyyyyyyyyy   (3.16) 0)1(, 2 2 )( 1,10,00 1 ,10,00 2 ,0,01,01,00,00,01,01,00,11      yy yyyyxyyxyxxyyy g pd g pr (3.17) 10,0,0)1()1( ,) ( ,11,111111 ,00.01,00,01 ,01,00,11,00,1,1 2 1,11    yat gm xy yyyxxyyy yyyxxyyyyyyyyyy    (3.18) 4. shear stress and nusselt number the shear stress on the channel walls is given by ly x v y u              ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel which in the nondimensional form reduces to 1 2 ,11,10,01,00 2 )](([ )(                yyyyyyyyy xxyy oecec a u      and the corresponding expressions are )()( 2 5431  odecddy  )()( 2 8761  odecddy  the local rate of heat transfer coefficient( nusselt number nu) on the walls has been calculated using the formula 1)( 1      y wm y nu   where    1 1 5.0 dym  and the corresponding expressions are )( )( )(, )( )( )( 6510 987 1 654 321 1 mecmm mmecm un mmecm mecmm un yy            5. discussion of the numerical results: the aim of this analysis is to discuss the effect of the dissipation on the convective flow and heat transfer of a viscous fluid through a porous medium confined in a vertical channel whose walls a travelling thermal wave is imposed. assuming the eckert number ec <<1 the coupled momentum and energy equations have been solved. the velocity and temperature distributions are analysed for different sets of the governing parameters. fig (1) variation of u with g fig (2) variation of u with m r=35, m=2, β=0.5, γ=2, x=π/4, n1=4, t=π/4 g=2x103, r=35, β=0.5, γ=2, x=π/4, n1=4, t=π/4 -1 .0 -0 .5 0 .0 0 .5 1 .0 -0 .3 5 -0 .3 0 -0 .2 5 -0 .2 0 -0 .1 5 -0 .1 0 -0 .0 5 0 .0 0 g = 1 0 3 g = 3 x 1 0 3 g = 5 x 1 0 3 g = -1 0 3 g = -3 x1 0 3 g = -5 x1 0 3 u  -1.0 -0.5 0.0 0.5 1.0 -0.55 -0.50 -0.45 -0.40 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 m=1.5 m=2.5 m=3 u  ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel -1 .0 -0 .5 0 .0 0 .5 1 .0 -0 .4 0 -0 .3 5 -0 .3 0 -0 .2 5 -0 .2 0 -0 .1 5 -0 .1 0 -0 .0 5 0 .0 0                     u  fig (3) variation of u with β fig (4) variation of u with r & γ g=2x103, r=35, m=2,γ=2, x=π/4, n1=4, t=π/4 g=2x103, m=2, β=0.5, x=π/4, n1=4, t=π/4 fig (5) variation of u with n1 fig (6) variation of u with x+ γ t g=2x103, r=35, m=2, β=0.5, γ=2, x=π/4, t=π/4 g=2x103, r=35, m=2, β=0.5, γ=5, n1=4, t=π/4 fig. (1) exhibits the variation of u with grashof number g. it is found that the axial velocity u is completely negative for all values of g with maximum occurring at the mid plane y=0 which drifts towards the upper plate for higher g(>0) and it drifts towards the lower boundary for |g|(<0). the magnitude of ‘u’ reduces in the lower half and enhances in the upper half with an increase in g, while a reversed effect is observed with increase in |g|(<0). the variation of ‘u’ with m shows that for lower values of the hartman number m we find reversed flow in the vicinity of both the boundaries and for higher m~o(1.5) the reversed flow in the vicinity of upper boundary disappears and reversal flow continues in the vicinity of lower boundary and for still higher values of m≤3 the reversal flow disappears in the entire flow region. for m=5.0 the reversal flow reappears in the vicinity of upper boundary. this shows that under the influence of strong magnetic field the free convection effect do not dominate over others. an increase in m~o(1.5) enhances ‘u’ in entire fluid region and for further increase in m(≥2.5) we find depreciation in |u| in the flow region. from fig. (3) we find that greater the dilation lesser the -1 .0 -0 .5 0 .0 0 .5 1 .0 -0 .3 5 -0 .3 0 -0 .2 5 -0 .2 0 -0 .1 5 -0 .1 0 -0 .0 5 0 .0 0 r = 3 5 ,   r = 7 0 r = 1 4 0         u  -1 .0 -0 .5 0 .0 0 .5 1 .0 -0 .4 0 -0 .3 5 -0 .3 0 -0 .2 5 -0 .2 0 -0 .1 5 -0 .1 0 -0 .0 5 0 .0 0 x +  t=   x +  t=   x +  t= 3   u  -1.0 -0.5 0.0 0.5 1.0 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 n 1 =1 n 1 =5 n 1 =10 n 1 =100 u  ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel magnitude of u. an increase in the reynolds number r leads to an enhancement |u| in the lower half and depreciation in the upper half. also an increase in the thermal wave velocity γ(≤10) depreciates |u| in the lower half and enhances in the upper half and for higher γ(≥15) a reversed effect is observed in the flow region (fig. (4)). the variation of ‘u’ with radiation parameter n1 shows a reversal flow in the mid region for smaller values of n1 and this reversal flow disappears for higher values of n1. |u| reduces in the lower half and enhances in the upper half with n1(≤1.0) and for higher n1(≥5) we notice a depreciation in |u| in the region abutting the boundaries and an enhancement in the mid region (fig.(5)). the variation of u with the phase x+γt of the boundary temperature curve, shows that |u| enhances with 4 7 x    t and for higher values of x+γt we notice an enhancement in the vicinity of the boundary and depreciation in the mid region. (fig.6) fig (7) variation of v with g fig (8) variation of v with m r=35, m=2, β=0.5, γ=2, x=π/4, n1=4, t=π/4 g=2x103, r=35, β=0.5, γ=2, x=π/4, n1=4, t=π/4 -1.0 -0.5 0.0 0.5 1.0 -0.35 -0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 g=10 3 g=3x10 3 g=5x10 3 g=-10 3 g=-3x10 3 g=-5x10 3 v  -1.0 -0.5 0.0 0.5 1.0 -0 .20 -0 .15 -0 .10 -0 .05 0.00 0.05 0.10 0.15 0.20 0.25 r =35,  r =70 r =140    v  -1.0 -0.5 0.0 0.5 1.0 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 m=2 m=5 m=10 v  -1 .0 -0 .5 0 .0 0 .5 1 .0 -0 .6 -0 .4 -0 .2 0 .0 0 .2 0 .4 0 .6       v  ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel fig (9) variation of v with β fig (10) variation of v with r & γ g=2x103, r=35, m=2, γ=2, x=π/4, n1=4, t=π/4 g=2x103, m=2, β=0.5, x=π/4, n1=4, t=π/4 fig (11) variation of v with n1 fig (12) variation of v with x+ γ t g=2x103, r=35, m=2, β=0.5, γ=2, x=π/4, t=π/4 g=2x103, r=35, m=2, β=0.5, γ=5, n1=4, t=π/4 the secondary velocity ‘v’ which arises due to the non-uniformity in the boundary has been depicted in figs. (7) (12) for different g,r,m,γ,β,n1 and x+γt. we notice that for smaller g(≤103) the secondary velocity in the left half and in the upper region it is directed towards the boundary. for higher g(≥3x103) the fluid in the left half is directed towards the boundary and the fluid in the upper half is directed towards the mid region, while for |g| (<0) the fluid in the left half is directed towards the mid region and the fluid in the upper half is directed towards boundary, for all |g|. the variation of ‘v’ with m shows that for smaller values of m~o(0.5) the fluid in entire flow region is directed towards the mid region, while for higher m(≥1.5) the fluid in the flow region is towards the boundary except in the vicinity of the left boundary is directed towards the mid region. the region where the transition takes place enhances its size with increase in m. an increase in m~o(3.5) we find a retardation in |v| and for further increase in m the velocity v in the left region experiences a depreciation and that in the right region experiences an enhancement and for still higher m(≥5) we find an enhancement in |v| in the entire flow region (fig. (8)). from fig. (9), it is found that greater the dilation β~o(3.5) larger the |v| and for higher β~o(0.5) we notice a depreciation in |v| in the left half and enhancement in |v| in the right half and for still higher β, larger |v| in entire flow region. an increase in r enhances |v|. also an increase in the thermal wave velocity γ enhances |v| in the upper half and reduces it in the lower half, and for further increase in γ we find an enhancement in |v| (fig. (10)). the variation ‘v’ with n1 shows that for small values of n1 the secondary velocity is directed towards the boundary and for higher values of n1 we find that v is towards the mid region for all n1. |v| enhances with increase in n1. (fig. (11)). the variation ‘v’ with phase x+γt of the boundary curve shows that for increasing 4 7 x    t the velocity in the left half reduces and that in the right region enhances with x+γt and for higher values of 4 11 x    t we find an increase in |v|. (fig.12) . -1.0 -0.5 0.0 0.5 1.0 -0.040 -0.035 -0.030 -0.025 -0.020 -0.015 -0.010 -0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 n 1 =0.5 n 1 =1 n 1 =5 n 1 =10 n 1 =100 v  -1.0 -0.5 0.0 0.5 1.0 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 x+ t= x+ t= x+ t=3 v  ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel fig (13) variation of θ with g fig (14) variation of θ with m r=35, m=2, β=0.5, γ=2, x=π/4, n1=4, t=π/4 g=2x103, r=35, β=0.5, γ=2, x=π/4, n1=4, t=π//4 fig (15) variation of θ with β fig (16) variation of θ with r & γ g=2x103, r=35, m=2, γ=2, x=π/4, n1=4, t=π/4 g=2x103, m=2, β=0.5, x=π/4, n1=4, t=π/4 -1 .0 -0 .5 0 .0 0 .5 1 .0 -0 .2 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 1 .8 2 .0 2 .2 2 .4 2 .6 2 .8 m = 0 .5 m =1 .5 m =2 .5 m =3 m =5   -1 .0 -0 .5 0.0 0.5 1.0 -1 .0 -0 .8 -0 .6 -0 .4 -0 .2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 r =35, r =70 r =140      -1 .0 -0 .5 0 .0 0 .5 1 .0 -1 .0 -0 .5 0 .0 0 .5 1 .0 1 .5 x+  t=  x+  t=  x+  t= 3     -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 g=10 3 g=3x10 3 g=5x10 3 g=-10 3 g=-3x10 3 g=-5x10 3   -1.0 -0.5 0.0 0.5 1.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 =0.1 =0.3 =0.5 =0.7 =0.9   -1.0 -0.5 0.0 0.5 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 n 1 =0.5 n 1 =1 n 1 =5 n 1 =10 n 1 =100   ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel fig (17) variation of θ with n1 fig (18) variation of θ with x+ γ t g=2x103, r=35, m=2, β=0.5, γ=2, x=π/4, t=π/4 g=2x103, r=35, m=2, β=0.5, γ=5, n1=4, t=π/4 the temperature distribution (θ) for different values of g,r,m,β,n1 and x+γt is shown in figs. (13) (18). the perturbation temperature in general is positive and hence contributes to the enhancement of actual temperature in the fluid region. fig (13) depicts behavior of θ for different |g|(><0). we notice that in a dilated channel the temperature increases/decreases with |g| according as with g(>0) or g(<0). an increase in m~o(2.5) we find an enhancement in θ. for higher m(≥3.0) the temperature in the left half enhances and that in the right half depreciates with increase in m. also higher the dilation larger the temperature in the left region and smaller the temperature in the upper region (fig.(15)). an increase in r decreases θ. the variation of ‘θ’ with γ shows that for γ≤5 the temperature is negative and for higher γ≤10, θ is positive. also an increase in γ≥10 leads to an enhancement in ‘θ’ and for further higher values of γ we notice a depreciation in θ. the effect of radiation on ‘θ’ is shown in fig. (17). an enhancement in the radiation parameter n1 results in an increase in the temperature in entire flow region. the variation of θ with phase x+γt of the boundary temperature curve shows that for 2 x    t the temperature is negative and for higher values of x+γt, θ is positive. for an increase 2 x    t we find depreciation in θ and for higher 2 3 x    t , θ enhances in entire flow field(fig.18). table.1 shear stress( ) at y = 1 , p=0.71 g i ii iii iv v vi vii viii 10 3 -27.32 -54.318 -86.19 -38.688 -49.176 -27.303 -49.182 -59.912 3x10 3 53.5904 -83.733 -106.441 -75.856 -88.87 53.501 -108.481 -127.611 5x10 3 84.3505 -122.711 -176.981 -100.541 -123.291 93.685 -236.881 -168.051 -10 3 -54.375 -62.796 -89.806 -43.128 -59.124 -54.393 -71.123 -66.969 -3x10 3 -76.578 -97.488 -102.951 -83.242 -108.591 -76.647 -177.041 -87.759 -5x10 3 -144.941 -167.721 -251.611 -159.061 -168.491 -128.711 -281.771 -196.321 table.2 shear stress( ) at y = -1 p=0.71 g i ii iii iv v vi vii viii 10 3 0.0857 0.5203 0.8311 0.6351 -1.1167 0.0494 -1.2606 0.5973 3x10 3 3.6351 -2.3797 -6.0387 -2.9492 -6.9704 3.2407 -9.6896 -3.1505 5x10 3 33.614 4.6238 -10.804 -3.5669 4.9161 32.0035 -7.2413 1.1336 -10 3 3.1566 3.6622 3.8092 2.3599 1.8977 3.1929 2.0418 3.7117 -3x10 3 -6.9607 -2.6153 -3.0992 -2.6016 -18.506 -6.5656 -15.785 -2.4721 -5x10 3 -50.075 -27.973 -25.889 -19.076 -82.909 -48.462 -70.746 -26.619 ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel table.3 average nusselt number(nu) at y = 1 p=0.71 table..4 average nusselt number (nu) at y = -1 p=0.71 g i ii iii iv v vi vii viii 10 3 4.1254 3.7439 3.6133 1.8222 2.2733 1.1622 1.4049 3.6995 3x10 3 5.7579 3.9748 3.9277 1.6329 1.7193 1.4569 1.6775 3.9398 5x10 3 2.6433 3.9312 3.8872 1.2358 -9.867 1.8127 2.0487 3.8663 -10 3 4.1209 3.7371 3.6112 1.7882 2.2721 1.1636 1.4041 3.6961 -3x10 3 5.7866 3.9729 3.9265 1.5031 1.7107 1.4549 1.6752 3.9381 -5x10 3 2.6557 3.9271 3.8852 0.8944 -10.557 1.8103 2.0461 3.8624 i ii iii iv v vi vii viii d 1 10 3 2x10 3 3x10 3 10 3 10 3 10 3 10 3 10 3  5 5 5 15 5 5 5 5  2 2 2 2 5 -2 -5 2 m 2 2 2 2 2 2 2 4 the shear stress () and the average nusselt number (nu) on the boundaries (y= 1) have been evaluated for different parameters and are given in tables (1)-(4). the shear stress is positive at y = -1 and negative at y = 1. it is found that  is observed to increase with an increase in g fixing the other parameters. higher the permeability of the medium larger the shear stress at both the boundaries. with reference to  we find that  increases for an increase in  for all g (tables 1 & 2). the average nusselt number measures the local rate of heat transfer across the boundary. we find from (tables.3 & 4) that the average nusselt number is positive at y = 1 and negative at y = -1 for all variations. the magnitude of nu at y=  1 increases with an increase in g > 0 and decreases with g<0 fixing the other paramerters. in axial heating case nu decreases with r and enhances with d-1 while a reversed effect is observed in the case of axial cooling. the rate of heat transfer (nu) declines with an increase in the amplitude  of the boundary temperature (tables.3&4). i ii iii iv v vi vii viii d 1 10 3 2x10 3 3x10 3 10 3 10 3 10 3 10 3 10 3  5 5 5 15 5 5 5 5  2 2 2 2 5 -2 -5 2 m 2 2 2 2 2 2 2 4 g i ii iii iv v vi vii viii 10 3 -0.9304 -1.2347 -1.6751 -0.7476 -1.0113 -5.6211 -1.6412 -1.3149 3x10 3 -0.6146 -0.9777 -1.3334 -0.5734 -0.6634 -3.9777 -3.0481 -1.0412 5x10 3 0.2125 -0.7692 -1.2206 -0.2668 0.5107 -3.5075 -2.7442 -0.8637 -10 3 -0.9312 -1.2373 -1.6772 -0.7435 -1.0099 -3.1893 -1.6391 -1.3176 -3x10 3 -0.6105 -0.9761 -1.3324 -0.5467 -0.6578 -5.5872 -2.0022 -1.0398 -5x10 3 0.2239 -0.7658 -1.2185 -0.1927 0.5286 -3.9744 -2.3682 -0.8607 ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 12 ijo international journal of mathematics (issn: 2992-4421 ) prakasha.p1* https://ijojournals.com/ volume 07 issue 04 || april, 2024 || mathematical analysis of effect of viscous dissipation on transient mhd convective heat transfer through a porous medium in a vertical channel 6.references [1] barletta,antonio , laminar mixed convection with viscous dissipation in a vertical channel., int.j.heat and mass transfer,41,no.22,pp.3501-3513,(1998). [2] barletta,antonio , combined forced and free convection with viscous dissipation in a vertical circular duct., int.j.heat and mass transfer,42,no.12,pp.2243-2253,(1999). [3] bulent yesilata , effect of viscous dissipation on polymeric flows between two rotating coaxial parallel discs., int.comm. heat and mass transfer, 29, no.29, no.5, pp.589600,(2002). [4] el-hakein, m.a., viscous dissipation effects on mhd free convection flow over a nonisothermal surface in a micropolar fluid. int.comm.heat and mass transfer ,27, no,4 pp.581590,(2000). [5] israel-cookey ,c. , influence of viscous dissipation and radiation on unsteady mhd free convection flow past an infinite heated vertical plate in a porous medium with time-dependent suction.,int.j.heat and mass transfer,46,no.13,pp.2305-2311,(2003). [6] rossie di schio,baletta,a,hahne,e and spindler,k , analysis of the effect of viscous dissipation for laminar flow in stadium –shaped ducts, int.comm.heat mass transfer,28, no.4, pp.449-459,(2001). [7] chen, c. h. laminar mixed convection adjacent to vertical, continuously stretching sheets. heat and mass transfer, 33(5-6), 471–476 (1998) [8] elbashbeshy, e. m. a. and bazid, m. a. a. heat transfer over an unsteady stretching surface.heat and mass transfer, 41(1), 1–4 (2004) [9] sharidan, s., mahmood, t., and pop, i. similarity solutions for the unsteady boundary layer flow and heat transfer due to a stretching sheet. international journal of applied mechanics and engineering, 11(3), 647–654 (2006) [10] tsai, r., huang, k. h., and huang, j. s. flow and heat transfer over an unsteady stretching surface with a non-uniform heat source. international communications in heat and mass transfer,35(10), 1340–1343 (2008) [11] ishak, a., nazar, r., and pop, i. hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. heat and mass transfer, 44(8), 921–927 (2008) [12] aziz, a. a similarity solution for laminar thermal boundary layer over a flat plate with a convective surface boundary condition. communications in nonlinear science and numerical simulation,14(4), 1064–1068 (2009) [13] chen, c. h. combined heat and mass transfer in mhd free convection from a vertical surface with ohmic heating and viscous dissipation. international journal of engineering science, 42(7),699–713 (2004) [14] pal, d. and hiremath, p. s. computational modelling of heat transfer over an unsteady stretching surface embedded in a porous medium. meccanica, 45(3), 415–424 (2010) [15] veena, p. h., subhas-abel, m., rajagopal, k., and pravin, v. k. heat transfer in a viscoelastic,fluid past a stretching sheet with viscous dissipation and internal heat generation. zeitschrift f¨ur,angewandte mathematik und physik (zamp), 57(3), 447–463 (2006) [16] palani, g. and ganesan, p. heat transfer effects on dusty gas flow past a semi-infinite inclined,plate. forsch ingenieurwes, 71, 223–230 (2007) ijo journals volume 07 | issue 04 | april 2024 | https://ijojournals.com/index.php/m/index 13 a mathematical model on cholera outbreak with vaccination bolanle adeola olokuntoye department of mathematics, obafemi awolowo university, ile-ife, nigeria abstract this study develops and analyzes a cholera transmission model of sirb type (susceptible–infected–recovered–bacteria) that incorporates vaccination. the main objective is to investigate the threshold conditions under which cholera can either be eradicated or persist in the community. the model formulation captures both direct person-to-person transmission and indirect infection through contaminated water. using standard dynamical systems techniques, the disease-free equilibrium (dfe) and endemic equilibrium (ee) were derived. the next-generation matrix approach was then applied to obtain the basic reproduction number, r0, which serves as the central threshold parameter governing disease dynamics. the analysis showed that the dfe is locally asymptotically stable whenever r0 < 1, implying cholera elimination under effective interventions, while the ee exists and is locally stable when r0 > 1, confirming sustained disease persistence. these results emphasize the importance of vaccination and improvements in sanitation as essential strategies to reduce r0 below unity and achieve long-term cholera control. keywords: cholera; sirb model; vaccination; equilibrium points; reproduction number; epidemiological modeling ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 23 1 introduction cholera remains one of the most devastating waterborne diseases, particularly in developing regions where inadequate sanitation and limited access to safe drinking water persist (ali et al., 2015; world health organization, 2023). caused by vibrio cholerae, cholera is characterized by acute watery diarrhea and can lead to severe dehydration and death if untreated (koelle and pascual, 2004). despite advances in treatment and prevention, cholera continues to pose significant public health challenges, with recurrent outbreaks reported in africa, asia, and parts of latin america (mukandavire et al., 2011; codeço, 2001). mathematical modeling has played a crucial role in understanding the transmission dynamics of cholera and evaluating the effectiveness of intervention strategies (anderson and may, 1991; capasso and paveri-fontana, 1979; hartley et al., 2006). early deterministic models, including classical sir frameworks, provided the foundation for analyzing epidemic thresholds and equilibrium behavior. extensions to incorporate environmental reservoirs have proven particularly important for waterborne diseases like cholera, where indirect transmission through contaminated water plays a key role (codeço, 2001; tien and earn, 2010). among the various interventions, vaccination has emerged as a promising strategy for cholera control (leung et al., 2012; world health organization, 2023). oral cholera vaccines (ocvs) have been deployed in both reactive and preventive campaigns, showing moderate to high effectiveness in reducing susceptibility and mitigating the severity of outbreaks (qadri et al., 2020). incorporating vaccination into mathematical models not only improves the biological realism of such frameworks but also provides decision-makers with quantitative tools to assess the optimal use of vaccination alongside sanitation, water treatment, and public health interventions (longini et al., 2007; chao et al., 2011). furthermore, the study of reproduction numbers remains central in mathematical epidemiology. the next-generation matrix method, formalized by van den driessche and watmough (2002), provides a rigorous framework for deriving the basic reproduction number r0, which determines whether a disease can invade a population. for cholera ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 24 models with vaccination, r0 captures both direct human-to-human and indirect waterborne transmission, as well as reductions due to vaccine-induced immunity. exploring the relationship between r0, vaccination coverage, and endemic equilibria is essential for developing effective control strategies (castillo-chavez and feng, 1997; mukandavire et al., 2011). mathematical models of cholera have evolved considerably since the early works of capasso and paveri-fontana (1979), who studied the 1973 cholera epidemic in the mediterranean region using an sir-type framework. later, codeço (2001) extended the model by explicitly incorporating an environmental reservoir of bacteria, leading to the widely known sirb framework. this innovation highlighted the role of aquatic environments in sustaining cholera transmission and explained the persistence of outbreaks beyond simple person-to-person dynamics. subsequent studies further enriched these frameworks. hartley et al. (2006) introduced the concept of *hyperinfectivity*, noting that freshly shed bacteria are significantly more infectious than older aquatic bacteria. tien and earn (2010) expanded the sirb model by including multiple transmission pathways, capturing the complexity of cholera spread in real-world settings. spatial extensions, such as those of bertuzzo et al. (2011), incorporated human mobility and hydrological transport, improving our understanding of cholera’s spatial dynamics during the haiti outbreak. the theoretical foundations for threshold analysis in such models are built on the reproduction number, r0. van den driessche and watmough (2002) established a systematic method for deriving r0 using the next-generation matrix approach, which has since become a standard in epidemiological modeling. applications of this method to cholera have clarified the conditions under which outbreaks fade out or become endemic (mukandavire et al., 2011; eisenberg et al., 2013). vaccination has increasingly been incorporated into cholera models. longini et al. (2007) and chao et al. (2011) examined vaccination strategies in endemic and epidemic settings, showing how oral cholera vaccines can reduce r0 and shift the stability of equilibria. immunological studies (leung et al., 2012) and large-scale vaccine campaigns (qadri ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 25 et al., 2020) provided empirical support for these models, demonstrating measurable impacts of vaccination on outbreak size and severity. more broadly, mathematical epidemiology has benefited from seminal works like anderson and may (1991) and hethcote (2000), which laid the theoretical groundwork for compartmental modeling, stability analysis, and disease control strategies. environmental and climatic drivers of cholera, such as those linked to el niño cycles, were explored by pascual et al. (2000) and koelle and pascual (2004), emphasizing the need to integrate ecological variability into cholera modeling. recent studies continue to refine these models by combining epidemiological, ecological, and immunological perspectives. for example, troeger et al. (2018) and world health organization (2023) highlighted the global burden of cholera and the urgency of integrating vaccination with water, sanitation, and hygiene (wash) measures. collectively, this body of literature underscores the importance of integrating vaccination into mathematical frameworks for cholera, not only for theoretical insights but also for informing evidence-based public health policies. in this study, we develop and analyze a mathematical model for cholera transmission that incorporates vaccination. the model builds on established sirb frameworks by including vaccination rate, allowing us to investigate how vaccination alters epidemic thresholds, disease-free equilibrium stability, and endemic persistence. 2 model formulation we consider a population divided into four main compartments: susceptible (s), infected (i), recovered (r), and bacteria concentration in the aquatic environment (b). the total population at time t is n(t) = s(t) + i(t) +r(t). the model is governed by the following system of nonlinear ordinary differential equaijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 26 tions: ds dt = µn − βpsi − βbsb − (µ+ ν + α)s, (1) di dt = βpsi + βbsb − (γ + µ+ δ)i + αs, (2) dr dt = γi − µr + νs + δi, (3) db dt = ξi − (µb + η)b, (4) (5) where: • µ is the natural birth/death rate, • βp is the transmission coefficient for direct human-to-human infection, • βb is the transmission coefficient for infection via the aquatic reservoir, • γ is the recovery rate, • δ is the disease-induced mortality rate, • ν is the vaccination rate, • α is the rate of waning immunity back to susceptibility, • ξ is the bacterial shedding rate from infected individuals, • µb is the bacterial natural death rate, and • η is the bacterial removal rate due to environmental sanitation. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 27 3 the equilibrium point 3.1 notation introduce compact notation: ρ := µ+ ν + α, σ := γ + µ+ τδ, := µb + aη, := βp + βb ξ τ . 3.2 disease-free equilibrium (dfe) at the disease-free equilibrium, i∗ = 0 and b∗ = 0. from (1): 0 = µn − ρs∗ =⇒ s∗ = µn ρ . from (3) (with i∗ = 0): 0 = −µr∗ + νs∗ =⇒ r∗ = ν µ s∗. thus, (s∗, i∗, r∗, b∗) = ( µn ρ , 0, νn ρ , 0 ) . however, substituting i∗ = 0 and b∗ = 0 into (2) gives 0 = αs∗. hence, for a true dfe to exist we require α = 0. therefore: if α = 0 : (s∗, i∗, r∗, b∗) = ( µn µ+ ν , 0, νn µ+ ν , 0 ) . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 28 3.3 endemic equilibrium (general case i∗ > 0) step 1. from the b equation: 0 = ξi∗ − τb∗ =⇒ b∗ = ξ τ i∗. step 2. combine the s and i equations: 0 = µn − βps ∗i∗ − βbs ∗b∗ − ρs∗, 0 = βps ∗i∗ + βbs ∗b∗ − σi∗ + αs∗. adding these gives 0 = µn − (ρ− α)s∗ − σi∗. since ρ− α = µ+ ν, σi∗ = µn − (µ+ ν)s∗ i, ∗ = µn − (µ+ ν)s∗ σ . step 3. substitute i∗ and b∗ into the s equation: 0 = µn − s∗ (βpi ∗ + βbb ∗)− ρs∗. using b∗ = ξ τ i∗ and defining a := βp + βb ξ τ , 0 = µn − as∗i∗ − ρs∗. substitute i∗ = µn−(µ+ν)s∗ σ : µn − as∗ σ ( µn − (µ+ ν)s∗)− ρs∗ = 0. step 4. rearranging: a(µ+ ν)s∗2 − ( aµn + ρσ ) s∗ + σµn = 0. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 29 this is a quadratic in s∗. step 5. solution: s∗ = aµn + ρσ ± √( aµn + ρσ )2 − 4a(µ+ ν)σµn 2a(µ+ ν) . then i∗ = µn − (µ+ ν)s∗ σ , b∗ = ξ τ i∗ r, ∗ = (γ + δ)i∗ + νs∗ µ . 3.4 special case: α = 0 when α = 0, the endemic equilibrium simplifies. from (2) (with i∗ > 0): βps ∗ + βbs ∗ ξ τ = σ =⇒ s∗ = σ a . define the basic reproduction number r0 := an σ = βpn σ + βbnξ στ . then s∗ = n r0 , i∗ = µn σ ( 1− 1 r0 ) , b∗ = ξ τ i∗. an endemic equilibrium (i∗ > 0) exists iff r0 > 1. 4 basic reproduction number 4.1 assumption (existence of dfe) the next-generation matrix method requires a disease-free equilibrium (dfe). a dfe with i∗ = b∗ = 0 exists only when there is no continuous import of infection; in other ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 30 words we set α = 0 for the threshold calculation. the dfe is then s∗ = µn µ+ ν i, ∗ = 0, r∗ = νn µ+ ν b, ∗ = 0. introduce the shorthand notations σ := γ + µ+ δ, τ := µb + η. step 1: infectious compartments and f , v partition choose infected variables x = (i, b)⊤. decompose the subsystem as ẋ = f(x) − v(x) where f collects new infection terms and v collects transitions and removals. from (2)–(4) with α = 0: f1 = βpsi + βbsb, f2 = 0, v1 = σi, v2 = τb − ξi. step 2: jacobians at the dfe compute jacobians of f and v with respect to (i, b) and evaluate at the dfe (s = s∗, i = b = 0): f = βps ∗ βbs ∗ 00  v, =  σ 0 −ξ τ  . step 3: next-generation matrix k = fv −1 since σ > 0, τ > 0 we may invert v : v −1 = 1 στ τ 0 ξ σ  . hence k = fv −1 =  βps ∗τ + βbs ∗ξ στ βbs ∗ τ 0 0  . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 31 step 4: spectral radius and r0 the eigenvalues of k are its diagonal entries. the spectral radius (dominant eigenvalue) is the (1, 1) entry, therefore r0 = βps ∗τ + βbs ∗ξ στ = s∗ σ ( βp + βb ξ τ ) . substituting s∗ = µn µ+ ν yields the explicit expression r0 = µn µ+ ν · 1 γ + µ+ δ ( βp + βb ξ µb + η ) . 4.2 interpretation the decomposition of r0 shows a direct transmission term and an environment-mediated term: r0 = βps ∗ σ︸ ︷︷ ︸ direct + βbs ∗ σ · ξ τ︸ ︷︷ ︸ environment . vaccination (rate ν) lowers s∗ and therefore reduces r0. enhancing environmental removal (increasing τ) or reducing shedding (ξ) lowers the environmental contribution. standard stability results imply the dfe is locally asymptotically stable if r0 < 1 and unstable if r0 > 1. 5 conclusion and recommendations 5.1 conclusion in this study, we proposed and analyzed a cholera transmission model of the sirb type incorporating vaccination and treatment. by formulating the system of nonlinear differential equations, we explicitly derived the disease-free equilibrium (dfe) and the endemic equilibrium (ee). the dfe corresponds to a population state where cholera infection cannot persist, while the ee describes a scenario in which cholera remains endemic under sustained transmission. the mathematical characterization of these equilibria provides important epidemiological insights. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 32 the stability of these equilibria was determined by computing the basic reproduction number r0 via the next-generation matrix approach. the explicit form of r0 is r0 = µn µ+ ν · 1 γ + µ+ δ ( βp + βb ξ µb + η ) , (6) which highlights the interplay between direct person-to-person transmission and environmentmediated transmission via the bacterial reservoir. the decomposition of r0 into direct and indirect pathways provides a useful threshold criterion: if r0 < 1, the dfe is locally asymptotically stable and the disease will eventually die out, while if r0 > 1, cholera can invade and persist, leading to the endemic equilibrium. from the derivations, several key epidemiological conclusions emerge: 1. impact of vaccination. the susceptible equilibrium level s∗ = µn µ+ν decreases with the vaccination rate ν, directly lowering r0. this confirms that increasing vaccine coverage is an effective means of driving r0 below unity and eliminating cholera. 2. role of environmental sanitation. the environmental contribution to r0 depends on both the bacterial shedding rate ξ and the bacterial removal rate τ = µb + η. improved sanitation, water treatment, and faster bacterial decay reduce the environmental load, thereby diminishing the potential for sustained outbreaks. 3. treatment and recovery. increasing the recovery rate γ and treatment efficacy reduces the average infectious period, thereby lowering r0. likewise, treatment interventions that reduce bacterial shedding can compound this effect. 4. threshold phenomenon. the model confirms the classical threshold property: cholera control is possible if and only if r0 < 1. the explicit dependence of r0 on epidemiological parameters provides a roadmap for targeted intervention strategies. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 33 recommendations based on these findings, we make the following recommendations: • strengthen vaccination campaigns. sustained and widespread vaccination of susceptible individuals is essential to lowering s∗ and hence reducing the overall reproductive potential of cholera. vaccination policies should prioritize high-risk communities with recurrent outbreaks. • improve water and sanitation infrastructure. investments in clean water supply, efficient sewage disposal, and bacterial removal measures (such as chlorination and filtration) are critical in reducing the environmental transmission pathway. • enhance early treatment and case management. effective case detection, rapid treatment, and supportive therapy increase recovery rates and reduce both morbidity and pathogen shedding, thus curtailing the force of infection. • integrate multi-intervention strategies. mathematical results suggest that no single intervention suffices when r0 is significantly above unity. a combined strategy involving vaccination, sanitation, and treatment will have synergistic effects, pushing the effective reproduction number below threshold. • policy implication. policymakers should use r0 not only as a theoretical threshold but as a measurable index of cholera control. targeting parameter domains that ensure r0 < 1 provides a rigorous, evidence-based criterion for evaluating the adequacy of public health strategies. in summary, this work demonstrates that vaccination coupled with environmental and clinical interventions is mathematically and epidemiologically sufficient to suppress cholera outbreaks. the derivation of equilibrium points and r0 offers both theoretical insight and practical guidance, showing that controlling cholera is contingent upon reducing susceptibility, minimizing environmental bacterial persistence, and shortening the infectious period through treatment. these results reinforce the importance of sustained, integrated control programs for cholera elimination. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 34 references ali, m., nelson, a. r., lopez, a. l., and sack, d. a. (2015). updated global burden of cholera in endemic countries. plos neglected tropical diseases, 9(6):e0003832. anderson, r. m. and may, r. m. (1991). infectious diseases of humans: dynamics and control. oxford university press. bertuzzo, e., mari, l., righetto, l., gatto, m., casagrandi, r., blokesch, m., and rinaldo, a. (2011). prediction of the spatial evolution and effects of control measures for the unfolding haiti cholera outbreak. geophysical research letters, 38(6). capasso, v. and paveri-fontana, s. l. (1979). a mathematical model for the 1973 cholera epidemic in the european mediterranean region. revue d’épidémiologie et de santé publique, 27(2):121–132. castillo-chavez, c. and feng, z. (1997). to treat or not to treat: the case of tuberculosis. journal of mathematical biology, 35(6):629–656. chao, d. l., halloran, m. e., and longini, i. m. (2011). vaccination strategies for epidemic cholera in haiti with implications for the developing world. proceedings of the national academy of sciences, 108(17):7081–7085. codeço, c. t. (2001). endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. bmc infectious diseases, 1:1. eisenberg, m. c., shuai, z., tien, j. h., and van den driessche, p. (2013). a cholera model in a patchy environment with water and human movement. mathematical biosciences, 246(1):105–112. hartley, d. m., morris, j. g., and smith, d. l. (2006). hyperinfectivity: a critical element in the ability of vibrio cholerae to cause epidemics? plos medicine, 3(1):e7. hethcote, h. w. (2000). the mathematics of infectious diseases. siam review, 42(4):599– 653. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 35 koelle, k. and pascual, m. (2004). disentangling extrinsic from intrinsic factors in disease dynamics: a nonlinear time series approach with an application to cholera. the american naturalist, 163(6):901–913. leung, d. t., rahman, m. a., mohasin, m., riyadh, m. a., patel, s. m., aktar, a., and qadri, f. (2012). comparison of immune responses to the killed oral cholera vaccine between bangladeshi children and adults. vaccine, 30(4):594–599. longini, i. m., nizam, a., ali, m., yunus, m., shenvi, n., and clemens, j. d. (2007). controlling endemic cholera with oral vaccines. plos medicine, 4(11):e336. mukandavire, z., liao, s., wang, j., gaff, h., smith, d. l., and morris, j. g. (2011). estimating the reproductive numbers for the 2008–2009 cholera outbreaks in zimbabwe. proceedings of the national academy of sciences, 108(21):8767–8772. pascual, m., rodó, x., ellner, s. p., colwell, r., and bouma, m. j. (2000). cholera dynamics and el niño–southern oscillation. science, 289(5485):1766–1769. qadri, f., wierzba, t. f., ali, m., and clemens, j. d. (2020). cholera in yemen—an old foe rearing its ugly head. new england journal of medicine, 382(6):568–571. tien, j. h. and earn, d. j. d. (2010). multiple transmission pathways and disease dynamics in a waterborne pathogen model. bulletin of mathematical biology, 72(6):1506– 1533. troeger, c., blacker, b. f., khalil, i. a., rao, p. c., cao, j., zimsen, s. r., and reiner, r. c. (2018). estimates of the global, regional, and national morbidity, mortality, and aetiologies of diarrhoea in 195 countries: a systematic analysis for the global burden of disease study 2016. the lancet infectious diseases, 18(11):1211–1228. van den driessche, p. and watmough, j. (2002). reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. mathematical biosciences, 180(1–2):29–48. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 36 world health organization (2023). cholera. https://www.who.int/news-room/ fact-sheets/detail/cholera. accessed: 2025-09-30. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 37 https://www.who.int/news-room/fact-sheets/detail/cholera https://www.who.int/news-room/fact-sheets/detail/cholera introduction model formulation the equilibrium point notation disease-free equilibrium (dfe) endemic equilibrium (general case i*>0) special case: =0 basic reproduction number assumption (existence of dfe) interpretation conclusion and recommendations conclusion ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" semigroup approach for the solution of boundary layer equation with sinc function term otobong j. tom1 &otobong g. udoaka2 1federal university of technology, ikot abasi. 2akwa ibom state university, ikot akpaden declarations 1. funding: not applicable 2. informed consent statement: not applicable 3. data availability: not applicable 4. conflict of interest statement: no conflict of interest. abstract boundary layer equation is crucial in fluid dynamics for modeling viscous flow near surfaces. it provides insight into flow behaviour, drag reduction, heat transfer and stability, making it essential in both theoretical and applied fluid mechanics. this paper examines the semigroup approach for solving the boundary layer equation incorporating a sinc function term.the addition of the sinc function term necessitates specialized functional analysis techniques and its effects are analyzed. we establish well-posedness in a suitable function space by analyzing the existence, uniqueness of the mild solution. we demonstrate the semigroup method's effectiveness in capturing boundary layer dynamics, contributing to the study of semigroup methods in fluid mechanics and partial differential equations. the influences of the sinc function term are analyzed and illustrative examples are included to validate the approach and to also highlight its applicability. key words: boundary layer equation, banach spaces, sinc function, strongly continuous semigroup, mild solution. ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 22 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" 1. introduction the boundary layer equation plays a crucial role in fluid dynamics, describing the behavior of viscous fluid flow near solid surfaces. prandtl [1] initiated the boundary layer theory, significantly advancing the study of viscous flows. this equation has been extensively studied due to its in aerodynamics, heat transfer, and hydrodynamics stability. the traditional method for solving the boundary layer equation include similarity transformations, perturbation methods, and numerical simulations. however, in recent years, the semigroup approach has emerged as a powerful tool for the existence, uniqueness, and stability of solutions to partial differential equations (pdes). this method leverages the properties of operator semigroups to study the time evolution of solutions in appropriate functional spaces. semigroup theory provides a robust framework for studying time-dependent pdes. hille and phillips [2] laid the foundation of semigroup analysis, later extended by pazy [3] to cover evolutionary equations in banach spaces. the semigroup approach has been successfully applied to fluid dynamics problems, including the navier-stokes equations [4]and parabolic pdes [5]. henry [6] applied semigroup method for nonlinear equations where he showed that the mild solution method could be used to establish the well-posedness of these equations in various function spaces. recently, the work of pruss [7] extended the semigroup approach to treat boundary boundary value problems with non-homogeneous conditions, which arise in practical fluid dynamics applications. husssian and kato [8] explored the use of semigroup approach to solve boundary layer equation numerically. their research showed that semigroup-based numerical methods offer significant advantages in terms of stability compared to other methods like finite difference schemes, particularly for complex flow configurations and high renolds numbers. the sinc function and its interpolation techniques have been widely studied in numerical analysis [10, 11,12]. the function is well known for its applications in signal processing and numerical analysis, introducing oscillatory behavior into the equation, which affects the solution properties. also, the function’s unique properties such as band-limited representation and rapid decay, makes it useful for solving differential equations. lund and bowers [10] have explored sinc-base methods for approximating pde solutions, but their interaction with semigroup methods remains an open area of research. this study aims to bridge this gap by developing a semigroup-theoretic approach for analyzing the well-posedness of boundary layer equation with sinc function term. the integration of sinc function term in the equation introduces additional complexities, requiring additional mathematical techniques for analysis and solution. our approach focuses on establishing well-posedess of the equation with the inclusion of the sinc function term. ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 23 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" 2. preliminaries in this section, we introduce the fundamental mathematical concepts and notations required for our analysis of the boundary layer equation with a sinc function term using the semigroup approach. 2.1. functional space and operators let x be a banach space, and consider a differential operator a on a dense subspace  d a x . we work within the frame work of semigroup theory, requiring the following definitions: 2.2. normed and banach spaces: a normed space  ,x  is a vector space if it is complete with respect to this norm. a normed space  ,x  is called banach space if every cauchy sequence in  ,x  converges. 2.3. hilbert space: a special case of banach space where the norm is induced by an inner product ,  . we primarily consider function spaces such as  pl ω and sobolevspaces  kh ω , which are crucial in studying pdes [ 5, 13, 14, 21]. 3. boundary layer equation with sinc term the general form of the boundary layer equation with a sinc function term can be written as:      , , 1 u au f x t s x t t      were :   ,u x t represents the velocity field,  a is a differential operator capturing the viscous effects,   ,f x t is an external forcing term, ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 24 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term"   ,s x t is a sinc function-modulated term affecting the solution structure. the boundary conditions and initial conditions depend on the physical context. 4. semigroup theory let x be a banach space, and let  :a d a x x  be a densely defined, closed linear operator. a strongly continuous semigroup (or 0c semigroup, or a semigroup of class 0c )   0t t t  on the banach space x is a family of bounded linear operators such that               0 0 the identity operator for all , 0 2 lim for all t t i t s t t s t t t s t t x x x x            the generator of the semigroup, denoted by a , is defined as   0 lim , t t t x x ax t   whenever the limit exists. the well-posedness of the equation depends on whether a generates a semigroup on x [3, 18, 19, 22, 24]. 4.1. spectral properties of the operators: the spectrum of a denoted by  a , plays a crucial role in determining the stability of the solutions. the resolvent operator     1 ,r a i a     helps analyze whether a an exponentially stable semigroup. 4.2. sinc function and its properties the sinc function is define as         sin sinc , 0, sinc 0 1 3 x x x x      ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 25 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" it has useful properties such as rapid decay and interpolation capabilities, making it a valuable tool in numerical method and spectral analysis [12]. 4.3. mild solution through semigroup approach: a mild solution to a differential equation is a solution that is defined through an integral equation involving a semigroup. for an evolution equation of the form  , du au f t dt   the mild solution is obtain using duhamel’s formula and is given by          0 0 4 t u t t t u t t s f s ds   where  t t is the semigroup generated by the operator a and 0u is the initial condition.the term mild is used because the solution is often less regular than classical solutions but still provides valuable information. also, if  t t is strongly continuous semigroup generated by a , then the mild solution can be expressed as             0 0 5 t u t t t u t t s f s s s ds    equation  5 provides insight into the existence, uniqueness, and stability of the solution, see [23, 24]. 4.4. mathematical formulation of the problem we begin by representing the boundary layer equation as follows      , , , , 0 6 u au f x t s x t x t t        ω where  ,u x t represent the unknown function (the state of the system) at time t and position x , a is a differential operator (often related to the laplacian or similar operator that models ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 26 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" diffusion or advection in fluid dynamics),  ,f x t and  ,s x t are external forcing terms. the initial condition for this equation is:    0,0 , ,u x u x x ω where  0u x represents the state of the system at time 0t  . 5. semigroup representation and solution method the solution is sought in the layout of semigroup theory, which provides a powerful method to deal with evolution equations. we assume that a generates a strongly continuous semigroup   0t t t  on a banachspace x . the semigroup approach allows us to express the solution in the form of equation  5 . here, the term   0t t u represents the evolution of initial condition, and the integral term capture the influence of the forcing    , ,f x t s x t over time, see [18, 24]. 5.1. well-posedness of the problem to ensure the problem is well-posed, we need to establish that equation  5 exists, is unique, and depends continuously on the initial conditions. this is done through the following steps:  existence: by the properties of semigroups, the integral equation for  u t is welldefined and yields a solution. the regularity of  ,f x t and  ,s x t ensure that the integral is finite and the solution ie well-behaved.  uniqueness: if two solutions  1u t and  2u t exist, we show that    1 2u t u t by employing the banach fixed-point theorem [3, 25], this relying on the fact that the operator  t t is strongly continuous and the forcing terms  ,f x t and  ,s x t are assume be identical for both solutions.  continuous dependence: since the solution depends continuously on the initial condition  0u x , small changes in the initial condition lead to small changes in the solution, which is a key property in proving stability. 5.2. influence of the sincfunctionterm a distinctive feature of this research is the inclusion of sinc function in the forcing term. the sinc function introduce oscillations into the system, which can affect both the regularity and ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 27 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" stability of the solution. in the context of boundary layers, the sinc function term might represent oscillatory external forces or disturbances that interact with the boundary layer dynamics.  oscillatory behavior: the sinc function’s oscillations may induce transient effects in the system, but under suitable condition (e.g., decaying forcing terms), these oscillations do not lead to unbounded growth of the solution.  damping effects: the decay of the sinc function ensures that its influence fades over time, which implies that the system will eventually settle to a steady state, as shown by the exponential decay established earlier. 5.3. regularity of solutions this is another crucial aspect of the research. regularity refers to the smoothness of the solution. if  u t belongs to a function space with sufficient differentiability (e.g., 1 2, ,c l etc.), it is said to be regular. the higher the regularity the smoother the solution, and this property often plays a crucial role in the stability and long-term behavior of solutions. the solution  u t is shown to belong to the class of 1c under suitable regularity assumptions on the initial data 0u and the forcing term    , ,f x t s x t . higher regularity ensures that the solution behaves smoothly over times and that its derivatives exist and are continuous. by applying standard semigroup theory, we can also show that higher derivatives of  u t exist and are bounded, which is important for numerical simulations and further theoretical analysis, see [18, 19, 22]. 6. results well-posedness of boundary layer equation: we consider the initial-boundary value problem:           0 , , , , 0 7 ,0 , u au f x t s x t x t t u x u x x          ω ω where a is as assumed in section 4. ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 28 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" theorem 5.1 (existence and uniqueness of mild solution): if a is as assumed in section 5, and if  ,f x t and  ,s x t satisfy suitable regularity conditions, then equation  7 has a unique mild solution given by equation  5 . proof: we apply duhamel’s formula [3, 15] to construct a mild solution of equation  5 . since the assumption of section 5 holds, it satisfies;   0 ,t i       t t s t t t s     tt t me for some , 0m   the existence of the integral form follows from the properties of semigroups and the assumed regularity of  ,f x t and  ,s x t , ensuring that the function inside the integral is welldefined. next, we prove the uniqueness of the solution. assume there exist two solutions  1u t and  2u t satisfying the integral equation                      1 2 1 2 1 2 1 20 0 0 t u t u t t t u u t t s f s f s s s s s ds        if    1 20 0u u and 1 2 1 2,f f s s  , then       1 2 0 0 0 t t s u t u t m e ds      thus, 1 2u u , proving the uniqueness. finally, we prove the continuity of the solution. by the strong continuity of  t t , we can show that  u t is continuous in x as follows: by assumption,  t t is strongly continuous semigroup, meaning that for all 0 ,u x ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 29 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term"   0 0 0 lim x t t u u x    this directly implies that   0t t u is continuous as function of t x . now, we need to prove the continuity of integral term by analyzing the integral term:          0 t i t t t s f s s s ds   to prove that  i t is continuous in x , we consider a small perturbation h in time and examine the difference:                   0 0 t h t i t h i t t t h s f s s s ds t t s f s s s ds            splitting the difference, we get               0 t i t h i t t t h s t t s f s s s ds                t h t t t h s f s s s ds     for the first term:           0 t t t h s t t s f s s s ds     since  t t is strongly continuous, for each fixed s,     0 lim h t t h s t t s      in x . since    f s s s is integrable, we can use the lebesgue dominated convergence theorem [16, 17] to conclude that:           00 lim 0. t h t t h s t t s f s s s ds        ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 30 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" for the second term.        0 lim t h th t t h s f s s s ds      as 0h , the interval  ,t t h shrinks to zero, and since    f s s s is integrable, this integral also vanishes. finally, since both terms vanish as 0h , we have     0 lim 0 h i t h i t     thus,  i t is continuous in x , and their sum  u t is also continuous in x . therefore, the mild solution  u t in equation  5 is continuous in x . proposition 5.2. (decay rate of solution): if  ,f x t and  ,s x t vanishes as ,t then   0u t  exponentially fast. proof: since  ,f x t and  ,s x t tend to zero, the integral term in        0 0 t t stu t me u m e f s s s ds      vanishes as t . thus,   0 0tu t me u  this shows exponential decay of solution. theorem 6.4. (higher regularity of solutions): if  0u d a , and    , ,f x t s x t are sufficiently smooth, then the solution satisfies ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 31 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term"   1, 0, , du au f s u c t x dt     . proof: first, since  0u d a , we apply a to the mild solution formula           0 0 tdu at t u a t t s f s s s ds f s dt       using the properties of semigroups [3, 19],    at t t t a , we write du au f s dt    this implies that u is differentiable with the stated regularity[6]. example 6.1. consider the boundary value problem:               2 2 0 sinc , 0,1 , 0 ,0 ,1 0, 0 0, sin . u u t x t t x u t u t t u x u x x                  v this is a linear boundary layer equation with sinc forcing. we interpret the sinc term as an external time-dependent forcing that is spartially uniform (i.e., acts identically across all spatial points). let 2 2 d a dx v with domain      2 1 00,1 0,1d a u h h   . then a generates a strongly continuous analytic semigroup   tat t e . the mild solution is given by: ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 32 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term"        0 0 t u t t t u t t s f s ds   . the eigenfunctions of a are:       2 2sin ,n ne x n x n   vλ . so the semigroup acts as:    0 01 ,nt n nn t t u e u e e x    λ . since      0 1 2 sin 2 u x x e x  , only the first mode is nonzero:     2 1 2 2 t t e e x v . now consider:           1 1 0 , sinc 1 sinc 1, , 2 2 , odd, 1, 2 sin 0, even. n nn n f s x s s e e x n e n x dx n n                 hence:          sinc 1 odd 2 2 n t s s n n n n t t s f s e e e n       λ . integrating:          1 0 0 odd 2 2 sincn t t t s n n n t t s f s ds e s ds e x n            λ . combining, we have the mild solution: ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 33 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term"             22 11 0 odd 2 2 2 , sinc 2 t n t st n n n u t x e e x e s ds e x n                vv . truncating to first odd mode  1n  , we have:         22 0 2 2 2 , sinc sin 2 t t stu t x e e s ds x              vv . this is a practical, computable low-mode mild solution. example 6.2. consideranother case given by:                 2 2 sinc , ,0 ,1 0, 0, sin , 1 . u u u u x t t x x u t u t u x x x x x                      v this is a semilinear case with sinc forcing. in this case, the mild solution is obtained as follows: let         sincxf u s u s u s s    , then:         0 0 t u t t t u t t s f u s ds   . as in example 5.1,     2 0 2 sin 2 tt t u e x  v . for the nonlinear term assume      , sinu t x t x  . then:     cosxu t x          2 sin cosxu u t x x     . ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 34 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" we project xu u onto    1 2 sine x x ;       1 2 2 1 0 , 2 sin cosxu u e t x x dx          1 2 0 1 1 1 sin cos 2 2 x x dx        so:    2 2 1 1 2 , 2 2 2 xu u e t t        . projection of forcing term:     1 1 30 4 , 2 1 sin 2e x x x dx       . putting everything together, the scalar integral equation for  t is:         22 2 30 2 2 4 2 sinc 2 2 t st t t e e s s ds                 vv . this a nonlinearvolterra equation of the second kind. remark:  the nonlinear term creates a coupling effect where past values of  s impact present  t .  the integral equation can be approximated numerically (e.g., trapezoidal rule or fixed point iteration).  the result is a reduced-order approximation to the original pde using the semigroup eigenfunction method. ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 35 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" 7. conclusion we have successfully establish a pattern for solving boundary layer equations incoperatingsinc function in the forcing term using semigroup theory. the major results include the wellposedness, regularity and the decay rate of the solutions, along with detailed analysis of the role of the sinc function term in these equations. this layout paves way for further exploration into numerical methods, such as the use of sinc functions in discretization schemes, and also have applications in modeling physical systems with oscillatory boundary conditions. illustrative examples have been shown to validate the approach and applicability. references [1] prandtl, l. on the motion of fluids with very little viscosity, in proceedings of the third international congress of mathematicians, heidelberg, germany, (1964), 484-491. [2] hille, e. and phillips, r. s, functional analysis analysis and semi-groups. ams colloquium publication, vol. 32 (1957). [3] pazy a., semigroups of linear operators and applications to partial differential equations, springer-verlag, applied math. sciences, vol. 44, 1983. [4] temam, r., infinite-dimensional dynamical system in mechanics and physics. springerverlag 1997. [5] l. c. evans, partial differential equations, graduate studies in mathematics, vol. 19, ams, providence, rhode island, 2002. [6] henry, d., geometric theory of semilinear parabolic equations. lecture notes in mathematics vol. 840, pringer-verlag, 1981. [7] pr�̈ss, j., evolutional integral equations and applications. birkhauser, (2015). [8] hussain, s., and kato, d., semigroup-based numerical approximations for boundary layer equations. journal of computational fluid dynamics, 40(3) (2022) 785-810. [10] lund, j., and bowers, d.,sinc method for quadrature and differential equations. siam (1992). [11] stenger, f. numerical methods based on sinc and analytic functions. springer, berlin, new york, (1993). [12] john, e. d. &ogbonna ,n. a double exponential sinc collocation method for volterrafredholm integral equations of the second kind, j. math. soc. 35 (2016) 408-423. [13] s. kesavan, topics in functional analysis and applications, new age international (p) limited, new delhi, (2003). [14] haim brezis, functional analysis, sobolev spaces and partial differential equations, universitext. springer, new york, (2011). ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 36 ijo international journal of mathematics (issn: 2992-4421 ) otobong j. tom1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || “semigroup approach for the solution of boundary layer equation with sinc function term" [15] michal rozanski, beatasikora, andriansmuda, and roman witula. on theoretical and practical aspect of duhamel’s integral. archives of control sciences, vol. 31(lxvii), no. 4, (2021) 815-847. [16] duchesne, g. w., lessard, jp. &takayasu. a. a rigorous integrator and global existence for higher-dimensional semilinear parabolic pdes via semigroup theory. j scicomput. 102(62) (2025). [17] royden, h. l., and fitzpatrick, p. m. real analysis. 4th ed., pearson, 2010. [18] klaus-jochenengel and rainernagel. one-parameter semigroups for linear evolution equations. springer(2000). [19] lunardi a. analytic semigroups and optimal regurity in parabolic problems. birkhauser, (1995). [20] kato, t. perturbation theory for linear operators. springer-verlag 1980. [21] de branges, l. hilbert spaces of entire functions and applications in fluid mechanics. cambridge university press, (2015). [22] triebel, h. interpolation theory, function spaces, differential operators. north-holland, (1978). [23] trefethen, l. n., and weideman, j. a. the exponentially convergent sinc method for integral equations. siam review, 56(3) (2014), 385-458. [24] tucsnak, m and weiss, g. observation and control for operator semigroups. birkhauser, (2009). [25] mannanmd. a., rahman md. r., akter h., nahar n., andmondal s. a study of banach fixed-point theorem and it’s applications. american journal of computational mathematics, 11 (2021), 157-174. ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 37 https://link.springer.com/book/10.1007/b97696 stability in semigroups of bounded linear operators: bridging algebraic and analytic notions 1otobong j. tom∗, and 2otobong g. udoaka. 1 fed. uni. tech., ikot abasi & akwa ibom state uni., ikot akpaden. 2akwa ibom state university, ikot akpaden. abstract stability is a cornerstone in the theory of semigroups, shaping the study of evolution equations, operator theory, and algebraic structures. yet, algebraic and analytic perspectives on stability have traditionally developed in isolation. this paper builds a novel bridge between the two. tom, udoaka and udo�a (2025) established that every strongly continuous (c0) semigroup of bounded linear operators is stable in the sense of koch and wallace (kw), a universal algebraic property that forces green's relations to collapse (d = j = l = r). this recognition is new in operator semigroup theory, where stability has typically been studied only in analytic terms. we further provide precise spectral conditions under which kw-stability aligns with analytic stability notions�strong, asymptotic, exponential, and uniform�thereby unifying algebraic semigroup stability with spectral/operator-theoretic stability. illustrative examples, including the translation, right shift, heat, and damped wave semigroups, demonstrate the stability �gap� and the exact conditions under which the two approaches coincide. the study is signi�cant because it supplies a universal structural property of operator semigroups, a spectral criterion for analytic decay, and practical insights for evolution equations, control design, and numerical discretization. keywords: semigroup theory; kw-stability; analytic stability; spectral bound; green's relations; evolution equations. 1 introduction the concept of stability is central to mathematics, capturing how systems behave under iteration, evolution, or perturbation. within algebraic semigroup theory, stability was formally introduced by koch and wallace (1956) [1], who de�ned a semigroup s to be stable if as ⊆ abs =⇒ as = abs, sa ⊆ sab =⇒ sa = sab. this condition enforces the collapse of green's relations�fundamental equivalence relations describing semigroup structure�so that d = j = l = r. stability in this sense ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 10 has deep structural implications, simplifying semigroup decompositions and embedding properties [4, 20, 21, 22, 23, 24, 25, 26]. in operator theory and functional analysis, a di�erent approach to stability has developed through the study of c0-semigroups, which arise naturally in solving the abstract cauchy problem du dt = uau, (0) = u0, where a is the generator [2, 6, 11, 12, 13, 15]. here, stability is measured analytically in terms of operator norms and spectral conditions. classical notions include asymptotic stability, strong stability, exponential stability, and uniform stability, each re�ecting different aspects of long-time behaviour. these notions are closely tied to the spectrum of the generator and results such as the gearhart�prüss theorem [5, 7, 8, 9, 10, 17, 32]. although both traditions revolve around the idea of stability, they have historically evolved in relative isolation: the algebraic approach is structural and norm-free, while the analytic approach is spectral and dynamical. a recent advance has begun to bridge this divide. tom, udoaka, and udo�a (2025) [3] introduced kw-stability into the setting of semigroups of bounded linear operators, proving that every strongly continuous (c0) semigroup on a banach space is stable in the sense of koch�wallace. this recognition provides a new link between classical semigroup stability theory and operator semigroup analysis, placing algebraic stability at the foundation of analytic operator theory. their work suggests further directions for stability research, including the study of unbounded operator semigroups, hypersemigroups, and semigroups arising in stochastic analysis [7, 9, 16, 32]. in what follows, we continue this line of investigation by examining how kw-stability interacts with analytic stability notions. in particular, we identify the spectral conditions under which algebraic and analytic stability coincide and illustrate this interplay with canonical examples such as translation, shift, heat, and damped wave semigroups. 2 preliminaries de�nition 2.1 (normed linear space). a normed linear space is a pair (x, ∥ · ∥) where x is a vector space over the �eld r or c, and ∥ · ∥ : x → [0,∞) is a function, called a norm, satisfying the following properties for all x, y ∈ x and all scalars α: 1. positivity: ∥x∥ ≥ 0, and ∥x∥ = 0 if and only if x = 0. 2. homogeneity (absolute scalability): ∥αx∥ = |α| ∥x∥. 3. triangle inequality: ∥x+ y∥ ≤ ∥x∥+ ∥y∥. de�nition 2.2 (banach space). a banach space is a vector space x over the �eld r or c together with a norm ∥ · ∥ : x → [0,∞) such that (x, ∥ · ∥) is complete; that is, every cauchy sequence {xn} in x converges to some x ∈ x with respect to the norm ∥ · ∥. formally, for every sequence {xn} in x, if lim m,n→∞ ∥xn − xm∥ = 0, then there exists x ∈ x such that lim n→∞ ∥xn − x∥ = 0. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 11 de�nition 2.3 (bounded linear operator). let x and y be normed linear spaces. a mapping t : x → y is called a linear operator if t (αx+ βy) = αt (x) + βt (y), ∀ x, y ∈ x, α, β ∈ r or c. the operator t is said to be bounded if there exists a constant m > 0 such that ∥t (x)∥y ≤ m∥x∥x , ∀ x ∈ x. equivalently, t is bounded if and only if it is continuous at 0 (and hence continuous everywhere). de�nition 2.4 (operator semigroup). let x be a banach space and b(x) the algebra of all bounded linear operators on x. a family {t (t)}t≥0 ⊆ b(x) is called a strongly continuous semigroup (c0-semigroup) if: 1. t (0) = i (the identity operator), 2. t (t+ s) = t (t)t (s) for all t, s ≥ 0, 3. for every x ∈ x, limt→0+ t (t)x = x. for more about this, the reader is referred to [3]. de�nition 2.5 (koch and wallace stability [1]). a semigroup s is called stable if for all a, b ∈ s: � (right stability): as ⊆ abs =⇒ as = abs, � (left stability): sa ⊆ sab =⇒ sa = sab. this de�nition was also given by east using green's relation in [3, 33]. equivalently, if a ≤j ab, then arab, and if a ≤j ba, then alba. 3 main results proposition 3.1 (kw-stability of c0−semigroups [3]). every c0-semigroup of bounded linear operators is stable in the sense of koch�wallace. proof. let s = {t (t) : t ≥ 0}. pick a = t (t), b = t (s) with t, s ≥ 0. by the semigroup law, ab = t (t)t (s) = t (t+ s) = t (s+ t) = t (s)t (t) = ba, so s is commutative. suppose as ⊆ abs. for x = t (u) ∈ s, (ab)x = t (t+ s)t (u) = t (t+ s+ u). by commutativity, (ab)x = at (s+u) ∈ as. thus (ab)s ⊆ as. combined with as ⊆ abs, we get as = abs. similarly, if sa ⊆ sab, then for x = t (u) ∈ s, x(ab) = t (u)t (t+ s) = t (u+ t+ s) = t (u+ t)t (s) = (xa)b ∈ sa, so sab ⊆ sa, hence sa = sab. therefore s satis�es kw-stability. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 12 theorem 3.2 (equivalence of green's relations [3]). for a c0-semigroup of bounded linear operators, d = j = l = r. proof. since s is commutative, left and right ideals coincide: sa = as, so l = r. for any a ∈ s, sas = {xay : x, y ∈ s}. but commutativity gives xay = (xy)a ∈ sa, so sas ⊆ sa. conversely, for xa ∈ sa, write xa = (x)a · i ∈ sas. thus sa = sas, so j = l. finally, d = l ◦ r and l = r imply d = l = r = j . 4 illustrative examples example 4.1 (translation semigroup). let x = c0(r), the banach space of continuous functions on r vanishing at in�nity, equipped with the supremum norm ∥f∥∞ = sup x∈r |f(x)|. de�ne a family of operators {t (t)}t≥0 by (t (t)f)(x) = f(x+ t), f ∈ x, t ≥ 0, x ∈ r. for strong continuity. we verify that {t (t)}t≥0 is a strongly continuous semigroup. for �xed f ∈ x, ∥t (t)f − f∥∞ = sup x∈r |f(x+ t)− f(x)|. since f is uniformly continuous on r (as every f ∈ c0(r) is uniformly continuous), the right-hand side tends to zero as t → 0. hence, lim t→0+ ∥t (t)f − f∥∞ = 0, so {t (t)}t≥0 is a c0-semigroup. for in�nitesimal generator. let a denote the generator of {t (t)}t≥0. by de�nition, af = lim t→0+ t (t)f − f t f, ∈ d(a), where the domain consists of those f ∈ x for which the above limit exists in x. a direct computation shows that t (t)f(x)− f(x) t = f(x+ t)− f(x) t . thus, the limit exists precisely when f is continuously di�erentiable with derivative vanishing at in�nity. therefore, af = f ′, d(a) = {f ∈ c0(r) : f ′ ∈ c0(r)}. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 13 for kw-stability. by proposition 3.1, every c0-semigroup on a banach space is stable in the sense of koch�wallace. explicitly, for f ∈ x and t, s ≥ 0, t (t+ s)f = t (t)t (s)f, and the kw-condition t (t)x ⊆ t (t+ s)x ⇒ t (t)x = t (t+ s)x is satis�ed. thus the translation semigroup is kw-stable. for analytic behavior. we compute the operator norm: ∥t (t)∥ sup= ∥f∥∞=1 ∥t (t)f∥∞. but for any f ∈ x, ∥t (t)f∥∞ = sup x∈r |f(x+ t)| = sup y∈r |f(y)| = ∥f∥∞. hence, ∥t (t)∥ = 1 for all t ≥ 0. therefore, there is no decay as t → ∞, and the semigroup fails to be analytically stable (in the sense of uniform exponential stability). graphical interpretation. the operator t (t) acts as a horizontal shift of the function graph. for example, if f(x) = e−x2 is a bell-shaped curve centered at the origin, then t (1)f(x) = f(x+1) is the same curve shifted left by one unit. importantly, the height of the curve is unchanged, so ∥t (t)f∥∞ = ∥f∥∞ for all t ≥ 0. this shows why the semigroup is kw-stable (algebraically the orbits are preserved) but not analytically stable (no decay in norm). x f(x) f(x) = e−x2 t (1)f(x) = f(x+ 1) 0−1 figure 1: translation semigroup illustrated on f(x) = e−x2 . the original function (blue) and its translated version (red dashed) have identical amplitude but shifted position, explaining why kw-stability holds while analytic stability fails. example 4.2 (right shift on ℓ2). let x = ℓ2(n) with norm ∥x∥2 = ∑ k≥1 |xk|2. de�ne t (n) for n ∈ n by t (n)(x1, x2, x3, . . . ) = (0, . . . , 0︸ ︷︷ ︸ n , x1, x2, x3, . . . ). for semigroup property. for m,n ∈ n, t (m)t (n) = t (m+ n), ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 14 so {t (n)}n∈n is a (discrete) semigroup. for kw-stability. the orbit structure is preserved under shifts: t (m)t (n) = t (m+ n) and the kw-condition t (n)x ⊆ t (n+m)x ⇒ t (n)x = t (n+m)x holds. for isometry / norm. for any x ∈ ℓ2, ∥t (n)x∥2 = ∑ k≥1 |(t (n)x)k|2 = ∑ k≥1 |xk|2 = ∥x∥2, so ∥t (n)∥ = 1 for all n: each t (n) is an isometry. for analytic behaviour. since there is no decay (∥t (n)∥ = 1 always), the semigroup is not analytically (exponentially) stable. interpretation. the right-shift moves entries to the right while preserving total ℓ2 energy. algebraic stability (kw) holds but analytic decay does not. x1 x2 x3 · · · 0 x1 x2 x3 · · · original after t (1) figure 2: schematic of the right shift t (1) on ℓ2: each component moves one box to the right and a 0 is inserted at the left. example 4.3 (heat semigroup). let x = l2(rn), the hilbert space of square-integrable functions on rn. de�ne, for t > 0, (t (t)f)(x) = (gt ∗ f)(x), gt(x) = (4πt)−n/2e− |x|2 4t . here gt is the gaussian heat kernel, representing the fundamental solution of the heat equation. for strong continuity. for each f ∈ l2(rn), the convolution t (t)f = gt ∗ f de�nes a continuous function of t. as t → 0+, gt tends to the dirac delta distribution δ, so t (t)f → f in l2, ensuring lim t→0+ ∥t (t)f − f∥2 = 0. hence {t (t)}t≥0 is a strongly continuous semigroup (a c0-semigroup). for the generator. we recall that t (t) solves the cauchy problem for the heat equation: ∂u ∂t = ∆ uu, (0, x) = f(x). thus, the generator is the laplacian operator af = ∆f, d(a) = h2(rn) = {f ∈ l2(rn) : ∆f ∈ l2(rn)}. this follows by di�erentiating t (t)f at t = 0 under the fourier transform. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 15 for kw-stability. by proposition 3.1, every c0-semigroup is kw-stable in the algebraic sense. in particular, for the heat semigroup, {t (t)f : t ≥ 0} = {t (s+ t)f : t ≥ 0}, for all s ≥ 0, re�ecting the invariance of reachable states. for analytic stability. the fourier transform of gt is given by ĝt(ξ) = e−t|ξ|2 , so in the fourier domain, t (t)f has the multiplier e−t|ξ|2, which decays exponentially in t for each ξ ̸= 0. since the spectrum of ∆ is σ(∆) = (−∞, 0], the spectral bound is strictly negative. therefore, there exists ω > 0 such that ∥t (t)f∥2 ≤ e−ωt∥f∥2, ∀t ≥ 0. hence the semigroup is not only contractive but also exponentially stable. interpretation. in this example, algebraic stability (kw-stability) and analytic stability (exponential decay of norms) coincide. unlike the translation and shift semigroups, which preserve norm without decay, the heat semigroup smooths and dissipates initial data over time. physically, this corresponds to the di�usion of heat: local peaks �atten, energy spreads out, and the system relaxes exponentially fast. −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 0 0.2 0.4 x g t( x ) t = 0.5 t = 1 t = 2 figure 3: gaussian heat kernel gt(x) at di�erent times t = 0.5 (blue), t = 1 (red), and t = 2 (green). graphical interpretation (figure 3). the curves show the gaussian kernel gt(x) for di�erent times: � at t = 0.5 (blue): the kernel is tall and narrow, concentrated near x = 0. heat is still localized. � at t = 1 (red): the peak is lower but wider, showing partial di�usion and �attening of the initial concentration. � at t = 2 (green): the kernel is very �at and spread out, indicating that heat has dissipated signi�cantly. thus, as t → ∞, the peak decays while the width grows like √ t. this illustrates exponential stability in the semigroup sense and di�usion in the physical sense. for detailed expositions of heat semigroups and their stability properties, see [16, 18, 19, 27, 28, 29, 30, 31]. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 16 4.1 example 3.4 (damped wave equation) we consider the damped wave equation on the bounded interval (0, l) with homogeneous dirichlet boundary conditions: utt(x, t) + αut(x, t)− uxx(x, t) = 0, x ∈ (0, l), t > 0, u(0, t) = u(l, t) = 0 t, ≥ 0, u(x, 0) = u0(x) u, t(x, 0) = v0(x) x, ∈ (0, l), (1) where α ∈ r is the (constant) damping coe�cient. state space and energy. set x = h1 0 (0, l)× l2(0, l), with state variable u(t) = (u(·, t), ut(·, t))⊤. we equip x with the energy inner product 〈 (u1, v1), (u2, v2) 〉 x := ∫ l 0 u′ 1(x)u ′ 2(x) dx+ ∫ l 0 v1(x)v2(x) dx, and corresponding norm ∥(u, v)∥2x = ∥u′∥2l2(0,l) + ∥v∥2l2(0,l). the physical energy associated to a solution of (1) is e(t) = 1 2 ( ∥ut(·, t)∥2l2 + ∥ux(·, t)∥2l2 ) . first-order formulation and generator. write (1) as a �rst-order system u ′(t) = au(t) by setting u = (u, v)⊤ with v = ut. de�ne a ( u v ) = ( v uxx − αv ) , d(a) = ( h2(0, l) ∩h1 0 (0, l) ) ×h1 0 (0, l). equivalently, in matrix form, a = ( 0 i ∂xx −αi ) , d(a) = (h2 ∩h1 0 )×h1 0 . generation of a c0-semigroup (sketch). the operator ∂xx with dirichlet boundary conditions is self-adjoint and has compact inverse on l2(0, l). standard results for second-order hyperbolic operators with bounded damping (see [7, 9, 10, 32]) imply that a with domain above is the generator of a c0-semigroup {t (t)}t≥0 on x. in particular: for α ≥ 0 the semigroup is contractive with respect to a suitable equivalent energy norm (damping is nonnegative). for α < 0 the operator has a component that can generate growth (negative damping gives energy injection). we therefore treat {t (t)} as the evolution operator for (1). ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 17 modal decomposition and spectrum. let {φn}n≥1 denote the dirichlet laplacian eigenfunctions φn(x) = sin (nπx l ) , −φ′′ n = ω2 nφn, ωn = nπ l , n ∈ n. expand the solution as u(x, t) = ∑ n≥1 qn(t)φn(x). each modal coe�cient satis�es the scalar ode q′′n(t) + αq′n(t) + ω2 nqn(t) = 0. the characteristic equation is λ2 + αλ+ ω2 n = 0 with roots λ± n = −α± √ α2 − 4ω2 n 2 . hence ℜ(λ± n ) ≤ −α 2 for every n ≥ 1, and the spectral bound of a satis�es s(a) := sup{ℜλ : λ ∈ σ(a)} = −α 2 . (here we used that the full spectrum of a consists of these modal eigenvalues due to compactness of the spatial resolvent and separation of variables.) energy identity and exponential decay for α > 0. multiply (1) by ut and integrate over (0, l) to obtain the standard energy balance: d dt e(t) = −α ∫ l 0 |ut(x, t)|2 dx ≤ 0. thus energy is nonincreasing. to obtain exponential decay we combine this dissipation with a coercivity (poincaré) inequality: for u ∈ h1 0 (0, l), ∥u∥l2 ≤ 1 ω1 ∥u′∥l2 , ω1 = π l . using the energy e(t) and the modal spectral gap one can show (standard multiplier or resolvent estimates; see [27, 28, 29, 30, 31]) that there exist constants m ≥ 1 and γ > 0 (depending on α and l) such that ∥t (t)∥l(x) ≤ me−γt, t ≥ 0. a simple modal estimate gives a concrete lower bound γ ≥ α/2 in the case of uniform damping (constant α); more careful resolvent estimates may yield the optimal rate γ = α/2 when the poincaré constant is accounted for. non-decay when α = 0. if α = 0 equation (1) reduces to the undamped wave equation. modal eigenvalues are purely imaginary λ± n = ±iωn, so s(a) = 0. energy is conserved (de/dt = 0) and no decay of the norm occurs in general (solutions persist as undamped oscillations). thus analytic stability fails when α = 0. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 18 instability for α < 0. if α < 0 the modal real parts satisfy ℜ(λ± n ) ≥ −α/2 > 0 (note sign), and high modes may exhibit growth; hence the semigroup is not stable and solutions typically grow exponentially [6, 7, 11, 19]. remarks. � the exponential decay argument above uses that damping is uniform (constant α > 0) and the spatial domain is bounded so the laplacian has compact resolvent. for localized damping (e.g. α(x) ≥ 0 supported only on a subregion) exponential decay may fail or require geometric control/observability conditions (see bardos� lebeau�rauch-type results). � the modal description also explains why s(a) = −α/2 in this uniform case: the real parts of all modal eigenvalues are bounded above by −α/2. thus the spectral criterion s(a) < 0 is equivalent to α > 0 here. conclusion for example 3.4. with the state space x = h1 0 (0, l) × l2(0, l) and generator a de�ned above, the semigroup {t (t)} satis�es: � kw-stability for all α ∈ r (algebraic property of the one-parameter family). � exponential (analytic) stability if and only if α > 0 (spectral bound negative). � conservation of energy (no decay) when α = 0. � instability when α < 0 [6, 11, 12, 13, 24]. for more about pde: see [8, 14, 18, 19] for generation results, spectral mapping, and standard energy/multiplier proofs; for localized damping and geometric control see the survey by lebeau and rauch and the literature cited therein. 5 conditions for coincidence theorem 5.1 (coincidence of stability). let {t (t)} be a c0-semigroup with generator a. kw-stability and analytic stability yield the same conclusion i� s(a) < 0 and σ(t (t)) \ {0} = etσ(a). proof. if s(a) < 0, then r(t (t)) = ets(a) < 1 for t > 0. by spectral mapping and gearhart�prüss theorem, exponential stability follows. conversely, if s(a) ≥ 0, analytic decay fails although kw-stability holds (examples 4.1, 4.2). thus both s(a) < 0 and spectral mapping are necessary and su�cient. 5.1 spectral-bound table spectral bound s(a) kw-stabilityanalytic conclusionspectral mapping s(a) < 0 always trueexponentially stableholds s(a) = 0 always trueneutral (no decay)holds s(a) > 0 always trueunstable (growth)holds ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 19 stability hierarchy diagram5.2 exponential stability uniform stability strong stability asymptotic stability koch�wallace stability (always holds) figure 4: hierarchy of analytic stability notions with koch�wallace stability in parallel. 6 conclusion this work has revealed a fundamental connection between algebraic and analytic stability in semigroups of bounded linear operators. we showed that every c0−semigroup is stable in the sense of koch�wallace, a universal algebraic property that enforces the collapse of green's relations. at the same time, we identi�ed precise spectral conditions under which this algebraic stability coincides with analytic stability in the form of decay properties such as strong, asymptotic, and exponential stability. the examples of the translation semigroup, right shift, heat semigroup, and damped wave semigroup illustrate the subtle boundary between structural invariance and spectral decay, giving rise to what may be described as a stability gap. this recognition clari�es why operator semigroups can exhibit robust algebraic structure while displaying very di�erent analytic behavior depending on their spectral placement. beyond its theoretical interest, the study o�ers insight into the analysis of evolution equations, the design of stable control systems, and the assessment of numerical schemes where stability properties are decisive. by showing that kw-stability is always present while analytic stability is conditional, we provide a uni�ed framework that advances semigroup theory and strengthens its applications in mathematics, physics, and engineering. references [1] s.b. koch and a.d. wallace, �stability in semigroups,� duke math. j., vol. 23, pp. 193�202, 1956. [2] e. hille and r. s. phillips, functional analysis and semi-groups, american mathematical society, 1957. [3] o. j. tom, o. g. udouaka, and e. s. udo�a kw�stability in semigroup of bounded linear operators, international journal of applied science and mathematical theory eissn 2489-009x p-issn 2695-1908, vol. 11 no. 7, pp. 9�14 2025 www.iiardjournals.org [4] j. east and p. higgins, stability in semigroups: green's relations and beyond, semigroup forum, 101:1�25, 2020. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 20 [5] a. bátkai and s. piazzera, semigroups for delay equations, research notes in mathematics 10, a k peters, 2005. [6] j. glück and a. mironchenko, stability criteria for positive semigroups on ordered banach spaces, j. evol. equ, vol. 25, no. 12, 1424-3199/25/010001-49, 2024. https://doi.org/10.1007/s00028-024-01044-8. [7] j. mui, spectral properties of locally eventually positive operator semigroups, semigroup forum, vol. 106, no. 2, pp. 460�480, 2023. [8] r. c. penney, self-dual cones in hilbert space, j. funct. anal., vol. 21, pp. 305�315, 1976. [9] h. h. schaefer, halbgeordnete lokalkonvexe vektorräume, iii. math. ann., 141, pp. 113�142, 1960. [10] h. h. schaefer, invariant ideals of positive operators in c(x), i, ill. j. math., 11, pp. 703�715, 1967. [11] h. vogt, stability of uniformly eventually positive c0−semigroups on lp−spaces, proc. am. math. soc., vol. 150 no. 8, pp. 3513�3515, 2022. [12] l. weis, the stability of positive semigroups on lp spaces, proc. am. math. soc., vol. 123, no. 10, pp. 3089�3094, 1995. [13] l. weis, a short proof for the stability theorem for positive semigroups on lp(µ). proc. am. math. soc., vol. 126, no. 11, pp. 3253�3256, 1998. [14] a. w. wickstead, compact subsets of partially ordered banach spaces, math. ann., 212, pp. 271�284, 1975. [15] p. p. zabre��ko and s. v. smickih, a theorem of m. g. kre��n and m. a. rutman, funktsional. anal. i prilozhen, vol. 13, no. 3, pp. 81�82, 1979. [16] b. simon, the bound state of weakly coupled schrödinger operators in one and two dimensions, ann. phys., 97, pp. 279�288, 1976. [17] josep martinez and josé m. mazon. c0−semigroups norm continuous at in�nity, semigroup forum, vol. 52, no. 2, pp. 213�224, 1996. [18] desheng li and mo jia, a dynamical approach to the perron-frobenius theory and generalized krein-rutman type theorems, j. math. anal. appl., vol. 496(2):paper no. 124828, 22, 2021. [19] matthias keller, daniel lenz, hendrik vogt, and radosª aw wojciechowski, note on basic features of large time behaviour of heat kernels, j. reine angew. math., 708, pp. 73-�95, 2015. [20] o. g. udoaka. rank of some semigroups, international journal of applied science and mathematical theory, vol. 9 no. 3, pp. 90-100, 2023. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 21 [21] m. n. john, and o. g. udoaka, algebraic and topological analysis of enveloping semigroups in transformation groups: proximal equivalence and homomorphic image, ijo international journal of mathematics, vol. 6, no. 12, pp. 9�23, 2023. http://ijojournals.com/index.php/m/article/view/769, doi; https://doi.org/10.5281/zenodo.10443958. [22] r. u. ndubuisi, o. g. udoaka, k p shum, and r b abubakar, on homomorphisms (good homomorphisms) between completely j◦simple semigroups, canadian journal of pure and applied sciences, vol. 13, no. 2, pp. 4793-4797, 2019. [23] r. u. ndubisi and o. g. udoaka, a structure theorem for left restriction semigroups of type f, international journal of semigroup theory appl., vol. 2, 2018. [24] o. j. tom, and o. g. udoaka, semigroup approach for the solution of boundary layer euation with sinc function term, ijo international journal of mathematics, vol. 8, issue 4, pp. 22�37. https://ijojournals.com/index.php/m/article/view/1057. [25] a.h. cli�ord and g.b. preston, the algebraic theory of semigroups, vols. i & ii, ams mathematical surveys, 1961/1967. [26] j.a. green, �on the structure of semigroups,� annals of mathematics, vol. 54, pp. 163�172, 1951. [27] d. daners, j. glück, and j. b. kennedy, eventually positive semigroups of linear operators, j. math. anal. appl., 433(2), pp. 1561�1593, 2016. [28] m. d. donsker and s. r. srinivasa varadhan, on a variational formula for the principal eigenvalue for operators with maximum principle, proc. natl. acad. sci. usa, 72, pp. 780�783, 1975. [29] m. d. donsker and s. r. srinivasa varadhan. on the principal eigenvalue of secondorder elliptic di�erential operators, commun. pure appl. math., 29, pp. 595�621, 1976. [30] s. friedland, characterizations of the spectral radius of positive operators, linear algebra appl., 134, pp. 93�105, 1990. [31] s. friedland, the collatz-wielandt quotient for pairs of nonnegative operators, appl. math., praha, 65(5), pp. 557�597, 2020. [32] j. mui, spectral properties of locally eventually positive operator semigroups, semigroup forum, 106(2), pp. 460�480, 2023. [33] j. east and p. m. higgins, �green's relations and stability for subsemigroups,� semigroup forum , vol. 101, no 1, pp. 77�86, 2020. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 9 | september 2025 | http://ijojournals.com/index.php/m/index 22 introduction preliminaries main results illustrative examples example 3.4 (damped wave equation) conditions for coincidence spectral-bound table stability hierarchy diagram conclusion ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || advancements in linear multi-step method for solving third order ordinary differential equations etim, uduak james department of mathematics akwa ibom state university, nigeria eno john department of general studies akwa ibom state polytechnic, ikot osurua, nigeria dr. tombotamunoa w. j. lawson department of mathematics/statistics, ignatius ajuru university of education, port harcourt, nigeria. udo ukemeobong monday department of mathematics akwa ibom state university, nigeria abstract this work addresses the development of four step linear multi-step methods for the solution of third order ordinary differential equations. the approach requires the construction of a truncation error term and expanding it in taylor’s series. the resulting four step method are analysed to show that it is consistent, zero stable and hence convergent with good interval of absolute stability.thus the new method satisfies the minimum condition for a linear multi-step method to be acceptable. the technique of derivation employed in this work is easier and more adaptable than those of collocation keywords: four-step method, third-order ordinary differential equations, truncation error, taylor's series, consistency, zero stability, convergence, absolute stability, numerical analysis. doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 1 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || 1. introduction ordinary differential equations (odes) are fundamental in modeling dynamic systems across various scientific disciplines. [1]’s paper discusses the construction of linear multistep methods, providing insights into the techniques used for their development and analysis.burden and faires' [2] textbook is a comprehensive resource for numerical analysis. chapter discussions on multistep methods offer foundational knowledge in the field.butcher's book [3] is a classic in the field, providing a deep understanding of various numerical methods for ordinary differential equations, including multistep methods. [4]’s book covers the computational aspects of ordinary differential equations, providing valuable insights into the development and analysis of numerical methods. the first volume of [5] series delves into the numerical solution of nonstiff ordinary differential equations, offering relevant information for the development of multistep methods.lambert's work [6] is a foundational resource on computational methods for ordinary differential equations, providing a historical context for the development of numerical techniques.shampine and gordon's book [7] is a classic in the field, providing practical insights into the numerical solution of ordinary differential equations, including the development of multistep methods. this paper presents a novel contribution to the field by introducing a fourstep linear multi-step method for solving third-order odes. the methodology involves the construction of a truncation error term, which is then expanded using taylor's series. the resulting four-step method undergoes a thorough analysis to establish its key properties. we demonstrate its consistency, ensuring an accurate representation of the underlying differential equation. moreover, we prove its zero stability, indicating reliable behavior, and establish its convergence with a substantial interval of absolute stability. this work represents a significant doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || advancement in numerical methods for odes, providing an efficient and acceptable solution to the complex challenges posed by third-order equations. 2. methodology this section describes the development of a foursteplinear multi-step method for the solution of initial value problems of ordinary differential equation. methodsof derivation of the new linear multi-step 2.1 i. the l. h. s is expanded by taylor’s series about h ii. the r. h. s i.e. ���� is expanded in taylor’s series about h. iii. replace the function ���� with ���� ��� and then expand in taylor’s about h iv. put the system in matrix form v. determine the values of �′� and �′� vi. finally form the new linear multi-step method vii. test the linear multi-step method for convergence. doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || from ���� = � �� � ��� ���� + ℎ� � �� ���� � ��� (2.1) implies ���� = ���� + ������+������ + ������ + ℎ�[���� ��� + ������ ��� + ������ ��� + ������ ��� + ������ ��� ](2.2) expanding the l.h.s.by taylor’s series, about h and when k = 4, we obtain; ���� = (4ℎ)� 0! �� + (4ℎ)� 1! �� � + (4ℎ)� 2! �� �� + (4ℎ)� 3! �� ��� + (4ℎ)� 4! �� �� + (4ℎ)� 5! �� � + (4ℎ)� 6! �� �� + (4ℎ)� 7! �� ��� + (4ℎ)� 8! �� ���� + ⋯ ���� = 4�ℎ��� + 4�ℎ��� � + ���� � �� �� + ���� � �� ��� + ���� �� �� �� + ���� ��� �� � + ���� ��� �� �� + ���� ���� �� ��� + ���� ����� �� ���� + ������ �� ������ + ⋯ ���� = ℎ��� + 4ℎ�� � + 16ℎ� 2 �� �� + 64ℎ� 6 �� ��� + 256ℎ� 24 �� �� + 1024ℎ� 120 �� � + 4096ℎ� 720 �� �� + 16384ℎ� 5040 �� ��� + 65536ℎ� 40320 �� ���� + 262144ℎ��� �� 362880 + (2.3) expanding the coefficients of ��, ��,��,��, ��,��, ��, ����� ��by taylor’s series about h as in equation(2.1), we obtain; doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || ����ℎ� + �� ���ℎ� + �� � ℎ + �� �� 2! ℎ� + �� ��� 3! ℎ� + �� �� 4! ℎ� + �� � 5! ℎ� + �� �� 6! ℎ� + �� ��� 7! ℎ� + �� ���� 8! ℎ� + �� �� 9‼ ℎ� + �� � 10! ℎ�� … � + �� ���ℎ� + 2�� � ℎ + 4�� �� 2! ℎ� + 8�� ��� 3! ℎ� + 16�� �� 4! ℎ� + 32�� � 5! ℎ� + 64�� �� 6! ℎ� + 128�� ��� 7! ℎ� + 256�� ����ℎ� 8! + 512�� ��ℎ� 9! + 1024�� � 10! ℎ�� … � + �� � (3ℎ)��� 0! ℎ� + (3ℎ)��� � 1! ℎ� + (3ℎ)��� �� 2! ℎ� + (3ℎ)��� ��� 3! ℎ� + (3ℎ)��� �� 4! ℎ� + (3ℎ)��� � 5! ℎ� + (3ℎ)��� �� 6! ℎ� + (3ℎ)��� ��� 7! ℎ� + (3ℎ)��� ���� 8! ℎ� + (3ℎ)��� �� 9! ℎ� + (3ℎ)���� � 10! ℎ�� + ⋯ � + ℎ� ����� ���ℎ� + �� ��� ���ℎ� + �� ��ℎ + �� � 2! ℎ� + �� �� 3! ℎ� + �� ��� 4! ℎ� + �� ���� 5! ℎ� + �� �� 6! ℎ� + �� � 7! ℎ� + ⋯ � + �� ��� ���ℎ� + 2�� ��ℎ + 4�� � 2! ℎ� + 8�� �� 3! ℎ� + 16�� ��� 4! ℎ� + 128�� �ℎ� 7! + ⋯ � + �� ��� ���ℎ� + 3�� ��ℎ + 9�� � 2! ℎ� + 27�� �� 3! ℎ� + 81�� ��� 4! ℎ� + 243�� ���� 5! ℎ� + 729�� �� 6! ℎ� + 2187�� � 7! ℎ� + ⋯ � + �� � (4ℎ)��� ��� 0! + (4ℎ)��� �� 1! + (4ℎ)��� � 2! + (4ℎ)��� �� 3! + (4ℎ)��� ��� 4! + (4ℎ)��� ���� 5! + (4ℎ)��� �� 6! + (4ℎ)��� � 7! + ⋯ �� doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || by clearing the bracket, we have; ����ℎ� + ����ℎ� + ���� � ℎ + �� �� �� 2 ℎ� + �� �� ��� 6 ℎ� + �� �� �� 24 ℎ� + �� �� � 120 ℎ� + �� �� �� 720 ℎ� + �� �� ��� 5040 ℎ� + ⋯ + ����ℎ� + 2���� � ℎ + 2���� ��ℎ� + 8���� ��� 6 ℎ� + 16���� �� 24 ℎ� + 32���� � 120 ℎ� + 64���� �� 720 ℎ� + 128���� ��� 5040 ℎ� + ��ℎ��� + 3��ℎ�� � + 9 2 ��ℎ��� �� + 27 6 ��ℎ��� ��� + 81 24 ��ℎ��� �� + 243 120 ��ℎ��� � + 729 720 ��ℎ��� �� + 2187 5040 ��ℎ��� ��� + 6561 40320 ��ℎ��� ���� + 19683 362880 ��ℎ��� �� + 59049 3628800 ��ℎ���� � … + ���� ���ℎ� + ���� ��ℎ� + ���� � 2 ℎ� + ���� �� 6 ℎ� + ���� ��� 24 ℎ� + ���� ���� 120 ℎ� + ���� �� 720 ℎ� + ���� � 5040 ℎ�� + ℎ������ �� 40320 + ℎ������ ��� 362880 + ℎ������ ���� 3628800 + ⋯ + ���� ���ℎ� + 2���� ��ℎ� + 2���� �ℎ� + 8�� �� 6 ℎ� + 16���� ��� 24 ℎ� + 32���� ���� 120 ℎ� + 64���� �� 720 ℎ� + 128���� � 5040 ℎ�� + ⋯ + ���� ���ℎ� + 3���� ��ℎ� + 9���� � 2 ℎ� + 27���� �� 6 ℎ� + 81���� ��� 24 ℎ� + 243���� ���� 120 ℎ� + 729�� ���� 270 ℎ� + 2187���� � 5040 ℎ�� + ⋯ + ℎ����� + 4ℎ����� � + 16 2 ℎ����� �� + 64 6 ℎ����� ��� + 256ℎ����� �� 24 + 1024ℎ����� � 120 + 4096ℎ����� �� 720 + (2.4) doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || equating equation (2.3) and equation (2.4), we have; ���� = ℎ��� + 4ℎ�� � + 16ℎ� 2 �� �� + 64ℎ� 6 �� ��� + 256ℎ� 24 �� �� + 1024ℎ� 120 �� � + 4096ℎ� 720 �� �� + 16384ℎ� 5040 �� ��� + 65536ℎ� 40320 �� ���� + 262144ℎ��� �� 362880 + ⋯ = ����ℎ� + ����ℎ� + ���� � ℎ + �� �� �� 2 ℎ� + �� �� ��� 6 ℎ� + �� �� �� 24 ℎ� + �� �� � 120 ℎ� + �� �� �� 720 ℎ� + �� �� ��� 5040 ℎ� + ⋯ + ����ℎ� + 2���� � ℎ + 2���� ��ℎ� + 8���� ��� 6 ℎ� + 16���� �� 24 ℎ� + 32���� � 120 ℎ� + 64���� �� 720 ℎ� + 128���� ��� 5040 ℎ� + ⋯ + ��ℎ��� + 3��ℎ�� � + 9 2 ��ℎ��� �� + 27 6 ��ℎ��� ��� + 81 24 ��ℎ��� �� + 243 120 ��ℎ��� � + 729 720 ��ℎ��� �� + 2187 5040 ��ℎ��� ��� + 6561 40320 ��ℎ��� ���� + 19683 362880 ��ℎ��� �� + 59049 3628800 ��ℎ���� � + ���� ���ℎ� + ���� ���ℎ� + ���� ��ℎ� + ���� � 2 ℎ� + ���� �� 6 ℎ� + ���� ��� 24 ℎ� + ���� ���� 120 ℎ� + ���� �� 720 ℎ� + ���� � 5040 ℎ�� + ⋯ + ���� ���ℎ� + 2���� ��ℎ� + 2���� �ℎ� + 8�� �� 6 ℎ� + 16���� ��� 24 ℎ� + 32���� ���� 120 ℎ� + 64���� �� 720 ℎ� + 128���� � 5040 ℎ�� + ⋯ + ���� ���ℎ� + 3���� ��ℎ� + 9���� � 2 ℎ� + 27���� �� 6 ℎ� + 81���� ��� 24 ℎ� + 243���� ���� 120 ℎ� + 729�� �� 270 ℎ� + 2187���� � 5040 ℎ�� + ⋯ + ℎ����� + 4ℎ����� � + 16 2 ℎ����� �� + 64 6 ℎ����� ��� + 256ℎ����� �� 24 + 1024ℎ����� � 120 + 4096ℎ����� �� 720 + ⋯ by comparing the coefficient in powers of h, we obtain; i. ℎ� ⇒ �� + �� + �� + �� = 1 ii. ℎ� ⇒ �� + 2�� + 3�� = 4 iii. ℎ� ⇒ �� � + 2�� + � � �� = �� � iv. ℎ� ⇒ �� � + �� � � + �� � ��+ �� + �� + �� + �� + �� = �� � v. ℎ� ⇒ �� �� + �� �� �� + �� �� �� + �� + 2�� + 3�� + 4�� = ��� �� vi. ℎ� ⇒ �� ��� + �� �� ��� + ��� ��� �� + �� � + 2�� + �� � � + �� � �� = ���� ��� vii. ℎ� ⇒ �� ��� + �� �� ��� + ��� ��� �� + �� � + �� � � + �� �� � + �� � �� = ���� ��� viii. ℎ� ⇒ �� ���� + ����� ���� + ������ ���� + �� �� + ���� �� + ���� �� + ����� �� = ����� ���� ix. ℎ� ⇒ �� ����� + ����� ����� + ������ ����� + �� ��� + ���� ��� + ����� ��� + ������ ��� = ����� ����� (� − ��)(2.5) doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || transforming the above into matrix form, yields ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 1 1 1 0 1 2 0 1 2 4 2 1 0 0 3 0 0 9 2 0 0 0 0 0 0 0 0 0 0 0 0 1 6 8 2 0 1 24 16 24 0 1 120 32 120 27 6 1 1 81 24 0 1 243 120 0 1 2 1 1 1 2 3 4 4 2 9 2 16 2 0 1 170 64 720 0 1 5040 128 5040 0 1 40320 256 40320 729 720 0 1 6 2187 5040 0 1 24 6561 40320 0 1 120 8 6 27 6 64 6 16 24 81 24 256 24 32 120 243 120 1024 120 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ �� �� �� �� �� �� �� �� ��⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ 1 4 16 2 64 6 256 24 1024 120 4096 720 16384 5040 65536 40320⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ … (2.6) doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || 3. analysis of basic properties of the four step method this section seek to establish the basic properties of the linear multistep method as stated in chapter one. properties of the four step method 3.1 order and error constant from equation (2.5), we obtain the follow: �� = 1 − �� − �� − �� − �� substituting 1, -2, 0,and 2 for ��, ��, �� ��� �� above we have �� = 1 − 1 − (−2) − 0 − 2 �� = 1 − 1 + 2 − 2 �� = 0. �� = 4 − �� − 2�� − 3�� whichimplies that �� = 4 − (−2) − 2(0) − 3(2) �� = 0 �� = 16 2 − �� 2 − 2�� − 9 2 �� doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || which implies that �� = 16 2 − 1 2 (−2) − 2(0) − 9 2 (2) taking the l.c.m, we have �� = 16 + 2 − 18 2 therefore �� = 0. �� = 64 6 − 1 6 �� − 8 6 �� − 27 6 �� − �� − �� − �� − �� − �� by substitution and for � ��� , � �� , �� �� , � �� , , � ��� = ��, ��, ����, �� we have �� = 1280 + 40 − 1080 − 1 − 56 − 126 − 56 − 1 120 therefore �� = −7 15 �� = 256 24 − �� 24 − �� 16 24 − 81 24 �� − �� − 2�� − 3�� − 4�� doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || substituting thevalue of the coefficient matrix, we have �� = 1280 + 10 − 3105 − 56 − 252 − 168 − 4 120 therefore �� = −2295 120 �� = 1024 120 − �� 120 − �� 32 120 − 243 120 �� − �� 2 − 2�� − �� 9 2 − 16 2 �� substituting the values of the coefficientmatrix, we have �� = 2048 + 4 − 972 − 560 − 504 − 504 − 16 120 and therefore �� = ��� �� . �� = 4096 720 − �� 720 − �� 64 720 − 729 720 �� − �� 6 − �� 8 6 − 27 6 − 64 6 �� by substitution, we have �� = 4096 720 + 2 720 − 1458 720 − 7 90 − 168 120 − 189 90 − 64 720 therefore we have that doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || �� = 0 . �� = 16384 5040 − �� 5040 − 128�� 5040 − 2187�� 5040 − �� 24 − 16�� 24 − 81�� 24 − 256�� 24 similarly upon substitution, we obtain that �� = 0. finally we have �� = 65536 40320 − �� 40320 − 256�� 40320 − 6561�� 40320 − �� 120 − 32�� 120 − 243�� 120 − 1024�� 120 substituting the value of the coefficient matrix, we have �� = �� ����� = ���� ≠ 0. hence we can deduce that our four step method is of order p = 5 with error constant ���� = �� ����� . stability and consistency 3.2 form � ������ = ℎ� � ������ � ��� ��� ��� doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 12 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || , for � = 4, if we substitute the coefficients 1,-2,0, 2, � ��� , � �� , �� �� , � �� , � ��� respectively, for ��, ��, ��, ��,��, ��, ��, ��,��, we have ���� = ���� + ������ + ������+�� + ℎ�[���� + ������ + ������ + ������ + ��] which implies, that ���� − 2���� + 2���� − �� = ℎ� � 1 120 ���� + 7 15 ���� + 21 20 ���� + 7 15 ���� + 1 120 ��� taking the l.c.m of the terms inside the square bracket, we have ���� − 2���� + 2���� − �� = 1 120 [���� + 56 ���� + 126 ���� + 56 ���� + ��] … (3.1) from equation(3.1), the first characteristic polynomial denoted by �(�) is: �(�) = �� − 2�� + 2�� − �� … (3.2) whichimplies that �(�) = �� − 2�� + 2� − 1 (3.3) and the second characteristic polynomial denoted by �(�) is: �(�) = 1 120 [�� + 56�� + 126�� + 56�� + ��](3.4) where ℎ� is ignored, which implies that �(�) = 1 120 [�� + 56�� + 126�� + 56� + 1](3.5) doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 13 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || consistency has it that a linear multi-step method must satisfies the following properties: (1) �(1) = 0 and (2)����(1) = 3! �(1). we now test conditions (1) and (2)by using(3.3) as follows: 1. �(1) = 1� − 2(1�) + 2(1) − 1 which implies that �(1) = 1 − 2 + 2 − 1 and therefore �(1) = 0 , this verifies condition 1. taking the first derivative of (3.3), we have ��(�) = 4�� − 6�� + 2 (3.6) for r = 1 we have ��(1) = 4. 1� − 6(1�) + 2 which implies that ��(1) = 4 − 6 + 2 therefore, ��(1) = 0 2. ����(1) = 3! �(1) we need the second and third derivatives of (3.6),thus doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 14 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || ���(�) = 12�� − 12� implying, ����(�) = 24� − 12, then ����(1) = 24 − 12 = 12. and �(�) = � ��� [�� + 56�� + 126�� + 56� + 1] 3! which implies that �(1) = � ��� [1� + 561� + 1261� + 56 + 1] 3! �(1) = 1 120 [1440] without loss of generality �(1) = 12 ����(1) = �(1) = 12. this result verifies condition 2. it has been established that our fourstep method satisfies the conditions1 ��� 2, hence it is convergent. the first characteristic polynomial �(�) = �� − 2�� + 2� − 1, gives the possible values of r when �(�) is zero to be (� − 1)� = 0. doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 15 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || which implies that � = 1,1,1. therefore, the four-step method satisfies the root condition, hence it is zero-stable. interval of absolute stability 3.3 we now seek to obtain the interval of absolute stability; this is done by applying the boundary locus method which is define as: ℎ(�) = �(�) �(�) (3.7) where�(�) is the first characteristic polynomial and �(�) is the second characteristic polynomial. from equation(3.3)and (3.5) �(�) = �� − 2�� + 2� − 1 and �(�) = � ��� [�� + 56�� + 126�� + 56� + 1]. then from (3.7) , we have ℎ(�) = 120[�� − 2�� + 2� − 1] �� + 56�� + 126�� + 56� + 1 (3.8) bydemoiveries’s theorem �� = ���� = ����� + ������ where ����� is the real part of ���� and ����� is the imaginary part of����, but here we will replace �� with �� . so �� = ����� + ������ doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 16 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || for n = 1 we have �� = ���� + ����� which implies that � = ���� + ����� (3.9) for n = 2, we have �� = ���2� + ����2� (3.10) for n = 3, we have �� = ���3� + ����3� (3.11) and for n = 4 we obtain �� = ���4� + ����4� (3.12) substituting equation(3.9)– (3.12)into(3.8), we have ℎ(�) = 120[���4� + ����4� − 2(���3� + ����3�) + 2(���� + �����) − 1] ���4� + ����4� + 56(���3� + ����3�) + 126(���2� + ����3�) + 56(���� + �����) + 1 which implies that ℎ(�) = 120[(���4� − 2���3� + 2���2� − 1) + �(���4� − 2���3� + 2����)] �����4� + 56���3� + 126���2� + 56���� + 1 + �(���4� + 56���3� + 126���2� + 56����)�� doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 17 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || by rationalization, we have ℎ(�) = 120[(���4� − 2���3� + 2���� − 1) + �(���4� − 2���3� + 2����)] [(���4� + 56���3� + 126���2� + 56���� + 1) − �(���4� + 56���3� + 126���2� + 56����)] [(���4� + 56���3� + 126���2� + 56����) + �(���4� + 56���3� + 126���2� + 56����)] [(���4� + 56���3� + 126���2� + 56���� + 1) − �(���4� + 56���3� + 126���2� + 56����)] which implies that ℎ(�) = 120[(���4� − 2���3� + 2���� − 1 )(���4� + 56���3� + 126���2� + 56���� + 1 )] −120�[(���4� − 2���3� + 2���� − 1 )(���4� + 56���3� + 126���2� + 56����)] +120� � (���4� − 2���3� + 2���� )(���4� + 56���3� + 126���2� + 56���� + 1 ) +120[(���4� − 2���3� + 2����)(���4� + 56���3� + 126���2� + 56���� )] � � (���4� + 56���3� + 126���2� + 56���� + 1 )(���4� + 56���3� + 126���2� + 56���� + 1 ) +�(���4� + 56���3� + 126���2� + 56���� + 1)(���4� + 56���3� + 126���2� + 56���� ) −�(���4� + 56���3� + 126���2� + 56���� + 1)(���4� + 56���3� + 126���2� + 56���� ) +(���4� + 56���3� + 126���2� + 56����)(���4� + 56���3� + 126���2� + 56���� ) � opening the bracket of the numerator and that of the denominator, we have doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 18 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || ℎ(�) = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ����������������������������������������������������������������������������� ����������������������������������������������������������������� ������������������������������������������������������������������������ ����������� �������������������������������������������������������������� ����������������������������� ���������������������������������� ���������������� ������������������������������ ������������������������� ���������������������������������������������� ���������������� �������������������������������� �������������������������������� ���������� ����������������� ��������������������� ����� ������������������������������ ���������������� ������������������������ ���� ��������������������� ���������������� ���������������������������� ������������������������������������������� ������������� ��������������� ������������������������� ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ��������������������������������������������������� ������������� ������������������������������������������������� �������������������������������������������������� �������������������������������������������������������������������������� ���������������������������������� �������������������������������������� ������������������������� ���������������������������� ������������������������������������������������������������������� �������������������������������������� ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ . since we are interested only on the real part, then we collect like terms only on the real part as follows: �(�) = 120 � ����4� + 54���4����3� + 126���4����2� + 58���4����� −112����3� − 252���3����2� − 58���� + 252�������2 + 1125����� −54����� − 126���2� − 1 + ����4� + 54���4����3� + 126���4����2� +58����4� − 112����3� − 252���3����2� + 252���2������ + 56����� � ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ����4� + 112���4����3� + 252���4����2� + 112���4����� +2���4� + 3136����3� + 14112���2����� + 6272���3����� +112���3� + 15876����2� + 14112���2����� + 112���� +126���2� + 1 + ����4� + 112���4����3� + 252���4����2� +112���4����� + 3136����3� + 14112���3����2� + 6272���3����� +15876����2� + 14112���2����� + 3136����� ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ we now evaluate �(�)for the interval 0 ≤ � ≤ 180 as follows: doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 19 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || for � = 0 we have �(�) = 120 ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ����0 + 54���0���0 + 126���0���0 +58���0���0 − 112����0 − 252���0���0 −58���0 + 252���0���0 + 1125����0 −54���0 − 126���0 − 1 + ����0 +54���0���0 + 126���0���0 + 58����0 − 112����0 −252���0���0 + 252���0���0 + 56����0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ����0 + 112���0���0 + 252���0���0 + 112���0���� +2���0 + 3136����0 +14112���0���0 + 6272���0���0 + 112���0 + 15876����0 +14112���0���0 + 112���0 + 126���0 + 1 + ����0 +112���0���0 +252���0���0 + 112���0���0 + 3136����0 + 14112���0���0 +6272���0���0 + 15876����0 + 14112���0���0 + 3136����0 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ �(�) = 120 � 1 + 54 + 126 + 58 − 112 − 252 − 58 + 252 +112 − 54 − 126 − 1 + 0 + 0 + 0 + 0 − 0 − 0 + 0 + 0 � � 1 + 112 + 252 + 112 + 2 + 3136 + 14112 + 6272 + 1 + 112 + 252 +112 + 2 + 3136 + 14112 +6272 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 15876 � therefore �(�) = 0. and for � = 180 we have doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 20 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || �(�) = 120 ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ����720 + 54���720���540 + 126���720���360 +58���720���180 − 112����540 − 252���540���360 − 58���180 +252���180���360 + 1125����180 − 54��180 − 126���360 −1125���180 − 1 + ����720 + 54���720���540 +126���720���360 + 58����720 − 112����540 + 252���540���360 +252���360���180 + 56����180 ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ ����720 + 112���720���540 + 252���720���360 +112���720���180 + 2���720 + 3136����540 + 14112���360���180 +6272���540���180 + 112���540 + 15876����360 + 14112���360���180 +112���180 + 126���360 + 1 + ����720 + 112���720���540 +252���720���360 + 112���720���180 + 3136����540 +14112���540���360 + 6272���540���180 + 15876����360 + 14112���360���180 +3136����180 ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ therefore �(�) ≅ −5. therefore our four step method has an interval of absolute stability[−5,0]. 4. conclusion the developed four-step linear multi-step method stands as a noteworthy solution for the numerical approximation of third-order ordinary differential equations. its consistency, zero stability, and convergence properties validate its reliability and effectiveness. the substantial interval of absolute stability further enhances its applicability to a wide range of dynamic systems. notably, the derivation technique employed in this work is characterized by its simplicity and adaptability, setting this method apart from traditional collocation-based approaches. this research offers a valuable contribution to numerical analysis, providing a practical and efficient tool for solving complex odes in scientific and engineering applications. doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 21 ijo international journal of mathematics (issn: 2992-4421 ) etim, uduak james* https://ijojournals.com/index.php/index volume 07 issue 01 || january., 2024 || references [1] ismail, m. s., el-tawil, m. a., & el-danaf, t. s. (2014). on the construction of linear multistep methods.abstract and applied analysis, 2014, 1-7. [2] burden, r. l., & faires, j. d. (2016). numerical analysis.cengage learning. [3] butcher, j. c. (2008). numerical methods for ordinary differential equations.john wiley & sons. [4] ascher, u. m., & petzold, l. r. (1998). computer methods for ordinary differential equations and differential-algebraic equations.siam. [5] hairer, e., nørsett, s. p., &wanner, g. (1993). solving ordinary differential equations i: nonstiff problems.springer-verlag. [6] lambert, j. d. (1973). computational methods in ordinary differential equations.john wiley & sons. [7] shampine, l. f., & gordon, m. k. (1975). computer solution of ordinary differential equations: the initial value problem.w. h. freeman and company. doi 10.5281/zenodo.10571505 ijo journals volume 07 | issue 01 | january 2024 | https://ijojournals.com/index.php/m/index 22 ijo international journal of mathematics (issn: 2992-4421 ) jyothi.mj 1 * https://ijojournals.com/ volume 08 || issue 04 || april, 2025 || *on sio4 molecular topological characterization of chemical structures* on sio4 molecular topological characterization of chemical structures jyothi. mj* , k. shivashankara** *department of mathematics, maharanis science college for women, mysore 570005, india **department of mathematics, yuvaraja’s college, university of mysore, mysore 570005, india abstract in this paper, our aim is to study valency-based molecular invariants for sio4 in a chain network. we compute the harmonic polynomial, atom bond connectivity polynomial, forgotten polynomial, geometric arithmetic polynomial, randic polynomial, reciprocal randic polynomial, symmetric division polynomial, inverse symmetric division polynomial, sigma polynomial, sombor polynomial, and their degree-base topological indices for sio4 embedded in a silicate chain network for various conditions. physio-chemical properties of chemical compounds, such as formation enthalpies, boiling points, chromatographic retention times, vapour pressure, and surface areas, can be determined using our investigated results, such as the h-index, abc-index, f-index, ga-index, r-index, rr-index, sdd-index, isddindex, s-index, and so-index. we also create graphical representations of the results that describe the dependence of topological indices on polynomial structure parameters. keywords: sio4 in a chain, abc polynomial and abc index, geometric arithmetic polynomial, randic index and reciprocal randic polynomial, sigma and sombor index 2020 mathematics subject classification: 05c07, 05c09, 05c31, 05c76, 05 c 99 ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 1 introduction one of the standard procedures used in the study of structure-property relations is the use of structure descriptors. the ability to correlate and predict physical, chemical, and biological activity (property) from a molecular structure is a challenging problem in theoretical and computational chemistry [1, 2]. a topological index is a number that describes the graph’s topology. it is one of the best quantification methods because it can be computed quickly for a large number of molecules and can be obtained directly from molecular structures. wiener, a chemist, used a topological index for the first time in 1947 while studying the relationship between molecular structure and the physical and chemical properties of certain hydrocarbon compounds [3, 17]. liu et al. discussed several aspects of graph theory in [5]-[12]. mathematical chemistry describes how to use polynomials and functions to offer instructions concealed in the symmetry of molecular graphs, and graph theory has many applications in modern chemistry, particularly organic chemistry. the atoms and bonds of a molecular structure are represented by vertices and edges, respectively, in chemical graph theory. many applications of topological indices are employed in theoretical chemistry, [13, 14], particularly qspr/qsar research. many famous researchers have studied topological indices to get information about different families of graphs [4, 15]. in qualitative structure-property relationships (qspr) and qualitative structure-activity relationships (qsar), topological indices are used directly as simple numerical descriptors in comparison with physical, biological, or chemical characteristics of molecules, which is a benefit. many researchers have worked on various chemical compounds and computed topological descriptors of various molecular graphs during the last few decades [18]. in chemical graph theory, a molecular graph is a simple connected graph that contains chemical atoms and bonds, which are often referred to as vertices and edges, respectively, and there must be a linkage between the vertices set vg and edges set eg.if two atoms have an atom-bond, then it is denoted by e ∼ f, the valency of every atom of g is actually the total number of atoms connected to f of g and it is denoted by df, [16, 19]. several polynomials closely related to degree-based indices are also introduced. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 2 in 2012, zhang introduced harmonic index [31]. the harmonic polynomial corresponding to harmonic index is defined as �(�, �) = ∑ � � ����� ��∈�(�) & �(�) = ∑ � ����� ��∈�(�) (1) in 1998, estrada et.al introduced atom bond connectivity index [21]. the abc polynomial corresponding to the abc indices is expressed as ���(�, �) = � � � ��+�� −2 ��+�� ��∈�(�) & ���(�) = � � ��+�� −2 ��+�� (2) ��∈�(�) in 2015, formula and gutmann introduced forgotten topological index or f-index [22]. the forgotten polynomial and index are defined as �(�, �) = � �[ (��)��[(��)�] ��∈�(�) & �(�) = � [ (��)� + [(��)�] (3) ��∈�(�) the first ga-index was proposed by vukicevic [23]. the geometric arithmetic polynomial and index are defined as ��(�, �) = � � ���+�� � ��+�� ��∈�(�) & ��(�) = � ���+�� � ��+�� (4) ��∈�(�) the randic polynomial and index, [24] are defined as �(�, �) = � � � ���+�� ��∈�(�) & �(�) = � 1 ���+�� (5) ��∈�(�) the reciprocal randic polynomial and index [25] are defined as ��(�, �) = � � ��� �� ��∈�(�) & ��(�) = � ��� �� (6) ��∈�(�) the symmetric division degree polynomial and index [26] are defined as ���(�, �) = � � [ (��)��[(��)�] (��)(��) ��∈�(�) & ���(�) = � [ (��)� + [(��)�] (��)(��) (7) ��∈�(�) the inverse symmetric division degree polynomial and index [27] are defined as ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 3 ����(�, �) = � � (��)(��) [ (��)��[(��)�] ��∈�(�) & ����(�) = � (��)(��) [ (��)� + [(��)�] (8) ��∈�(�) sigma polynomial and index [28] are defined as �(�, �) = � � (�����)� ��∈�(�) & �(�) = � (��−��)� (9) ��∈�(�) the concept of sombor index was recently introduced by gutman [29]. the sombor polynomial and index are defined as �(�, �) = � � �[ (��)��[(��)�] ��∈�(�) & ��(�) = � �[ (��)� + [(��)�] (10) ��∈�(�) in this study, the atom-bond partition set of sio4 in a chain network, which is partitioned according to the valencies of their si and o2 atoms, is used to generate the ten polynomials mentioned above and their corresponding indices. 1 chain of sio4 a sio4 tetrahedron, the fundamental building block of silicates, is created by fusing metal oxides or mixing metal carbonates with sand. the sio4 tetrahedron is present in almost all silicates. as shown in figure 1, a tetrahedron sio4 is a pyramid with a triangular base (a single tetrahedron sio4), and the silicon atom si is bonded with evenly spaced oxygen atoms. the resulting sio4, a silicate tetrahedron that connects with other sio4 horizontally, forms a single chain. similar to this, when two sio4 molecules join corner to corner, each one shares its o2 atoms with the other, as shown in figure 1. these two molecules of sio4 can be joined with two other molecules once this sharing process is finished. we now have a silicate chain, scqp, where p and q stand for the total number of sio4 atoms in one silicate chain and the number of silicate chains that were formed, respectively. the pq number of sio4 tetrahedrons used in the chain of sio4 scqp is shown in figure 1. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 4 1.1 result and discussion here, we have observed that there are three types of atom bonds on the bases of the valency of each atom of scqp in a chain of types of atoms, vi and vj, with valencies of respectively. three different types of atom based on the valencies (3 and 6) of atoms. table 1 provides the division of the set of atom bonds based on valency. table 1: atom type of atom-bond number of atom bonds theorem 2.1. for p >1 and p (3�� + 3� − 4)� � � + (3�� − 6� figure 1: 1 e, we have observed that there are three types of atom bonds on the bases of the in a chain of sio4 scqp. as a result, there are two different , with valencies of and dvi = 3 and three different types of atom-bonds (3 ∼ 3), (3 ∼ 6), and (6 ∼ 6) in sc based on the valencies (3 and 6) of atoms. table 1 provides the division of the set of table 1: atom-bond partition of scqp, for p = q 3 = de ∼ df = 3 3 = de ∼ df = 6 6 = de ∼ d 3p + 2 3(pq + q) − 4 3(pq − 2q and p = q, the harmonic polynomial ofscqp, is ( + 2)� � � e, we have observed that there are three types of atom bonds on the bases of the . as a result, there are two different = 3 and dvj = 6, ∼ 3), (3 ∼ 6), and (6 ∼ 6) in scqp are based on the valencies (3 and 6) of atoms. table 1 provides the division of the set of df = 6 q) + 2 (3� + 2)� � � + ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 5 proof. using table”1” enter the following formula harmonic polynomial (1), we get ��sc� � , �� = � � � ��� ����~���� + � � � ��� ����~���� + � � � ��� ����~���� this gives ��sc� � , �� = (3� + 2)� � � + (3�� + 3� − 4)� � � + (3�� − 6� + 2)� � � by taking the first derivative of the polynomial in theorem 2.1 at y = 1, we get the harmonic index of silicate network sc� � as follows: corollary 2.2. for p >1 and p = q, the harmonic index of sc� � is ���������� �� theorem 2.3. for p >1 and p = q, the abs polynomial ofsc� � is (3� + 2)� � � + 3�� + 3� − 4)� � �� + (3�� − 6� + 2)� � �� proof. using table”1” enter the following formula abc polynomial (2), we get ����sc� � , �� = � � � ����� (�)(�) ����~���� + � � � ����� (�)(�) ����~���� + � � � ����� (�)(�) ����~���� this gives ����sc� � , �� = (3� + 2)� � � + (3�� + 3� − 4)� � �� + (3�� − 6� + 2)� � �� by taking the first derivative of the polynomial in theorem 2.3 at y = 1, we get the abc index of chain ���� (sc� � ) of as follows: corollary 2.4. for p >1 and p = q, the abc index of ���������� �� theorem 2.5. for p >1 and p = q, the forgotten topological polynomial of sc� � is (3p + 2) y18+ (3p2 + 3p − 4) y45+ (3p2 − 6p + 2) y72. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 6 proof. using table”1” enter the following formula forgotten topological polynomial (3), weget ��sc� � , �� = � �[�����] ����~���� + � �[�����] ����~���� + � �[�����] ����~���� this gives ��sc� � , �� = (3� + 2)��� + (3�� + 3� − 4)��� + (3�� − 6� + 2)��� by taking the first derivative of the polynomial in theorem 2.5 at y = 1, we get the forgotten index of chain of sio4 (scpp) as follows: corollary 2.6. for p >1 and p = q, the forgotten topological index of sc� � is 351p2 −243p. theorem 2.7. for p >1 and p = q, the geometric arithmetic polynomial of sc� �(3� + 2)� √� � + (3�� + 3� − 4)� � � + (3�� − 6� + 2)� � √� proof. using table”1” enter the following formula geometric arithmetic polynomial (4), we get ���sc� � , �� = � � √���� ��� ����~���� + � � √���� ��� ��~� + � � √���� ��� ����~���� this gives ���sc� � , �� = (3� + 2)� √� � + (3�� + 3� − 4)� � � + (3�� − 6� + 2)� � √� by taking the first derivative of the polynomial in theorem 2.7 at y = 1, we get the geometric arithmetic index of chain of sio4 (sc� � ) as follows: corollary 2.8. for p >1 and p = q, the geometric arithmetic index ofsc� � �� (3� + 2)� √� � + (3�� + 3� − 4)� � � + (3�� − 6� + 2)� � √� theorem 2.9. for p >1 and p = q, the randic polynomial of sc� � �� (3� + 2)� � � + (3�� + 3� − 4)� � √� � + (3�� − 6� + 2)� � � ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 7 proof. using table”1” enter the following formula randic polynomial polynomial (5), we get ��sc� � , �� = � � � �(�)(�) ����~���� + � � � �(�)(�) ����~���� + � � � �(�)(�) ����~���� this gives ��sc� � , �� = (3� + 2)� � � + (3�� + 3� − 4)� � √� � + (3�� − 6� + 2)� � � by taking the first derivative of the polynomial in theorem 2.9 at y = 1, we get the randic polynomial index of chain of sio4 (sc� � ) as follows: corollary 2.10. for p >1 and p = q, the randic polynomial index ofsc� � is √���������.√�� ����.√� � theorem 2.11. for p >1 and p = q, the reciprocal randic polynomial ofsc� � is(3� + 2)�� + (3�� + 3� − 4)��√� + (3�� − 6� + 2)�√�. proof. using table “1” enter the following formula reciprocal randic polynomial polynomial (6), we get ���sc� � , �� = � ��(�)(�) ����~���� + � ��(�)(�) ��~� + � ��(�)(�) ����~���� this gives ���sc� � , �� = (3� + 2)�� + (3�� + 3� − 4)��√� + (3�� − 6� + 2)�√�. by taking the first derivative of the polynomial in theorem 2.11 at y = 1, we get the reciprocal randic polynomial index of chain of sio4 (sc� � ) as follows: corollary 2.12. for p >1 and p = q, the reciprocal randic polynomial index of 3(3� + 2) + 3√2(3�� + 3� − 4) + √6(3�� − 6� + 2). theorem 2.13. for p>1 and p=q. the symmetric degree polynomial of sc� � is(3�� − 3� + 4)�� + (3�� + 3� − 4)� �� � ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 8 proof. using table enter the following formula symmetric division degree polynomial (7), we get ����sc� � , �� = � � [(�)��(�)�] (�)(�) ����~���� + � � [(�)��(�)�] (�)(�) ����~���� + � � [(�)��(�)�] (�)(�) ��~� this gives ����sc� � , �� = (3� + 2)�� + (3�� + 3� − 4)� �� � + (3�� − 6� + 2)�� = (3�� − 3� + 4)�� + (3�� + 3� − 4)� �� � by taking the first derivative of the polynomial in theorem 2.19 at y = 1, we get the symmetric division degree index of chain of sio4 (sc� � as follows: corollary 2.14. for p >1 and p = q, the symmetric division degree index of sc� � is2(3�� − 3� + 4) + ������������ � theorem 2.15. for p >1 and p = q, the inverse symmetric division polynomial of sc� � is (3�� − 3� + 4)� � � + (3�� + 3� − 4)� � �� proof. using table “1” enter the following formula inverse symmetric division degree polynomial (8), we get �����sc� � , �� = � � [(�) �(�) ] (�)��(�)� ����~���� + � � [(�)��(�)�] (�)(�) ��~� + � � [(�)��(�)�] (�)(�) ����~���� this gives ����sc� � , �� = (3� + 2)� � � + (3�� + 3� − 4)� � �� + (3�� − 6� + 2)� � � = (3�� − 3� + 2)� � � + (3�� + 3� − 4)� � �� by taking the first derivative of the polynomial in theorem 2.15 at y = 1, we get the inverse symmetric division degree index of chain of sio4 sc� � as follows: corollary 2.16. for p >1 and p = q, the inverse symmetric division degree index of sc� � is ����������� �� . ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 9 theorem 2.17. for p >1 and p 3�� − 3� + 4. proof. using table “1” enter the following formula sigma polynomial (7), we get �(g, �)�sc� � , �� = � this gives ����sc� � , �� = ( = ( by taking the first derivative of the polynomial in theorem 2.19 at sigma index of chain of corollary 2.18. for p >1 and p theorem2.19. for p >1 and√ p (3�� + 3� − 4)��√� + (3�� − 6 proof. using table “1” enter the following formula somber polynomial (7), we get ���sc� � , �� = � ����~���� this gives ���sc� � , �� = (3� + 2 by taking the first derivative of the polynomial in theore somber index of chain of corollary 2.20. for p >1 and p = q, the somber index of 3� − 4)√5. and p = q, the sigma polynomial of sc� � ��(3�� + 3� 1” enter the following formula sigma polynomial (7), we get � � �(���)� ����~���� + � �(���)� ����~���� + � �( ��~� � (3� + 2) + (3�� + 3� − 4)�� + (3�� − 6� + 2 (3�� + 3� − 4)�� + 3�� − 3� + 4 by taking the first derivative of the polynomial in theorem 2.19 at y ) as follows: and p = q, the sigma index of sc� �p is 9(3p2 + 3p − 4). p = q, the somber polynomial of sc� � is (3� + 2)� 6� + 2)��√� 1” enter the following formula somber polynomial (7), we get ��(�)��(�)� � + � ��(�)��(�)� ����~���� + � �� ��~� 2)��√� + (3�� + 3� − 4)��√� + (3�� − 6� + 2 by taking the first derivative of the polynomial in theorem 2.19 at y ) as follows: q, the somber index ofsc� � �� 9(2�� − 3� + 2)√ � − 4)�� + 1” enter the following formula sigma polynomial (7), we get (���)� 2) y = 1, we get the . )��√� + 1” enter the following formula somber polynomial (7), we get �(�)��(�)� 2)��√� y = 1, we get the )√2 + 3(3�� + ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 10 1.2 results for p<q and p is odd here, in chain of sio4 (sc� � ), we observed for p < q and p is odd, atom-bonds on the bases of valency of every atom of sc� �changed. so, on the base of valency, table 2 provides the partition of the set of atom-bonds. table 2: atom-bond partition of sc� � , for p is odd and p < q type of atom-bond 3 = de ∼ df = 3 3 = de ∼ df = 6 6 = de ∼ df = 6 number of atom bonds 3(p + 1) 3pq + p + 2q − 5 3pq − 2(2p + q − 1) theorem 2.21. let p be odd and p < q. then the harmonic polynomial ofsc� � �� 3(� + 1)� � � + (3�� + � + 2� − 5)� � � + (3�� − 2(2� + � − 1)� � � proof. using the atom-bond partition from table 2, in the formula of harmonic polynomial (1), we get ��sc� � , �� = � � � ��� ����~���� + � � � ��� ��~� + � � � ��� ����~���� this gives ��sc� �, �� = 3(� + 1)� � � + (3�� + � + 2� − 5)� � � + (3�� − 2(2� + � − 1)� � � by taking the first derivative of the polynomial in theorem 2.21 at y = 1, we get the harmonic index of silicate network sc� � as follows: corollary 2.22. let p be odd and p < q. then the harmonic index of sc� � �� � + ������������� �� + 1. theorem 2.23. let p be odd and p < q. then the abc polynomial ofsc� � �� 3(� + 1)� � �(3�� + � + 2� − 5)� � �� + (3�� − 2(2� + � − 1)� � �� proof. using the atom-bond partition from table 2, in the formula of abc polynomial (2) , we get ����sc� �, �� = � � � ����� (�)(�) ����~���� + � � � ����� (�)(�) ����~���� + � � � ����� (�)(�) ����~���� ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 11 this gives ����sc� �, �� = 3(� + 1)� by taking the first derivative of the polynomial in theorem 2.23 at index of chain of sio4 (sc� �) as follows: corollary 2.24. let p be odd and p < q. then the abc index of theorem 2.25. let p be odd and p < q. then the forgotten topological polynomial of 3(p + 1)y18 + (3pq + p + 2q − 5)y proof. using the atom-bond partition from table 2, in the formula of forgotten topological polynomial (3), we get ��sc� � , �� = � ����~�� this gives ��sc� � , �� = 3(� + 1)��� by taking the first derivative of the polynomial in theorem 2.25 at forgotten topological index of chain of corollary 2.26. let p be odd and p < q. then the forgotten topological i − 189p − 54q − 27. theorem 2.27. let p be odd and p < q. then the geometric arithmetic polynomial of 3(� + 1)� √� � + proof. using the atom-bond partition from table 2, in the formula of geometric polynomial (4), we get ����sc� �, �� = � )� � � + (3�� + � + 2� − 5)� � �� + (3�� − 2(2� + by taking the first derivative of the polynomial in theorem 2.23 at y = 1, we get the ) as follows: let p be odd and p < q. then the abc index of(sc� �) is ������� �� let p be odd and p < q. then the forgotten topological polynomial of y45 + (3pq − 2(2p + q − 1))y72. bond partition from table 2, in the formula of forgotten topological �[�����] �� + � �[�����] ����~���� + � �[ ����~���� + (3�� + � + 2� − 5)��� + 2�3�� − 2(2� + by taking the first derivative of the polynomial in theorem 2.25 at y forgotten topological index of chain of sio4 scqp as follows: let p be odd and p < q. then the forgotten topological index of let p be odd and p < q. then the geometric arithmetic polynomial of + (3�� + � + 2� − 5)� � � + 2(2� + � − 1)� � √� bond partition from table 2, in the formula of geometric � � � �√��� ��� ����~���� + � � �√��� ��� ��~� + � � �√� �� ����~���� + � − 1)� � �� = 1, we get the abc �������� �� let p be odd and p < q. then the forgotten topological polynomial of scqp is bond partition from table 2, in the formula of forgotten topological [�����] � − 1)��� y = 1, we get the ndex of is 351pq let p be odd and p < q. then the geometric arithmetic polynomial of is bond partition from table 2, in the formula of geometric arithmetic ��� �� ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 12 this gives ���sc� �, �� = 3(� + 1)� √� � by taking the first derivative of the polynomial in theorem 2.27 at geometric arithmetic index of chain of corollary 2.28. let p be odd and p < q. then the geometric arithmetic index of 2(� + 3�� + 2� − 5) + √3�3�� − 2(2 3. theorem 2.29. let p be odd and p < q. then the randic polynomial polynomial of 3(� + 1)� � � + (3�� + � + 2� − proof. using the atom-bond partition from table 2, in the formula of randic polynomial polynomial (5), we get ��sc� �, �� = � ����~� this gives ��sc� � , �� = 3(� + 1)� � � + (3�� by taking the first derivative of the polynomial in theorem 2.29 at randic polynomial index of chain of corollary 2.30. let p be odd and p < q.then the randic polynomial index o (3�� + � + 2� − 5)� � �√� + 2�3�� theorem 2.31. let p be odd and p < q. then the reciprocal randic polynomial polynomial of sc� �is 3(� + 1)�� + (3�� + � proof. using the atom-bond partition from table 2, in the polynomial polynomial (6), we get ��sc� �, �� = � ����~ ) � � + (3�� + � + 2� − 5)� � � + 2�3�� − 2(2� + by taking the first derivative of the polynomial in theorem 2.27 at y = 1, we get the geometric arithmetic index of chain of sio4 sc� � as follows: let p be odd and p < q. then the geometric arithmetic index of (2� + � − 1)� + 3 √6 � + 3√6 let p be odd and p < q. then the randic polynomial polynomial of 5)� � �√� + 2�3�� − 2(2� + � − 1)�� � � bond partition from table 2, in the formula of randic polynomial � � � �(�)(�) ���� + � � � �(�)(�) ����~���� + � �� ����~���� ( �� + � + 2� − 5)� � �√� + 2�3�� − 2(2� + � − 1) by taking the first derivative of the polynomial in theorem 2.29 at y randic polynomial index of chain of sio4 (sc� �)as follows: let p be odd and p < q.then the randic polynomial index of sc � �� − 2(2� + � − 1)�� � �. let p be odd and p < q. then the reciprocal randic polynomial polynomial � + 2� − 5)��√� + �3�� − 2(2� + � − 1)��� bond partition from table 2, in the formula of reciprocal randic polynomial polynomial (6), we get � ��(�)(�) ~���� + � ��(�)(�) ��~� + � ��(�) ����~���� � − 1)�� � √� = 1, we get the let p be odd and p < q. then the geometric arithmetic index of sc� � is let p be odd and p < q. then the randic polynomial polynomial of is bond partition from table 2, in the formula of randic polynomial � �(�)(�) )�� � � y = 1, we get the c� �is let p be odd and p < q. then the reciprocal randic polynomial polynomial � formula of reciprocal randic � )(�) ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 13 ��sc� � , �� = 3(� + 1)�� + (3�� + � + 2� − 5)��√� + �3�� − 2(2� + � − 1)��� by taking the first derivative of the polynomial in theorem 2.31 at y = 1, we get the reciprocal randic polynomial index of chain of sio4 sc� �as follows: corollary 2.32. let p be odd and p < q. then the reciprocal randic polynomial index of sc� � �� 9�2�� + 18�� + 3�2� − 15� + 6√2� + 21 − 12� − 15√2. theorem 2.33. let p be odd and p < q. then the sigma polynomial of sc� �is(3�� − � − 2� + 1)�� + (4�� + � + 2� − 5)� �� � proof. using the atom-bond partition from table 2, in the formula of symmetric division degree polynomial (7), we get ����sc� � , �� = � � [(�)��(�)�] (�)(�) ����~���� + � � [(�)��(�)�] (�)(�) ����~���� + � � [(�)��(�)�] (�)(�) ��~� this gives ����sc� � , �� = 3(� + 1)�� + (3�� + � + 2� − 5)� �� � + (3�� − 2(2� + � − 1))�� = (3�� − � − 2� + 1)�� + (3�� + � + 2� − 5)� �� � by taking the first derivative of the polynomial in theorem 2.33 at y = 1, we get the symmetric division degree index of chain of sio4 (sc� � ) as follows: corollary 2.34. let p be odd and p < q. then the symmetric division degree index of sc� � is � � (69�� + 7� + 14� − 567). theorem 2.35. let p be odd and p < q. then the inverse symmetric division polynomial of sc� �is(3�� − � − 2� + 1)� � � + (3�� + � + 2� − 5)� � ��. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 14 proof. using the atom-bond partition from table 2, in the formula of inverse symmetric division polynomial (8), we get �����sc� �, �� = � ����~ this gives �����sc� �, �� = 3(� + 1)� � � + = (3�� − � − 2 by taking the first derivative of the polynomial in theorem 2.39 at inverse symmetric division index of chain of corollary 2.36. let p be odd and p < q. then the inverse symmetric division index of . theorem 2.37. let p be odd and p < q. then the sigma polynomial of y9+ 3pq − p − 2q + 1. proof. using the atom-bond partition from table 2, in the formul we get ��sc� � , �� = � ���� this gives ����sc� � , �� = 3(� + 1) + = (3�� by taking the first derivative of the polynomial in theorem 2.39 at sigma index of chain of corollary 2.38. let p be odd and p < q. then the sigma index of bond partition from table 2, in the formula of inverse symmetric polynomial (8), we get � � (�)(�) [(�)��(�)�] ~���� + � � (�)(�) [(�)��(�)�] ��~� + � �[ ����~���� + (3�� + � + 2� − 5)� � �� + (3�� − 2(2� − 2(2 2� + 1)� � � + (3�� + � + 2� − 5)� � ��. by taking the first derivative of the polynomial in theorem 2.39 at y inverse symmetric division index of chain of sio4 sc� � as follows: let p be odd and p < q. then the inverse symmetric division index of let p be odd and p < q. then the sigma polynomial of sc� �is (3 bond partition from table 2, in the formula of sigma polynomial (8), � �(���)� �~���� + � �(���)� ����~���� + � �(��� ��~� ) + (3�� + � + 2� − 5)�� + (3�� − 2(2� − 2(2 ( + � + 2� − 5)�� + 3�� − � − 2� + 1. by taking the first derivative of the polynomial in theorem 2.39 at y as follows: let p be odd and p < q. then the sigma index of sc� �is (3pq + p bond partition from table 2, in the formula of inverse symmetric (�)(�) [(�)��(�)�] (2� + � − 1))� � � y = 1, we get the let p be odd and p < q. then the inverse symmetric division index of is (3pq + p + 2q − 5) a of sigma polynomial (8), �)� (2� + � − 1)) y = 1, we get the p + 2q − 5). ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 15 theorem 2.39. let p be odd and p < q. then the somber polynomial of sc� � is 3(� + 1)��√� + (3�� + � + 2� − 5)��√� + (3�� − 2(2� + � − 1)��√�. proof. using the atom-bond partition from table 2, in the formula of somber polynomial (8), we get ���sc� � , �� = � ��(�)��(�)� ����~���� + � ��(�)��(�)� ����~���� + � ��(�)��(�)� ��~� this gives ���sc� � , �� = 3(� + 1)��√� + (3�� + � + 2� − 5)��√� + (3�� − 2(2� + � − 1)��√� by taking the first derivative of the polynomial in theorem 2.39 at y = 1, we get the somber index of chain of sio4 (scqp) as follows: corollary 2.40. let p be odd and p < q. then the somber index of sc� � �� (3�� + � + 2� − 5)3√5 + (6�� − 5� − 4� + 5))3√2 conclusion in this article, two important silicon tetrahedron compound structures are considered, and the accurate formulas of some important valency-based topological indices are calculated using the technique of atom-bonds partitioning of these molecular structures. our investigated results, such as the h-index, abc-index, f-index, ga-index, r-index, rrindex, sdd-index, isdd-index, s-index and so-index are useful for determining physio-chemical properties of chemical compounds, as in 2005, zhou explain in [34], such as formation enthalpies, boiling points, chromatographic retention times, vapour pressure, and surface areas. the obtained results are also innovative and noteworthy contributions to network science, providing a foundation for understanding the deep topology of these important networks. these findings, may also be useful in determining the role of silicon-carbon in electronics and industry. we also present a numerical comparison of topological characterizations for p = q for the sio4 chain in (scqp) in table 3 and a graphical comparison in figure 2. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 16 table 3: topological characterizations of p=q h abc f 2 6.11 9.56 918 3 12.61 20.39 2430 4 21.44 35.22 4644 5 32.61 54.06 7560 6 46.11 76.89 11178 7 61.94 103.72 15498 8 80.11 134.56 20520 9 100.61 169.39 26244 10 123.44 208.22 32670 figure 2: graphical comparison open problems for the characterization of the chain of discuss or research the these open problems. table 3: topological characterizations of and p is odd ga r rr sdd isdd 17.02 6.30 88.30 72.5 8.73 36.66 13.04 195.71 164 19.53 63.78 22.20 343.27 290 34.93 98.34 33.77 530.99 450.5 54.93 11178 140.38 47.76 758.86 645.5 79.53 15498 189.89 64.16 1026.89 875 108.73 20520 246.86 82.97 1335.07 1139 142.53 26244 311.29 104.20 1683.39 1437.5 180.93 32670 383.18 127.84 2071.88 1770.5 223.93 figure 2: graphical comparison for the characterization of the chain of sio4 the followers are invited to discuss or research the these open problems. is odd isdd so 8.73 144.83 19.53 354.67 34.93 655.67 54.93 1047.84 79.53 1531.16 108.73 2105.65 142.53 2771.30 180.93 3528.11 223.93 4376.08 the followers are invited to ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 17 references [1] chu, yu-ming, et al. ”computation of zagreb polynomials and zagreb indices for benzenoid triangular & hourglass system.” polycyclic aromatic compounds (2022): 110. [2] mekapati, suresh babu, and corwin hansch. ”comparative qsar studies on bibenzimidazoles and terbenzimidazoles inhibiting topoisomerase i.” bioorganic & medicinal chemistry 9.11 (2001): 2885-2893. [3] wiener, harry. ”structural determination of paraffin boiling points.” journal of the american chemical society 69.1 (1947): 17-20. [4] costa, paulo cs, et al. ”chemical graph theory for property modeling in qsar and qsprcharming qsar & qspr.” mathematics 9.1 (2020): 60. [5] j.-b. liu, c. wang, s. wang, and b. wei, zagreb indices and multiplicative zagreb indices of eulerian graphs, bulletin of the malaysian mathematical sciences society, vol. 42, no. 1, pp. 6778, 2019. [6] j. b. liu, j. zhao, j. min, and j. cao, ,e hosoya index of graphs formed by a fractal graph, fractals, vol. 27, no. 8, article id 1950135, 2019. [7] j.-b. liu and s. n. daoud, number of spanning trees in the sequence of some graphs, complexity, vol. 2019, article id 4271783, 22 pages, 2019. 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[14] zhang, ying-fang, et al. ”connecting sio 4 in silicate and silicate chain networks to compute kulli temperature indices.” molecules 27.21 (2022): 7533. [15] sourav mondal, arindam dey, nilanjan de, and anita pal, qspr analysis of some novel neighbourhood degree-based topological descriptors, complex & intelligent systems 7 (2021), no. 2, 977996. [16] alam, ashraful, et al. ”degree-based entropy for a non-kekulean benzenoid graph.” journal of mathematics 2022 (2022). [17] al-ahmadi, bashair, anwar saleh, and wafa al-shammakh. ”downhill zagreb topological indices and m dn-polynomial of some chemical structures applied for the treatment of covid-19 patients.” open journal of applied sciences 10.04 (2021): 395. [18] anton b zakharov, dmytro k tsarenko, and vladimir v ivanov, topological characteristics of iterated line graphs in the qsar problem: a multigraph in the description of properties of unsaturated hydrocarbons, structural chemistry (2021), 111. [19] natarajan, vanasundaram, et al. ”effect of electron-phonon interaction and valence band edge shift for carrier-type reversal in layered zns/rgo nanocomposites.” journal of colloid and interface science 586 (2021): 39-46. [20] zhong, lingping. ”the harmonic index for graphs.” applied mathematics letters 25.3 (2012): 561-566. [21] estrada, ernesto, et al. ”an atom-bond connectivity index: modelling the enthalpy of formation of alkanes.” (1998). ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 19 [22] furtula, boris, and ivan gutman. ”a forgotten topological index.” journal of mathematical chemistry 53.4 (2015): 1184-1190. [23] vukicevic, damir, and boris furtula. ”topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges.” journal of mathematical chemistry 46.4 (2009): 1369-1376. [24] li, xu, et al. ”bounds on general randic index for f-sum graphs.” journal of mathematics 2020 (2020). kulli, v. r., b. chaluvaraju, and h. s. boregowda. ”connectivity banhatti indices for certain families of benzenoid systems.” journal of ultra chemistry 13.4 (2017): 81 [25] farrukh, fatima, rashid farooq, and mohammad r. farahani. ”calculating some topological indices of sio2 layer structure.” journal of informatics and mathematical sciences 8.3 (2016): 181-187.. [26] ghorbani, modjtaba, samaneh zangi, and najaf amraei. ”new results on symmetric division deg index.” journal of applied mathematics and computing 65.1 (2021): 161176. [27] aguilar-snchez, r., et al. ”analytical and computational properties of the variable symmetric division deg index.” arxiv preprint arxiv:2106.00913 (2021). [28] shpiz, grigory b., and alexander p. kryukov. ”the method of colored graphs for simplifying expressions with indices.” programming and computer software 47.1 (2021): 25-28. [29] gutman, ivan. ”geometric approach to degree-based topological indices: sombor indices.” match commun. math. comput. chem 86.1 (2021): 11-16. [30] kulli, v. r. ”on the product connectivity reverse index of silicate and hexagonal networks.” rn 55 (2017): 7. [31] hu, min, et al. ”on distance-based topological descriptors of chemical interconnection networks.” journal of mathematics 2021 (2021). [32] kulli, v. r. ”the sum connectivity revan index of silicate and hexagonal networks.” annals of pure and applied mathematics 14.3 (2017): 401-406. [33] kulli, v. r., and ivan gutman. ”computation of sombor indices of certain networks.” ssrg int. j. appl. chem 8.1 (2021): 1-5. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 20 [34] zhou, bo, and ivan gutman. ”further properties of zagreb indices.” match commun. math. comput. chem 54.1 (2005): 233-239. ijo international journal of mathematics (issn: 2992-4421 ) ijo journals volume 08 | issue 04 | april 2025 | https://ijojournals.com/index.php/m/index 21 introduction 1 chain of sio4 1.1 result and discussion 1.2 results for p<q and p is odd conclusion references ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis abstract: diphtheria is a bacterial infectious disease that can lead to severe complications and even deaths. this work presents the caputo-fabrizio fractional derivatives of the aged-structured deterministic model of diphtheria infection. the existence and the uniqueness of the solution of the model are investigated and established using the contraction principle. the stability of the model is investigated with the help of the well-known ulem-hyers and the generalized ulem-hyers theorems. analyzing the model using the laplace adomian decomposition methods, the system’s analytical solution, in the form of an infinite series that converges quickly to it exact value is obtained. keywords: diphtheria, caputo-fabrizio, adomian decomposition, aged-structured, contraction principle introduction 1.1 introduction diphtheria is one of the respiratory diseases raphaging the population in recent time. it is a bacterial (corynebacteriumdiptheriae) infectious disease that can lead to severe complications such as respiratory failure, heart problems and even deaths if it is not detected early. this infection that mostly affects the throat and the nose can be prevented by vaccination. case-fatality occurs only in places where there is poor sanitation condition and inadequate vaccination coverage as a result of low resources [1,2,3] diphtheria is a highly contagious infection that spreads primarily through person-to-person contact via respiratory droplets(coughing, sneezing or spitting) or direct contact with infected skin lessions or any material (like clothe) that has been in contact with the bacteria. it is possible to get diphtheria more than once. anyone who is not protected by diphtheria vaccine and comes in close contact with diphtheria is susceptible. some of the symptoms/signs of diphtheria are: throat pain, weakness/ fatigue, fever, swollen neck glands, problems breathing due to tissues obstructing nose and throat, nerve, kidney or heart problems ( if the bacteria enters the blood stream). the incubation period is one to ten days after exposure( 4, 5, 6,7,8) thomas, henry sylvester and udofia, ekere sunday akpan, ubong dominic department of mathematics, aksu uwakwe, joy ijeoma, department of mathematics,alvan ikoku university of education, owerri department of mathematics akwaibom state university, ikotakpaden, akwaibom state, nigeria ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 38 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" to effectively contend the spread of diphtheria, different control measures, such as isolation of patient, maintenance of one meter between patients, keeping patient care areas with good ventilation, the use mask that is medically prepared and cover any wound/lesions on patient’s body by patient who may have to move out of the isolation areas, population subgroup such as young children under five years of age, school children, elderly who are at greater risk and have close contact with diphtheria infection and health workers should highly prioritized with treatment and vaccination, epidemiological surveillance ensuring early detection of diphtheria outbreak, administering antitoxin to neutralize the toxin and antibiotics to kill the bacteria, reducing complication and mortality should be implemented [6,7,8]. understanding, describing and analyzing the dynamics of infectious diseases [ 9, 10, 11, 12, 13, 14, 15,16, 17,18] have been key in guiding decisions and policies in public health system. in recent time, mathematicians and epidemiologist have demonstrated great effort in understanding and describing the dynamics of diphtheria infection. [19], presented the mathematical model of diphtheria diseases that categorizes the individuals based on susceptibility, vaccination, infected and recovery status. the stability of the system wa confirmed, the basic reproductive ratio was calculated. they also converted the deterministic model into caputo-fabrizio fractional order model, analyzed it for existence and uniqueness of solution using appropriate principle. adomian decomposition method was applied for numerical solution of the model. in the research under consideration, we seek to transform the aged-structured deterministic model of diphtheria infection [8,9] into the caputo-fractional order of aged-structured model of diphtheria infection. the existence and the uniqueness of the solution of the model shall be investigated and established using the contraction principle. the stability of the model shall be investigated with the help of the well-known ulem-hyers and the generalized ulem-hyers theorems. analyzing the model using the laplace adomian decomposition methods, the system’s analytical solution, in the form of an infinite series that converges quickly to it exact value shall be obtained. 2.0 model formulation 2.1 assumptions 1. the control of diphtheria is based on primary prevention of disease by ensuring high population immunity of the infant ((0-1 year) and school children by vaccination 2. isolation of detected cases (that is confirmed cases are not allowed to interact with the population freely. 3. epidemiological surveillance ensuring early detection through contact tracing is carried out 4. secondary prevention spread by the rapid investigation of close contacts to ensure prompt treatment of those infected ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 39 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" 5. the total population of human at time � under consideration denoted by �� is split into mutually exclusive sub-population of �� susceptible infant at time � (0-1years), �� , susceptible school children population at time �, �, vaccination population at time �, �, exposed population at time � , ��, asymptomatic infection population at time �, ��., symptomatic infection population at time �, �., recovered population at time �. ��., detected infectious humans at time �(asymptomatic and symptomatic) population through testing, �� = �� + �� + � + � + �� + �� + � + �� 2.2 state variables �� −the total population of human at time � under consideration �� − susceptible infantat time � (0-1years), �� − susceptible school children populationat time �, � −vaccination populationat time �, � − exposed populationat time , ��. − asymptomatic infection populationat time �, ��. − symptomatic infection populationat time �, � −recovered populationat time �. ��. − detected infectious humansat time �(asymptomatic and symptomatic) population through testing, 2.3 parameters �� − �������� ���� ���� ������� �� ��, asymptomatic infection population �� − �������� ���� ����� ������� �� ��, symptomatic infection population � − �������� �� ��� ��������� �ℎ�� ��� ��, asymptomatic 1 − � − �������� �� ��� ��������� �ℎ�� ��� ��, symptomatic ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 40 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" �� − ��������� ������������ ���� ���� �� �� − ��������� ������������ ���� ���� �� �� − ����������� �������� ��� �� �� − ����������� �������� ��� �� � − ������� �������� � − �������� ���� ���� �� �� �� � − ��� ������ ����ℎ ���� �� ℎ����� ��� − ������������ ���� �� �� ��� − ������������ ���� �� �� � − ���� �� ����������� �� − ��������� ���� (��� ������� ������� ) ��� �� �� − ��������� ���� (��� ������� ������� ) ��� �� � − ����ℎ ��� �� ��������� � − ������� ����ℎ ���� ��(�) = ��(���� + ����) �� − �� ��(�) = ��(���� + ����) �� − �� 2.4 model equations ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 41 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" using the above described state variables and parameters together with the schematic diagram in figure 1, the model of the diphtheria infection transmission dynamics results in the following system of deterministic non-linear first order differential equations ��� �� = ��� − ��(���� + ����) �� − �� �� − ��1�1 − ��� − ��1 (2.1) ��� �� = ��� − ��(���� + ����) �� − �� �� − ��2�2 − ��2 (2.2) �� �� = ��1�1 + ��2�2 − �� (2.3) �� �� = ��(���� + ����) �� − �� �� + ��(���� + ����) �� − �� �� − �1�� − �2�1 − ��� − �� (2.4) ��� �� = �1�� − ���1 − �1�1 − ��1 − ��1 (2.5) ��� �� = �2�1 − ��� − ���2 − �2�2 − ��2 − ��2 (2.6) ��� �� = �1�1 + �2�2 − ���� − ��� − ��� (2.7) �� �� = ���1 + ���2 + ���� − �� (2.8) 2.5 fractional order model thecaputo-fabrizio order derivatives of (2.1) –(2.8) is given as follows �� � ��(�)� �� = ��� − ��(���� + ����) �� − �� �� − ����� − ��� − ��� (3.1) �� � ��(�)� �� = ��� − ��(���� + ����) �� − �� �� − ����� − ��� (3.2) �� � �(�)� �� = ����� + ����� − �� (3.3) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 42 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" �� � �(�)� �� = ��(���� + ����) �� − �� �� + ��(���� + ����) �� − �� �� − ���� − ��(1 − �)� − �� (3.4) �� � ��(�)� � = ���� − ���� − ���� − ��� − ��� (3.5) �� � ��(�)� �� = ��(1 − �)� − ���� − ���� − ��� − ��� (3.6) �� � ��(�)� �� = ���� + ���� − ���� − ��� − ��� (3.7) �� � �(�)� �� = ���� + ���� + ���� − �� (3.8) ssss with the initial conditions, ��(0) = ���, ��(�) = ���, �(�) = ��, �(�) = ��, ��(�) = ���, ��(�) = ���, , ��(�) = ����(�) = �� ��(�) + ��(�) + �(�) + �(�) + ��(�) + ��(�) + ��(�) + �(�) = 1 �� � � �� represents the caputo-fabrizio fractional derivative of order � �[0, 1] 3.0 analysis of the model 3.1 existence and uniqueness of solution we shall use some basic fixed point theorem to establish the existence and uniqueness of the solution of (3.1) –(3.8) �� � �(�)� �� = ℒ(�, �(�) (3.9) �(0) = �� �(�) = ���(�), ��(�), �(�), �(�), ��(�), ��(�), ��(�), �(�)� � ∈ ℜ� ��� � ∈ [0. ����] denotes the state of the model and ℒ represent the continuous vector given below ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 43 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ℒ = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ ℒ2 ℒ3 ℒ4 ℒ5 ℒ6 ℒ7 ℒ8 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ ��� − ��(���������) ����� �� − ����� − ��� − ��� ��� − ��(���������) ����� �� − ����� − ��� ����� + ����� − �� − �� ��(���������) ����� �� + ��(���������) ����� �� − ���� − ��(1 − �)� − �� ���� − ���� − ���� − ��� − ��� ��(1 − �)� − ���� − ���� − ��� − ��� ���� + ���� − ���� − ��� − ��� ���� + ���� + ���� − �� ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ (3.10) the initial condition of the variable of the model is denoted by ���(0), ��(0), �(0), �(0), ��(0), �20, ��0, �0� where �(0) = ���(0), ��(0), �(0), �(0), ��(0), ��(0), ��(0), �(0)� � ��� , �� = (���, ���, ��, ��, ���, �20, ��0, �0� in addition, we define ℒ: [0. ����] × ℜ� → ℜ� is said to satisfy lipschitz condition in the second argument, if we have: �ℒ��, ��� − ℒ��, ���� ≤ ���� − ��� ∀ � ∈ [0, ����] ∀ ���, �� ∈ ℝ� �ℎ��� � > 0, ���� �� ����� ���� … … … … … … (3.11) the existence of a unique solution to the model (3.1) (3.8)is established in the following theorem: theorem 3.1. there exists a unique solution to the initial value problem (3.9) on �([0, ����], ℝ�), provided that (3.11) and � 2(1 − �)� (2 − �)ℱ(�) + 2�� (2 − �)ℱ(�) ����� < 1 … … … … … … . . (3.12) are satisfied. proof: if we apply the caputo-fabrizio fractional integral on each sides of (3.9), then we havethecaputo-fabrizio time-fractional integral of the function �(�) of order � is defined by �(�) = �� + 2(1 − �) (2 − �)ℱ(�) �(�, �) + 2� (2 − �)ℱ(�) � ℒ(�, �(�)�� � 0 (3.13) let us defined the operator κ: �([0, ����], ℝ�) → �([0, ����], ℝ�) by ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 44 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" κ[�](�) = �(�), �, v ∈ ��[0, ����], ℝ�� (3.14) where �(�) = � 0 + 2(1 − �) (2 − �)ℱ(�) �(�, �) + 2� (2 − �)ℱ(�) � ℒ(�, �(�)�� � � the supremum norm on ��[0, ����], ℝ�� is given by: ‖�(�)‖ = ‖�(�)‖ �∈[�,����] ��� , ∀ � ∈ �([0, ����], ℝ�) clearly, �([0, ����], ℝ�) equipped with ‖. ‖ is a banach space. suppose, ℘ is the fixed point of the operator κ: �([0, ����], ℝ�) → �([0, ����], ℝ�), then ℘ becomes the solution of the initial value problem (3.9), and κ℘(�) = ℘(�) where ℘(�) = � 0 + 2(1 − �) (2 − �)ℱ(�) ℒ(�, ℘(t)) + 2� (2 − �)ℱ(�) � ℒ(�, ℘(�)�� � � consider ‖k[�](�) − k[℘](�)‖ = �� 0 + 2(1 − �) (2 − �)ℱ(�) ℒ(�, v(t)) + 2� (2 − �)ℱ(�) � ℒ(�, �(�)�� � � − �� 0 + 2(1 − �) (2 − �)ℱ(�) ℒ(�, ℘(t)) + 2� (2 − �)ℱ(�) � ℒ(�, ℘(�)�� � � �� ‖k[�](�) − k[℘](�)‖ ≤ � 2(1 − �) (2 − �)ℱ(�) 1 γ(�) �ℒ��, v(t)� − ℒ��, ℘(t)�� + 2� (2 − �)ℱ(�) �[ℒ(�, �(�) − ℒ(�, ℘(�)]�� � � � ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 45 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ≤ 2(1 − �) (2 − �)ℱ(�) ��ℒ��, v(t)� − ℒ��, ℘(t)��� + 2� (2 − �)ℱ(�) ��[ℒ(�, �(�) − ℒ(�, ℘(�)]�� � � � (3.15) since the operator ℒsatisfies the lipschitz condition (eq. 3.11), we have that ‖k[�](�) − k[℘](�)‖ ≤ 2(1 − �) (2 − �)ℱ(�) �‖v(t) − ℘(t)‖ + 2�� (2 − �)ℱ(�) �‖v(t) − ℘(t)‖�� � � ‖k[�](�) − k[℘](�)‖ ≤ 2(1 − �) (2 − �)ℱ(�) � ‖v(t) − ℘(t)‖ �∈[�,����] ��� + 2�� (2 − �)ℱ(�) � ‖v(t) − ℘(t)‖�� � ��� � �∈[�,����] (3.16) ≤ 2(1 − �)� (2 − �)ℱ(�) + 2�� ∫ �� � � (2 − �)ℱ(�) ‖v − ℘‖ ≤ 2(1 − �)� (2 − �)ℱ(�) + 2������ (2 − �)ℱ(�) ‖v − ℘‖ thus if the condition (3.12)holds then, ‖k[�](�) − k[�](�)‖ ≤ 2(1 − �)� (2 − �)ℱ(�) + 2������ (2 − �)ℱ(�) ‖v − w‖ hence, the operator k becomes a contraction. therefore k has a unique fixed point which is a solution to the initial value problem (3.9)and hence asolution to the system ((3.1)-(3.8)). ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 46 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ulam-hyersstability theulam-hyers (uh) stability and generalized uh stability [20,21]for the fractional system (3.1) –(3.8) using the caputo-fabrizio operator is discussed in this section.lets = �([0, ����]: ℝ�) be the space of all continuous functions from [0, ����]toℝ�, endowed with the norm:‖�‖ = ‖�‖ �∈[�,����] ��� , consider �� � � �� �(�) = ℒ��. �(�)� (3.17) �(�) = � 0 also let ε > 0. consider the following inequality: � �� � � �� ��(�) − ℒ��. ��(�)�� ≤ �, � �ℑ, � = max(��)� , � = 1,2,3, … ,8, �� ∈ s (3.18) remark 3.1. “a function � ∈ s satisfies the inequality (3.18)if and only if there exists a function p ∈ s, having the following properties; (�)|�(�)| ≤ �, � = max���� , � �ℑ (ii) �� � � � ��(�) = ℒ��. ��(�)� + �(�), � �ℑ definition 3.1. the fractional model ((3.1)-(3.8))or the transformed system (3.17)is uh stable if for every ε > 0 there exists k > 0, such that for any solution φ ∈ s of the inequality (3.18), there exists a unique solution � ∈ s, of the fractional system (3.17)such that the following inequality is satisfied: ‖��(�) − �(�)‖ ≤ ��, � �ℑ� = max���� , � = 1, 2, 3, … , 8 (3.19) where �(�) = ���(�), ��(�), �(�), �(�), ��(�), ��(�), ��(�), �(�)� � ��(�) = ���̅ � (�), �� � (�), ��(�), ��(�), �� �(�), �� �(�), �� � (�), ��(�)� � �(0) = ���(0), ��(0), �(0), �(0), ��(0), ��(0), ��(0), �(0)� � definition 3.2. themodel system (3.17) is generalized uh stable if there exists a continuous function�: ℝ+ → ℝ+satisfying�(0) = 0, such that for anysolution � ∈ s of system (3.18), there exists a unique solution � ∈ s such that the following inequality is satisfied: ‖��(�) − �(�)‖ ≤ �(�), � �ℑ� = max ���� � , � = 1, 2, 3, … , 8 (3.20) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 47 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" theorem 3.4. if φ ∈ ε satisfies the system (3.17), then we have the following: ���(�) − �� � (�) − 2(1 − �) (2 − �)ℱ(�) �(�, �) + 2� (2 − �)ℱ(�) � ℒ��. ��(�)��� � 0 � ≤ ωε �ℎ��� ω = 2(1 − �) (2 − �)ℱ(�) + 2� (2 − �)ℱ(�) � �� � 0 (3.19) proof: using remark 3.1(ii) �� � � �� ��(�) = ℒ��. ��(�)� + �(�), � �ℑ, which on applying the caputofabrizio integral gives ��(�) = ��� (�) + 2(1 − �) (2 − �)ℱ(�) �(�, �) + 2� (2 − �)ℱ(�) � ℒ��. ��(�)��� � � + 2(1 − �) (2 − �)ℱ(�) �(�) + 2� (2 − �)ℱ(�) � �(�)�� � � by rearranging, applying norm on both sides and using remark 4.1 (i), it follows that; ���(�) − ��� (�) − 2(1 − �) (2 − �)ℱ(�) �(�, �) − 2� (2 − �)ℱ(�) � ℒ��. ��(�)��� � � � ≤ 2(1 − �) (2 − �)ℱ(�) �(�) + 2� (2 − �)ℱ(�) �|�(�)|�� � � ≤ ωε theorem 3.5. supposeℒ: ℑ × ℝ� → ℝ� satisfies the lipschitz condition, with lipschitz constant � > 0 and(1 − ω) � > 0, then the model (3.17)is generalized uh stable. proof: suppose that �∈s satisfies the inequality in (3.18)and �∈s is a unique solution of (3.17). then ∀ε > 0; ��[0, ����], we have ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 48 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" |��(�) − �(�)| = ���(�) − ��� (�) − 2(1 − �) (2 − �)ℱ(�) ��(�, �) − 2� (2 − �)ℱ(�) � ℒ��. ��(�)��� � � � ��[�,����] ��� ≤ ���(�) − ��� (�) − 2(1 − �) (2 − �)ℱ(�) ��(�, �) − 2� (2 − �)ℱ(�) � ℒ��. ��(�)��� � � � ��[�,����] ��� + �� 2(1 − �) (2 − �)ℱ(�) ��(�, �) − 2(1 − �) (2 − �)ℱ(�) �(�, �)�� ��[�,����] ��� + 2� (2 − �)ℱ(�) ��ℒ��. ��(�)� − ℒ��. �(�)�� � � �� ��[�,����] ��� ≤ ���(�) − ��� (�) − 2(1 − �) (2 − �)ℱ(�) ��(�, �) − 2� (2 − �)ℱ(�) � ℒ��. ��(�)��� � � � ��[�,����] ��� + 2(1 − �) (2 − �)ℱ(�) |��(�, �) − �(�, �)| ��[�,����] ��� + 2� (2 − �)ℱ(�) �|��(�) − �(�)| � � �� ��[�,����] ��� ≤ �ω + ωℳ|�� − �| thus, we have ‖�� − �‖ ≤ ��, �ℎ��� � = � ���ℳ (3.20) hence equating �(�) = ��, so that �(0) = 0we conclude that the model (3.16),is both uh and generalized uh stable. 4.0 iterative schemes involving the caputo fractional operators this section, we shall study an iterative scheme using the caputo-fabrizio fractional derivative. we seek to derive explicit expressions for the unknown functions,��(�), ��(�), �(�), �(�), ��(�), ��(�), �(�), ��(�) using series representation approach based on ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 49 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" laplace adomian decomposition method. the system is transformed to algebraic equations by the application of laplace transform to the caputo-fabrizio fractional order derivative (3.1) – (3.8). laplace adomian decomposition method empowers us to construct a convergent series solution for ��(�), ��(�), �(�), �(�), ��(�), ��(�), �(�), ��(�)which can be evaluated numerically to obtain accurate approximations. for the solution of themodel(3.1)-(3.8), we shall adopt the laplace adomian decompositionmethod. applying the laplace transform of the caputo-fabrizio fractional operator toboth sides of the system (3.1)-(3.8), we have ℒ � �� � ��(�)� �� � = ℒ ���� − ��(���� + ����) �� − �� �� − ����� − ��� − ���� (4.1.1) ℒ � �� � ��(�)� �� � = ℒ ���� − ��(���� + ����) �� − �� �� − ����� − ��� � (4.1.2) ℒ � �� � �(�)� �� � = ℒ{����� + ����� − �� − ��} (4.1.3) ℒ � �� � �(�)� �� � = ℒ � ��(���� + ����) �� − �� �� + ��(���� + ����) �� − �� �� − ���� − ��(1 − �)� − ��� (4.1.4) ℒ � �� � ��(�)� �� � = ℒ{���� − ���� − ���� − ��� − ���} (4.1.5) ℒ � �� � ��(�)� �� � = ℒ{��(1 − �)� − ���� − ���� − ��� − ���} (4.1.6) ℒ � �� � �(�)� � � = ℒ{���� + ���� + ���� − �� } (4.1.7) ℒ � �� � ��(�)� �� � = ℒ{���� + ���� − ���� − ��� − ��� } (4.1.8) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 50 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" using the property of laplace transform for caputo-fabrizio fractional derivatives, we obtain following the definition of laplace transform for the caputo-fabrizio derivative, (laplace transform of the caputo-fabrizio derivative of functions) the laplace transform of the caputo-fabrizio derivative is given by ℒ{ �� � � �� �(�, �)}(�) = (2 − �)ℱ(�) 2 �ℒ{�(�, �)} − �(�, 0) � + �(1 − �) �ℒ{�1(�)} − �1(0) � + �(1 − �) = ℒ ���� − ��(���� + ����) �� − �� �� − ����� − ��� − ���� (4.2.1) �ℒ{�2(�)} − �2(0) � + �(1 − �) = ℒ ���1 − � 2 ��1�1 + �2� 2 � �ℎ − �� �2 − ��2�2 − ��2 � (4.2.2) �ℒ{�(�)} − �(0) � + �(1 − �) = ℒ{��1�1 + ��2�2 − �� − ��} (4.2.3) �ℒ{�(�)} − �(0) � + �(1 − �) = ℒ � � 1 (�1�1 + �2� 2 ) �ℎ − �� �1 + � 2 (�1�1 + �2� 2 ) �ℎ − �� �2 − �1�� − �2(1 − �)� − ��� (4.2.4) �ℒ{�1(�)} − �1(0) � + �(1 − �) = ℒ��1�� − � 1 �1 − �1�1 − ��1 − ��1� (4.2.5) �ℒ{�2(�)} − �2(0) � + �(1 − �) = ℒ��2(1 − �)� − � 2 �2 − �2�2 − ��2 − ��2� (4.2.6) �ℒ{��(�)} − ��(0) � + �(1 − �) = ℒ��1�1 + �2�2 − � 3 �� − ��� − ��� � (4,2.7) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 51 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" �ℒ{�(�)} − �(0) � + �(1 − �) = ℒ�� 1 �1 + � 2 �2 + � 3 �� − �� � (4.2. 8) ℒ{�1(�)} = �1(0) � + � + �(1 − �) � ℒ ���ℎ − � 1 ��1�1 + �2� 2 � �ℎ − �� �1 − ��1�1 − ��1 − ��1� (4.3.1) ℒ{�2(�)} = �2(0) � + � + �(1 − �) � ℒ ���1 − � 2 ��1�1 + �2� 2 � �ℎ − �� �2 − ��2�2 − ��2 � (4.3.2) ℒ{�(�)} = �(0) � + � + �(1 − �) � ℒ{��1�1 + ��2�2 − �� − ��} (4.3.3) ℒ{�(�)} = �(0) � + � + �(1 − �) � ℒ � � 1 (�1�1 + �2� 2 ) �ℎ − �� �1 + � 2 (�1�1 + �2� 2 ) �ℎ − �� �2 − �1�� − �2(1 − �)� − ��� (4.3.4) ℒ{�1(�)} = �1(0) � + � + �(1 − �) � ℒ��1�� − � 1 �1 − �1�1 − ��1 − ��1� (4.3.5) ℒ{�2(�)} = �2(0) � + � + �(1 − �) � ℒ��2(1 − �)� − � 2 �2 − �2�2 − ��2 − ��2� (4.3.6) ℒ{��(�)} = ��(0) � + � + �(1 − �) � ℒ�� 1 �1 + � 2 �2 + � 3 �� − �� � (4.3. 7) ℒ{�(�)} = �(0) � + � + �(1 − �) � ℒ��1�1 + �2�2 − � 3 �� − ��� − ��� � (4.3.8) according to the adomian decomposition method, the solution will be in the following series type ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 52 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ��(�) = ∑ ���(�)� ��� , ��(�) = ∑ ���(�)� ��� , �(�) = ∑ ��(�)� ��� , �(�) = ∑ ��(�)� ��� , ��(�) = ∑ ���(�)� ��� , ��(�) = ∑ ���(�)� ��� ��(�) = ∑ ���(�)(�)� ��� , r(�) = ∑ ��(�)� ��� (4.4) the nonlinear term involved in the model are ��(�)��(�), ��(�)��(�), ��(�)��(�) , ��(�)��(�). these are decomposed by adomian decomposition polynomial as ��(�)��(�) = ∑ ��� � ��� , ��(�)��(�) = ∑ ��� � ��� , ��(�)��(�) = ∑ ��� � ��� , ��(�)��(�) = ∑ ��� � ��� (4.5) where �� is adomian polynomial defined as: �� = � �(���) �� ��� �∑ ℎ��� � ��� ∑ ℎ��� � ��� � � �� (4.6) �� = ���� , �� = ���� + ����, �� = ���� + ���� + ����, �� = ���� + ���� + ���� + ����,. . . (4.7) applying equation (4.4)-(4.7) into the system (4.3.1)-(4.3.8), we have ℒ �� ���(�) � ��� � = �1(0) � + � + �(1 − �) � ℒ ���ℎ − − � 1 �1 ∑ �1� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 − � 1 �2 ∑ �2� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 − (��1 + � + �) � �1� ∞ �=0 � (4.8.1) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 53 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ℒ �� ���(�) � ��� � = �2(0) � + � + �(1 − �) � ℒ �� � �1� ∞ �=0 − � 2 �1 ∑ �3� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 − � 2 �2 ∑ �4� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 − (��2 + �) � �2� ∞ �=0 � (4.8.2) ℒ �� ��(�) � ��� � = �(0) � + � + �(1 − �) � ℒ ���1 � �1� ∞ �=0 + ��2 � �2� ∞ �=0 − (� + �) � �� ∞ �=0 � (4.8.3) ℒ �� ��(�) � ��� � = �(0) � + � + �(1 − �) � ℒ � � 1 �1 ∑ �1� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 + � 1 �2 ∑ �2� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 + � 2 �1 ∑ �3� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 + � 2 �2 ∑ �4� ∞ �=0 �ℎ − ∑ ��� ∞ �=0 − (�1� + �2(1 − �) + �) � �� ∞ �=0 � (4.8.4) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 54 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ℒ �� ���(�) � ��� � = �1(0) � + � + �(1 − �) � ℒ ��1� � �� ∞ �=0 − �� 1 + �1 + � + �� � �1� ∞ �=0 � (4.8.5) ℒ �� ���(�) � ��� � = �2(0) � + � + �(1 − �) � ℒ ��2(1 − �) � �� ∞ �=0 − (� 2 + �2 + � + �) � �2� ∞ �=0 � (4.8.6) ℒ �� ���(�) � ��� � = ��(0) � + � + �(1 − �) � ℒ ��1 � �1� ∞ �=0 + �2 � �2� ∞ �=0 − ( � 3 + � + �) � ��� ∞ �=0 � (4.8. 7) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 55 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ℒ �� ��(�) � ��� � = �(0) � + � + �(1 − �) � ℒ �� 1 � �1� ∞ �=0 + � 2 � �2� ∞ �=0 + � 3 � ��� ∞ �=0 − � � �� ∞ �=0 � (4.8.8) using initial value condition, ��(0) = ���, �(0) = ��, ��(0) = �����(0) = ���, �(0) = ��, �(�) = ��, ��(0) = ���, ��(�) = ���, matching the items on both sides of (4.8.1) –(4.8.8) and applying ���(�)���(�) = ���, ���(�)���(�) = ���, ���(�)���(�) = ���, ���(�)���(�) = ��� the general term of the model is given below � ��(���)(�) � ��� = ℒ�� � � + �(1 − �) � ℒ ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)����� (4.9.1) � ��(���)(�) � ��� = ℒ�� � � + �(1 − �) � ℒ 1 �� ℒ ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)����� (4.9.2) � ����(�) � ��� = ℒ�� � � + �(1 − �) � ℒ{������ + ������ − (� + �)��}� (4.9.3) � ����(�) � ��� = ℒ�� � � + �(1 − �) � ℒ � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)���� (9.4) � ��(���)(�) � ��� = ℒ�� � � + �(1 − �) � ℒ{����� − (�� + �� + � + �)���}� (4.9.5) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 56 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" � ��(���)(�) � ��� = ℒ�� � � + �(1 − �) � ℒ{��(1 − �)�� − (�� + �� + � + �)���}� (4.9.6) � ��(���)(�) � ��� = ℒ�� � � + �(1 − �) � ℒ{����� + ����� − ( �� + � + �)���}� (4.9.7) � ����(�) � ��� = ℒ�� � � + �(1 − �) � ℒ{����� + ����� + ����� − ���}� (4.9. 8) let � = 0 ���(�) = ℒ�� � � + �(1 − �) � ℒ ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)����� (4.10.1) ���(�) = ℒ�� � � + �(1 − �) � ℒ 1 �� ℒ ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)����� (4.10.2) ��(�) = ℒ�� � � + �(1 − �) � ℒ{������ + ������ − (� + �)��}� (4.10.3) ��(�) = ℒ�� � � + �(1 − �) � ℒ � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)���� (4.10.4) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� − (�� + �� + � + �)���}� (4.10.5) ���(�) = ℒ�� � � + �(1 − �) � ℒ{��(1 − �)�� − (�� + �� + � + �)���}� (4.10.6) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� − ( �� + � + �)���}� (4.10.7) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 57 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ��(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� + ����� − ���}� (4.10. 8) ���(�) = ℒ�� � � + �(1 − �) � ℒ ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)����� (4.10.1) ���(�) = ℒ�� � � + �(1 − �) � ℒ 1 �� ℒ ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)����� (4.10.2) ��(�) = ℒ�� � � + �(1 − �) � ℒ{������ + ������ − (� + �)��}� (4.10.3) ��(�) = ℒ�� � � + �(1 − �) � ℒ � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)���� (4.10.4) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� − (�� + �� + � + �)���}� (4.10.5) ���(�) = ℒ�� � � + �(1 − �) � ℒ{��(1 − �)�� − (�� + �� + � + �)���}� (4.10.6) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� − ( �� + � + �)���}� (4.10.7) ��(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� + ����� − ���}� (4.10. 8) applying the inverse laplace transform, we obtain ���(�) = ℒ�� � � + �(1 − �) � ℒ ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)����� (4.10.1) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 58 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ���(�) = ℒ�� � � + �(1 − �) � ℒ 1 �� ℒ ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)����� (4.10.2) ��(�) = ℒ�� � � + �(1 − �) � ℒ{������ + ������ − (� + �)��}� (4.10.3) ��(�) = ℒ�� � � + �(1 − �) � ℒ � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)���� (4.10.4) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� − (�� + �� + � + �)���}� (4.10.5) ���(�) = ℒ�� � � + �(1 − �) � ℒ{��(1 − �)�� − (�� + �� + � + �)���}� (4.10.6) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� − ( �� + � + �)���}� (4.10.7) ��(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� + ����� − ���}� (4.10. 8) ���(�) = ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)���� {1 + �(� − 1)} (4.11.1) ���(�) = ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)���� {1 + �(� − 1)} (4.11.2) ��(�) = {������ + ������ − (� + �)��}{1 + �(� − 1)} (4.11.3) ��(�) = � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)��� {1 + �(� − 1)} (4.11.4) ���(�) = {����� − (�� + �� + � + �)���}{1 + �(� − 1)} (4.11.5) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 59 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ���(�) = {��(1 − �)�� − (�� + �� + � + �)���}{1 + �(� − 1)} (4.11.6) ���(�) = {����� + ����� − ( �� + � + �)���}{1 + �(� − 1)} (4.11.7) ��(�) = {����� + ����� + ����� − ���}{1 + �(� − 1)} (4.11. 8) let � = 1 ���(�) = ℒ�� � � + �(1 − �) � ℒ ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)����� (4.12.1) ���(�) = ℒ�� � � + �(1 − �) � ℒ ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)����� ( 4.12.2) ��(�) = ℒ�� � � + �(1 − �) � ℒ{������ + ������ − (� + �)��}� (4.12.3) ��(�) = ℒ�� � � + �(1 − �) � ℒ � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)���� (4.12.4) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� − (�� + �� + � + �)���}� (4.12.5) ���(�) = ℒ�� � � + �(1 − �) � ℒ{��(1 − �)�� − (�� + �� + � + �)���}� (4.12.6) ���(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� − ( �� + � + �)���}� (4.12.7) ��(�) = ℒ�� � � + �(1 − �) � ℒ{����� + ����� + ����� − ���}� (4 .12. 8) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 60 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ���(�) = ���� − � 1 �� − [{����� + ����� − ( �� + � + �)���}{1 + �(� − 1)}] ������{����� − (�� + �� + � + �)���}{1 + �(� − 1)}� − �����{��(1 − �)�� − (�� + �� + � + �)���}{1 + �(� − 1)}�� − (��� + � + �)� ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)���� {1 + �(� − 1)}� {1 + �(� − 1)} 4.13.1 ���(�) = = �� ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)���� − 1 �� − {����� + ����� − ( �� + � + �)���}{1 + �(� − 1)} [����{����� − (�� + �� + � + �)���} − ����{��(1 − �)�� − (�� + �� + � + �)���}] − (��� + �)� ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)���� {1 + �(� − 1)}{1 + �(� − 1)}(4.13.2) ��(�) = ���� ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)���� + ��� ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)���� − (� + �){������ + ������ − (� + �)��}� {1 + �(� − 1)}{1 + �(� − 1)} (4.13.3) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 61 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ��(�) = � 1 �� − {����� + ����� − ( �� + � + �)���}{1 + �(� − 1)} [����{����� − (�� + �� + � + �)���} ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)���� + ����{��(1 − �)�� − (�� + �� + � + �)���} ���� − ���������� �� − ��� − ���������� �� − ��� − (��� + � + �)���� + ����{����� − (�� + �� + � + �)���} ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)���� + ����{��(1 − �)�� − (�� + �� + � + �)���} ����� − ���������� �� − ��� − ���������� �� − ��� − (��� + �)����]{1 + �(� − 1)} − (��� + ��(1 − �) + �) � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)��� {1 + �(� − 1)}� {1 + �(� − 1)} (4.13.4) ���(�) = ���� � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)��� − (�� + �� + � + �){����� − (�� + �� + � + �)���}� {1 + �(� − 1)}{1 + �(� − 1)} (4.13.5) ���(�) = ���(1 − �) � ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� + ���������� �� − ��� − (��� + ��(1 − �) + �)��� − (�� + �� + � + �){����� − (�� + �� + � + �)���}� {1 + �(� − 1)}{1 + �(� − 1)} (4.13.6) ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 62 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" ���(�) = ���{����� − (�� + �� + � + �)���}{1 + �(� − 1)} + ��{��(1 − �)�� − (�� + �� + � + �)���}{1 + �(� − 1)} − ( �� + � + �){����� + ����� − ( �� + � + �)���}� ��(�) = ���{����� − (�� + �� + � + �)���} + ��{��(1 − �)�� − (�� + �� + � + �)���} + ��{����� + ����� − ( �� + � + �)���} − �{����� + ����� + ����� − ���}�{1 + �(� − 1)}{1 + �(� − 1)} (4.13. 8) hence the required solution ��(�) = ���(�) + ���(�) + ���(�) + . . . ��(�) = ���(�) + ���(�) + ���(�) + . . . �(�) = ��(�) + ��(�) + ��(�) + . . . �(�) = ��(�) + ��(�) + ��(�) + . . . ��(�) = ���(�) + ���(�) + ���(�) + . . . ��(�) = ���(�) + ���(�) + ���(�) + . . . ��(�) = ���(�) + ���(�) + ���(�) + . . . �(�) = ��(�) + ��(�) + ��(�) + . . . summary diphtheria remains a re-emerging public health concern despite vaccination programs, with children and adults exhibiting different levels of susceptibility and transmission potential. classical integer-order models often fail to capture the memory effects inherent in disease transmission, such as waning immunity and delayed intervention impact, and few models incorporate age structure, which is crucial for diphtheria dynamics. motivated by these limitations, this study develops a novel age-structured fractional-order model of diphtheria using the caputo–fabrizio derivative, which accounts for non-locality and memory effects in disease spread. the model stratifies the population into susceptible children, susceptible adults, vaccinated, exposed, infectious children, infectious adults, isolated infectious, and recovered individuals. rigorous analysis establishes the existence and uniqueness of solutions via the contraction mapping principle and proves ulam–hyers ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 63 ijo international journal of mathematics (issn: 2992-4421 ) thomas, henry sylvester * https://ijojournals.com/ volume 08 || issue 09 || september, 2025 || “caputo-fabrizio fractional drivatives of age-structured dipptheria infection model with laplace adomian decomposition analysis" stability, confirming the robustness of the system under perturbations. to obtain approximate analytic– numerical solutions, the laplace adomian decomposition method (ladm) is applied, yielding a rapidly convergent series representation. results show that the fractional order significantly alters outbreak intensity and timing, reflecting the role of memory in diphtheria persistence and control. this work achieves three key outcomes: (i) the formulation of a new fractional-order, age-structured diphtheria model; (ii) provision of rigorous mathematical guarantees for solution behavior; and (iii) demonstration of efficient analytic–numerical solutions through ladm. the study contributes to knowledge by introducing a more realistic modeling framework that integrates age heterogeneity and memory effects, offering deeper epidemiological insights and practical guidance for sustaining vaccination and isolation strategies in diphtheria control. 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(ijmsea) issn 0973-9424, vol. 9 no. i (march, 2015), pp. 1-17 www.ascent-journals.com 10. ekere s. udofia, (2023), mathematical model of male circumcision in hiv/aids preventions, international journal of innovative science and research technology, volume 8, issue 8, august – 2023 issn no: -2456-2165 www.ijisrt.com 11. onuoha joy ljeoma, inyama simeon chioma and udofia sunday ekere(2014) mathematical model of the transmission dynamics of swine flu with the vaccination of newborns, international j. of math. sci. &engg. appls. (ijmsea) issn 0973-9424, vol. 8 no. v (september, 2014), pp. 217-229 12. udofiaekere sunday and inyama simeon chioma (2012), application of optimal control to the epidemiology of fowl pox transmission dynamics in poultry, journal of mathematics and statistics 8 (2): 248-252, issn 1549-3644, © 2012 science publications 13. udofia, ekere sunday and inyama, simeonchioma mathematical model of structural strategy( delayed first intercourse) in hivaids prevention, journal of the nigerian association of mathematical physics volume 24 (july, 2013) pp 257-260 © j. of namp 14. udofia, ekere sunday, sampson, marshal imeh (2014) mathematical model for the epidemiology of fowl pox infection transmission that incorporates discrete delay, iosr journal of mathematics (iosr-jm) e-issn: 2278-5728, p-issn: 2319-765x. volume 10, issue 4 ver. v (july aug. 2014), pp 08-16 www.iosrjournals.org 15. ia agwu, sc inyama, ra umana, a omame, n ukanwoke, a ofomata, hi mbachu, es udofia, ji uwakwe (2018), determining the impact of variation of harvesting effort on the qualitative behaviour of a coexistence steady state solution and its stability in prey-predator fishery model, academic journal of applied mathematical sciences vol. 4 issue 10 pages 119-128, 2018 16. udofiaekere sunday and amos amosidungafa (2018), mathematical model of bacteria-nutrient harvesting in a cultured environment, journal of the nigerian association of mathematical physics volume 46 (may, 2018 issue), pp115 –118, 2018 © j. of namp 17. udofia, ekere s, and etukudoidorenyin a (2019), optimal allocation of biscuit ingredient in the production process an invariant property based algorithm approach, mathematical theory and modeling www.iiste.org issn 2224-5804 (paper) issn 2225-0522 (online) doi: 10.7176/mtm vol.9, no.4, 2019 18. i. j. udom, e. s. udofia, s. a. nta, g. a. usoh, e. o. sam (2023)’ prediction of piggery wastewater nutrient attenuation by constructed wetland in a humid environment’ international journal of innovative science and research technology issn no:-2456-2165, volume 8, issue 9, september – 2023,www.ijisrt.com 19. olayiwola, m. o, alaje, a.i. (2024) mathematical modelling of diphtheria transmission and vaccine efficacy using nigeria, modelearth syst.environ.10, 3941-3967 https://doi.org/10.1007/s40808-02401976-7 20. ulam sm, a collection of mathematical problems; new york; 1960, p.29 21. ulam sm, problems in modern mathematics; courier co; 2004 ijo journals volume 08 | issue 09 | september 2025 | https://ijojournals.com/index.php/m/index 65 http://www.ascent-journals.com/ http://www.ijisrt.com/ http://www.iosrjournals.org/ http://www.ijisrt.com/ https://doi.org/10.1007/s40808-024-01976-7 https://doi.org/10.1007/s40808-024-01976-7 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load i.u. udo-akpan1 and j. u. chukwuchekwa2 1department of mathematics and statistics, university of port harcourt, port harcourt, rivers state, nigeria. 2department of mathematics, federal university of technology, owerri, imo state, nigeria. abstract in this investigation, we extend our search on the dynamic buckling loads of some elastic structures to that of a clamped column lying on a nonlinear (cubic) elastic foundation but impacted upon axially by a step load. in order to ensure a uniformly valid solution, we employ multi–scaling two–timing regular perturbation procedures in asymptotic expansions of the variables. it is shown that (a) clamped columns buckle at higher buckling loads than columns with simply–supported ends irrespective of whether the columns are loaded statically or dynamically and whether damped or undamped, (b) specifically, the inequalities satisfied by the static buckling load �� and dynamic buckling load�� in the clamped case are respectively given as 1 < �� < 2.125 and 1 < �� < 2.125 as against 0 < �� < 1 and 0 < �� < 1 for simply– supported end conditions, (c) at low values of the static buckling load ��, there is no appreciable change in the values of the dynamic buckling load �� but at higher values of ��, �� increases sharply with increased static buckling load ��. however, the increase seems to decrease with increased damping. we are able to mathematically relate the dynamic buckling load �� to the static buckling load �� and thereby by–passing the labour of repeating the entire process for different imperfection parameters. thus, given either �� or ��, we can predict either value without the actual knowledge of the size of the small imperfection parameter. keywords: dynamic buckling, viscously damped clamped column, elastic structures, step load, two-timing perturbation 2010 mathematical subject classification: 74b20, 74h10, 34e10 1. introduction investigations into the static or dynamic stability (or otherwise) of columns (finite or infinite) are age old problems that have been embarked upon by researchers for some years now. as observed by [1] and [2], buckling of structures is one of the structural instabilities that have been known for centuries ever since the equation to derive the critical buckling of a column was derived by leonhard euler [3]. as a result of this, previous studies on the subject matter are indeed enormous and varied, and include investigations by [4]-[7], among others. we must stress that columns, are in themselves, indispensable structural materials and their utility cuts across all cultures in human history. this investigation is concerned with analytical determination of the dynamic buckling load of a viscously damped but clamped imperfect column that rests on a nonlinear (cubic) elastic foundation, where the column is struck by a step load. the viscous https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 1 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load damping, though small in magnitude, is not in any way related either physically or mathematically, to an equally small imperfection that is stress-free and twicedifferentiable. the formulation therefore contains two small but dimensionally independent parameters upon which asymptotic expansions are initiated using a twotiming multi-scaling perturbation procedure. the analysis contained here is an extension of a similar study espoused by [8], where, in that study, damping was taken to be related in some way, to the imperfection �� (�), so that once the imperfection was fixed, the damping was equally fixed. however, we note, from physical reasoning, that damping need not in any way, be related to the imperfection in all probabilities. similar studies were done by [9]-[12], among others. apart from addressing the phenomenon of viscous damping in a special fashion, this investigation is related, in spirit, to similar studies by [13]-[25]. 2. formulation of the problem as in [6], the dimensional differential equation satisfied by the deflection �(�, �) of a finite viscously damped column lying on a nonlinear (cubic) elastic foundation but struck by a load �(�) is ���,�� + ��,� + ���,���� + 2�(�)�,�� + ��� − ����� = −2�(�) ���� ��� , � > 0, (2.1a) �(�, 0) = �,�(�, 0) = 0, (2.1b) � = �,� = 0, �� � = 0, π, � > 0 (2.1c) here, �� is the mass per unit length, � is the damping coefficient, ei is the bending stiffness, where e and i are the young’s modulus and the moment of inertia respectively. the nonlinear elastic foundation exerts a force per unit length given by ��� − ����� on the column, where �� and �� are constants such that ��>0, ��>0, and � is the imperfection-sensitivity parameter which is such that for � = 1, the nonlinear elastic foundation is said to “softening”, whereas for � = -1, the foundation is said to be “hardening”. we have here excluded all nonlinearities of �(�, �) higher than the cubic and have also excluded all nonlinear derivatives of�(�, �). 3. non-dimensionalization of the governing equation we shall non-dimensionalize the equations (2.1a c) by using the following nondimensional quantities � = � �� �� � � � �, � = � �� �� � � � �, ��(�) = �(�) 2(����) � � , ��� = � �� �� � � � �� , 2� = � (����) � � , � = � �� �� � � � �, 0 < � << 1, 0 < δ ≪ 1; 0 < � < 2.125; https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 2 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load on introducing these non-dimensional quantities into (2.1a-c) and simplifying, we get �,�� + 2��,� + �,���� + 2��(�)�,�� + � − ��� = −2���(�) ���� ��� , � > 0, 0 < � < π (3.1a) �(�, 0) = �,�(�, 0) = 0, 0 < � < π (3.1b) � = �,� = 0, �� � = 0, π, � > 0 (3.1c) here, a subscript following a comma indicates partial differentiation and �(�) indicates the actual time dependence of the load having its magnitude as λ. in our case, �(�) is a step load characterized by �(�) = � 1, � > 0 0, � < 0 � (3.2) 4. classical buckling load, �� the classical buckling load �� is the load that the associated linear perfect column buckles statically. the equations required are obtained from (3.1a) as �,���� + 2��,�� + � = 0, 0 < � < � (4.1a) � = �,� = 0, �� � = 0, π. (4.1b) we note that the deflection � at this stage depends only on�. to solve (4.1a,b), we let �(�) = ∑ (1 − ���2��)�� � ��� (4.2) on substituting (4.2) into (4.1a),multiplying by ���2�� for fixed� and integrating form 0 to �, we see that for � = �, we get (16�� − 8��� + 1)�� = 0 (4.3) where, m is a fixed value of n. according to [18], the condition for static buckling is �� �� = 0 (4.4) where, � is the displacement (or deflection). this gives the classical buckling load �� as λ� = ����� � ��� (4.5a) the least value of �� is when � = 1 and in this case, we get λ� = �� � = 2.125 (4.5b) in retrospect, a similar column with simply-supported end conditions that satisfy the same equation as (4.1a) but, instead of (4.1b), it would satisfy the conditions � = �,�� = 0, �� � = 0, π (4.5c) has the classical buckling load as �� = 1 which is different from (4.5a,b). thus, a clamped column has a higher classical buckling load than the same column with simply – supported end conditions. https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 3 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load 5. static buckling load, �� this is the load that the column would require to buckle statically. the required differential equation is �,���� + 2��,�� + � − ��� = −2�� ���� ��� , 0 < � < � (5.1a) � = �,� = 0, �� � = 0, π, (5.1b) to determine the displacement (or deflection) �(�) in (5.1a,b), we set �(�) ≡ 1and let �� = ���(1 − ���2��), │���│ ≪ 1 next, we let ∑ �(�)� ��� �� ; �(�) = �(�)(�) (5.2) by substituting (5.2) into (5.1a) and equating the coefficients of powers of ��, � = 1, 2, 3, …, we get o(ϵ):��(�) ≡ �,���� (�) + 2��,�� (�) + �(�) = −8���������2�� (5.3) o(��):��(�) = 0 (5.4) o(��): ��(�) = ���(�)� � (5.5) etc. �(�)(�) = �,� (�) �� � = 0, π (5.6) we seek for the solutions of (5.3) – (5.6) by letting �(�)(�) = 2 ∑ �� (�)� ��� sin� �� = ∑ (1 − ���2��)�� (�)� ��� (5.7) on substituting (5.7) into (5.3), we get ∑ ��� (�)(8��� − 16��)���2�� + �� (�)(1 − ���2��)�� ��� = −8���������2�� (5.8) on multiplying (5.8) through by cos2mx and integrating from 0 to π, we see that for � = �, (16�� − 8��� + 1)�� (�) = 8������ this gives �� (�) = ������� ����� ������ = � (5.9a) ∴ �(�) = �� (�)(1 − ���2��) (5.9b) on substituting (5.7) into (5.4), we easily get �� (�) = 0 (5.10) next, we substitute (5.7) in (5.5) and use (5.9a,b) to get ���� (�)(8��� − 16��)���2�� + �� (�)(1 − ���2��)� = ���(�)� � (1 − ���2��)� � ��� = ���(�)� � � � � − �� � ���2�� + � � ���4�� − � � ���6���(5.11) we multiply (5.11) through by cos2mx and integrate from 0 to π, to get �� (�) = ����� �(����� ������) (5.12) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 4 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load next, we multiply (5.11) by cos4mx and integrate from 0 to π to get ��� (�) = ����� �(������ �������) (5.13) lastly, we multiply (5.11) by cos6mx and integrate from 0 to π and get, ��� (�) = � ��� �������������� � (5.14) thus, we get �(�) = �� (�)(1 − ���2��) + ��� (�)(1 − ���4��) + ��� (�)(1 − ���6��)(5.15) so that �(�) = ��� (�)(1 − ���2��) + ����� (�)(1 − ���2��)� + ��� (�)(1 − ���4��) �+ ��� (�)(1 − ���6��)� + … (5.16) before determining the static buckling load, we need to evaluate (5.16) at � = � �� . this is informed by the fact that eventually, we shall need to determine the associated dynamic problem at � = � �� (which also means finding the maximum of (5.16)). such evaluation yields � = 2��� (�) + 2����� (�) − ��� (�) � + … (5.17) we can write (5.17) as � = ��� + ���� + … (5.18) where, �� = 2�� (�) , �� = 2��� (�) − ��� (�) � (5.19) as in [18], the static buckling load is obtained by first reversing the series (5.18) in the form, � = ��� + ���� + … (5.20) on substituting for w from (5.18) in (5.20) and equating the coefficients of powers of ϵ, we get �� = � �� , �� = ��� �� � (5.21) the maximization (4.4) is now easily executed from (5.20) to yield �� + 3���� � = 0 (5.22) where, �� is the value of � evaluated at static buckling. from (5.18), we get �� = � ��� ��� = � √� � �� � �� � � � (5.23) next, we determine (5.20) at static buckling and get � = ��� + ���� + … = ��(�� + ���� �) = � �√� � �� �� � � � (5.24) on substituting in (5.24) for �� and ��, we get (16�� − 8���� + 1) � � = 18√5��� � ������� �1 + � �� � ����� ������� ��������������� �� � � (5.25) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 5 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load where �� is the static buckling load. the least value of �� is obtained when � = 1, and for this, we get (17 − 8��) � � = 18√5��� � ����� �1 + � �� � ��� ��� ����� ��� �� � � (5.26) if it is required that the buckling mode be strictly in the shape of imperfection, then the results corresponding to (5.25) and (5.26) respectively become (16�� − 8���� + 1) � � = 18√5����� � ����� (5.27) and (17 − 8��) � � = 18√5��� � ����� (5.28) by way of comparison, we can perform a similar analysis on the same column but with simply-supported end conditions and the results corresponding to those of (5.25), (5.26), (5.27) and (5.28) are respectively given by (�� − 2���� + 1) � � = � � ��������� � � �1 + � ��� ������� ������������� �� � � (5.29) (1 − ��) � � = � �√� ������� � � �1 + � ���� ������ �� � � (5.30) (�� − 2���� + 1) � � = � � ��������� � � (5.31) and (1 − ��) � � = � �√� ������� � � (5.32) the result (5.32) was first obtained by [6]. so far, we conclude that the static buckling load of structures largely depends, among other things, on the type of end constraints of the structures. 6. the dynamic problem the associated dynamic problem follows from (3.1a) which we now recast as �,�� + 2��,� + �,���� + 2��(�)�,�� + � − ��� = −2���(�) ���� ��� , � > 0, 0 < � < π (6.1a) �(�, 0) = �,�(�, 0) = 0, 0 < � < π (6.1b) � = �,� = 0, �� � = 0, π, � > 0 (6.1c) henceforth, we shall substitute for the step load �(�) as in (3.2). we let � = ��, (6.2a) �̂ = � + � ��(�)��� ��(�)���⋯ � � (6.2b) where, ��(0) = 0, i = 1, 2, 3, …, �� = ��(�) we stress that ϵ and δ are small and unrelated parameters. thus, from (6.2a,b), we get �� �� = �� ��̂ ��̂ �� + �� ��̂ ��̂ �� �� �� + �� �� �� �� = (1 + �� � �� + �� � �� + ⋯ )�,�� + ��,� (6.3a) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 6 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load where,(… )� = �(… ) �� and a subscript after a comma indicates partial differentiation. ∴ ��� ��� = (1 + �� � �� + �� � �� + ⋯ )��,���� + ���,�� + 2�(1 + �� � �� + �� � �� + ⋯ )�,��� + �(�� ���� + �� ���� + ⋯ )�,�� (6.3b) on substituting (6.3a,b) into (6.1a) for �(�) = 1, we get �(1 + �� � �� + �� � �� + ⋯ )��,���� � + ���,�� + 2�(1 + �� � �� + �� � �� + ⋯ )�,��� +��(�� ���� + �� ���� + ⋯ )�,��� + 2��(1 + �� � �� + �� � �� + ⋯ )�,�� + ��,�� + �,���� + 2��(�)�,�� + � − ��� = −2���(�) ���� ��� (6.4) next, we adopt the asymptotic series � = ∑ ∑ �(��)(�, �̂� ��� � ��� , �)���� (6.5) and substitute same into (6.4), and afterwards, equate powers of ���� to get �(�): ��(��) = �,���� (��) + �,���� (��) + 2��,�� (��) + �(��) = −2��(�) ���� ��� (6.6) �(��): ��(��) = −2 ��,��� (��) + �,�� (��) � (6.7) �(���): ��(��) = −2 ��,��� (��) + �,�� (��) � − �,�� (��) (6.8) �(��): ��(��) = 0 (6.9) �(���): ��(��) = −2 ��,��� (��) + �,�� (��) � (6.10) �(����): ��(��) = −2 ��,��� (��) + �,�� (��) � − �,�� (��) (6.11) �(��): ��(��) = ���(��)� � − 2�� � �,���� (��) (6.12) �(���): ��(��) = 3���(��)� � �(��) − 2 ��,��� �� + �,�� (��) � − �� ���,�� (��) − 2�� � �,�� (��) (6.13) �(����): ��(��) = 3� ��(��)��(��)� � + ��(��)� � �(��)� − �,�� (��) −2 ��,��� (��) + �� � �,��� (��) � − �� ���,�� (��) − 2 ��,�� (��) + �� � �,�� (��) + �,� (��) � (6.14) the initial conditions, which are evaluated at �̂ = 0 = � are �(��)(�, 0, 0) = 0, � = 1, 2, 3, … ; � = 0, 1, 2, 3, … (6.15) ο(ϵ): � ,�� (��)(x, 0, 0) = 0 (6.16) ο(ϵδ): � ,�� (��)(x, 0, 0) + �,� (��)(x, 0, 0) = 0 (6.17) ο(ϵδ�): � ,�� (��)(x, 0, 0) + �,� (��)(x, 0, 0) = 0 (6.18) in general, we have � ,�� (��)(x, 0, 0) + �,� �(���)(x, 0, 0) = 0, k = 1, 2, 3, … (6.19) ο(ϵ�): � ,�� (��)(x, 0, 0) = 0 (6.20) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 7 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load in general, we have � ,�� (��)(x, 0, 0) + �,� �(���)(x, 0, 0) = 0, k = 1, 2, 3, … (6.21) ο(ϵ�): � ,�� (��)(x, 0, 0) + �� � (0)� ,�� (��)(x, 0, 0) = 0 (6.22) o(ϵ��): � ,�� (��)(x, 0, 0) + �� � (0)� ,�� (��)(x, 0, 0) + �,� (��)(x, 0, 0) = 0 (6.23) ο(ϵ���): �,�� ��(x, 0, 0) + �� � (0)� ,�� (��)(x, 0, 0) + �,� (��)(0, 0) = 0 (6.24) generally, we have �,�� (��) (x, 0, 0) + �� � (0)�,�� (��) (x, 0, 0) + �,� ��(���)� (x, 0, 0) = 0 , k = 1, 2, 3, … (6.25) the boundary conditions are �(��) = u,� (��) = 0 �� � = 0, �. for solution to all the systems of equation involved here, we set �(��)(x, t̂, τ) = 2 ∑ u� (��)(t̂, τ)sin�nx = ∑ u� (��) (t̂, τ)(1 − cos 2��)� ��� � ��� (6.26a) we shall assume ��(�) = ���(1 − cos 2��) (6.26b) for �, fixed. on substituting (6.26a,b) into (6.6) and simplifying, we get � �u�,���� (��) + u� (��) � (1 − cos 2��) � ��� + �(8��� − 16��)u� (��) cos 2�� � ��� = −8���������2�� (6.27a) next, we multiply (6.27a) by ���2�� and integrate from 0 to π , and note that for � = �, we get u�,���� (��) + ��u� (��) = 8������ (6.27b) u� (��)(0,0) = 0, u�,�� (��) (0,0) = 0 (6.27c) where, �� = (16�� − 8��� + 1) > 0, ∀ � (6.27d) the solution of (6.27b-d) is u� (��) (t̂, τ) = ��(�) cos ��̂ + ��(�) sin ��̂ + � (6.28a) � = ������� �� = ������� ����� ������ (6.28b) where, ��(0) = −�, ��(0) = 0 (6.28c) this means that �(��) = u� (��) (1 − cos 2��) (6.28d) we next substitute (6.28d) into (6.7), using (6.25), and thereafter, multiply by cos 2�� and note that for � = 2�, we get u �,���� (��) + ��u� (��) = −2 � u�,��� (��) + u�,�� (��) � (6.29a) u� (��)(0,0) = 0, u�,�� (��)(0,0) + u�,� (��)(0,0) (6.29b) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 8 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load on substituting for u� (��) in (6.29) from (6.28a), we ensure a uniformly valid solution in �̂ by equating to zero the coefficients of cos ��̂ and sin ��̂ and getting �� � + �� = 0, �� � + �� = 0 (6.30a) the solution of (6.30a) using (6.28c) yields ��(�) = 0, ��(�) = −���� (6.30b) the remaining equation in (6.29a,b) is solved to get u� (��) = ��(�) cos ��̂ + ��(�) sin ��̂ (6.31a) ��(0) = 0, ��(0) = � � (6.31b) we however note that �� �(0) = b, �� ��(0) = −b (6.31c) we equally note at this stage that (6.32) �(��) = u� (��) (1 − cos 2��) on substituting from (6.32) and (6.28d) in (6.8), multiplying thereafter by cos 2�� and integrating from 0 to π, we get, (for � = �) u �,���� (��) + ��u� (��) = −2 � u�,��� (��) + u�,�� (��) � − u�,�� (��) (6.33a) u� (��)(0,0) = 0, u�,�� (��) (0,0) + u�,� (��)(0,0) (6.33b) next, we substitute in (6.33a) for u� (��) and u� (��) from (6.31a) and (6.28a) and ensure a uniformly valid solution in �̂ by equating to zero the coefficients of cos ��̂ and sin ��̂ and so, get, respectively �� � + �� = ��� �� �� and �� � + �� = 0 (6.33c) on solving (6.33c), we get ��(τ) = ��� �∫ ��� ��(�) �� � � �� + ��(0)�, ��(τ) = 0 (6.33d) the remaining equation in the substitution into (6.33a) is solved to yield u� (��)(t̂, τ) = ��(�) cos ��̂ + ��(�) sin ��̂ (6.34a) where, ��(0) = 0, ��(0) = 0 (6.34b) in passing, we note from (6.33c) that �� � (0) = −��(0) ��� ��(�) �� = ��� �� , �� ��(0) = � � (6.34c) we conclude that �(��) = u� (��) (1 − cos 2��) (6.35) on solving equations (6.9) to (6.11), using the appropriate initial and boundary conditions, we get �(��)(�, �̂, �) = �(��)(�, �̂, �) = �(��)(�, �̂, �) = 0 (6.36) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 9 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load we next substitute on the right hand side of (6.12), using (6.25) and simplify to get ��(��) = � ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + � �� � cos 3��̂ 4 � � 5 2 − 15 cos 2�� 4 + 3 2 cos 4�� − 1 4 cos 6��� −2�� � u�,���� (��) (1 − cos 2��) (6.37) next, we assume �(��) = ∑ u� (��) (1 − cos 2��) � ��� (6.38) on substituting (6.38) into (6.37) and first multiplying through by cos 2��, and thereafter integrating from 0 to π, we see that, for � = �, we get u�,���� (��) + ��u� (��) = 15� 4 ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + � �� � ��� ���� � � − 2�� � u�,���� (��) (6.39a) u� (��)(0,0) = 0, � �,�� (��)(0, 0) + �� � (0)� �,�� (��)(0, 0) = 0 (6.39b) next, if in the substitution in (6.12), we multiply through by ���4�� and thereafter integrate from 0 to π, we get (for� = 2�) u��,���� (��) + ��� � u�� (��) = −3� 2 ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + � �� � ��� ���� � � (6.40a) u�� (��)(0,0) = 0, � ��,�� (��)(0, 0) = 0 (6.40b) where, ��� � = (256�� − 32��� + 1) > 0, ∀ � (6.40�) lastly, if in the substitution into (6.12), we multiply through by cos 6��, and integrate from 0 to π, we see that, for � = 3�, we get u��,���� (��) + ��� � u�� (��) = � 4 ���� + 3�� �� 2 � + 3 ����� + �� � 4 � cos � �̂ + 3��� � cos 2��̂ 2 � + ��� � ��� ���� � � (6.41a) u�� (��)(0,0) = 0, � ��,�� (��) (0, 0) = 0 (6.41b) where, ��� � = (1296�� − 72��� + 1) > 0, ∀ � (6.41�) to solve (6.39a,b), we substitute for u� (��) from (6.28a), (noting (6.28c)), and ensure a uniformly valid solution in �̂ by equating to zero, the coefficient of cos � �̂ and get �� � (�) = ���� �� ��� + �� � � �, �� � (0) = ������� ���� (6.42a) �� ��(0) = ����� ���� (6.42b) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 10 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load the remaining equation in (6.39a) is now solved to get u� (��)(�̂, �) = �� cos � �̂ + �� sin � �̂ + 15� 4 � �� �� − 3��� � cos 2��̂ 2�� � − ��� � cos 3��̂ 32�� � (6.43�) where ��(�) = ��� + ��� �� � � (6.43b) ��(0) = ���� ����� , (0) = 0 (6.43c) ��(0) = ��� � , �� �(0) = −3��, �� ��(0) = 6�� (6.43d) turning to (6.40a,b), we solve to get u�� (��) (�̂, �) = �� cos ��� �̂ + �� sin ��� �̂ − 3� 2 � �� ��� � + �� cos ��̂ ��� � − �� + 3��� � cos 2��̂ 2(��� � − 4��) � + � �� � ��� ���� ����� � ����� � (6.44a) where, ��(�) = 3 ����� + �� � � �, ��(0) = ����� � , �� �(0) = ���� � , �� ��(0) = ����� � (6.44b) ��(0) = ������ � , ��(0) = 0 (6.44c) �� = � ���� � + �� ����� � ����� − � ��� � ���� + � ����� � ����� (6.44d) we next solve (6.41a) and get u�� (��)(�̂, �) = �� cos ��� �̂ + �� sin ��� �̂ + � 4 � �� ��� � + �� cos ��̂ ��� � − �� + 3��� � cos 2��̂ 2(��� � − 4��)� + � �� � ��� ���� ����� � ����� � (6.45a) where, ��(0) = ����� 4 , �� = � 1 ��� � + 15 ��� � − �� − 3 ��� � − 4�� � + � 1 2(��� � − 9��) �, ��(0) = 0 (6.45�) thus, we have �(��) = u� (��)(1 − cos 2��) + u�� (��)(1 − cos 4��) + u�� (��)(1 − cos 6��) (6.45�) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 11 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load we next substitute on the right hand side of (6.13) and get ��(��) = 3� 4 ��� ���� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� � 5 2 − 15 cos 2�� 4 � ��+ 3 cos 4�� 2 − cos 6�� 4 �� − 2 �u�,��� (��) (1 − cos 2��) + u��,��� (��) (1 − cos 4��)� + u��,��� (��) (1 − cos 6��) + u�,�� (��) (1 − cos 2��) + u��,�� (��) (1 − cos 4��) �+ u��,�� (��) (1 − cos 6��)� − �� �� u�,�� (��) (1 − cos 2��) −2�� � u�,�� (��) (1 − cos 2��) (6.46) let �(��) = � u� (��) (1 − cos 2��) (6.47) � ��� on substituting (6.47) into (6.46), first multiplying through by cos 2�� and integrating from 0 to π, we get, for � = � u�,���� (��) + ��u� (��) = 45� 4 ��� ���� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� − �� �� u�,�� (��) − 2�� � u�,�� (��) − 2 �u��,��� (��) + u��,�� (��) � (6.48a) u� (��)(0,0) = 0, � �,�� (��)(0, 0) + �� � (0)� �,�� (��)(0, 0) + ��,� (��)(0, 0) = 0 (6.48b) next, we multiply (6.46) by cos 4��, integrate from 0 to π, we get, for � = 2� u��,���� (��) + ��� � u�� (��) = −9��� 8 ����� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� − 2 �u��,��� (��) + u��,�� (��) � (6.49a) u�� (��)(0,0) = 0, � ��,�� (��)(0, 0) + ���,� (��) (0, 0) = 0 (6.49b) similarly, we multiply (6.46) by cos 6��, integrate from 0 to π, we get, for � = 3� u��,���� (��) + ��� � u�� (��) = 3��� 16 ����� + �� � 4 � sin � �̂ + ��� sin2 � �̂ − �� � 4 sin3 � �̂�� − 2 �u��,��� (��) + u��,�� (��) � (6.50a) u�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) = 0 (6.50b) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 12 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load to solve (6.48a), we substitute for terms there and after, ensure a uniformly valid solution by equating to zero the coefficients of cos � �̂ and sin � �̂. this gives, for cos � �̂ �� � + �� = 0 (6.51�) forsin � �̂: �� � + �� = �� 2 (�� �� + 2�� � ) − 45� 32� (�� + �� �)�� (6.51�) on solving (6.51a,b), we get �� = 0; �� = ��� ���(0) − � �� � �� 2 (�� �� + 2�� � ) + 45� 32� (�� + �� �)��� � � ��� (6.51�) �� �(0) = �������� ����� (6.52d) the remaining equations, having ensured a uniformly valid solution in (6.48a) are u�,���� (��) + ��u� (��) = �� sin2 � �̂ + ��sin3 � �̂ (6.52�) u� (��)(0,0) = 0, � �,�� (��)(0, 0) + �� � (0)� �,�� (��)(0, 0) + ��,� (��)(0, 0) = 0 (6.52b) where, �� = �������� �� − ���� �� {(�� �)� + (�� �)} (6.52c) �� = −45������ � 64 − 15� 32� {(�� �)� + (�� �)} (6.52�) (0) = 75��� 16� , ��(0) = −225��� 64� (6.52�) we note the following r� � (0) = −345��� 128� , r� � (0) = 465��� 32� , r� � (0) = −45��� 128� (6.52�) on solving (6.52a,b), we get u� (��) = �� cos � �̂ + �� sin � �̂ − �� sin2 � �̂ 3�� − ��sin3 � �̂ 8�� (6.53�) where, ��(0) = 0 (6.53�) ���(0) − 2��(0) 3� − 3��(0) 8� + ��(0)���(0) + 15� 4 � �� � �� − ��� ��� �� − 3�� ��� � �� � │��� = 0 this yields ��(0) = −5695�� 512�� (6.53�) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 13 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load to ensure a uniformly valid solution in terms of �̂ in (6.49a), we equate to zero the coefficients of cos2 ��� �̂and sin2 ��� �̂ and respectively get �� � + �� = 0 (6.53�) and �� � + �� = � ���� � � �� � ��� + �� � � � + ����� �� ��� ��� � � �� � (6.53e) on solving (6.53d, e), we get �� = 0, �� = ������(0) + ∫ ���� � � ��� (6.53f) where, �� = � ���� � � �� � ��� + �� � � � + ����� �� ��� ��� � � �� � (6.53g) from (6.53e), we get �� � (0) = �����, �� = � ���� � �� ��� + �� ����� � � ��� � − ��� � (6.53h) the remaining equation in (6.49a) is now written as u ��,���� (��) + ��� � u�� (��) = �� sin � �̂ + �� sin2 � �̂ + ��sin3 � �̂ (6.54a) u�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) = 0 (6.54b) where, �� = ����� � ��� + �� � � � + ������ �� ��� ��� � � �� (6.54c) �� = �������� � − �������� �� � � ��� ��� ��� � � ��� (6.54d) �� = ���� ��� �� − ������� �� � � ��� ��� ����� � � ���� (6.54e) where, ��(0) = �����, �� = ��� ��� + �� ����� � � ��� (6.54f) ��(0) = �����, �� = 9 � � �� + � ��� � � ��� � (6.54g) ��(0) = ������, ��� = � ��� − �� ����� � � ���� (6.54h) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 14 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load on solving (6.54a,b), we get u�� (��) = �� cos ����̂ + �� sin ����̂ + �� ��� ��� ��� � � �� + �� ��� ���� ��� � � ��� + �� ��� ���� ��� � � ��� (6.55a) ��(0) = 0, ��(0) = − 1 2��� � ��� ��� � − �� + 2��� ��� � − 4�� + 3��� ��� � − 9�� + �� � (0)� = ��− �� � � �� � ��� � + �� � ��� � � �� + ���� � �� ��� � � ��� + ��� ��� ����� � � ���� ��� ��� (6.55b) that is ��(0) = ������, (6.55c) ��� = − 1 2��� �� � �� ��� � − �� + 2�� ��� � − 4�� + 3��� ��� � − 9�� �� + �� � � � ��� � − �� ����� � � ��� + � ��� � � ��� − � ��� � � ����� (6.55d) next we ensure a uniformly valid solution in �̂ in (6.50a) by first substituting for the relevant terms there, and equating to zero the coefficients of cos ����̂ and sin ����̂ to get �� � + �� = 0, �� � + �� = 0 (6.56a) on solving (6.56a), we get ��(�) = 0, ��(�) = ��(0)��� (6.56b) where, �� � (0) = −��(0), �� ��(0) = ��(0) (6.56c) the remaining equation in (6.50a) is now written as u��,���� (��) + ��� � u�� (��) = �� sin � �̂ + �� sin2 � �̂ + ���sin3 � �̂ (6.57�) u�� (��)(0,0) = 0, � ��,�� (��)( 0, 0) + ���,� (��) (0, 0) = 0 (6.57b) where, �� = ���� �� ��� + �� � � � + �������� � � ��� ��� � � � ����� � � ��� (6.57c) �� = ������� �� + ������� � ��� �� �� ����� � � ���� (6.57d) ��� = ����� ��� �� + ��� � (�� ��� � + �� �) (6.57e) ��(0) = �����, �� = �� ��� + �� ����� � � ��� (6.57f) ��(0) = ����� ��� , ���(0) = ����� �� (6.57g) �� �(0) = ������ ���� − ����� ����� � � ��� (6.57h) �� �(0) = ��� � �� ��� − �� ��� � � ��� (6.57i) ��� � (0) = ���� ��� (6.57j) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 15 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load on solving (6.57a,b), we get u�� (��) = �� cos ����̂ + �� sin ����̂ + �� ��� ��� ��� � � �� + �� ��� ���� ��� � � ��� + ��� ��� ���� ��� � � ��� (6.58a) with ��(0) = 0, (6.58b) ��(0) = − 1 ��� � � � �� ��� � − �� + 2�� ��� � − 4�� + 3��� ��� � − 9�� � + �� � � + ��� � � �� � ��� � + �� � ��� � � �� + ����� �� � �� ����� � � ���� + ��� ��� � ����� � � ���� ��� ��� (6.58c) we note from (6.56a) that �� � (0) = −��(0) = ������ � (6.58d) on simplifying (6.58c), we get ��(0) = �����, (6.58e) �� = − 1 ��� �� � �� ��� � − �� − 3 8(��� � − 4��) − 9 56(��� � − 9��) �� − �� 8 + �� � � � ���� � + �� ����� � � ��� − � ��� � � ��� + � ����� � � ���� �� (6.58f) we next substitute on the right hand side of (6.14) and get ��(��) = 3� ��� (��) ��� (��) � � + ��� (��) � � �� (��) � (1 − cos 2��) − ���,�� (��)(1 − cos 2��) + ���,�� (��) (1 − cos 4��) + ���,�� (��) (1 − cos 6��)� −2 ���,��� (��) (1 − cos 2��) + ���,��� (��) (1 − cos 4��) + ���,��� (��) (1 − cos 6��)� + ���,�� (��) (1 − cos 2��) + ���,�� (��) (1 − cos 4��) + ���,�� (��) (1 − cos 6��)� + �� � ��,��� (��) (1 − cos 2��) − �� ����,�� (��) (1 − cos 2��) −2 ��� � ��,�� (��) (1 − cos 2��) + ��,� (��)(1 − cos 2��) + ���,� (��) (1 − cos 4��)� +����,� (��) (1 − cos 6��)� (6.59a) by letting �(��) = ∑ u� (��) (1 − cos 2��) � ��� (6.59b) and substituting same into (6.59a), multiplying the resultant equation by cos 2�� and integrating from 0 to π, we see that, for � = �, we get u�,���� (��) + ��u� (��) = 45� 4 ��� (��) ��� (��) � � + ��� (��) � � �� (��) � − ��,�� (��) − 2 ���,��� (��) + ��,�� (��) + �� � ��,��� (��) � − �� ����,�� (��) − 2�� � ��,�� (��) − 2��,� (��) (6.60a) u� (��)(0,0) = 0, � �,�� (��)(0, 0) + ��,� (��)(0, 0) + �� � ��,�� (��) = 0 (6.60b) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 16 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load for � = 2� in the substitution into (6.59a), we get u��,���� (��) + ��� � u�� (��) = −9� 4 ��� (��) ��� (��) � � + ��� (��) � � �� (��) � − ���,�� (��) − 2 ����,��� (��) + ���,�� (��) � − 2���,� (��) (6.61a) u�� (��)(0,0) = 0, � ��,�� (��)(0, 0) + ���,� (��) (0, 0) (6.61b) for � = 3� in the substitution into (6.59a), we get u��,���� (��) + ��� � u�� (��) = 3� 4 ��� (��) ��� (��) � � + ��� (��) � � �� (��) � − ���,�� (��) −2 ����,��� (��) + ���,�� (��) � − 2���,� (��) (6.62a) u�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) (6.62�) if we substitute for terms on the right hand side of (6.60a), we get u �,���� (��) + ��u� (��) = 45� 4 ��� � � �� cos ��̂ 4 + � 2 (1 − cos 2��̂) − �� 4 cos 3��̂�� + ������ + �� ��� + ��� � � � cos ��̂ + �� ��� + �� � � � ��� ��̂� + ������� cos 2��̂ + �������� 2��̂ + ���� � � cos 3��̂ + ���� � � sin 3��̂�� − ��� �� cos ��̂ + ��� � � �� �� �� − ���� �� �� ��� cos 2��̂ − ��� �� �� ���� cos 3��̂�� −2 ����� � cos ��̂ − ��� � sin ��̂ − 2�� � 3� cos 2��̂ − 3�� � 8� cos 3��̂�� −��� � �� � � cos ��̂] − �� ��� ��cos ��̂ − 2�� � ��� cos ��̂ −2 ��� � cos ��̂ + ��� � � �� � �� − ���� �� � ��� cos 2��̂ − ��� �� � ���� cos 3��̂�� (6.63a) to ensure a uniformly valid solution in �̂ as far as (6.63a) is concerned, we equate to zero the coefficients of cos ��̂ and sin ��̂ and respectively get �� � + �� = −ℎ�(�) = − � �� �� ��� + �� � � � (6.63b) and �� � + �� = ℎ�(�) = − � �� � ��� � � ���� � � + �� ��� + ��� � � �� − 2�� � − �� �� − �� ����� � −��� � �(2�� + �� � )] (6.63c) on solving (6.63b,c), we get �� = ��� ���(0) − � ℎ�(�)���� � � � �� = ��� ���(0) + � ℎ�(�)���� � � � https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 17 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load the remaining equation in (6.60a) is u �,���� (��) + ��u� (��) = ��� + ��� cos 2��̂ + ��� sin 2��̂ + ��� cos 3��̂ + ��� sin 3��̂ (6.64�) u� (��)(0,0) = 0, � �,�� (��)(0, 0) + ��,� (��)(0, 0) + �� � ��,�� (��) = 0 (6.64b) where, ��� = ��� � ������ − ��� � � � − ���� �� ��� − ���� ��� ��� (6.64c) ��� = ��� � ������ − ��� � � � + ������ �� �� ��� + ����� �� �� �� + ������� �� � ��� (6.64d) ��� = �������� � (6.64e) ��� = ��� �� (�� ��� − �� ���) + ������ �� �� ����� + ���� ����� �� (6.64f) ��� = ������� � �� + ������ �� � ���� (6.64g) ���(0) = ������ ��� , ���(0) = ������ ���� , ���(0) = ������ �� , (6.64h) ���(0) = ������� ����� , ���(0) = ������ ���� (6.64i) on solving (6.64a), using (6.64b) we get u� (��)(�̂, �) = ��� cos ��̂ + ��� sin ��̂ + ��� �� − 1 3�� (��� cos 2��̂ + ��� sin 2��̂) − � ��� (��� cos 3��̂ + ��� sin 3��̂) (6.65a) where, ���(0) = −���(0) �� + ���(0) 3�� + ���(0) 8�� ���(0) = ��������� ����� , ���(0) = 0 (6.65b) as was in the case of u� (��) in (6.60a) which eventually led to (6.65a), we now substitute for terms into (6.61a) and to ensure a uniformly valid solution in �̂, we equate to zero the coefficients of cos ����̂ and sin ����̂ and respectively get �� � + �� = �� �� (�� �� + 2�� � ) (6.66a) and �� � + �� = �� �� (�� �� + 2�� � ) (6.66b) on solving, we get �� = ��� ���(0) − � �� ∫ ��(�� �� + 2�� � )�� � � � (6.66b) �� = ��� ���(0) − � �� ∫ ��(�� �� + 2�� � )�� � � � (6.66c) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 18 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load the remaining equation in the substitution into (6.61a) is u��,���� (��) + ��� � u�� (��) = ��� + ��� cos � �̂ + ��� sin � �̂ + ���cos 2� �̂ + ��� sin 2� �̂ + ���cos 3� �̂ + ���sin 3� �̂ (6.67a) u�� (��)(0,0) = 0, � ��,�� (��)(0, 0) + ���,� (��) (0, 0) (6.67b) where, ��� = ��� � (��� � + �����) + ��� �� ��� � + ��� ��� ���� � (6.67c) ��� = ��� � � ���� � � + �� � ��� � � + ���� + ���� �� ����� � ���� + ���� � ����� � ���� − ����� ����� ��� � ��� (6.67d) ��� = ����� � � ��� � � + ��� (6.67e) ��� = −9� � ��� � � + ������ + ������ �� �� ����� � ����� + ������ �� � ����� � ����� − ����� ����� ��� � ���� (6.67f) ��� = ������� � (6.67g) ��� = −9� � �� ��� � − ���� � � + ����� �� �� ����� � ����� + ����� �� � ����� � ����� − ����� ����� ��� � ���� (6.67h) ��� = ������� � � (6.67i) on solving (6.67a,b), we get u�� (��)(�̂, �) = ��� cos ����̂ + ��� sin ����̂ + ��� ��� � + � 1 ��� � − �� � (��� cos ��̂ + ��� sin ��̂) + � � ��� � � ���� (��� cos 2��̂ + ��� sin 2��̂) + � � ��� � � ���� (��� cos 3��̂ + ��� sin 3��̂) (6.68a) we may not need ���(0) and ���(0). we next substitute into (6.62a) and to ensure a uniformly valid solution in �̂, equate to zero, the coefficients of cos ����̂ and sin ����̂ and respectively get �� � + �� = �� ���� (�� �� + 2�� � ) (6.68b) and �� � + �� = � ���� (�� �� + 2�� � ) (6.68c) on solving, we get �� = ��� ���(0) − � ���� ∫ ��(�� �� + 2�� � )�� � � � (6.68d) �� = ��� ���(0) + �� ���� ∫ ��(�� �� + 2�� � )�� � � � (6.68e) the remaining equation in the substitution into (6.62a) is u��,���� (��) + ��� � u�� (��) = ��� + ��� cos � �̂ + ��� sin � �̂ + ���cos 2� �̂ + ��� sin 2� �̂ + ���cos 3� �̂ + ���sin 3� �̂ (6.69a) u�� (��)(0,0) = 0, � ��,�� (��) (0, 0) + ���,� (��) (0, 0) (6.69b) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 19 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load where, ��� = ����� � � + ������� � − � ��� ��� �� �� ���� � (6.69c) ��� = ���� ��� �� + ���� � � ��� � � + ��� − ��� �� ����� � ���� − ����� �� ��� ��� � ��� − ���� � ����� � ���� (6.69d) ��� = 3��� ��� − �� � � � (6.69e) ��� = ������ � � + ������� � − ������ �� �� ����� � ����� − ������ �� � ��� � ���� − ����� ����� ��� � ���� (6.69f) ��� = ������� � (6.69g) ��� = − ������� � � + ��� ���� �� − ���� �� �� ����� � ����� − ���� �� � ����� � ����� − ������ � ����� ��� � ���� (6.69h) ��� = ������ � �� (6.69i) ���(0) = ���� ��� , ���(0) = 3������ (6.69j) ��� = �� ���� + �� ������ � ���� + ��� ��� � ��� � �� ���� + �� ����� � ����� � − �� ����� � ���� (6.69k) ���(0) = 0, ���(0) = 3������ (6.69l) ��� = �� ��� − � ����� � ���� − �� ��� � ��� � �� ��� − �� ��� � ���� + � ��� � ��� (6.69m) ���(0) = 0, ���(0) = 3������, ���(0) = 0 (6.69n) ��� = � ���� + �� ������ � ����� (6.69o) on solving (6.69a,b), we get ��� �� (�̂, �) = ��� cos ����̂ + ��� sin ����̂ + ��� ��� � + � 1 ��� � − �� � (��� cos ��̂ + ��� sin ��̂) + � � ��� � � ���� (��� cos 2��̂ + ��� sin 2��̂) + � � ��� � � ���� (��� cos 3��̂ + ��� sin 3��̂) (6.70a) where, ���(0) = 3������, (6.70b) ��� = − � � ������ � + ��� ��� � � �� + ��� ��� � � ��� + ��� ��� � � ���� (6.70c) ���(0) = 0 (6.70d) so far, we write the deflection as �(�, �̂, �) = ���� (��) + ��� (��) + ���� (��) + ⋯ �(1 − cos 2��) +����� (��)(1 − cos 2��) + ��� (��)(1 − cos 4��) + ��� (��)(1 − cos 6��)� +���� (��)(1 − cos 2��) + ��� (��)(1 − cos 4��) + ��� (��)(1 − cos 6��)� +����� (��)(1 − cos 2��) + ��� (��)(1 − cos 4��) + ��� (��)(1 − cos 6��)� �+ ⋯ ] + ⋯ (6.71) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 20 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load 7. critical values of the dependent variables at maximum deflection as in [18, 19], the dynamic buckling load ��, which is defined as the largest load parameter for the solution of the problem to be bounded, is obtained from the maximization �� ��� = 0 (7.1) where, �� is the maximum deflection whose maximum values of the dependent variables we shall now determine. let ��, �̂�, �� be the values, at maximum displacement of �, �̂and � respectively and let us now assume the following asymptotic series �̂� = �̂� + ��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ (7.2a) �� = �� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ �� = ��� = �[�� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ ] (7.2b) as a function of �, �̂, �, the conditions for �(�, �̂, �) to have a maximum are �,� = 0, (1 + �� � �� + �� � �� + ⋯ )�,�� + ��,� = 0 (7.3) on substituting (6.71) into the first of (7.3), we get the value of � at maximum deflection, namely ��, as �� = � �� , � = 1, 2, 3, … (7.4) on evaluating (6.71) at � = �� = � �� , we get � = 2���� (��) + ��� (��) + ���� (��) + ⋯ � + 2�� ��� (��) + ��� (��) + ���� (��) + ��� (��) �� +�� ���� (��) + ��� (��) � + ⋯ � (7.5) we shall now expand each of the terms of (7.3), (which is evaluated at (��, �̂�, ��)) by using (7.2a-c), as well as (7.4) and (7.5). thus, we get ���,�� (��) = � ���,�� (��) + {��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ }��,���� (��)� +�{�� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + ⋯ }� �,��� (��) + 1 2 �{��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ }�� �,������ (��) � +2�{��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ } × {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� �,����� (��) +��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ } �����,���� (��) ��� (���,�) (7.6a) ��� �,�� (��) = ���� �,�� (��) + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + … )}� �,���� (��)� +�{�� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ }� �,��� (��) + 1 2 �{��̂�� + … + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ }���,������ (��) � +2�{��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ } × {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� �,����� (��) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 21 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load +��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ } �����,���� (��) ��� (���,�) (7.6b) �����,�� (��) = ��� ���,�� (��) + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + … )}��,���� (��)� + �{�� + ⋯ + ��(��� + ���� + ⋯ ) + ⋯ } ��� �,��� (��) + ⋯ �� (���,�) (7.6c) �� ���,�� (��) + ���,�� (��) � = �� ����,�� (��) + ���,�� (��) � + { ��̂�� + ���̂�� + ⋯ } ���,���� (��) + ���,���� (��) �� + �{�� + ���� + ⋯ } ���� �,��� (��) + � ��,��� (��) � + ⋯ �� (���,�) (7.6d) ����� �,�� (��) + � ��,�� (��) � = ��� ����,�� (��) + ���,�� (��) � + { ��̂�� + ⋯ } ���,���� (��) + ���,���� (��) �� + �{�� + ���� + ⋯ } ���� �,��� (��) + � ��,��� (��) � + ⋯ �� (���,�) (7.6e) ������ �,�� (��) + � ��,�� (��) � = ���� ���� �,�� (��) + � ��,�� (��) � + ⋯ �� (���,�) (7.6f) ���� � � �,�� (��) = �� ��� � � �,�� (��) + �� � {��̂�� + ���̂�� + ⋯ }� �,���� (��) + �{�� + ���� + ⋯ }��� � � �,�� (��) � ,� � + � � {�̂�� + ⋯ }��� � � �,���� (��) + �{�� + ���� + ⋯ }{��̂�� + ���̂�� + ⋯ }��� � � �,���� (��) � ,� + � � ��(�� + ⋯ )� ����� � � �,�� (��) � ,�� + ⋯ �� (���,�) (7.6g) ����� � ��,�� (��) = ��� ��� � ��,�� (��) + (��̂�� + ⋯ )�� � ��,���� (��)� + ���(�� + ⋯ )��� � � �,���� (��) � ,� + ⋯ �� (���,�) (7.6h) ������ � � �,�� (��) = ���� ���� � � �,�� (��) + ⋯ �� (���,�) (7.6i) ����,� (��) = �� ���,� (��) + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + … )}��,��� (��) � + � � �{��̂�� + … + ��(�̂�� + ��̂�� + ⋯ ) + ⋯ }�� �,����� (��) � +2�{��̂�� + ⋯ + ��(�̂�� + … ) + ⋯ } × {�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} ���� �,���� (��) + ⋯ ��� (���,�) (7.6j) �����,� (��) = ������,� (��) + ⋯ + {��̂�� + ⋯ + ��(�̂�� + ��̂�� + ⋯ )}� ����,��� (��) + ⋯ �� (���,�) (7.6k) ������,� (��) + ���,� (��) � = ��� ����,� (��) + ���,� (��) � + �(��̂�� + … ) ���,��� (��) + ���,��� (��) � + ⋯ �� ��+�(�� + … )���,�� (��) + ���,�� (��) � + ⋯ �� (���,�) (7.6l) �������,� (��) + ���,� (��) � = ������,� (��) + ���,� (��) �│(���,�) + ⋯ (7.6m) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 22 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load by substituting (7.6a – m) into the second equation of (7.3) and equating the coefficients of ����, we get �(�): ��,�� (��) (�̂�, 0) = 0 (7.7a) �(��): �̂��� �,���� (��) + ��� �,��� (��) + ��,� (��) = 0 (7.7b) �(���): �̂����,���� (��) + �����,��� (��) + (�̂��)� 2 ��,���� (��) + �̂������,����� (��) + (��)� 2 ��,���� (��) +�̂����,���� (��) + ����,��� (��) + ��,�� (��) + �̂����,��� (��) + ����,�� (��) + ��,� (��) (7.7c) �(��): �̂��� �,���� (��) + �� �,���� (��) + � ��,���� (��) � + �� � (0)� �,�� (��) = 0 (7.7d) �(���): �̂����,���� (��) + �����,��� (��) + 1 2 �2�̂���̂����,������ (��) + 2�̂������,����� (��) � + �̂����,���� (��) + �̂�� ���,���� (��) + ���,���� (��) � + �� ���,��� (��) + ���,��� (��) � + ���,�� (��) + ���,�� (��) � + �̂��� �,��� (��) + ���,� (��) + ���,� (��) � = 0 (7.7e) etc., where equations (7.7a) to (7.7e) are evaluated at (�̂�, 0). from (7.7a), we get ��̂� = ��, � = 1, 2, 3, … ∴ �̂� = � � , (� = 1) (7.8a) where we have taken � = 1. on substituting (7.8) in (7.7b) and simplifying, we get �̂�� = � ��� �,�� (��) � ��,� (��) � � �,��� (��) � � �,���� (��) � (���,�) = 0 (7.8b) on substituting for terms in (7.7c) and simplifying, we get �̂�� = �������,�� (��) � �,���� (��) � (���,�) = ��� �� = � �� (7.8c) we next substitute into (7.7d) and get �̂�� = � ��� �,���� (��) � � ��,���� (��) � � �,���� (��) � (���,�) (7.8d) now, ��,���� (��) (�̂�, 0) = ����� �� , ���,���� (��) (�̂�, 0) = ������, (7.8e) ��� = � � �� � ���� � − �� ����� � � ��� + � ����� � � ���� − � ����� � � ���� �� − �� ���� ����� � � ��� + ���� ���� � � ���� − ��� ����� � � ���� �� (7.8f) � �,���� (��)(�̂�, 0) = −��� (7.8g) on substituting for terms in (7.8d), we get �̂�� = ������, ��� = ���� + �� �� � (7.8h) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 23 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load after substituting for terms in (7.7e) and simplifying, many terms vanish and the remaining terms give �̂�� = �� �� � �,���� (��) ��̂���� + � �,����� (��) + � �,�� (��) + � ��,�� (��) + ��,� (��) + ���,� (��) ��� (���,�) (7.8i) we now evaluate each of the terms in (7.8i) as follows: ��,�� (��) (�̂�, 0) = ������� ��� (7.9a) ���,�� (��) (�̂�, 0) = ������ (7.9b) ��� = �� cos ����̂� − ��� ����� � � ��� − � ������ � � ���� − � ������ � � ���� (7.9c) ��,� (��)(�̂�, 0) = ������ ���� (7.9d) ���,� (��) (�̂�, 0) = ������ (7.9e) ��� = − �� ��� ������ � − � � � � ��� � + �� ����� � � ��� + � ������ � � ���� + � ������ � � ���� � (7.9f) � �,����� (��) (�̂�, 0) = ���� �(0) = ��� (7.9g) on substituting for terms in (7.8i), we get �̂�� = ������, ��� = � �� �������� + ���� ��� + ��� − ����� ����� + ���� (7.10) later, we shall also need terms like ��, ���, ���, ��� ��� ��� which we now evaluate directly from (6.2b) (evaluated at maximum values of the variables). thus, at maximum deflection, (6.2b) becomes �̂� = �� + � ��(��)��� ��(��)���⋯ � � (7.11) by using (7.2a – c), we can write (7.11) as �̂� + ��̂�� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ ) + ⋯ = �� + ���� + ����� + ⋯ + ��(��� + ���� + ����� + ⋯ ) + �� ���(0) + ���� � (0) + ��� � 2 �� ��(0)� +�� ���(0) + ���� � (0) + ��� � � �� ��(0) + ⋯ � + ⋯ (7.12) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 24 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load we note the fact that ��(0) = 0 and equally note that �� can be expanded using (7.2b). by equating coefficients of ���� in (7.12), we get �(�): �̂� = � � = �� ∴ �� = � � (7.13a) �(��): �̂�� = 0 = ��� ∴ ��� = 0 (7.13b) �(���): �̂�� = �� � = � �� = ��� ∴ ��� = � �� (7.13c) �(��): �̂�� = ������ = ��� + �� � (0)�� ∴ ��� = ������ − �� � (0)�� = ������ (�) (7.13d) ��� (�) = ��� + ���� ���� (7.14) �(���): �̂�� = ������ = ��� + ����� � (0) + �� � 2 �� ��(0) ∴ ��� = ������ − ����� � (0) − �� � � �� ��(0) = ������ (�) (7.15a) ��� (�) = ��� − ���� ���� (7.15b) 8. maximum deflection, �� to determine the maximum deflection w�, we evaluate (7.5) at �̂�, �� ��� ��. thus, we get w� = 2����� (��) + ���� (��) + ����� (��) � + 2�� ����� (��) + ���� (��) � + ����� (��) + ���� (��) �� + ������� (��) + ���� (��) � + ⋯ � + ⋯ (8.1) where,��� (��) = �� (��) (�̂�, ��). we now expand each term of (8.1) using (7.2a – c). thus, we have ���� (��) = ��� (��)(�̂�, ��, ��) = ���� (��) + {�̂��� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + ⋯ )}� + � � {{�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}�� + �{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} × {�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}��,�� (��) ��+��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� + ⋯ ]|(���,�) (8.2) ����� (��) = �� ��� (��) + {�̂��� + ���̂�� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}��,�� (��)� + �{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} + � � {{�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}�� + 2�{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )} × {�̂��� + ⋯ + ��(�̂�� + ��̂�� + ���̂�� + … )}��,��� (��) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 25 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load ��+��{�� + ���� + ⋯ + ��(��� + ���� + ⋯ )}� + ⋯ ]|(���,�) (8.3) ������ (��) = ��� ��� (��) + {�̂��� + ⋯ + ��(�̂�� + ⋯ )}��,�� (��)� ��+�{�� + ⋯ + ��(��� + ���� + ⋯ )} + ⋯ ]|(���,�) (8.4) ����� (��) + ��� (��) � = �� ���� (��) + ��� (��) � + {�̂��� + ⋯ }��� (��) + ��� (��) � ,�� � ��+�{�� + ⋯ + ��(��� + ���� + ⋯ )}���,� (��) + ���,� (��) ��� (���,�) (8.5) ������ (��) + ��� (��) � = ��� ���� (��) + ��� (��) � + {�̂��� + ⋯ }��� (��) + ��� (��) � ,�� � ��+ �(�� + ⋯ )���,� (��) + ���,� (��) ��� (���,�) (8.6) ������� (��) + ��� (��) � = ���������,� (��) + ���,� (��) ��� (���,�) (8.7) on substituting (8.2) – (8.7) into (8.1), we observe that most terms vanish and the remaining ones are given as w� = 2� ����� (��) + �����,� (��) + ���� � 2 ��,�� (��) ��� (���,�) +2�� ���� (��) + ��� (��) � + � ������,� (��) + �̂����,�� (��) + �����,� (��) + ���,� (��) ��� +��������,� (��) + �̂���̂��� �,���� (��) + �������,�� (��) + �̂��� �,�� (��) + ���̂��� �,��� (��) � ���+ �� � � ���,� (��) + ���,� (��) � + ��� (��) + ��� (��) ���� (���,�) + ⋯ (8.8) we however note that �� (��)(�̂�, 0) = ����� �� , (8.9a) ��� (��)(�̂�, 0) = ����� (8.9b) �� = � � � �(����� ������) ��� + �(����� ������) ����� � ���� + �(����� ������) ����� � ����� + (����� ������) ����� � ����� � (8.9c) �� (��)(�̂�, 0) = ������ ��� , (8.10a) ��� (��)(�̂�, 0) = ������ (8.10b) ��� = � �(����� ������) ���� � �� − ���(����� ������) ��� � ��� + ���(����� ������) ��� � ���� − ���(����� ������) ����� � ����� � (8.10c) on substituting (8.9a) – (8.10c) into (8.8) and simplifying, we get w� = 4�� �1 − �� �� + � � � �� � � � � + ������� �� ��1 + ���� �� � + ���� + ������ + ⋯ (8.11a) where, ��� = �� �� � � � � ��� ���� + ���� − ��� (�) − ���� (8.11b) ��� = �� �� �� � � � ��� (�) − ��� (�) − � � � � ��� − ��� + � � � � ��� � ��� ���� + ���� − �� ��� + ���� (8.11c) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 26 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load 9. dynamic buckling load, �� having determined the maximum deflection, w� as in (8.11a – c), we shall now determine the dynamic buckling load �� from the maximization (7.1). we shall first reverse the series (8.10a) which we now write as w� = ��� + ���� + ⋯ (9.1a) where, �� = 4 �1 − �� �� + � � � �� � � � � (9.1b) �� = ����� �� ��1 + ���� �� � + ���� + ������ (9.1c) thus, for the reversal, we write � = ��w� + ���� � + ⋯ (9.2) upon substituting in (8.13) for w� from (8.12a), and equating the coefficients of �, we get �� = � �� , �� = − � �� �� �� (9.3a) the maximization (7.1) is now easily executed from (9.2) to yield �� + 3����� � = 0, (9.3b) where w�� is the maximum value of the deflection at dynamic buckling. thus, we have w�� = � ��� ��� (9.4) on substituting from (9.3a) in (9.4) we get w�� = � √� � �� � �� � (9.5) if we next evaluate (9.2) at dynamic buckling, we get � = ��w�� + ����� � + ⋯ = w��( �� + 3����� � ) (9.6) on substituting in (9.6) for ��, �� ��� w��, we get � = � �√� � �� �� � � � (9.7) on substituting in (9.7), we get the equation for determining the dynamic buckling load λ� as (16�� − 8��λ� + 1) � � = 18√10(����)��λ�� � � � ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � � � � (9.9) equation (9.9) gives an implicit formula for determining the dynamic buckling load λ�. by using equation (5.25), we can eliminate the imperfection parameter ϵ and hence relate λ� to λ�. this gives � ������������ ������������ � � � = √2 � �� �� � ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � �� � �� � ����� ������� ��������������� � ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ � � (9.10) https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 27 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load the least value of the dynamic buckling load λ� is obtained when � = 1 and for this value, equations (9.9) and (9.10) respectively become (17 − 8λ�) � � = 18√10(����)λ�� � � � ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � � � � (9.11) and � ������ ������ � � � = √2 � �� �� � ⎣ ⎢ ⎢ ⎢ ⎢ ⎡ ���� ���� �� �� ����� ������ �� �� �� � � � � �� � � � �� � �� � ������ ��������� � ⎦ ⎥ ⎥ ⎥ ⎥ ⎤ � � (9.12) where the right hand sides of (9.11) and (9.12) are to be evaluated at � = 1. 10. analysis of results the graphical plots of the results were done using q-basic codes and the results are hereby presented in fig. 1, fig. 2 and fig. 3. fig. 1: relationship between the static buckling load, �� and imperfection factor, ϵ using eqn. (5.26), eqn.(5.28), eqn.(5.30) and eqn.(5.32). 0 0.5 1 1.5 2 2.5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 st a ti c b u c k li n g l o a d imperfection factor static buckling load (eqn 5.28) static buckling load (eqn 5.30) static buckling load (eqn 5.32) static buckling load (eqn. 5.26) 〖λ�〗^ ϵ https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 28 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load fig. 2: relationship between the dynamic buckling loads and imperfection parameters at some fixed values of the damping factor, δ, using eqn. (9.11). fig. 3: relationship between the dynamic buckling loads and the static buckling loads at some fixed values of the damping factor, δ, using eqn. (9.12). from fig. 1, we observe that the static buckling load of a clamped column is always higher than that of the same column with simply–supported boundary conditions. in general, the static buckling load of a column with either clamped or simply-supported boundary conditions and whose deflections are strictly in the shape of imperfection always has the least static buckling load. however, while the static buckling load of a clamped column satisfies the inequality, 1 < �� < 2.125, a similar column with simply– supported boundary supports satisfies the inequality 0 < �� < 1. fig. 2 shows that the dynamic buckling load �� decreases with increased imperfection for any value of the damping parameter. from fig. 3, we observe that at low values of the static buckling load �� (precisely for 1 < �� < 1.45), there is no significant difference in the value of the dynamic buckling load �� of the column, whether damped or undamped. however, at higher values of �� (i.e, 1.45 < �� < 2.125), the dynamic buckling load �� rises with �� and the highest of 0 0.2 0.4 0.6 0.8 1 1.2 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.2 d y n a m ic b u c k li n g l o a d imperfection factor dynamic buckling load, δ = 0 dynamic buckling load, δ = 0.01 dynamic buckling load, δ = 0.02 dynamic buckling load, δ = 0.03 λd ϵ 0 0.5 1 1.5 2 2.5 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1d y n a m ic b u c k li n g l o a d static buckling load dynamic buckling load , δ = 0 dynamic buckling load , δ = 0.01 dynamic buckling load , δ = 0.02 dynamic buckling load , δ = 0.03 λd λs https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 29 ijo international journal of mathematics (issn: 2992-4421 ) i.u. udo-akpan1 * https://ijojournals.com/ volume 07 issue 02 || february, 2024 || on a-two-parameter dynamic buckling of a viscously damped but clamped column stressed by a step load such rise is the undamped case. it is not clear whether such a result is specific to clamped boundary conditions or whether it is general. as in the static loading case, the inequality satisfied by the clamped column is 1 < �� < 2.125. we thus observe that generally, whether in the static or dynamic loading cases, clamped columns buckle at much more higher loads than similar columns with simply–supported boundary supports. for reasons attributed to nonlinearity and imperfection, clamped columns on nonlinear elastic foundations buckle at lower values of buckling loads than the corresponding classical buckling load of the column. we observe that while the buckling modes split into three distinct modes proportional to (1 − cos 2��), (1 − cos 4��) and (1 − cos 6��), it is only the buckling modes in the shapes of (1 − cos 2��) and (1 − cos 6��) that eventually contribute to dynamic buckling. the mode in the shape of (1 − cos 4��) does not contribute. lastly, as seen in equations (8.20) and (8.22), we are able to directly relate �� to �� even without the knowledge of the size of the associated imperfection. in this way, we have circumvented the process of repeating the arduous manipulations for different imperfection parameters. 11. conclusion we have carried out an analytical investigation of the buckling of an imperfect clamped column lying on a nonlinear elastic foundation but struck axially by a step load. it is our contention that similar enquiries can be extended to other loading conditions apart from step load and to other structures apart from columns. acknowledgements: authors acknowledge the valuable contribution their mentor, prof. a.m. ette (fnms), for his guardian and support on this work. references [1] timoshenko, s.p. and gere, j.m., theory of elastic stability, dover publications, london. 2012. 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(2020); a multitiming perturbation analysis of the deformation and dynamic buckling of a viscously damped toroidal shell segment stressed by a step load, iosr journal of mathematics (iosr-jm), 16(4), 40-59. [21] a. m. ette, j. u. chukwuchekwa, n.o. onuoha, and i. u. udo-akpan (2020); on the deformation and static buckling of a toroidal shell segment using a two-term fourier series imperfections, international journal of mathematics trends and technology (ijmtt), 66(8), 100-114 [22] g. ozoigbo, a.m. ette, j. chukwuchekwa, w. osuji, i.u. udo-akpan (2022); on the analysis of a pre-statically loaded nonlinear cubic structure pressurized by an explicitly time dependent slowlyvarying load, american journal of mechanics and applications, 10(1), 1-15 [23] g.e. ozoigbo, a.m. ette, j.u. chukwuchekwa, w.i. osuji, i.u. udo-akpan (2023); nonlinear analytical investigation of dynamic buckling of spherical shell trapped under a periodic load, mechanics of solids, springer link, 58, 202-215. [24] obong, h.p. and udoakpan, i.u. (2023); a review of tensor interaction in the theory of potential models, scientia africana, 22(3), 11-22. [25] o.g. udoaka, u.j. etim and i.u. udoakpan, (2024) efficient solution of nonhomogeneous linear differential equations with constant coefficients: the method of undetermined coefficient approach, ieee-sem, 12(1), 21-36 https://zenodo.org/doi/10.5281/zenodo.10726874 ijo journals volume 07 | issue 02 | february 2024 | https://ijojournals.com/index.php/m/index 32 ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals the polynomials [ n.. knots–figures ] edge points vibration & m-geometry by markos georgallides 1larnaca–cyprus (expelled from varosha-famagusta town occupied by the barbaric turks , in aug 1974) cyprus , civil structural o engineer (natua) , athens . abstract the interactions : one of the most important concept in geometry is , distance , which is the quanta in egeometry , while in material-geometry the composition of opposite , where the material point is the quanta in chemistry and physics . as in algebra zero ,0, is the master-key number for all positive and negative numbers and this because their sum and multiplication becomes zero, and the same on any coordinate-system where ± axes pass from zero . in pns space , the rolling of the positive ⊕ constituent on the negative ⊝ constituent , creates the neutral material point which equilibrium . angular-momentum is identical with spin and consists the first-discrete-energy-monad which occupies , discrete value and direction , in contradiction to the point which is nothing , dimensionless and without any direction . quaternions [(+)↻↺(-)] ≡ box 𝐁 𝐑 =𝐀𝐁 carries the principal stress 𝛔 𝐀 , 𝛔 𝐁 between points a(+) , b(-) which σ , as centripetal-acceleration is the minimum energy becoming from the in-storage ab acceleration and is equal to the gravity force g . [108 – 110] . because of the revolving and periodic acceleration of gravity g ≡  σ exists as the first energy-box-𝐁 𝐑 , while in the second box 𝐁𝐏 is followed the local-extreme-case where gravity g ≡  σ , and is altered locally by changing the principal-stress σ with an local-uniform-pressure → 𝐠𝐋 ≡ g k = g . [ force/area ] = g ← i.e. the minimum local energy acceleration is the known , universal gravitational-constant g = g k = 𝐤𝐄 g = 𝐤𝐋 σ , such for macrocosm and for microcosm , obeying the newton`s laws of motion . g ⏊ σ this energy in hydrogen-cave as e-m , conductor ≡ edge points vibration ≡ the pin of atom → plug into their sockets , which are the orbit – bracket–hooks ≡ the hands of atoms ← i.e. the atoms plug with their pins into the other atoms-drains = holes , and so bond and carry informations . 1 https://doi.org/10.5281/zenodo.16737369 mailto:georgallides.marcos@cytanet.com.cy 2 the fundamental particles origination mechanism in mfmf pns caves . this resonance frequency of hydrogen is common to all atoms and to all compounds in this cosmos . the energy-quaternion w ̅̅ ̅, b̅, monad-magnitudes exist as dualnature for any { ⊕ , ⊝ }, { position , motion },{ universe , black-holes } , { gravity ,antigravity} , {action → . ← reaction} , edge points vibration creating electric = [⊕] and magnetic = [⊝] forces as [⊕↔⊝] ,the light and others .the stplline conductors on the [stpl]mechanism , are the physical-rotors for the origination of the cosmic –particles which transfer informations as the signals spectrum .[110] from mechanics-physics , all systems possessing elasticity ≡ motion and reaction to the motion , the called mass , are capable of free vibration or vibration ≡ periodic motion taking place in the absence of external excitation .this principle issues for both systems closed {the atoms nucleus} or open systems {the orbitals}. for instance , in order that shifting of an u-d-quarks from an anti-proton into a proton , the spin-pair requires extra input of energy in (mev) , so that would the proton paired with a neutron and be stable. transfer informations as the signals . vibration in a cave ∆ , means the wave pattern . cave–spin-�̅� of pns space is �̅� = r m v and is the first monad occupying 4-spaces , i.e. from number n , of the equilibrium number of masses m 𝐧 in a system. a.. the n spaces of monad �̅� are the polygons 𝐒 𝐧 with n = 1 ≈ ∞ knots , b.. the n anti-spaces of monad �̅� are the polygons 𝐒 𝐧 with n =1≈ ∞ knots , c.. sub-spaces of monad ≡ ± √ �̅� 𝒏=𝟏−∞ are the polygons with n = 1 ≈ ∞ knots all n-regular polygons end to equations of n-degree segment , by finding a suitable value of the segment , x , that is we have in the general case to solve one or two equations of the form : a .r0. xn b .r2. xn−2 + c .rn−6. x³ – d .rn−4. x² + e .rn−2.x1– f. rn. x0 = 0 for the even polygons , and a .r 2. xn−2 b.rn−2. xn−3 + c.r2(n−4). x³ -d.r2(n−3). x² +e.r2(n−2). x1– f.r2(n−1).x0 = 0 for the odd polygons , where a , b , c , d are constants . the presented geometrical method is the solution of the above equation in the general case . because , the nth degree equations are → the vertices (n) and the sides (𝐚𝐧= 𝛌𝐧) of the n-polygon in circle ← number , π , is their common mould . [ 62 ] . the natural mechanism continuously originates the elementary particles and compounds , atoms and molecules with the one action from opposite . article [111] encloses many paragraphs of [110] , in order to distinguish and prove the way of energy and the stressespaths in the spaces . for the regular polygons is given such the geometrical solution as well the algebraic . [106] = programming the atoms and compounds , is an program which solves the problem of the n-knots figures and gives the energy spectrum of any complex-forced-vector . article [114] an way of deceptioning the cells is prepared . keywords : the figure n-knots , the polynomials n-knots , spaces and figure n-knots , the waves of spaces . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 3 the fundamental particles origination mechanism in mfmf pns caves . content abstract page ….................1 , a… the structure of the euclidean geometry 1a.. : the points , vectors , lines , planes , volumes , n-spces page ….................4 , 2a.. : the fundamental principles of e-geometry page ……….4.fig-1 , 3a.. : the quantization of the energy – space – universe page …………....7 , 4a.. : the correlation of spaces and the primary motion page ………11.fig-2 , 5a.. : the figure-eight knot & the figure n = 1 ≈ ∞ knots page ………13.fig-3 , 6a.. : the origination of the primary motion in spaces page …… ..21.fig-4 , 7a.. : the mechanism and charges for the dual originations page …...............23 , 8a.. : the material geometry e-geometry physicks and chemistry page ……....24.fig-5 , [a].. the { s tp l } natural mechanisms page …..26.fig-6,7.7a,7b,7c , [b].. the { tvpm } natural mechanisms page ... 32.fig-8a , 8b , [c].. the { tc i c } natural mechanisms page ..35.fig-9,9a,b,9c,9d,9e , [d].. the { tobm } natural mechanisms page ......... 43.fig-10 , 9a.. : the flow plan of the space energy universe page ……..45.fig-11 , 10a..:the e-spaces and the energy regular-polygons page.49.fig-12a,12b,12c,12d , b… : general : 1b.. : the spaces and the energy states page …... 54.fig-12, 2b.. : the complex numbers and quaternion. page …............. 56 , 3b.. : the stability of space anti-space , in the neutral-space page …….59.fig-13 , 4b.. : the quantum interference of the duality-photon page …….59.fig-14 , c.. : the origination of the physical loops ≡ caves 1c.. : the gravitation force g , and the kick-start page ….............. 62 , 2c.. : the united-coulomb-newton law for interactions page ….............. 69 , 3c... : the origination of photon`s light velocity c page ……..........69 , 4c... : the origination of electron-charge e̅ ≡ q̅ page ……..........72 , 5c... : the origination of the hydrogen cave h & explanations page ……..........73 , 6c.: the epilogues of prior page…78.fig-17a,17b,17c,17d,17e , d... the origination of the fundamental particles 1d… : the balancing of the space and anti-space page ……..99.fig-18 2d… : the vibration of particles in all levels page ……100.fig-19 5d… : the natural electromagnetic e-tetrahedron , cube , spheres page …..102.fig20.b 8d.. : the stability of fundamental particles conductors & atoms page ……108.fig26 9d.. : the origination of proton-neutron funta-particles , objectivity page ……110.fig28 e.. : references page…….……113 , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 4 the fundamental particles origination mechanism in mfmf pns caves . a… the structure of the euclidean geometry . [6 -12 ] a1.. point is nothing , has not any position and dimension , and may be anywhere in the known space therefore , the primary point a , being nothing also in no space , it is the only point and no-where ( i.e. primary point is the only space and from this all the others ) . [ 6 ] a2.. straight line is 0 , positive , negative , consisted of ± ∞ points which are nothing , and since it is composed of infinite points which are filling the line , then nature of line is that of point i.e. ( the all is one for lines and for points ). a3.. plane is positive , negative , ± neutral lines and since points are imaginary lines it is composed of infinite straight lines which are filling plane , so then , nature of plane is that of line and that of points ( i.e. the all is one for planes , lines and points ) . a4.. space is + positive , negative , ± neutral planes and since points are imaginary planes it is composed of infinite planes which are filling space , so then , nature of space is that of plane and that of points (i.e. the all is one for spaces , planes , lines and points) . 2a… the fundamental principles of e-geometry . 2a1.. the first dimensional unit ab is of two points a , b not coinciding which is the geometrical shape that has as position the point of direction ab⃗⃗⃗⃗ ⃗ , ba⃖⃗⃗⃗⃗⃗ , and as magnitude (the length |ab| = 0 → n → ∞ ) . |ab| is a straight line through points a , b . ( f.1b) , [ 6 ] ↓ a c b a c b ca + cb = ab , where , ▫ ▫ ▫ —▫—— ▫ ————▫— f.1(a) f.1(b) ds = an infinitely small increment of length ab in the direction ab⃗⃗⃗⃗ ⃗ or ba⃖⃗⃗⃗⃗⃗ . ∞ = an infinitely great magnitude |ab| in the directions ab⃗⃗⃗⃗ ⃗ , ba⃖⃗⃗⃗⃗⃗ ) . any point c is on straight line ab , only when exists the equation ca + cb = ab , i.e. the whole ab is equal to the parts ca and cb . ( as equation f 1.b ) . in case that ca + cb > ab then point c is not on line ab , and this is the main difference between euclidean and non-euclidean geometries . in definition 2 ( a line ab is breathless length) is altered as for any point c on line ab exists ca + cb =ab. edge points a , b not coinciding on monad ab , keep the properties of complex numbers with imaginary part which differs between the infinite point [11] – [13] unit ab creates spaces with two basic elements , the position ( which are the three directions ab⃗⃗⃗⃗ ⃗ , ba⃖⃗⃗⃗⃗⃗ , ab⃖⃗⃗⃗ ⃗ ) and the dimension ( which is the magnitude n such that n = 0 → |ab| → ∞) , and when exists , a = b = the principle of the equality . a ≠ b = the principle of the inequality . a→↔← b = ∞ the principle of virtual displacements . w = ∫ p.ds = 0 , ds = ∂w / ∂p ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 5 the fundamental particles origination mechanism in mfmf pns caves . pa⃗⃗⃗⃗ ⃗ + pb⃖⃗⃗⃗⃗⃗ = 0 the principle of stability pa⃗⃗⃗⃗ ⃗ + pb⃖⃗⃗⃗⃗⃗ > < 0 the principle of motion . 2a2.. the monads = quantum = 𝐝𝐬⃗⃗⃗⃗ = ab / ( n = ∞ → 0 ) = [ a ± b i ] = 0 → ∞ , are simultaneously ( actual infinity ) and also ( potential infinity ) in complex number form and this defines , infinity which exists between all points which are not coinciding and where ( ds > 0 ) , and because ds comprises any two edge points with imaginary part then this property differs between the infinite points. the plank length is a monad ds = 1,62 x 𝟏𝟎−𝟑𝟓 m , for the two points a , b , and for the moment is accepted as the smallest possible size . monad is also infinitely divided because edge points a , b are not coinciding i.e. ... ds = 1x𝟏𝟎−𝑵 < ∞ , where n is any number . 1.. spaces of unit-ab are (in plane) the infinite (+) circumscribed-regular-polygons on the circles with ab as side (repetition of unit ab) the nth space , the nth unit tensor of the n equal finite elements ds , and for the ∞ spaces , the line ab ↔ . ( fig-1 ) the diameter of this circles extends to infinity ( it is of potential nature ) . [11] 2.. anti spaces of unit ab are ( in the three dimensional space ) the symmetrically infinite (-) circumscribedregular-polygons on the circles or ( a solid in cube in sphere with ab as side of the solid ) . the harmonic repetition of unit ba , symmetrical to ab ) is the nth anti-space , the nth unit tensor of the n equal finite anti elements , and for the ∞ spaces , the sphere through line ba .the diameter of this spheres extends to the infinity ( it is of potential nature ) . [11] 3.. subspaces of unit ab are ( in plane) the infinite inscribed-regular-polygons in the circle with ab as diameter ( are the harmonic repetition of the roots in unit ab ) and in nth sub-space , the nth unit tensor of the n finite roots and in case of the ∞ elements are the points on the circle , and for 3d-space , the points on sphere n.ab ) . the superposition of spaces , anti-spaces and sub-space layers of unit ab is shown in figure . remark : (+) spaces , (-) anti -spaces , ( ± ) sub-spaces , of a unit ab are between magnitude ( point = 0 = nothing ) and the infinite magnitude ( ↔ = ab = ± ∞ = infinite ) which means that all spaces exist in one space only . because in spaces and anti-spaces , the ∞ spaces of unit ab is sector ab ↔ , and in sub-spaces the ∞ sub-spaces of unit ab are the points on the circle with ab as diameter , then this ordered continuum for points on the circle of unit ab and on line ab shows the correlation of spaces in unit ab . i.e. monads ds = 0 → ∞ are simultaneously, actual infinity (because for n = ∞ then ds = [ ab / n = ∞ ] = 0 , or a point ) and , potential infinity , ( because for n = 0 then ds = [ ab / n = 0 ] = ∞ is the straight line through ab . infinity exists between all points ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 6 the fundamental particles origination mechanism in mfmf pns caves . which are not coinciding , and because monads ds comprises any two edge-points with imaginary part , then this property is the difference between the infinite points . [11] 4.. neutral-spaces of unit-ab are ( in plane) the infinite circles with ab as diameter ( the harmonic repetition of the roots in unit ab ) and in nth sub-space , the nth unit tensor of the n finite roots , and in case of the ∞ elements are the points on the circle , and for the 3d-space , the points on the unit sphere ab ) . since spaces , anti – spaces and sub-spaces are created from unit ab , and are property of this unit only , therefore these are a restrained system ( s ) presupposition for unit |ab| = ds (displacement ds) is point a to move at the new position b (a ≡ b) which means an impulse ( p ) transfers point a to b . since in each restrained system ( s ) the work done ( w) by impulse ( p ) on a virtual displacement ( ds > 0 ) is zero , or w = ∫ 𝑃𝑑𝑠 𝐵 𝐴 = 0 therefore , each unit |ab| = |ds| > 0 exists , by this inner impulse ( p ) which means that , the position and dimension of all points which are connected across the universe and that of spaces , exist because of this static inner impulse p between a and b , on the contrary , should be one point only ( primary-point = black-hole → ds = 0 → p = ∞) impulse p = ∞ , and may be vacuum , momentum or potential or induced potential and it is a type of effect of push nature . 2a3..the superposition of plane space , anti-space layers and sub-space layers the simultaneously co-existence of spaces , anti-spaces and sub-spaces of any unit |ab| , unit ab = 0 → ∞ , ( a ≡ b ) i.e. , euclidean , elliptic , spherical , parabolic , hyperbolic, geodesics , metric and non-metric geometries , exist in euclidean model as an sub-case within . the interconnection of homogeneous and heterogeneous bounded spaces anti-spaces and subspaces of the universe . the unity of opposites is also the quantized property of euclidean geometry < all is one > as it is discrete ( for monads ab ) and continuous (for points a , b ) . for primary point a it is the only space [17] the unique case where p = 0 and δs > 0 = ab → ∞ is the primary planck`s space [ ps ] the spherical connection of the same points in ps , is related on the influence ( p = 0 → ab⃗⃗⃗⃗ ⃗ → ∞) of the infinite ( ∞ ) equilibrium primary-spaces and anti-spaces . for every point in [ ps ] exist the three spatial dimensions ( x , y , z ) and the infinite dimensions of the ( i ) layers at this point existing from the other layers ( at small scale using coordinate system ) of primary anti-space and sub-space where i = 1 → ∞ . it has been also shown that , every point m of primary planck`s space [ ps ] exists in this space with a deficit of impulse p = 0 → ∞ , due to the influence of the other equilibrium primary spaces . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 7 the fundamental particles origination mechanism in mfmf pns caves . the unique case where p = 0 and δs = 0 → 0 is the primary neutral space [ pns ] the ⊕ material point ≡ the space , continually rotates around the ⊝ material point ≡ the anti-space , on 2r circle . [pns] space is a this property of points in neutral space is the cause of their continuously survive . the quantization of points becomes through vector unit d�̆� = |ab| , and for energy is done in bound states ( loops ) which withhold the diffusion ( flow ) .the two fundamental dimensions (quanta of points (ds) and quanta of energy (d p) , are connected on| ab|, so at any point m exist the centripetal force of two opposite [⊕↻↺⊝] .all points of [pns] are homogenous ( i.e. all points of pns space are equivalent ≡ ) and isotropic ( i.e. all directions , ⇉ , of ab are equivalent ) → [ a ≡ a ] , for the primary dipole and for any dipole ai-bi issues , the position of dipole in the equilibrium space antispace creates charge = momentum and angular momentum ( spin = the intrinsic twist of space , anti-space ) which inextricably unify geometry of space and motion . [10 ] figure – 1 . the 4-primary-spaces of euclidean-geometry and the primary rotational quantized motion ≡ the unit-angular-momentum-energy in pns spaces . 3a… the quantization of the → { energy – space universe } ← in a conservative system the total energy ≡ motion is constant and the differential equation of motion can be established by the principle of conservation of energy . for the free vibration of an undamped system , the energy is party kinetic = 𝐄 𝐊 which is stored in ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 8 the fundamental particles origination mechanism in mfmf pns caves . the mass by virtue of its velocity and party potential = 𝐄 𝐏 which is stored in the form of strain energy in elastic deformation , or the work done in a forced-field such as gravity . newton`s second law is the first basis for examining the motion of a free – vibration in a system . when the deformation of a spring in the static equilibrium position is δ and the spring force k.δ equal to the gravitation al force mg , then issues k.δ = mg …. (a) applying newton`s second law to the mass m , or mass can be replaced by a mass moment of inertia denoting the resistance to motion δ , and then m �̈� = k x for an x – x axis . the natural – period of the oscillation is established from relation wn t = 2 π where issues [7] t = 2 π √ m 𝐤 , 𝐟 𝐧 = 1 / t = 1 2π √ k 𝐦 , 𝐟 𝐧 = 1 / t = 1 2π √ g δ ….…(b) i.e. the kinetic energy 𝐄 𝐊 is applicable in a single dof system including rotation . since issues e k = e p so and e k + e p = constant . from the theory of vibrations , { all systems possessing mass= m and elasticity = e = 1 / k are capable of free vibration in the absence of external excitation which is its natural frequency of vibration = 𝐰 𝐧 and the equation of motion is → �̈� + 𝐰 ²𝐧 x = 0 ←where w ²n= k m , gravitational force g = m g } i.e. since the hydrogen-cave system h , as e-m conductor , possesses mass 𝐦 𝐇 , and light-velocity c , then vibrates in – out the space-positions (x , y , z) of all universe with its natural frequency 𝐰𝐇 =√ k 𝐦 𝐇 . or and frequency 𝐟 𝐇 = 𝐰 𝐇 𝟐𝛑 , and is a wave – function providing the complete description of the physical reality . from , work = w = force 𝐅 𝐢,𝐣,𝐤 x displacement 𝐝𝐬 𝐢,𝐣,𝐤 = ( 𝐅 𝐢,𝐣,𝐤 x 𝐝𝐬 𝐢,𝐣,𝐤 ) = ( f x ds ) , and for the unit of time dt , → w / dt = f x ( ds / dt ) = ( f x �⃡� i , j , k ) = power = p . i.e. power ≡ force x velocity or ( p = f x �⃡� ) remarks , from mechanics physics : 1…energy e = 1 2 m h . c ² , or → 2e = 𝐦 𝐇 . c ² ← and velocity = the light-velocity c , 2…wave length λ h = 𝟐𝛑 𝐜 𝐰 𝐇 , and wave amplitude 𝐀 𝐇 = λ h 𝟐𝛑 , 3…hydrogen power 𝐏 𝐇 = 0,5. 𝐦 𝐇 . [ 𝐀 𝐇 ] ² . [ c ] ² 4…the hydrogen stress-vector 𝛔 𝐇 is related to c as , c = 𝚽 . 𝛔 𝐇 5…since hydrogen cave is an e-m conductor then it is a wave with natural frequency , and since occupies the property of →atoms-plug ≡ pin and atoms-drain = hole so are bonded ← in a multi-dof system at the stationary conditions . 6…to find the eigenvalues and eigenvectors of a multi-dof system , iteration method of motion is formulated by either the flexibility matrix [a] or the stiffness matrix. 7…momentum ≡ ± spin ≡ [gravity and antigravity] because the light velocity �̅� = [ 𝐆 𝚽 𝐀 ] , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 9 the fundamental particles origination mechanism in mfmf pns caves . and precedes that of a-momentum  �̅� = r m �̅� . exists vector b ⊥ r-c plane . 8…since angular-velocity |w̅| , a vector , rotates in one axis k with respect to one variable only and this happens in {pns-space ,atoms-space ,objective-reality and in universe} then quaternion momentum , b̅ & w̅ , and w ≡ motion are stabilized in the dual – type , such in microcosm , as in macrocosm , or both as omnipresent . 1…the dual-momentum b̅ & w̅, as real ≡ existing-universe , imaginary≡ black-holes 2…the dual photon , as real ≡ space ≡ particle and, imaginary ≡{ e m wave }. 3…the dual spin b̅ , as real ≡ space ≡ gravity and, imaginary ≡ { anti gravity }. 4…the dual – matter b̅ , as real ≡ space ≡ atoms and, imaginary ≡{ e m -wave }. 5…the dual-refraction of energy as real ≡ the space angle φ , imaginary ≡ �̅� 1, �̅� 2 { the energy ≡ w from the different velocities , �̅� 1, �̅� 2 }. the results in hydrogen cave : 1…hydrogen-cave , in – out universe occupies mass 𝐦 𝐇 , velocity �⃡� , and power 𝐏 𝐇 2…electron in hydrogen-cave precesses and nutates due to the gravitational constant g. the produced work is stored in form of→ stress energy as hydrogen-bracket-hook← the electron precesses from the continuous and immense-communication to gravity, g . electron-spin is the angular-momentum-vector �̅� and rotates according to equation db dt 3…the stationary → tetrahedron , in-sphere , cube , ex-sphere ←construction of atoms permits the space – coordinate structure of atoms , the wave-eigenfunctions of many non-commuting physical operators as momentum power, from the quantum-mechanical description of the physical-reality is complete i.e. if-known the physical operators , their coordinates are simultaneous physical reality. 4…the interactions of two or more systems with known status can be calculated any time by the bioelectronic-spectrum of →{ carrier-modulating-modulated , demodulation process mechanism } ← using markos program < programming atoms-bonding and their compounds > so , energy ≡ motion is of wave nature which enters the energy caves and becomes a particle or wave or both . in case of photons exists this dual property , wave – particle . thus the historical doubt of einstein podolsky rosen for the q-mechanics completion vanishes . [100,104] . 5… the programming of atoms bonding is the quantization of atoms -wave -energy to all possible equilibrium positions of the [⊕↔⊝] constitutes reactions . the wave-energy as vibration travels at 75-90 % of the light speed c , while the wave-energy in black holes is n π c times of the light speed . [100-101] 6… from all the possible reactions in compounds , the bonding or the releasing of energy is the vital rule of the theory of vibrations .the program programming the atoms and their compounds , analyses the interactions of two or more energy systems with known ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 10 the fundamental particles origination mechanism in mfmf pns caves . status using the → carrier–modulating–modulated–demodulation waves process ← some following wave properties that are defined : 6a... the equilibrium mode – shapes , φ , diagrams . 6b1..the circular – frequencies , 𝐖 𝐑 , in 1015 hz , 6b2..the natural – frequencies , 𝐟 𝐑 , in 1015 hz , 6c.. the resultant energy – state , 𝐄 𝐑 , in e-volt , 6d...the electric–magnetic field , 𝐄 𝐑 𝐌 𝐑 , in 10−12 ampere -10−6 tesla , 6e.. the intensity of the λ-electric–field , 𝐈 𝝀 , in 10−12 ampere , 6f.. the intensity of magnetic–field , 𝐌 𝝀 , in 10−6 tesla , 6g.. the wavelength , λ , in .10−10 meters , 6h.. the wave velocity , �̅� , in 105 meters / s , 6i1..the resultant voltage at wavelength-sides , 𝐕 𝝀 , in volt , 6i2..the s-bands voltages at wavelength-sides , 𝐕 𝝀 , in volt , 6j.. the radius of the helical motion , r = 𝐀 𝐑 in .10−10 meter , 6k.. the total carrier power , 𝐏 𝐂𝐓 , in 10−20 watt , 6l.. the total side-bands power , 𝐏 𝐓 , in 10−20 watt , 6m.. the side-bands amplitude , 𝐀 𝐁 , in 10−10 meter , 6n.. the side-bands coefficients , a m , in n-n , 6o.. the modulating phase shift , 𝛗 , in rad /2π , 6p.. the modulating factor , 𝒎 , in n-n , 6q.. the total energy-status , a m , in h-w , figure-2.. the ± energy in hydrogen and atoms exists from hydrogen common-atom . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 11 the fundamental particles origination mechanism in mfmf pns caves . the two hydrogen bonding → h + h = h1 1 + h1 1 = h1 1−1 h1−1 1 = h1 0 + h0 1 = h2 water h2o = o1 h2= o2 2 h2 2 = t 4 4 = o2 2−2h2−2 2 = o2 0 h0 2 = es 2=t/2 2=t/2 , still. [102] 4a.. the corelations of spaces and the primary motion . euclidean geometry spaces occupy position (x , y , z) and dimension |⇅| at any point p, cauchy stress tensor defines the state of stress (σ , τ) to the traction-vector (te= e. σ) the position of spaces is defined from any 3-dimension coordinate (x , y , z) or from angular-system ( φ , θ , ω ) or any other system with axis from any fixed point o . the dimension of spaces is defined as the distance between any two points o , p , and from any fixed point o the magnitude |op| = 𝐧 |op| , where n⃡ is the unit line length direction and is identified by vector n⃡ ≡ |ab| = the unity monad . in fig-1-1 , on ab diameter = 2r = the monad of neutral-circle exist the spaces with their sides 𝛌𝐧= 2,√r2 − a²n , and their critical-sides 𝐚𝐧 = 1 2 √4r2 − λ²n , as below 1… for spaces , the polygon side is λ n(s) = 2 .√r2 − a²n(s) , 2… for anti-spaces , the polygon side is λ n(as) = 2 .√r2 − a²n(as) , 3… for sub-spaces , the polygon side is λ n(ss) = 2 .√r2 − a²n(ss) , 4… for neutral spaces , the polygon side is λ n(ns) = 0 , while in material-geometry becomes from the centripetal force fc = σ = m.v³ r = j w² r , where the unity monad r ≡ monad r = σ j.w² = 𝛔 �̅� = stress angular−momentum̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ , and polygon side 𝛌 𝐧(𝐍𝐒) = 2 √r2 − a²n , all becoming from pythagorean relation [ r ] ² = [ an] ² + [ λn 2 ] ² . in fig-3 , is shown the geometrical construction of the inscribed and the circumscribed n-regular polygons . with 4 , 5 , 6 , different vibrating masses for each space and by using the argand diagram for a vector of amplitude ab = 2r = the monad , then the 4 spaces are measured using the euler equation for exponential functions as follows , on the spaces , the vector of amplitude can be represented as the complex quantity on → s  z + n = + 2r.𝑒𝑖.𝑛𝜔𝑡 = + 2rn.[ cosw. t + i. sinw. t ] = x + i .y on → as  z − n = 2r.𝑒𝑖.𝑛𝜔𝑡 = 2rn.[ cosw. t + i. sinw. t ] = x + i .y on → ss  z ± 1/𝑛 = ±2r 1/n. ei.θ/n = ± √2r n . √[ cosw. t + 𝐢. sinw. t ] n = x + i .y where the space vector–amplitude 2r = √ λ n(s) 2 − a²n(s) , for the n-masses . the anti-space vector–amplitude 2r = √ λ n(as) 2 − a²n(as) , for the n-masses . the sub space vector–amplitude 2r = √ λ n(ss) 2 − a²n(ss) , for the n-masses . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 12 the fundamental particles origination mechanism in mfmf pns caves . remarks : a…from argand diagram the vector z + n =2r.𝑒𝑖.𝑛𝜔𝑡=+2rn.[cosw. t + i. sinw. t] = x + i .y meaning the wave pattern of spaces in planck`s cave 10−35 m .the vector amplitude 𝐀𝐁̅̅ ̅̅ = 2r = the monad ,is the complex quantity 𝐀𝐁̅̅ ̅̅ ≡ �̅� ≡ x + i .y , where x = 2r → the geometry of pns space and y = 𝑒𝑖.𝑛𝜔𝑡→ the motion in planck`s space . b…since anti-space z − n ≡ z + n , means the conjugate space to the space z + n . c…since from wave-action of spaces issues z − n x z + n = [2r] 2n . e i.2nθ , this means that ,the conservation of energy = work , between the spaces and the anti-spaces happens by doubling the number , n , of the vertices of the regular polygon . d... the natural frequency of an x-x linear vibration , [⊕↔⊝], becomes from 𝐰𝐧= 2π.f n meaning that the infinite frequencies f n are stored in subspaces by doubling the number , n , of the vertices with 2n-sides to a regular polygon of initial n.sides . e... the natural frequency of an rotational ↻ r ↺ dof vibration system becomes from �̅� = r.m.v n= r.m.v n= r.m.w n.r = m.w n.r ² = m.r².w n , meaning the natural identities of the ∞ monads in our universe . the equations of motion in a material point related to the inner motion was referred before where the in-between-magnitude circumscribed – inscribed polygon , the part , becomes the outer ab as the ,whole or the self-growth .this augmentation-property , of these two golden-ratios , exists in the material-point and on frequency fn , which motion ≡ growth , and spread in two-closed-transverse planes as is in the propagating electromagnetic wave where e ⊥ h . from planck e = h.fn ≡ [ 1+√5 2 ] hσ 2πr = [ nσ 8 r² ].b̅ , 𝐟𝐑 ≡ [ f1=n ,f2 ,f3 , f r = w²n ] , euler`s 𝐞−𝐢.𝐀( 𝛑 𝟐 +𝟐𝐤𝛑).𝐛 = e−i.a( π+4kπ 2 ).b = a.cos[ π+4kπ 2 ]. b – i . asin[ π+4kπ 2 ].b , l = [ b̅/2].w , so then w = √2π. fr = 2l / b̅ , and the wave equation is y = 2a.sin( 2π.x λ ).cos . wt ....(a) f... the natural frequency of the unit-energy-vectors , �̅�.�̅� = m.r².w ²n = m.v²n= ke . from momentum �̅� = r m v = m �̆� r² = ± [ 𝛑.𝐫𝟐 𝟐 ].�̅� x �̅� , �̅�.�̆� = ± [ π.r2 2 ].c̅ ² , c̅ = σ φ , i.e. the primary space is originated from the stress  σ , and from the angular momentum  �̅� , �̅� , produced from the two opposite [⊕↔⊝] components. g...since monads are complex quantities as ab ≡ �̅� ≡ x + i. y , then are quaternions where material-points ≡ quaternions , which carry the → motion ≡ energy ← from the two edge-poinds [ �̅�↔�̅� ] , as this is from point a , to point b circularly . simultaneousley on ab conductor is created an electric field 𝐄 𝐅 between a–b points and ⊥ on ab axis directed to the motion , and an transverse (⊥ab) magnetic field as , 𝐌 𝐅 ⊥ 𝐀𝐁̅̅ ̅̅ ⊥ 𝐄 𝐅 . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 13 the fundamental particles origination mechanism in mfmf pns caves . 5a.. the figure-eight knot & the figure n = 1 ≈ ∞ knots : figure -3 : the geometrical construction of the 4-spaces and the , spaces waves of pns . in argand diagram ab = 2r → x = -1 , o , +1 , y = i , exist the vector of amplitude �̅� ≡ x + i . y , with angle θ ᶱ = [ z – oc] ᶱ , [i = √−1] .x . the wave's " strength " and " direction ", where x is a number that represents the magnitude or length of the vector z , and from definition of the complex vector space , is the set of vectors of length (the n vertices which form the n polygon sides , or the norm of the complex-vectors ) in (1) , (2) , are the 4-geometrical spaces consisting the geometrical mould . in (2) , are the vectors of amplitude on the 4 -spaces consisting the e-form-mould . it is constructed in space → the inscribed and the circumscribed to the circle of diameter ab , the → regular tetra pleuron ← it is constructed in anti-space → the inscribed and the circumscribed to the circle of diameter ab , the → regular penta pleuron ← it is constructed in sub-space → the inscribed and the circumscribed to the circle of diameter ab , the → regular hexa pleuron ← in (3) , are the norm (magnitudes) of the 4 -vectors consisting the strength -mould . in fig-3 the time-harmonic function �̅� (t)≡wave , x is the amplitude, w the angular frequency , and phase of �̅� (t) to be determined by the number n , n+1 of vertices . the complex – vector ± �̅� (t) uses the norm iz̅i = 2.r , and the number n , of the space which determines the number of knots and sides of regular polygons as , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 14 the fundamental particles origination mechanism in mfmf pns caves . 1…the space is consisted of 4 masses [99] which produce the space wave = 2r.𝑒𝑖.𝑛𝜔𝑡 , 2…the anti-space is consisted of 5 masses producing the anti-space–wave= 2r.𝑒𝑖.𝑛𝜔𝑡 3…sub-space is consisted of 6 masses producing the sub-space–wave = ± 2r 1/n. ei.θ/n 5a1.. the physical notion of the regular polygones : [ 62 ] according to archimedes , geometric means , speaking of numbers , whether solid or square , observes that , between plane one mean suffices , but to connect two solids two – means are necessary .this denotes that between two square numbers there is one mean proportional number and between two cubes there are two means proportional numbers . it was proved that odd numbers become from any two consequent even numbers , so the sum of two irrationals may be either rational or irrational . the cattle – problem of archimedes may be further analyzed reaching to equations of any degree. it was shown before that , all n-regular polygons end to equations of n-degree segment , by finding a suitable value of the segment , x , that is we have in the general case to solve one or two equations of the form : a . r 0. x n b . r 2. x n−2 + c . r n−6. x ³ – d . r n−4. x ² + e . r n−2. x1 – f . r n. x0 = 0 for the even polygons , and a .r 2. x n−2 b .r n−2. x n−3 + c .r2(n−4). x ³ d .r 2(n−3). x ² + e .r 2(n−2). x1 – f .r 2(n−1). x0 = 0 for the odd polygons , where a , b , c , d are constants . the presented geometrical method is the solution of the above equation in the general case . because , the nth degree equations are → the vertices (n) and the sides (𝐚𝐧) of the npolygon in circle ← so number , π , is their mould . the knot is a closed – curve in space ( x , y , z ) , that does not intersect itself . a simple parametric representation of the figure-eight knot is as the set of all ( x , y , z ) points where , x = ( 2 + cos 2𝑡 ). cos(3𝑡) , y = ( 2 + cos 2𝑡 ). sin(3𝑡) , z = sin(3𝑡) , for the cave-spin , �̅� = r m v , which is a monad and is consisted of 4-spaces exists → (( the spaces n.�̅� polygons , the anti-spaces n.�̅� polygons , the sub-spaces ± √𝐧. �̅� 𝒏=𝟏−∞ polygons )) ← , with n = 1 ≈ ∞ knots it has been proved [62] that , projecting the vertices of the regular n-polygon on any tangent of the circle , then the sum of the heights 𝐲 𝐧 is equal to , n  r . this is a linear relation between heights , h , and the radius of the circle , the monad . this property on the circle yields to the geometrical construction ( as resemblance ratio of areas which is controlled ) and the algebraic measuring of the regular polygons as follows, when : r = the radius of the circle , with a random diameter iab i . 𝐚𝐧 = the side of the regular n -polygon inscribed in the circle n = number of sides , a , and knots of the n -polygon , then exists : ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 15 the fundamental particles origination mechanism in mfmf pns caves . n . r = 2 . r + 2 . y1 + 2 . y2 + 2 . y3 +……… 2 . 𝐲𝐧 …………. (n) the heights yn are as follows yb = [ 2 . r ] , y1 = [ 4.r ² a ² ] /. ( 2 r) …(y) equation (y) is a space-wave depended on 𝐚𝐧 , meaning n informations on 𝐲𝐁 height . y2 = [ 4.r4 – 4. r2. a2 + a4 ]/(2. r3) y3 = [ 8. r6 – 10. r4. a2 + 6 . r2. a4 – a6 ] –a2 . 64. r 896. r6.a²+52. r4. a4 –12.r². a6 +a8 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ 2. r5 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ yn = [ …………….. ] / 2. rn figure – 3a in fig-1-1 , on ab diameter = 2r = the monad of neutral-circle exist the spaces with their sides 𝛌𝐧 = 2,√r2 − a²n , and their critical-sides 𝐚𝐧 = 1 2 √4r2 − λ²n , as below 1… for spaces , the polygon side is λ n(s) = 2 .√r2 − a²n(s) , 2… for anti-spaces , the polygon side is λ n(as) = 2 .√r2 − a²n(as) , 3… for sub-spaces , the polygon side is λ n(ss) = 2 .√r2 − a²n(ss) , 4… for neutral spaces , the polygon side is λ n(ns) = 0 , while in material-geometry becomes from the centripetal force fc = σ = m.v³ r = j w² r , where the unity monad r ≡ monad r = σ j.w² = 𝛔 �̅� = stress angular−momentum̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ , and polygon side 𝛌 𝐧(𝐍𝐒) = 2 √r2 − a²n , all becoming from pythagorean relation [ r ] ² = [ an] ² + [ λn 2 ] ² . 5a2 .. the general equations of motion in a polygon-vector–conductor d = side 𝛌 𝐧(𝐍𝐒) the displacement at point x of a conductor d , is x + dx and strain will be u + ( ∂u ∂t ).dx , and the unit strain is ( ∂u ∂x ) . from hook`s law , σ = e u , the ratio of unit-stress to unit-strain is equal to the `modulus of elasticity e and a = the cross section area of the conductor , then issues ∂u ∂t = force a .e = f a .e and by differentiation becomes → a .e. 𝛛²𝐮 𝛛𝐱² = 𝛛𝐄 𝛛𝐱 …….(1) applying newton`s law of motion for the element dx and equate the unbalance-force ∂p ∂x , to the product of the mass and acceleration of the elements , and then ∂e ∂x dx = ρ a dx ∂²u ∂t² ….(2) eliminating ∂u ∂x then we have the partial differential equation ∂²u ∂t² = [ e ρ ] ∂²u ∂ x² .…(3) and 𝛛²𝐮 𝛛 𝐱² = [ 𝛒 𝐄 ] . 𝛛²𝐮 𝛛𝐭² ≡ [ 𝟏 𝐜² ] . 𝛛²𝐮 𝛛 𝐱² …(4) where �̅� =√ e ρ , and it is the velocity of propagation of the displacement dx , along the conductor or → is the stress–wave ← applying equation (4) in an conductor [ d ≡ 𝛌 𝐧(𝐍𝐒) ] of space anti-space equilibrium formation of polygon stpl line with both ends free , then the natural frequencies must ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 16 the fundamental particles origination mechanism in mfmf pns caves . be zero because charges ⊕ , ⊝ interchange at both ends , a , ka , and issues ∂u ∂t = 0 such at x = 0 , and as at , x = d = λ n(ns) = aka . the two equations of the below (5) correspond to these , stretched ≡ linear , boundary conditions which are as follows , { ∂u ∂x } 𝑥 =0 = a 𝟏 𝐜 [c sin wt + d cos wt ] = 0 { ∂u ∂x } 𝑥 =0 = 𝐰 𝐜 .[ a cos | 𝐰 .𝐀𝐊𝐀 𝐜 | b sin| 𝐰 .𝐀𝐊𝐀 𝐜 | ].[ c sin wt + d cos wt ] …..(5) equation (5) must be true for any time t , therefore from the first equation a = 0 . from the second equation b must be finite in order to have vibration and is satisfied when , sin | w . aka c | = 0 , or , | w.n . aka c | = | 𝐰 𝐧 𝐝 𝐜 | = 𝟐.𝐧 �̅� 𝐜 = π , 2π , 3π , ... nπ , .…(6) where , n = the order of the mode and the equation of motion in conductor becomes d = λ n(ns) =aka which is an longitudinal-vibration u = 𝐮𝟎 cos [ 𝐧 𝛑 𝐝 ] x sin [ 𝐧 𝛑 𝐝 ].c ..(7) a cosine-wave having n nodes in pipe d . the equation of motion for an longitudinal wave in a conductor d = λ n(ns) =aka is → y(x ,t) = n. sin [ kx wt + φ ] ← …(8) where , c = the light velocity-vector , n = the amplitude , w = the angular velocity-vector , φ = the phase angle , and k = 2π λ = π rl = propagating-energy constant =wave-number , λ =the wave length , �̅� = w k = λ t =√ t m =√ e ρ = the light-velocity in conductors . the frequency of the longitudinal wave is → f l = [ n ] �̅� 2λ = 0 , 1.c 2λ , 𝟐.c 2λ , 𝟑.c 2λ …. . . 𝐧.c 2λ , the equation of motion of a standing-wave in conductor d = λ n(ns) =aka ≡ 2 |ra−k𝐴| ≡ r l ≡ λ is as , y(x ,t) = 2n sin[k x].cos [(w =2πc/λ) t] ...(9) where longitudinal frequency f l = ( n + 1 2 ) �̅� 2λ . the amplitudes become zero at y = 0 when k x = 0 , which implies k x = n π , n = 0 , 1 , 2 , 3 , …n . and since the sine-function repeats itself after every 2π change in angle which is the wavelength λ , of the wave , and from equation → sin[k x] ≡ sin[k x +2nπ] ≡ sink.[x+ 2n.π k ] ← then for n =1 , the wavelength λ = [ 2.1.π k ] = 2π k , and is the-unit-energy , the quantum , k = 𝟐𝛑 𝛌 .....(10) and wave-equation → sin[k x] ≡ sin[k x +2nπ] ≡ sink.[x+ 2n.π k ] ≡ sin[ 𝟐𝛑 𝛌 ].[ x + λπ ] ....(10a) since , k λ = 2π constant and λ , is the displacement for n =1 , then implies that k , is force and because force x displacement = work ≡ energy ≡ motion , therefore k λ ≡ e ≡ constant ≡ the unit-energy ≡ energyquantum . in case of energy equilibrium, then frequency in conductor d = aka= λ / 2 and , f l = (n + 1 2 ) �̅� 2λ ≡ 0,5 c 2λ , 1,5.c 2λ , 2,5.c 2λ , 3,5.c 2λ ,,,,, (2n+1) .c 2 2λ = 1 2 [ c 2λ ] , 3 2 [ c 2λ ] , 5 2 | c 2λ | ,,,,,, 2n+1 2 | c 2λ | the wave is a force 𝐐 aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = f = |⊕ => aka| , an energy-vector | ⊕ 𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ |≡| ↔ | that propagates from the place where it was created , the point a , so when is found an obstacle at end-point 𝐊𝐀 then the particles of the conductor medium oscillate perpendicular |↕|| , because it is the only exit , to the direction of the wave propagation , and by this way the motion continues transverse as longitudinal wave in conductor d = aka as , |⇉ ⇈| ≡ ® , or in conductor d = λ n(ns) =aka , the two perpendicular energy-vectors , |k 1 x k 2| = k 1┴2 = k = 𝟐𝛑 𝛌 = 𝟐𝛑𝐟=𝐰 𝐜 and longitudinal wave -energy → c { k 1 x k 2 } = 2π.f , or | 𝐤 𝟏 𝐱 𝐤 𝟐 𝟐𝝅 |.c = 𝐜 𝛌 = f . in figure -3 and -9 ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 17 the fundamental particles origination mechanism in mfmf pns caves . the obstacle at edge 𝐊𝐀 point , is the presentation of the , ⊕ constituent and the lack of ⊝ constituent . the transverse propagating-wave , or the moving energy -vector |⊕ 𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ | = |↕| at node ka , and with the energy-vector | ⊕ 𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ | ≡ |↔| at node a , consist the electromagnetic wave in d = aka= 2r conductor . the energy in this stationary wave of wavelength λ = 2d is , e = h f l = h 2n+1 2 | c 2λ | = 2n+1 2 | hc 2λ | = | 3.h.c 4λ | ……… (11) and from relation λ = c t = c f = 2π k , then k = 2π λ = 2πf c = w c = | v̅ c |.| 1 r | = | v̅ c |.| 2 λ | , or the propagating-transverse-wave carries the standing-wave-energy-constant-magnitude constant-motion k = 2π λ = 2πf c = w c = | v̅ c |.[ 2 r[a−ka] ] .…...(12) the two perpendicular energy-vectors , |𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔ | and |𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ | ,in conductor d ,of this moving energy form the energy-square , {| ↔ |.| ↕ |≡|𝐐 ⇉ |}.{| 𝐐 ⇈|} ≡ u0 , on conductors aka . with conductors pivot-axis propelled on their merging -resultant -axis of direction |⇉ ⇈ ≡ ⤱| . in case of an closed conductor above motion with energy = k = 2π λ = 2πf c = w c , and with velocity �̅� = w k = λ t =√ t m =√ e ρ , consists the stationary–energy-storage with fundamental frequency f 1 = (n + 1 2 ) �̅� 2λ ≡ 3.�̅� 4 λ ≡ 3.𝛔 𝚽 4 λ ….…(13) . the obstacle at ka point is the ⊕ constituent which executes an coulomb force f [+ ↔ +] = [⊕↔⊕] r² = [σ.σ] r² = | σ. r | 2 =| e. r | 2 = | 2πf φ | 2 = | w φ | 2 . the energy k at point ka is the total mass mt of conductor d = aka = r = λ n(ns) equal to the moment of inertia jt = m.r² 3 ,where m is the stiffness per meter of conductor r . the impact of this mass at the end-point is the energy = j w and impact → m v = j w r = [ m.r² 3 ] . wr , or velocity after impact velocity → v = [ r³ 3 ].2πf = [ 2πf.r³ 3 ] .......(14) coulomb force with fundamental frequency becomes f [+↔+] = | 2πf. φ | 2 = [ 2π φ | 3.�̅� 4 λ |] ² = | 3πv 2λφ | 2 …….(15) placing velocity after impact (14) in equation (15) then the concentrated energy at point ka is f [+↔+] = { 3π 2λφ [ 2πf.r³ 3 ]}² = | π2r4f λr.φ | 2 =| b=(π2r4f) λr.φ | 2 =| �̅� λr.φ | 2 ≡ angular-momentum ≡ spin …...(16) in short-distance conductor d = 6,195543.10−7m , the magnetic-field b̅ = m.v q̅ r = 9.10−31kg.2,9979.108 𝑚/𝑠 1,6.10−19c.[6,195543.10−7] = 2,718088.1013 telsa , which is perpendicular , normal , to the conductor`s axis . ⇉ i.e. in a closed energy-conductor 𝐫 [+↔+] , the fundamental frequency 𝐟 𝟏 becomes the spin. the energysquare prism , |aka| .{𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } = |qx.qy|.|aka|=kx.ky|.|aka| = |↔|.|↕|.|aka| …..(17) , where |↔| = |kx| the conductor`s axial-energy , |↕| =| ↺ | the conductor`s rotation-energy = spin , and |↔|.|↕| the conductor`s resultant-energy-vector is an cruise-missile . when a conductor d = aka = r = λ n(ns) has a hole at an point 𝐀𝐄 such that aae= 2 r ,{ a -2r 𝐀e -r𝐊a}= d then the energy k at point a e is the total mass mt of conductor aae = 2 r equal to the moment of inertia jt = | m.4r² 3 | , where m is ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 18 the fundamental particles origination mechanism in mfmf pns caves . the stiffness per meter of conductor 2r . from equation (15) coulomb force f [+↔+] =| 2πf. φ | 2 = [ 2π φ | 3.�̅� 4 λ |]² =| 3πv 2λφ | 2 becomes as f[+⇅ ⇵+] ≡ f[+↔+] , equal and perpendicular and {𝐐 ⇉ aae⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } = {𝐐 ⇈ aae⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } , |qx| = |qy| , |kx| = |ky| , |↔| = |↕| …....(17a) transverse motion |↕| , presupposes an |←| impact of the mass at end-point a e and is the energy = j w and because lack of obstacle impact → mv = jw² = [ m.r² 3 ].w² , or velocity after impact → v = [ 2πf.r³ 3 ] = [ v.r² 3 ] ..(14a). placing velocity after impact (14a) in equation (15a) then the concentrated energy at edge point ka is f[+↔+] = { 3πw 2λφ [ −2πf.r² 3 ]}² = | 𝑤[π2r4f ] rλφ | 2 = | w.b=(π2r4f) r.λφ | 2 = w².| �̅� rλφ | 2 ≡ | 2l= w.�̅� rλφ | 2 ≡ the total energy 2l to the twin circles squared , or to angularmomentum ≡ spin squared …….(16) ⇉ i.e. in an energy – slit conductor 𝐫 +{ 𝐀 −2r− 𝐀e −r− 𝐊a}= 𝐝 + , the fundamental frequency 𝐟 𝟏 becomes the spin , or , the carrier of the total-energy 2l of the moving system . at ae point energy square-prism is → |aae|.{𝐐 ⇉ aae⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{𝐐 ⇈ aae⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } = |qx.qy|.|ae ka | = |kx.ky|.|ae ka | = ≡ |↔|.|↕|.|ae ka | .........(18) , where |↔| = |kx| the conductor`s axial-energy , |↕| = |ky| the conductor`s . transverse energy , |↺| the conductor`s rotational-energy = spin , where |kx.ky| = |↔|.|↕| ≡ ⤱ = the dot-product = area of vectors , which is a scalar magnitude 𝐀 𝐱.𝐲 = |𝐤𝐱|.|𝐤𝐲|.𝐜𝐨𝐬 𝛉 , where θ , is the angle between the two vectors , kx , ky and , the cross-product =the resultant-vector =®= �̅� = 𝐤𝐱 x 𝐤𝐲 which is perpendicular to kx,kyvectors where , r =√k²x + k²y , x = r.cos θ , y = r.sin θ , θ = tan−1( y x ) , and �̅� = 𝐤𝐱 x 𝐤𝐲 = 𝐰 𝐜 = 𝟐 𝛑 𝛌 the conductor`s resultant-energy-vector , ® = �̅� , is as an non-guided cruise – missile . from total energy relation 2l = b w ≡ e k + e u ≡ |kx|+ |ky| is seen that , kx , ky , consist the 2-line , plane-launcher of { energy ≡ motionmissile } into → two perpendicular and equal conductors , with conductors-pivot-axis propelled on their merging-resultant axis of the two directions ⇉ ⇈ ≡ ⤱ . in one-conductor , kx= za̅̅̅̅ , k y= 1,mm̅̅ ̅̅ ̅̅ ̅̅ , resultant r̅ = z1mo̅̅ ̅̅ ̅̅ ̅̅ , while for (2) → in two-conductors , kx= zb̅̅̅̅ , k y= za̅̅̅̅ , r̅ = z ,1_2_3 , mo̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅ ≡ ⤱ the total energy 2l = b w ≡ e k + e u ≡ |kx|x|ky|x|aka| ≡ |kx|x|ky|x|λ| ≡ an propagatingwave in λ , i.e. energy-square-prism is a moving-energy-volume and because from cauchy equations of stresses in three dimensions , the energy stress remains flat only when the plane section becomes a circle , and because from mechanics the sphere occupies the least resistance to the motion , then the square prism is altered to the equivalent , energy-sphere cone → ⸿ = k πr ³ ← fig-9a ,9b , 9c. where the cube [| ↔ |.| ↕ |.|𝐀𝐊𝐀|] = x ³ becomes an sphere–cone ≡ 𝐐.𝐀𝐊𝐀 ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | ⇉ | ↑↓ 𝛌 | → k π r³ = x ³ . from the max-sin | w .aka c | = 1 , of equation (6) then u = [ c sin wt + d cos wt ].sin|. 𝐰 .𝐀𝐊𝐀 𝐜 | ……..…(14a) the condition u = y (aka , t ) = 0 , leads to the equation → sin | 𝐰 .𝐀𝐊𝐀 𝐜 | = 0 , or to …(14) for the duality-photons issues , v̅ . [ fn̅ + fn ] ≡ | 𝐯 𝛑². 𝐫⁴𝐧 |. b̅n + | c̅ | fn ..……...(15a) or v̅. [ fn̅ ≡ (n + 1 2 ) �̅� 2.|aka| ] + [f n ≡ n �̅� 2λ ] ≡ → �̅� { | �̅� 𝟐.𝛌 | (n + 𝟏 𝟐 ) + 𝐧 𝟐𝛌 } ..........(15) ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 19 the fundamental particles origination mechanism in mfmf pns caves . e-vectors |𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔| x |𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ |x|aka| ≡ |qx.qy|.|aka| ≡ | ↔ |x| ↕ |y| λ | ≡ k π r³. from wave-energy-constant k = 2π λ = 2πf c = w c = | v̅ c |.{ 1 r = 2 r|a−k𝐴| }so c= w k = λ f , or | �̅� 𝟐.|𝐀𝐊𝐀| |≡ f i.e. frequency f aka of pipe |aka| is the stationary-photon energy-storage fn̅ ≡ | �̅� 𝟐.|𝐀𝐊𝐀| | ≡ | �̅� 𝛌 | ...(16) and photon v̅. [ fn̅ + fn ] ≡ �̅�. { | �̅� 𝛌 | + 𝐟𝐧 } i.e. , as particle ≡ v̅. | �̅� 𝛌 | ≡ e u ≡ potential energy , and as wave ≡ v̅. fn ≡ e k ≡ kinetic energy . since energyconstant-magnitude , k , denotes the total motion then needs a scalar -magnitude for measurements and a vectormagnitude for directions . in conductor d = |aka| ,| d | ≡ the scalar magnitude between points , which gives also the direction of motion with points , a , ka , and aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ vector the direction which defines the vector aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ . the total energy k = | v̅ c |.[ 2 r[a−ka] ] of photon-wave is split as, e u+ e k= k ≡ potential + kinetic energy where e u ≡ v̅. | �̅� 𝛌 | ≡ particle ≡ �̅� { | �̅� 𝟐.𝛌 | (n + 𝟏 𝟐 ) } ≡ | 𝐯 𝛑². 𝐫⁴𝐧 |. b̅n = | 𝐯 𝛑². 𝐫⁴𝐧 | r².q.b̅m ≡ | 𝐪.𝐯 𝛑². 𝐫⁴𝐧 | x b̅m is the stationary-storage ≡ particle ≡ magnetic-field with velocity , v , in cave λ = 2r = 2π / k and e k ≡ v̅.fn ≡ wave ≡ kinetic energy = 2π λ = q k.a = q k.πr ² = σ.φ k.3rb̅m = { σ.φ k.3r } 𝟏 b̅m = 𝐜/𝐄 b̅m , and �̅�𝐄 ≡ c�̅�𝐌 , and the moving electromagnetic-wave is , e̅e = |e̅0|.sine [kz wt] , b̅m = |b̅0|.sine [kz wt] , k = 2π λ , motion is kept in caves as electric , magnetic fields and as frequencies and conserved in them when the space-points become energy-conductors . the in caves energy-import is shown in figures . examples : 1.. in an conductor d =aka= 2r , with ⊕ obstacle at ka point occupies zero velocity and ⊕ charge is accelerated . the initial light-velocity is accelerated as v² = v̅o²+2a.r , or a = −v̅o² 2.r = [2,9979.108]² 2.r = 8,9874044.1016 2.r m/s² . for conductors , r =10−16,100,1016 then , a1 = −v̅o² 2.r = 8,9874.1016 2.10−16 = 4,4937.1032 m s² , a2 = −v̅o² 2.r = 8,9874044.1016 2.100 = 4,4937022.1016 m/s², a3 = −v̅o² 2.r = 8,9874044.1016 2.1016 = 4,4937022.100 m/s². force on charge becomes from the electric field as f̅k = m.a̅x= q.e̅f =+e. e̅f and e̅f= m.a q = e , or e̅f = m.a e = m e [a] = 9.10−31 1,6022.10−19 [4,4937022.1032] = 25,242366. 1020 n c e̅f = m.a e = m e [a] = 9.10−31 1,6022.10−19 [4,4937022.1016] = 25,242366. 10 4 n c e̅f = m.a e = m e [a] = 9.10−31 1,6022.10−19 [4,4937022.100] = 25,242366. 10−12 n c i.e. the ⊕ charge is accelerated opposite to initial direction of ⊝ charge in an electric-field e̅f . this property of acceleration , creates the magnetic-fields in electric conductors as atoms are . 2.. uniform circular motion of electron in caves constitutes a current i = e t = e.f , and t= 2πr c , i = e.c 2πr = 1,6022.10−19.2,9979108. 2π.r = 7 ,644811.10−12. [ 1 r ] , and for r =𝟏𝟎−𝟏𝟔, 𝟏𝟎𝟎, 𝟏𝟎𝟏𝟔 m then i = e.c 2πr =7 ,644811.10−12. [ 1 r =10−16, ] = 7 ,644811.104 7 ,644811.10−12 – and 7 ,644811.10−28 ampere. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 20 the fundamental particles origination mechanism in mfmf pns caves . 3.. the electric-potential v = 𝐪 𝐫 = 8,99.109 nm² c² [ 1,602.10−19. r = 10−16 ]=14,4038.106 =1,44038.107volt v = 𝐪 𝐫 = 8,99.109 nm² c² [ 1,6022.10−19. r = 100,10 16 ] = 14 ,403778.10−10 v , 14,403778.10−26 v , 4.. electric-field �̅�𝐅 = 𝚫𝐕 𝐝 = [ 14,403778.106. r =10−16,100,,10 16 ] =14,403778.1022, 14,403778.10−10 v 5.. force on electron is f = q .�̅�𝐅 =1,6022.10−19.{14 ,403778.1022}= 23,077733.103 n and f = q .�̅�𝐅 =1,6022.10−19.{14 ,403778.10 6−16}=23,07.10−13 n,23 ,077733.10−35 n , and 6.. the produced work is w = f.ds = f r = 23 ,077733.103.{10−16 } = 23 ,077733.10−13 j . w = f.ds = f r = 23 ,077733.103.{100,1016 } = 23 ,077733.103, 23 ,077733.1019 j . 𝑒 0 7.. intensity i e−r is , 𝐈 𝐄−𝐫= c.e² 2 = [ 2,998.108.1,602210−19(14 ,4038.1022)² 2 ]=498,26077.1033w/m2 𝐈 𝐄−𝟐= c.e² 2 = 498 ,261.101w= 4,9826077.103w and 𝐈 𝐄−𝟑= c.e² 2 = 4,9826077.10−29 w/m² 8.. the power 𝐏 𝐄−𝐫= e² z=πr² = (14,403778.1022)² π.(10−16)² = 6,60394.1077w , 6,60394.1057, 6,604.1013w 9.. two-capacitors conductor of area a and charge q charge density is on plates σ = q a and electric field e = σ 𝑒 0 = q a 𝑒 0 and from potential difference v= e d then e = v 𝑒 0 and v d = q a 𝑒 0 or v = q.[ d a .e 0. ] = q a = e d . from above → a = q.d v = [ 1,602210−19(10−16) 14,403778.(r=106) ] = m2 1,11235.10−42 ,1,11235.10−26,1,11235.10−10 m2 ,stress σ = q a = 1,44038.1023 ,7 ,−9 t m² remarks : 1.. the stress-strain relationship , from static theory of elasticity solves the problem of static indeterminacy. that is , the stress parameters σ x , σ y , are bound in , (1) by the stereo static conditions (2) and by the constraint that the strain accompanying the stress s x , s y , remains geometrically continuous . this property of the stresses allows .the transportation of the informations on knots to the polygon-sides. 2.. when a body contains potential energy , during any unloading it gives off mechanical work , since the points of application of the gradually withdrawn loads are displaced. if the unloading is abrupt , then kinetic energy is converted into potential and after is kinetic , converted back into kinetic and so on . that is, the body performs an oscillating motion until the mechanical energy is extinguished due to a state of equilibrium or alters . 3.. the action of the complex–vectors create the waves of motion ≡ energy waves , which are closed curves and transported . the pattern of disturbances with the informations , propagates from the one-edge-point to the other edge-point of conductors , or is on the [sub-units] . action of vectors exist on polygon–conductor`s d = side λ n(ns) of polygons. 4.. action of a signal which strikes the tetrahedron–cube-spheres mechanism . unloading happens on the three tetrahedron sides which withdrawn loads are displaced on the anti-parallel-strand . if the unloading is abrupt, then kinetic energy of 1-strand is converted in the 2-strand as potential and so on . that is , cubes perform an oscillating motion until the mechanical energy is extinguished to their vertices , which are the stores of information . 5.. the harmonic displacement of support-points on a regular polygon vertices . the projections of heights y n of a regular polygon are as 2.y n= n r , which allows to the vectorforce polygon to follow the space-vector-wave = 2r.𝑒𝑖.𝑛𝑤𝑡 = 2r.[cos w t + i .sin wt] . i.e. complex frequency response , happens on the regular polygon vertices only . 6a.. the origination of primary motion in e spaces . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 21 the fundamental particles origination mechanism in mfmf pns caves . the stress state (σ , τ) in spaces is defined from the cauchy stress tensor where , exists at any point inside a material in the deformed state placement or configuration. the second order tensor consists of nine components and relates a unit length direction vector 𝐧 to the traction-vector 𝐓𝐧 across an imaginary surface (in m-geometry is φ) perpendicular to n⃡ , tn⃖ = n⃡ σ , or tj n⃖ = ∑σ i j n i , obeying to the tensor transformation law for stress analysis . a contact force ∆f exerted at point p of element area ∆a , the ratio ∆f/∆a becomes df/da and the couple stress vector ∆m vanishes and prior tensor becomes tj n⃖ = lim ∆s=0 ∆fi/∆s = dfi 𝑑𝑆 . equation means that the stress-tensor depends on the position (location) in body and the orientation of the plane on which it is acting . from spaces (4) in material-geometry of pns -space (fig-1-3) the eternal rotation of the two opposite [⊕↻↺⊝] constitutes , and because of their in between stress σ which is equal to the centripetal-force then issues fc = σ = 𝐦.𝐯³ 𝐫 = j w² r = �̅� r ,where the monad r ≡ monad r = σ j.w² = 𝛔 �̅� i.e. the stress σ , which is created between the opposite [⊕↔⊝] , and through the only possible rotational-motion , originates the angular-velocity �̆� = 𝛔.𝚽 𝐫 = 2πr.f and the angular momentum �̅� = 𝛔. 𝐫 = 𝛔. 𝐑 , which when placed on ab diameter = 2r = the monad consist the neutral – spaces energy. the planck`s cave , 10−35 m , is the safety-valve ≡ the critical state for the stress ellipsoids , b̅ w̅, vectors in order that this total-energy to superflow and be still into vacuum ≡ [10−62-10−35m] . this vacuum between the |gravity cave planck`s cave| consists the black holes chaos , as the recycling–energy into which → the { energy – space monad |�̅�| ≡ |ab| } , decomposes ← {fig-14} the rolling of polhode unit-space into the herpolhode space-monad cone. figure -4the central axial-ellipsoid of ⊕ constituent rotating through constant point o . in (1-2) the ⊕ material point ok , {⊕ ≡ o,[ok = r] ≡ the space}, continually rotates around the ⊝ material point ok, { ⊝ ≡ o , [ok] ≡ the anti-space }, on 2r circle . their common circle center , forms the material angle φ = φp.t = ( vp √c²−r² ).t , which then ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 22 the fundamental particles origination mechanism in mfmf pns caves . creates at any contact point p the centripetal force 𝐅𝐂 = mv² r = 𝜋r² 2r [w². r²] = 𝝅.𝐫³ 𝟐 [w ²]. the vector �̆� of angular velocity w , formulates the polhode cone of base-radius a , and of op = s side forming the 𝐰-angle with z axis , while the vector �̆� of angular momentum b , formulates the herpolhode cone of base-radius r , of op = s side and 𝛉-angle with z axis in (3) , is the mpmp space where time is not existing t = 0 , and the same for motion also . in pns-space the 3-dimension motion of the primary constituents [⊕↻↺⊝] the time is as period t , and momentum is the angular-velocity �̆� and angular momentum �̅� , where the momentum �̅� = r m v = m �̆� r² = ± [ 𝛑.𝐫𝟐 𝟐 ].�̅� x �̅� and ± [ 𝛑.𝐫𝟐 𝟐 ] ≡ the hemi circle area . the polhode is of radius a , and rotates with an angular velocity w̆ = 2π.fp in the cone πa².s . the herpolhode is of radius r , and is the stationary cone πr².s . with an angular momentum b̆ = r m v = r m (w r) = m �̆� r² . the polhode cone is rolling on op and contact common line , into the herpolhode cone without friction . i.e. the motion in pns-space exists as the rotation of space point p (+) on [ o ,ok ] circle anti-point p ` (-) on [ o ,ok ] of anti-circle and defines the angular-momentum �̅� , so the position of spaces points p (+) and points p (-) and momentum are simultaneously determined and measured , consisting a direct violation of the uncertainty principle . the primary rigid -body analysis : a.. the position of the herpolhode cone is , fig-4. x = a.sin θ. coswt , y = a.sin θ. sinwt , z = a a.cos θ = a.[ 1 cos θ ] where exists , a = the polhode cone base-radius , r = the herpolhode cone base-radius [ x , y . z ] = coordinate system such that z-axis ⊥ [kok] line , and angle 𝛉 between op , oz axis . b.. the energies of the polhode herpolhode cones are from lagrange`s principle where for any particle , the kinetic 𝐄 𝐊 and potential 𝐄 𝐔 energy are measured as below , 𝐄 𝐊 = 1 2 m.[ ẋ ² + ẏ ² + ż ² ] = 1 2 m a².[ θ̇ ² + w².sin ² θ ] , and 𝐄 𝐔 = m g z = m g a.cos θ the lagrangian 𝐋 𝐊𝐔 = m.a².[ θ̇ ² 2 v effective ] , where the effective potential v effe is v effe = 1 𝑚,𝑎² [ m g a.cos θ + 1 2 m a².w².sin ² θ ] = [ g .cosθ 𝑎 + w².sin ²θ 𝑎 ] …..(e) the equation of motion for the bead becomes from lagrangian which is �̈� = [ ∂v effe ∂θ ] ..(b) since θ is constant then issues θ̇ = θ ̈ = 0 , and for the bead-herpolhode motion ∂v eff ∂θ = 0 ..(c) equation (c) is stationary at positions which satisfy relation g.𝐬𝐢𝐧𝛉 = a.w².𝐬𝐢𝐧𝛉.𝐜𝐨𝐬 𝛉..(c1) 1…the solutions of (c1) are at position sin θ = 0 where issues 𝛉 = 0 2…the solutions of (c1) are at position sin θ.cos θ = sin 2θ = 0 and is 𝛉 = π & 2𝛉 = 2π 3…the solutions of (c1) are at position cos θ = g a,w ² and for θ = 0 then w ² = 𝐠 𝐚 remarks : 1… solution (1) for θ = 0 exists always and from figure-4-(2) the velocity vector �̆� is applied at the center of mass 𝐊 𝟎 , anti-clockwise spinning . fig-4-(3) 2… solution (2) for θ = π exists always and from figure-4-(2) the velocity vector �̆� is ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 23 the fundamental particles origination mechanism in mfmf pns caves . applied at the center of mass 𝐊 𝟎 , clockwise spinning . fig-4-(3) 3… solution (3) where since θ = 0 exist the prior and from figure-4-(2) velocity vector �̆� is applied at the center of mass 𝐊 𝟎 , and the ± spinning is very fast from the center and is such that w ² ≥ 0 , and happening when { from relation g a } the radius a  0 . 4…the solution w ² ≥ 𝐠 𝐚 , determines the stability of the polhode system . the solution w ² < 𝐠 𝐚 and θ = 0 , determines stability at point o position and−v effe. the solution w ² < 𝐠 𝐚 and θ = π , determines stability at point o position and+v effe. for solution w ² > 𝐠 𝐚 and θ = 0 , the position is unstable at point o with −v effe. for solution w ² > 𝐠 𝐚 and θ = π , the position is unstable at point o with +v effe. meaning the black – holes origination . from solutions w ² = 𝐠 𝐚 , the θ = 0 = π =2π positions are unstable and below point o with+v effe. meaning an evolving and operational black-hole stable origination. 5…all above stable solutions result to the determination of the two vector magnitudes that of angular-velocity �̆� and that of angular momentum �̅� , differently black-hole. ---------------------------------------------------------------------------------------------------------- both magnitudes define the unit-energy vectors → { unit-e vector r = σ j.w² = 𝛔 �̅� } ← when the unit-e.vector r is placed on [ab≡2r s.diameter of spaces-neutral-circle] then the energy-monad |�̅�| exists on space-monad |ab| ≡ |�̅�| as {e+s -monad |�̅�|}, i.e. is the { energy space monad |�̅�| } , with the magnitudes , spaces sides 𝛌𝐧= 2,√r2 − a²n , & , quantum critical-side 𝐚𝐧 = 1 2 √4r2 − λ²n , ---------------------------------------------------------------------------------------------------------- the quantization of energy in spaces , happens when follow the archimedes formula for doubling the vertices n , 𝛌𝟐𝐧=√2r2 − r√4r2 − λ²n & critical-sides 𝐚𝟐𝐧= 1 2 √4r2 − λ²2n where then energy is divided and spread on the double number of the polygon sides . monad r = σ j.w² = 𝛔 �̅� remains the same , while spaces sides change as above 𝛌𝟐𝐧 , 𝐚𝟐𝐧 . 7a.. the mechanisms & charges , for dual originations , the natural spaces of ab diameter is the { ab̅̅ ̅̅ unit complex vector } . 1… spaces ≡ the infinite (+) circumscribed-regular-polygons in ab̅̅ ̅̅ . [fig-1,3]. 2…anti spaces ≡ the infinite (-) circumscribed-regular-polygons in ab̅̅ ̅̅ .[fig-1,3]. 3…sub. spaces ≡ the infinite (+) inscribed-regular-polygons on diameter ab̅̅ ̅̅ .[f-1,3]. 4…neutral-spaces ≡ the infinite circles on diameter ab̅̅ ̅̅ .(the sphere on ab), [fig-1,3]. 5…{mfmf} ≡ the gravity cave = 10−62 m . 6…{ pns } ≡ the planck`s cave length = 10−35 m . 7…vacuum ≡ [ 10−62 -10−35m ] = the in-between distance  the origination-mechanisms in spaces & the origin of charges. 8…action-spaces ≡ the infinite quaternion monads → �̅� ≡ ( s + v̅.i ) ≡ 𝐀𝐁̅̅ ̅̅ ≡ ≡ √s2 + [ v̅. i ]² ] , or �̅� ≡ x + i . y ≡ 𝐀𝐁̅̅ ̅̅ ≡ moduli , r , is their magnitude [ r = | r | = √ x2 + y2 ] , on unit-diameter ab̅̅ ̅̅ . the [qmas] . for quaternion v = s 𝐳 * 𝐳 = (s + v̅.i )² = s² +[v̅]² +2.sv̅.i = s² – v̅ ² ± 2.s v̅ = |s|² -|s̅|² ± 2.|s|.|s̅| , for |v̅| = s . 8a.. the material geometry , e-geometry , physics and chemistry . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 24 the fundamental particles origination mechanism in mfmf pns caves . 8..[a].. the { stpl } mechanisms in spaces , from charges in under-planck`s-length ≡ the six triple points line mechanism ≡ the complex vector-spaces rotor , it is a mechanism on where the charges of opposites , [⊕↔⊝] , react as centrifugal force fc which in turn changes → (action-reaction ≡ + spin – anti spin) ←of [mfmf] space to rotational motion angular-velocity = �̅� & angular-momentum = �̅� . [fig-6]. 8..[b].. the {tvpm} mechanisms , from charges on two-perpendicular-vectors-edges ≡ markos two vectors three poles mechanism . [fig-8a , 8b]. in fig-8|4| , vector cp̅̅̅̅ ⏊ ca̅̅̅̅ vector , which changes the energy ≡ the stress σ of the | cp̅̅̅̅ x ca̅̅̅̅ | square , to 2 halve ± squares on vectors-axis for equilibrium . 8..[c].. the { tcic } mechanisms , from charges in three-vectors-conductors-edges ≡ markos tetrahedron cube inscribed-circumscribed circles mechanism . [fig-27-29]. in fig-6 , the tetrahedron conductors -vectors da̅̅ ̅̅ ⏊ db̅̅ ̅̅ ⏊ dc̅̅ ̅̅ , carry energy as electromagnetic-fields (forces-frequencies-stresses-any motion) to inscribed sphere (nucleus of atoms), to cube vertices (electrons-position) ,to circumscribed sphere (the electrons orbits) for equilibrium and for transferring information and signals . 8..[d].. the {tobh} mechanisms , from charges in two-orbits-pins & drain. markos atoms bonding bracket hook mechanism . [fig-2]. in fig-9 , the ⊝ electron precesses due to the gravity and produces work which is stored in a second orbit . this 2-orbit equilibrium exists from the existing ⊝ -electron ≡ drain and from the new ⊕ -nucleus ≡ the pin . the second -orbit called as the bracket – hook -mechanism , with which atoms and compounds are bonded .[99-104] the { lncs } mechanisms , from spaces action on quaternion monads 𝐀𝐁̅̅ ̅̅ is the lagrange`s non conservative systems for generalized coordinates , where the system is subjected to forces that do not have potential or , d (t + u) ≡ δw𝐍𝐏𝐅 = 0 , as exists in the gravitational force g and from which are created → the elementary particles using the stpl – mechanism , and all universe using the istccs – mechanism ← and the dual → the black holes ← as equation δw𝐍𝐏𝐅 = ∑ qi. δqin i=1 , [fig-3, 21 , 22]. 8..[e].. material-geometry & e-geometry in physics and chemistry figure – 5 : the relation between the [+] spaces , [-] anti-spaces , [+ -] sub-spaces ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 25 the fundamental particles origination mechanism in mfmf pns caves . of euclidean & material-geometry in a circle (r,oa) of three-dimensions. in (1) , the euclidean geometry , e-g , is defined upon the number of points which can define the specific space i.e. the → one-point in e-g is defining one-material-point-space , the m-p-s . the → two-points are defining the line-segment-space , the line-m-p-s . the → three-points are defining the plane-triangle-space , the plane-m-p-s .. the → four-points are defining the volume tetrahedron-space , the volume-m-p-s . the → five-points are defining the volume regular pentahedron-space the sv-m-p-s. and so on , representing the steady-stable-regular geometrical-formations. the corresponding to the material-geometry-units are for : 1.. one-point → two-units , [⊕],[⊝] with dimension as , the material-vector [⊕→ ⊝] 2.. two-points → four-units , [⊕],[⊝] with one dimension as , the material-line 3.. three-points → six-units , [⊕],[⊝] with two dimension as , the material-plane 4.. four-points → eight-units , [⊕],[⊝] with three dimension as the material-volume keeping the property of edge-points to be the vector ⊕ → ⊝ then bond is the potential to the unique-steady-stable-regular material-geometry-formations. in (2-3-4-5). all units in vectors follow quaternion q = [s+v̅.i] = ab̅̅ ̅̅ properties i.e. [⊕] → the positive constituent of quaternion at , point a . [⊝] → the negative constituent of quaternion at , point b . pythagorean relation [ r ] ² = [ an] ² + [ λn 2 ] ² . in euclidean-geometry e-vector ab̅̅ ̅̅ carries point a to point b as , vector a → b in material-geometry m-vector ab̅̅ ̅̅ carries energy , ⊕ , from point a to ⊝ , energy of point b as , ⊕ moves to ⊝ , which is as the periodic pattern . velocity �̅� = the rate of change ( m / s ) in ab , therefore units are formatted according to the steady-material-geometry-formations which are : the line-vector in cave , r ,which is the simplest with double number of 4-units the plane-regular-triangle in orbits , is the most stable shape of 6-plane-units the volume-regular-tetrahedron in space is the most stable shape of 8-volume units , the cube , which are the crystals . the regular n-hedron are for all others . this is the why the glue-bond , bond is the potential , between the 6-units formulates the regular-hexagon as the first steady plane formulation in nature as in fig-18 remark : the constraint , that strain accompanying the stresses s x , s y remains geometrically continuous is , because the material point is consisted of the two units ⊕ , ⊝ .only ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 26 the fundamental particles origination mechanism in mfmf pns caves . 8..[a].. the { stpl } physical mechanisms : the six triple points line mechanism . figure -6: the physical three – phase mechanism , producing the fundamental particles and the primary forces in absolute system a ≡ r . γ = r. 𝐬𝐞𝐜𝛗. [97] in –fig. abc is any right-angled triangle at , a = (the-space) , kakbkc = triangle is the (anti-space) ,, aebece = triangle is the (sub-space) respectively.the instaneous pole p of rotation is off the circle of diameter |bc| . the poles of rotation lie on{dapa} reference system .the reference system {𝐃𝐀𝐏𝐀}≡ [r] (x' , y', z', t') moves with velocity �̅�, parallel to , x x', axis with respect to the fixed and absolute system {𝐃𝐀o} ≡ [s]-(x , y , z , t ) . the geometrical expression of lorentz factor γ = 𝒄 𝒗 , is the 𝐬𝐞𝐜𝛗 = γ = o𝐃𝐀 : a𝐃𝐀 = ± 1 / [ √1– (v/c) ² ] = c / [√c² – v²] , which is the conchoidal of nicomedes {s = a + b. secφ} and which acquires the material angle , φ = 𝐯 √c²−r² , i.e. issues |a| ≡ |r| . γ [a]-1…the [stpl] line is an extreme spaces–physical -originating-wave pattern . extreme-spaces (are the extreme space-triangles abc≡ ⊕) meet anti-spaces (extreme anti-space-triangles , kakb kc≡ ⊝ ) through the only-gateway which is the circularly charged + , 0 , , or [ ⊕  ⊝ ],[ ⊝  ⊕ ],[ ⊕ ⊝  ] formulation which is a physical geometrical formation mechanism called the [stpl] line . figures-[ 6 ,7 ] → the [stpl] line is a physical – mechanism as , in-circle  triangle  ex-circle  ex triangle consisted from the three spaces , on which the three breakages [ s² = ⊕ , 2s ² =  , s ² = ⊝ ] are circularlycharged on the three – extreme triangles {a b c} , {ka kb kc} , {ae be ce} ← producing the energy-quantities ± 𝐐𝐩−𝐩 . this circular-charge from breakages ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 27 the fundamental particles origination mechanism in mfmf pns caves . on the-three -triangles is the thrust upon the energy-quantity produced , for each needed circular charge , to shake off quantity 𝐐𝐩−𝐩 . the 3-circular charges correspond to the 3-phase rotors produce the 3-phase physical-current , since common circle exists on their common sphere , then the infinite straightlines , launch the e quantity 𝐐 𝐩−𝐩 off the fundamental particles to all universe and into the planck`s confinement only . i.e. the physical-three-phasestpl mechanism is the only absolute-system [a] on which circularly exist the relative-system [r] ≡ [ ⊕  ⊝ ] ↻ [ ⊝  ⊕ ] ↻ [ ⊕ ⊝  ] of the three constituents [ ⊕  ⊝ ] , which produces the fundamental particles and the existing objectivity as [a] ≡ [r].γ , where γ = sec φ = [ 𝐎da 𝐀da ] = ± [ 𝟏 √1−(v/c)² ] = [ 𝐜 √c²−v² ] the material angle φ = 𝐯 √c²−r² , and {a ≡ dao}≡{r≡ da-pa}. γ , or |a| ≡ |r| . γ the euclidean geometry is the spring , for the geometrical relative -velocities . when the points [ a , b , c ] , [𝐀𝐄, 𝐁𝐄, 𝐂𝐄] , [ 𝐊𝐀, 𝐊𝐁 , 𝐊𝐂 ] are applied on the three lines (𝐊𝐀𝐊𝐁 , 𝐊𝐁 𝐊𝐂 ,𝐊𝐂𝐊𝐀) , then the six pairs of the corresponding lines , extended are concurrent at points pa , pb , pc for the triple pairs of lines ( the pascal`s perspectivity of points in euclidean geometry ) , [ a𝐀𝐄, 𝐁𝐂𝐄, 𝐂𝐁𝐄 ] , [b𝐁𝐄, 𝐂𝐀𝐄, 𝐀𝐂𝐄] , [c𝐂𝐄, 𝐀𝐁𝐄, 𝐁𝐀𝐄] and at points da , db , dc for the triple pairs of lines [𝐊𝐁 𝐊𝐂 , bc , 𝐁𝐄𝐂𝐄] , [𝐊𝐀𝐊𝐂 ,ac ,𝐀𝐄𝐂𝐄] and [𝐊𝐀𝐊𝐁 , ab , 𝐀𝐄𝐁𝐄] , ( desargues`s perspectivity of points in euclidean geometry ) and all the 18 common points lie on a straight line the  stpl mechanism this mechanism stabilizes the nucleus forces for the{ ⊕ space [ a b c] } and for the {⊝ anti-space [ ka kb kc ]} and this succeeded by the pascal`s –launched caves r through the stpl circuit of the ptolemy`s and the ceba`s geometry theorem for the complex vector forces which are , circularly and diagonally equilibrium .i.e. → stpl is the in-sphere , tetrahedron-cube , out-sphere physical mechanism ← for its 27-conductors ≡ electric magnetic fields , space-energy cpt is invariant . 8..[a]-2…the explanation of the absolute and relative motion. . fig – 6 because properties in and on [stpl] line , are relative to the only one equilibrium and also absolute system ± λ = r.mv̅ = r.m.w̅.r = mr².w̅ , so exists that what is called relativity. absolute system let it be the [a] ≡ {dao} ≡ the stpl mechanism , and as relative system ( reference , affine ) [r] ≡ {dao} the absolute motion happens on [a] ≡ {dao} and on [r] ≡{da-pa} of the two systems . it was shown , that in {dao}, (x ,y ,z , t ) , system �̅� , �̅� , vectors are isochrones i.e. period t = l / v = 2πr/v = 2π / [c/𝑟𝑐] = 2π/[v/𝑟𝑐] → c/rc = v/rv → c.rv= v. rc , where rv , rc are the radius of their intrinsic rolling circles . this relation is geometrically expressed as , 𝐬𝐞𝐜𝛗 = o da : a da = γ = 𝒄 𝒗 = ± 1 / [ √1–(v/c) ²] = c / [√ c² – v²] , and it is a ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 28 the fundamental particles origination mechanism in mfmf pns caves . geometrical cycloid property equal to lorentz`s , γ , factor . the newton`s laws are true into the reference system [r] ≡ {dapa} as follows proof , considering { dao},(x ,y ,z , t ) , as the fixed frame [a] of the coordinate system in the gravity cave (d = 2r) , and point a= (x, y , z) is fixed on circle (o,oa) which is rotating with a velocity v̅ = w ̅̅ ̅r , of angular velocity w̅ = 2π /t = 2πf , in where period of rotation t , is then also constant . since the acceleration ,a, for a quaternion z = (s + v̅. i) is a = [d²z/dt²] = (ds/dt.v̅. i) + s. d(v̅. i)/dt = 0 + s. d(w r)/dt = 0 + 0 , and this because w̅ is constant for both therefore velocity �̅� = constant also , i.e. → the centrifugal velocity of the absolute system [ a ] is any constant , �̅� , and this because angular velocity , �̅� , is constant also and thus , is not needed to accept a priori this constancy of velocity �̅� = 0 → �̅� → ∞ on circle (o, oa) and to exist in frame , ( it has been proved that this velocity �̅� ≡  σ φ ) so , automatically is defined the conversion factor t = time , between the conventional time units (second) and the length units (meter = a.da) or as c̅. rv = v̅. rc , → c̅ (v)(t/2π) = v̅(c).(t/2π) → c̅(v)/w = v̅(c) / w , which is happening with the same w , without setting any restrictions , in contradiction to the general relativity which is an axiom a priori . additional , this is the why that the conversion factor , t = time , has not any essence in all universe but it is only a meter of changes in all relative systems . stpl mechanism is the only absolute system on which circularly exist [ ⊕  ⊝ ],[ ⊝  ⊕ ],[ ⊕ ⊝  ] , the three constituents , [ ⊕  ⊝ ] , and originates this existing objective cosmos . this circularity of the constitutes exist because of their natural frequency of vibrations . because [stpl] line of the fixed frame becomes from this system [a] , then this relative frame [r} is common to the fixed one (common da) and let it be [r] ≡ (x' , y' , z' , t') . from figure , sin φ = (�̅� / �̅�) , meaning that the relative system , [r] ≡ (x' , y' , z', t') , ( the affine frame) is the projection of absolute frame [a] ≡ {da-o} (x , y , z , t) where exists as the simultaneity for all motions where issues , [r] ≡ {daa} ≡ [ (x' , y' , z' , t') ] , [a] ≡ {dao} ≡ (x , y , z , t) = [r] .γ ≡ ( x' , y' , z', t' ) . γ considering point da as the common center and [stpl] as the x-x axis of the two systems , then becomes da ( x , y=y ', z = z', t) and for all linear systems da ( x', y'= y , z '= z , t' ) this specific state of constancy , i.e., the centrifugal velocity of absolute system [a] to be a constant c̅, and the rectilinear motion with respect to one another defines the natural inertial frames , and because of uniformity of space and motion , therefore occupies the same meter for their changes , (i.e. the time) . since also points o , a remove to point da isochrones by their intrinsic property motion which are → their wavelengths are a stationary wave in cycloid ← following lorentz`s factor γ , then this following happens also to all frames which make this motion , and so issues for all the system relations { da – o } = γ.{ da – a } . fig-6 on this relative system da( x', y' = y, z' = z , t`) are conveyed , the breakages [ ± (wr)² , 2(wr)² ] of (o,oa) circle after the colliding with the rotating velocity v̅ = w̅.r of the [a] system , and are the fundamental particles , fermions and bosons, or by escaping consisting the rest field and gravity , or dark matter and dark energy as analytically was shown .[39] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 29 the fundamental particles origination mechanism in mfmf pns caves . the angular-momentum ellipsoid nibs on →{ herpolhode and polhode ellipsoids}. in any rotational motion , b̅xw̅ are constants and the angular-velocity-vector �̅� , describes the ellipsoid of angular velocity and its nib describes a cone of which plane-base is fixed . simultaneously .the angular-momentum �̅� describes the ellipsoid of angular momentum and its nib describes a cone also of which plane-base is also fixed . the nib of angular velocity-vector , �̅� , describes on the tangential-plane of the angular-momentum ellipsoid , the polhode , while the nib of angular-momentum �̅� , describes on the tangential-plane , of the angular – velocity ellipsoid , the herpolhode . or the fixed tangential-planes on �̅� , �̅� nibs which are alternately perpendicular to b̅ , and w̅ , common central axes of rotation , or the angular-momentum ellipsoid nibs , in herpolhode and on polhode ellipsoids as → electro magnetic fields ← and which are space-energy cpt invariants . the moving energy-vector is |⊕ 𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ | = |↕| →{ the electron-unit-charge }, and since charges correspond to the 3-phase rotors is ,the unit-charge complex-vector. remarks : material point a ≡ ± |(w̅.r)²| of the fixed system {dao} travels with velocity v̅ at point da , so geometrical distance a.da in the relative system [r] ≡ {dapa} is a.da = x'+ v̅ t', and because of the isochrones motion in the fixed system [a] ≡ {dao} , then holds , x = (x'+ v̅.t') .γ or x = (x'+ v̅.t').γ ≡ [ x'+ v̅.t'] / [√1-(v/c) ²] ……. (a) inversely , by using (a) , where [a] ≡ { da-a} ≡ {dao}/ γ , then if material point a of the fixed system { dao} travels with velocity v̅ at point da , where then the geometrical distance ada in the fixed system is , [a] ≡ {da o} is → a.da = x v̅.t and in the relative system , [r] ≡ { dapa} is → x' = (x-v t).γ = [x-v t] : [√1 (v/c)²] …...(b) fig-6 an application to gravity [72] and photon [95] , because gravitational force is equal to → the geometric-resultant of the light-velocity c , acting on electron-unit-charge �̅� ← or is , g = c √𝟐 �̅� , then electron charge is , q̅ = g c √2 = 6,680561 .10−11 1,41429.[2,9979346.10 8] = 1,58.10−19 coulomb , which is the known coulomb`s charge quantity . from gl ,kl universal constants the newton`s gravitational constant force g ≡ g.ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[gl kl ] ≡ 9,808238* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² the in depth position of electrons allows to the work produced in prior phase areas as golden ratio pattern , to be stored in the next . energy = motion / t ≡ ( v 2πr ) . [σ + σ φ] = �̅� . [ 𝛔 𝟐𝛑𝐫 + 𝛔𝚽 𝟐𝛑𝐫 ] ≡ �̅� . [. fn̅ + 𝐟𝐧 ] ≡ moving storage →[ �̅� . fn̅ ] ← + moving-frequency → [ �̅�.𝐟𝐧 ] ← ≡ the material-point . i.e. the energy produced in photon-cave is consisted of two-moving-storages , that travel as wave [�̅�.𝐟𝐧] →[ �̅� . fn̅ ] → [ �̅� = �̅� = λ f φ ] → [ s ≡ em-r ≡ f1=n , f2 , f3, fd ,,f n= w² ] , and as particle →[ f1= (e²+h²) = n (1+√5)σ 2πr = nb̅ π² r⁴ ] →{w≡em-r≡ [εe² + μb²] = 2.λc.sin.2φ} and is the duality of the energy-storage s ≡ { [⊕← 𝐫 →⊝] + motion m . [87-89] .the three elements [σ = n . h]  [s² = ⊕ , 2s²= , s² = ⊝] of this {stpl}-mechanism formulate the 3-knots figures ( n = 3 ) and determinates the elementary-particles vector mould . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 30 the fundamental particles origination mechanism in mfmf pns caves . 8.[a]-3..the circularly existing unit-charge complex-vector [⊕  ⊝],[⊝  ⊕],[⊕ ⊝ ], figure 7 : the , stpl , mechanism is the generator of the cosmic-particles . the unit-charge is as complex-vector velocity thrust ( v̅ = w̅.r ) , with and from the breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . 7-a = the positive breakage quantity q ≡ [ ⊕  ⊝ ] , for leptons and quarks . figure –7a : the , stpl , mechanism is the generator of the cosmic-particles . the unit-charge is as complex-vector velocity thrust ( v̅ = w̅.r ) , with and from the breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . v-trust ≡ [ ⊕  ⊝ ] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 31 the fundamental particles origination mechanism in mfmf pns caves . 7-b = the positive breakage quantity q ≡ [⊝  ⊕] , for anti-leptons , anti-quarks . figure -7b : the , stpl , mechanism is the generator of the cosmic-particles . the unit-charge is as complex-vector velocity thrust ( v̅ = w̅.r ) , with and from the breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . v trust ≡ [⊝  ⊕] 7-c = the positive breakage quantity q ≡ [⊕ ⊝ ] , for bosons origination . figure -7c : the , stpl , mechanism is the generator of the cosmic-particles . the unit-charge is as complex-vector velocity thrust ( v̅ = w̅.r ) , with and from the breakages → 𝐳 * 𝐳 = (s + v̅.i )² = s² + [v̅]² +2.sv̅.i . v trust ≡ [⊕ ⊝ ] , since the primary motion ≡ [pm] exist from the three breakages only , and stpl is the n =3 mechanism creating elementary particles  [n-knots=3] in 𝐕 gra = 7,593.1035 v. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 32 the fundamental particles origination mechanism in mfmf pns caves . 8..[b].. the { tvpm } physical mechanisms : two vectors three poles mechanism . figure -8a : the two-perpendicular complex-vectors , ca ⊥ cp , slide on (o,oa) circle , and through the three poles , a , c , p , carry the stress σ , of square , acpc’, to the two equal and opposite squares |coab| , |copb’| for halve area stress conservation b-1... the conservation of energy in geometry mechanism : [62] in(1)…on aa` diameter circle (o,oa) exists the inscribed -square oa x oa = oa², and the circumscribed square cp x cp = cp². since vector cp = [oc= oa].√𝟐 , therefore the inscribed square is halve the circumscribed . the same to the circles π [ 𝐂𝐏 √𝟐 ] ² = π [cp] ² = π [oa.√𝟐] ² = 2.π [oa]² . this linear expansion of archimedes-circles and squares passes from a point m , such that π r ² = cmnh square . [ 47 51 ] in(2)... on two-perpendicular vectors , ca ⊥ cp , point c` slides on (o,oa = op) circle forming the cac`p to cmnh to cbao squares with prior of (1) properties .this sliding-rotating method is the markos { 2 -vectors 3 -poles mechanism } which circles and squares as areas pass from the point m determined by the position 𝐀 𝐞 , such that π r ² = cmnh square . the squaring of the circle π [ 𝐂𝐀 √𝟐 ]² = cm² . [ 47 51 – 98a ] in(3).. if the two-perpendicular vectors are conductors with prior properties then square cac`p to cmnh to cbao square , are a rotating and changing-squares-system , cp ² ↑ = |∷| ,↻, → to 2-half and equal area squares cb ² = ⤭ =[⊕ ∷ 2 +⊝ ∷ 2 ] cb`² , by turns in oabc ⊥ opb`c planes , which equilibrium and consist , the neutral-space . placing stress in squares becomes the physical world as , physics mechanics and chemistry , by conservation of energy as stresses in areas . as in electricity , the flow of a current in an conductor , ca = d , represents the created around the conductor by the circular-lines of forces , magnetic field , the flow of ⊕→⊝-charge in d ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 33 the fundamental particles origination mechanism in mfmf pns caves . conductor , is the torsional force �̅�𝐅 = v⃗ .r = w⃗⃗⃗ .r ² = 2πf. r ² = σ φ r …....(1) . since edge-velocities v̅c = v̅a= 0 , therefore ca =d= ½ (at²) and acceleration , a = 2d /t² = 2d.f², and then the velocities diagram is energy-triangle → v.[δ-cap] = v.[½ (2d. |v⃗ |)] = v.[d w r ] = v d.2πf.r = v.[σ φ d] ..(2) the current flow in electric-field �̅�𝐂𝐀𝐏 of cap plane is transported into the perpendicular to ca axis opposite magnetic-field �̅�𝐏𝐀 in pa-plane due to the force �̅�𝐅= �⃗� .r = i = 𝟐𝛑𝐫 𝛍 �̅�𝐏𝐀= σ φ r .(3) stress σ = 𝐅𝐨𝐫𝐜𝐞 𝐀𝐫𝐞𝐚 = 𝐆 𝐯.𝐯 = g √2 .c = the �̅� electron charge = g √2 .c = 6,6736923 .10−11 1,41429.2,9979346.108 =1,574.10−19 c the spin of cave , r, is equal to the angular-momentum-vector → spin ≡ |�̅�| ≡ r σ φ ← which contains and it is the golden-radio-frequency φ as pressure , σ , in cave r . i.e. [a] the forces are spread on surfaces as stresses σ , and in case of area = 0 then are spread on the velocity – square – surfaces , and for the light c , on resultant √𝟐.c . b-2... for conductor-case which is an screw-motion where a torque is created in perpendicular plane of con-axis . markos { 2-vector 3-poles rotating mechanism },{ −a ∷ ↻ +𝐂 ↑ [⊕ ∷ 2 +⊝ ∷ 2 ]𝐏 } carries motion (+) → (-) from point a through an perpendicular-conductor ca , to point b , into an changeable and rotating-squares-system , pa ↑ = |∷| ,↻, → to 2-half and equal area squares cb = ⤭ = [⊕ ∷ 2 +⊝ ∷ 2 ] cb` , by turns in oabc ⊥ opb`c planes which equilibrium [⊕ ∷ 2 +⊝ ∷ 2 ]=0 (the neutral-space) . b-3..for the motion of the ⊕→⊝-charge of any conductor d = ca becomes stress σ in length d motion is created in conductors d ≡ [⊕← 2𝐫 →⊝] following the [⊕↔⊝] interactions .this motion is conserved and kept in caves as electric and magnetic fields and as frequencies which are conserved in them when the space-points become energy-conductors and energy k = 2π λ = 2π d . b-4...the in caves energy imports are in squares which equilibrium through mpm ≡ markos of ⊝ charge in an electric–field e̅f . for velocity-vectors at point c issues , cp⃖⃗⃗⃗ = ca⃖⃗⃗⃗ and cp⃖⃗⃗⃗ ⊥ ca⃖⃗⃗⃗ and the resultant is the velocity-vector c⃡c`ap , electric-vector = πσφ 4 [d]² , magnetic-vector = πσφ 4 [d]² . similarly on material vector cp⃖⃗⃗⃗ carries the maximum conjugate-force cc`= c⃡ka from e-square cp² to two equal and opposite squares – cb`po ,+ cbao from cac`p ≡ +𝐂𝐎𝐀𝐁 −𝐂𝐎𝐏𝐁 ` [⤭] which rest in the circumscribed circle π.oa² = 2π( 𝐎𝐀 √𝟐 )², to the screwdriver inscribed to , ca⃖⃗⃗⃗ , cd⃖⃗ ⃗⃗ circles , π[ 𝐂𝐀² 𝟒 ] , π [ 𝐂𝐏² 𝟒 ] . motion to rest , before the halve areas , circles pass through the equal to cmnh square where exist quadrature at position m and to half-area circle π [ 𝐎𝐁 √𝟐 ]² = π[ 𝐎𝐁 ² 𝟐 ] at position m b-5..this property of any two equal and perpendicular vectors at point c , issues for any point n , which has a screwdriver –material vectorcn . the quantization of e-geometry to its dimensional-moulds : theone , two , three vectors and three poles mechanism . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 34 the fundamental particles origination mechanism in mfmf pns caves . figure -8 b : the quantization of e-geometry to its moulds , the point , the linear , the plane , the space (volume) mould for e-geometry quantization , to opposite. b-6…for (1) monad ea , to antimonad ec–for (2) ab line to parallel line mm` for (3) ce radius to the cm square segment for (4) ka segment to the kd cube segment . from (3) the two-perpendicular vectors , ca ⊥ cp , slide on (o,oa) circle and through the three poles , a , c , p , carry the stress σ , of square acpc , to the two equal and opposite squares |coab| , |copb’| for still to the halve area stress conservation . for the rotating square cmnh exist as upper-still-state the square cac`p and as lower still-state the square cbao , related [ cac`p ] = 2.[ cbao ] . in(3)..on two-perpendicular vectors , ca ⊥ cp , point c` slides on (o,oa = op) circle forming the cac`p to cmnh to cbao squares with prior of (1) properties .this sliding rotating method is markos { 2-vectors 3-poles mechanism } which circles and squares as areas pass from the point m determined by position 𝐀 𝐞 , such that issues π r ² = cmnh square .the squaring of the circle is π [ 𝐂𝐀 √𝟐 ]² = cm² = |cm`|² , fig-15, [ 47-51 ] , [62] remarks : 1… the two vectors three poles mechanism is a mechanism from two ⊥ vectors in geometry , where the action of energy on their vectors square and is conservated . 2… the mechanism is a rotating and changing-squares system , on a circle which changes the inscribed to the circle square , changes to 2 –halve opposite and equal squares .the same procedure issues for the inscribed and circumscribed circles to the circle . 3… since from wave-action of spaces issues z − n x z + n = [2r] 2n . e i.2nθ , this means that ,the conservation of energy = work , between the spaces and the anti-spaces happens by doubling the number n , of the vertices of the regular polygon . i.e. the surplus energy is stored in 2.n vertices , and on 𝛌𝟐𝐧=√2r2 − r√4r2 − λ²n sides , where then energy is divided and spread on the double number of the polygon sides. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 35 the fundamental particles origination mechanism in mfmf pns caves . 8..[c].. the { tcic } mechanism : the tetrahedron , the cube , and the inscribed – circumscribed to the circle mechanism . figure 9: the active-conductors of elements and the space , anti-space formation . in figure za , zb , zc , are the three conductors on the three diagonals of cube with 1 , 2 , 3 , 4 , 5 , 6 , 7, 8 , vertices . at point z is acting a complex-vector z̅ and which transform the vibration to the three conductors in space pattern . it was shown [61] that the stresses of complex-vectors are spread in surfaces only for still states. the number of moulds of stationary-states are → 𝟐n=1−4 points ≡ 2 , 4 ,8 ,16 ≡ 4–types← (2) = for the proton = nucleus , (4) = tetrahedron , (8) = cube = the neutral-caves , (16) = the double number of vertices for the 8-stationary neutral-caves and called tcic-pattern →{ the sphere – in tetrahedron – in cube , in external sphere }← the active-conductors of elements and the space , anti-space formation . point , in e-geometry is considered nothing , without position and direction while in , mg ≡ material geometry [1 positive ⊕ and 1 negative ⊝] . from nothing (i.e. the point z ) to existence ( i.e. to be another spherical point p ) issues the zero virtual work law ,where zero work is the equilibrium of two equal and opposite forces on points and is [⊕ d ⊝] . thus space [s] is z-point ⊕ , anti-space p-point ⊝ , and d = zp = the conductor of points since ,the position and the number of points, create the spaces so the only three elements are → {⊕ , [⊕↔⊝] , ⊝} ≡ [+, 0 , -] ← for material geometry exist linearly in moulds . the stresses of complex-vectors act on a surface , a circle of sphere , between the sub units axially and rotationally following ptolemy`s and cebas theorem of equilibrium. the space anti-space and , neutral-space conductors are self-sustaining . stability : ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 36 the fundamental particles origination mechanism in mfmf pns caves . the number of moulds of stationary-states are as→ 2n=1−4 points ≡ 2 ,4 ,8 ,16 ≡ 4–types← i.e. monads are formed from the first 4moulds as → the 1-point which is everywhere, the 2points which form vectors , the 3points which form the planes , the 4 points which form the volumes . for n > 4 as this is the atoms structure, follows the sustainability of the spaces and anti-spaces. since space-energy structures follow the m-geometry stability these follow the mould of spaces 2 n = 3 = 8 elements which is the-cube as shown before . instances , 1.. in the constant , hydrogen conservative-cave-system , which creates only closed orbit shapes as circles , ellipses and eight-shape , ∞ , are placed the above elements , π g ≡ energy ≡ [meter of area * meter of force] ≡ electrons on orbits , and also the unit-space ≡ massive-complex-unit-space ≡ →[+�̅�.s²]← jointed through the neutral material-points [ (+) [↔] (-)] with the strong-force → s f = h .fn ≡ h .{ [s ≡ bp ≡ em-r ≡ f1=n , f2 , f3, fd ,,f n ] ≡ h.n (1+√5) σ 2πr ≡ h [ n.b̅ 4𝜋2. r4 ] ….(mg) equations (m) , (mg) are the energy-space-constituents in hydrogen-system . c-1.. the protons and the nucleus structure : from above mp ≡ [(+)[↔](-)] ≡ neutral = n , then a.. for 1 proton [(+)[↔](-)]↔(+) ≡ neutral → n = 1 p = 1 b.. for 2 proton (+)↔[(-)↔(+)][(+)↔(-)]↔(+) ≡ neutral → p n n p c.. for 3 proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)] ≡ neutral → [ 0 0 0 0 2⃡ 0 1e 0 0 ] d.. for 4 proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)](+)[(-)(+)] → [ 0 0 0 0 2⃡ 0 1e 1e 0 ] e.. for k proton (k1)↔[(-)(+)](k2)[(-)(+)](+)[(-)(+)](+)[(-)(𝐡𝐤)] → [ ke ke ke ke 𝑘 ∶ 8 − 18⃖⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ke ke ke ke ] f.. the inscribed to the tetrahedron-sphere consists the nucleus ≡ the ⊕ space , and in circumscribed to the tetrahedron-cube sphere consists the orbitals ≡ the ⊝ space . c-2.. the annihilation of the fundamental particles . considering the , matter antimatter , with the classical known theory of annihilation as the common constant energy then issue for a force ↔anti force , action ↔reaction electron = e−= ⊝→ mass 𝐦𝐞 = 9,11.10−31 kg → charge 𝐂𝐞 = 1,602.10−19 ev → cave a = 5,0.10−17 m 1.. electron positron  e−→ e+ ≡ 2 γ ≠ 0 the common me→p is from relation 1 mt = 1 me + 1 mp = 1 me + 1 me = 2 me = 2 9,11.10−31 = 1 4,555.10−31 from energy equation e = 1 2 me→p c² = 1 2 [4,555. 10−31].[2,998. 108]² = 6,827945. 10−15 j = j / (1,6022.10−19 ev) = 4,2616059. 10 4 ev = 42,616059 .kev the common frequency is f e→p = e h = 6,827945.10−15 j 6.62607.10−34 js. = 1,0304667. 10 19 h  i.e. 𝐟 𝛄 ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 37 the fundamental particles origination mechanism in mfmf pns caves . c-3.. the equilibrium of the three-spaces , the space anti-space , neutral-space the stability of elementary particles , conductors and the atoms -nucleus : [60] figure– 9a: the electric and magnetic field of 3 conductors on 2 anti-parallel-strands the waves motion ≡ energy , is transported and , the pattern of disturbances with informations , propagates from the one-edge-point to the other edge-point of conductors , or is on [ sub-units ] . to prove the stability of the system , the equations of motion become everywhere from any opposite space ≡ ⊕ , anti space ≡ ⊝ . during these motions as , ds . is done a constant-work , k , or and from their velocities �̅� and this work may be positive or negative for equilibrium . in f-23 cycloid anti-cycloid motion is as , d. the origination of gravity g , antigravity [g ] in fig-9a , conductor d = d = aka where ⊕ constituent attacks ⊝ by a hook`s law force as stress σ = e u , and an coulomb force f [+ ↔ −] = [⊕↔⊝ ] r² = [2 σ] r² , creating work ≡ motion ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 38 the fundamental particles origination mechanism in mfmf pns caves . as was shown it was shown that , the stability of nucleus forces , is succeeded through the pascal`s ceba`s triangle launched caves r , while leptons from their anti leptons existence in the same launched pascal`s caves r . [58] the equilibrium of the three-spaces is done on the three cyclic-quadrilaterals (abeaece) , (bcebeae) , (caecebe) , of the 6 conductors each for , [a , b , c , d , p , q ] x [3-abc] = 18 conductors . the stability is 1..linearly through the products of the diagonal connectors as , → → → [aae].[bece] = [ace].[aebe]+[abe].[aece] , [bbe].[ceae] = [bae].[bece]+[bce].[beae] , [cce].[aebe] = [cae].[bece]+[cbe].[ceae] and , 2..rotationally through the ratio of the diagonals ≡ the conductors | akb akc |x| bkc bka |x| cka ckb | = 1 or  rotational-equilibrium as , [akb].[cka].[bkc]=[akc].[bka].[ckb], of all opposites  tangential-electromotive forces . 3.. from mechanics-physics , all systems possessing elasticity ≡ motion and reaction to motion , the called mass , are capable of free vibration or vibration taking place in the absence of external excitation [7] .this principle issues for both closed or open systems. i.e. either for the nucleus where issues the pascal`s ceba`s triangle stability , or for the orbitals where issues the tetrahedron-cube -sphere-mould .a signal as the electric and magnetic field in 3-conductors is transformed on the 2 anti-parallel – strands . c-4... the natural electromagnetic energy-tetrahedron and , the 3conductors into the energy cube and spheres : [102] in fig9c the electric magnetic field of 3-conductors which lie on the dabc occurs into the {⊝ 8-anti-space cube , [da1a d1 , b b1cc1]}, and the positive → { ⊕ 8 space – cube , [da1a d1 , b b1cc1]} , into the {⊝ 16-anti-space [ 1 ,2 ,3 ,4 5 , 6, 7, 8 9 ,10 ,11, 12 , 13 ,14 , 15, 16] } sphere cube on vertices.= point [the point 16 ≡ |⇉|z|] consists the navel-string gate into the triangle-circle-system {z-(∆d,abc), ∆[t1t2 t3]} of the stationary bases which results on the 2 -anti-parallel-strands . the dual photon v̅ [ σφ 2πr + σ 2πr ] ≡ v̅.[ fn̅ + fn ] , occupies stresses = σ and velocities v̅ , in the tiny-caves r . the colours in light , are the still-sub-units in storage →[v̅ . fn̅ ]← and exist as frequencies of → violet , blue , green , with their complementary colours yellow , orange , red ← every 8-electrons are vibrations on atoms-cube-structure {a tetrahedron in cube in a sphere} whether it be sound or light . above structure is followed by all compounds. the molecules are systems consisted of the periferal ≡ skeleton 𝐌 𝟏𝟔 and the central ≡ fittings 𝐌 𝟖 , and the accessories ≡ 𝐌 𝟒 or 𝐌 𝟐 , 𝐌 𝟏 . this property of atoms and of the compounds is the critical valve of switching the motion as this happens in the electro-magnetic solenoid valves with high or low pressure and flow rates directly or not . remarks : work of elastic deformation . elastic deformation requires (1) that every new stress that is applied to a new infinite element is always added to the existing one .(2) be always unity work of deformation. when a body contains potential energy , during any unloading it gives off mechanical work , since the points of application of the gradually withdrawn loads are displaced. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 39 the fundamental particles origination mechanism in mfmf pns caves . if the unloading is abrupt , then kinetic energy is converted into potential and after is kinetic , converted back into kinetic and so on . that is, the body performs an oscillating motion until the mechanical energy is extinguished due to a state of equilibrium or alters . figure – 9b: the electric magnetic field of 3 conductors results on 2 -anti-parallel strands .the stability of →{ ⊕ 4-space [d a b c]} regular tetrahedron , in {⊝ 8anti-space cube , [ da1b d1 , c c1ab1]}, and the →{ ⊕ 8space [ da1bd1 , cc1ab1]} cube, into the {⊝ 16-anti-space [ 1 ,2 ,3 ,4 ,5 , 6, 7, 8 9 ,10 ,11, 12 , 13 ,14 , 15, 16]} sphere cube vertices . the point [16 ≡ | ⇉ |z|] consists the navel string gate in system . and are { z (∆d , abc) , ∆ [ t1 t2 t3 ] } stationary bases . the stability analysis in [88] c-5.. : the mater antimatter 3-d origination mechanism [⊕↔⊝] figure – 9c: the 4-dimentional spinning , tetrahedron cube – spheres mould . the stationary-states are → 2n=1−4 points ≡ 2 , 4 , 8 , 16 ≡ 4–types ← ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 40 the fundamental particles origination mechanism in mfmf pns caves . figure – 9d.. :the hydrogen , carbon , oxygen , atoms-conductors into the sphere-cube mould . deka hexahedron , octahedron ≡ cube , tetrahedron ≡ 4-points , triangle . the nucleus ≡ ( in-sphere ) , tetrahedron ≡ ( d a.b.c ) , cube ≡ ( o,2,a,4,b,6,c,8 ) , orbitals ≡ ( ex-sphere ) , mould . when vector ≡ 2-points ≡ the closed-conductor , 1-point ≡ the opened -conductor ≡ the bracket-hook . oxygen is the element which fills 8 2 = 6 vertices of cube as el → { 1𝑒 0𝑒 1𝑒 1 𝑒 2⃡ 0 𝑒 1𝑒 1𝑒 1𝑒 } ≡ (± 2 2 ) , where ⓪ is the first , zero-acting cube filled with 1e , therefore is the only steady element which occupies the first energy-level ① with 2 electron-positions as ① = 2e = 2⃡ positions at 1⃗ , 1⃗⃖ the electron being in the hydrogen precesses because of the different axis of rotation and nutation`s , from the continuous and immense-communication to the effect of gravity g . since electron is continually oscillating , with the nutation-frequency 𝐟 𝐍 , so produces a continuous and oscillating magnetic-field –mwhich in turn is the source of an oscillating electric-field – e which implies the regeneration of each other , i.e. it is a propagating electromagnetic-wave where e = b c , and with a quantum-energy e = h fn or e = 2μ.b , this energy e consists the hydrogen bracket hook [hbh] , or [bh⃗⃗⃗⃗ ⃗] and it is the only free monad which can bond to all energy structures . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 41 the fundamental particles origination mechanism in mfmf pns caves . the spheres-cube-structure of atoms in sphere center –oallows the stability of the system, per–two electrons in dipole positions , 1 ↔ 6 , 2 ↔ 5 , 3 ↔ 8 , 4 ↔7 , consisting the-positive dipole-[bh⃗⃗⃗⃗ ⃗] , to be able to move → into-inoutside it ← and allow the atoms bonding as this in energy levels . the cube-structure of atoms and orbits in the outsphere formulates +[bh⃗⃗⃗⃗ ⃗] to those dipole positions which lack of one electron and so can plug-in the loop of the others . the atoms with zero stable state are , the hydrogen bracket-hook into the carbon encloses the inscribed sphere for the nucleus , into the oxygen for the orbitals , into the sulfur enclosed into the circumscribed sphere for the atoms-conductors sustains . the first 4-dimentional spinning ,tetrahedron-cube–sphere-mould are , 9c--2 1… o-1-3-5-7 → the first solid tetrahedron structure in nature , 2… o-1-2-3-4-5-6-7-8 → the first cube with max. positions e = 2.n² = 8 , 3… r [ o , a , b , c ] → nucleus is in the first inscribed sphere into tetrahedron. 4… r-o-1, 2, 3, 4, 5, 6, 7, 8 → orbits are in enveloped cube-sphere of tetrahedron. deka hexahedron , octahedron ≡ cubes , tetrahedron ≡ 4-points , triangle ≡ 3-points ,vector ≡ 2-points ≡ the closed-conductor , 1-point ≡ the opened -conductor ≡the bracket-hook . the hydrogen nucleus is kept in the inscribed to the tetrahedron-cube-system where exist the protons and neutrons , while the electrons of hydrogen orbits are kept into the circumscribed , to the tetrahedron-cube-system . both equilibrium by following the menelaus ceba`s principles . [70] any two perpendicular energy-vectors , |𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔ | , |𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ | of this propagating energy , form the energy-square , |↔|.|↕|≡|𝐐 ⇉ |.| 𝐐 ⇈| ≡ q0 , on conductor aka and then energy-circular-prism = |aka| .{𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{ 𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } ≡ [�̅� 𝐱�̅� 𝐲].|aka| = | ↔ |.|↕|.|aka| , i.e. an energy-circular-cone which is a moving energy -volume , and because from cauchy equations of stresses are of three dimensions , then the energy stresses remains flat onlywhen the plane-section is a circle [𝐄𝟐 ] and the square becomes circle, anti-quadrature.. because from mechanics , the hydrogen nucleus is kept in the inscribed to the tetrahedron-cube-system where exist the protons and neutrons , while the electrons of hydrogen orbits are kept into the circumscribed , to the tetrahedron–cube-system . both equilibrium by following the menelaus-cebas principles . oxygen is the element which fills 8 2 = 6 vertices of cube and for neon 10 -2 = 8 , as ox → { 1𝑒 0| 0/ 1 𝑒 2|⃖⃗ ⃗/ 1 𝑒 1𝑒/ 1𝑒| 1𝑒 }≡ ( −2 +2 ) , ② ≡ 𝐍𝐞 1x8 2⃡ = 2⃡ + 8 = 10 → { 1 1 1 1 2⃡ 1 1 1 1 }, the possible orbits of oxygen are two as , → 1e /-2⃡/-/ 0 ← and → 1e |-2|⃖⃗ ⃗- 0 | ← for all the 8-cube positions . ⓪ is the first , zero-acting cube and is filled with 1e , therefore is the only steady element which occupies the first energy-level ① with 2 electron-positions as ① = 2e positions at 1⃗ , 1⃗⃖ . in the same way ① is the first one-acting cube and is filled ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 42 the fundamental particles origination mechanism in mfmf pns caves . with 2e, and so on , the electron being in the hydrogen -cave precesses because of the different axis of rotation and nutation`s , from the continuous and immense-communication to the gravity g . since electron is continually oscillating with the nutation-frequency 𝐟 𝐍 so produces a continuous and oscillating magnetic-field –mwhich in turn is the source of an oscillating electric-field – e which implies the regeneration of each other , i.e. it is a propagating electromagnetic-wave where e = b c and with a quantum-energy e = h fn or e = 2μ.b , this energy e consists the hydrogen bracket hook hbh , or [bh⃗⃗⃗⃗ ⃗] and it is the only free monad which can bond to all energy structures . the sphere-cube-tetrahedron structure of atoms in sphere center –oallows to issue the ceba`s theorem for 6-stresses on the inscribed tetrahedron which secure the stability for the nucleus-system , and for the outer electron-system the stability is obtained per–two electrons the dipole positions 1 ↔ 6 , 2 ↔ 5 , 3 ↔ 8 , 4 ↔7 , consisting thepositive dipole-[bh⃗⃗⃗⃗ ⃗] , to be able to move → into-inoutside it ← and allow the atomsbonding as this in energy levels . the cube-structure of atoms and orbits in the out-sphere formulates the +[bh⃗⃗⃗⃗ ⃗] to those dipole positions which lack of one electron and so can plug-in the loop of the others. conclusion : 1… the forces of atom`s nucleus , are holded from → the inscribed sphere in the tetrahedron , which follows the ptolemy`s and ceba`s vectors structure . 2… the forces of atom`s orbitals , are holded from → the into the cube , circumscribed–sphere complex vectors structure . 3… the forces of atom`s positions , are holded from → the on cube inscribed tetrahedron complex vectors structure . 4… the stability of the {tcic} mechanism follows the ptolemy`s & ceba`s complex vectors structure [62] the action of the complex–vectors create the waves of motion ≡ energy waves which are closed curves and transported . the pattern of disturbances with the informations propagates from the one-edge-point to the other edge-point of conductors , or is on the [ sub-units ] . coulomb force [⊕→⊝] = f [+ ↔ −] = [⊕↔⊝ ] r² = [ 𝟐 𝛔 ] 𝐫² = f = σ .a = [ 2πrf φ ].a = w r.[ 𝐀 𝚽 ] = v .[ 𝐀 𝚽 ] , and acts on a surface , a , between the sub-units axially and rotationally following ptolemy`s and cebas theorem of equilibrium . figure –9e remarks : the work of elastic deformation on the { tcic } mechanism : every new stress that is applied to d element is always added to the existing and the unity work of deformation is transformed to three tetrahedron sides. when a signal strikes the tcic – cube`s mechanism . the signal is energy-wave carrying motion . unloading happens on the three still-spaces in tetrahedron sides which withdrawn loads are displaced on the anti-parallel-strand . if the unloading is abrupt, then kinetic energy of 1-strand is converted in the 2-strand as potential and so on . that is , cubes perform an oscillating motion until the mechanical energy is extinguished to their vertices , which are the stores of information . anti-parallel-strand is the body containing the potential energy , during any unloading on cubes , it gives off mechanical-work , since the points of application of the gradually withdrawn loads are displaced . the stability of kinetic – potential energy is continues . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 43 the fundamental particles origination mechanism in mfmf pns caves . figure –10the hydrogen bracket – hook of 2 = orbitals  [ (+) [↔] (-)] stationary-state happens as → [⊝] ≡ orbit-1 ↔ [⊕] ≡ orbit-2 ← 8..[d].. the { tobh } mechanisms atoms bonding bracket hook mechanism . d-1.. : the precession and nutation of electron : gravitational constant g , is the pulling and the cohesive force on all the quantized energy structures which communicates with everything due to the periodic excitation on all spaces.[84] the newton`s laws issues for the masses and also the same to electrons in caves as below , g ≡ g.ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[gl kl ] ≡ 9,8078925* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² the electron being in hydrogen-cave precesses because of the different axis of rotation and nutation`s , from the continuous and immense-communication to the gravity , g . electron-spin is the angular-momentum-vector b̅ and rotates according to equation db dt = [u̅b̅] = u b.[k̅k̅`] in gravitational potential ug = [ mg] .s.cos θ = s q . [k̅k̅`] so the change of angular momentum b̅ is → db dt = u = s.q b = s.q j3.w3. ← and from the 1-degree equation of motion u , become �̈� + w ² u = 0 , with solution a period of nutation t = 2π u = 2π.j3w sq , and its natural frequency f n = sq 2π.j3w ….(1) with total energy of nutation e n = j1 2 [w1² + w2²] + j3 2 w3² ….(2) [70] . for material-points , the chains of spins due to periodic excitation [↔] is created in orbit a magnetic field due to lrc-circuit and which is tuning to the critical quantum critical-state 𝐠𝐆 . the light velocity vector �̅� = �̅� , by acting on cave , r = 𝐋 𝐏 , finds the impedance m g , and becomes the centrifugal-force 𝐅𝐠 of cave equal to gravity g . the chains of spins for the , one-way pointy vibrating , is the resonance – frequency fr = (1+ √5 ] ) .σ 4πr = σ 𝛷 2πr of vectors v̅ and b̅ , of the-stationary-photon-cave , where b̅ ≡ s̅ ≡ spin , i.e. the fr is common to both . the moving electron in orbit of charge �̅� ≡ ⊝ , energy e with the orbit -velocity vector �̅� =√ 2 𝑚 [e − { 𝐤 𝐫 + 𝐋 𝟐 𝟐𝒎 𝒓𝟐 }] which creates in orbit , r , the varying and perpendicular magnetic ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 44 the fundamental particles origination mechanism in mfmf pns caves . field �̅� , which in time-turn creates an electric-field �̅� ⊥ �̅� , with the resultant force f acting on electron velocity �̅� and which is composed of the 𝐕𝐩 , perpendicular to the magnetic-circles o ⊥ b , and 𝐕 𝐯 , parallel to the magnetic-field-vector |�̅�| , is tending in such way that l ≡ s ≡ spin . the resulting motion of electron in trajectory is the helical motion and work produced during this motion . the conservation of energy exist in orbit between protons and electrons [p↔ e] therefore energy as frequency exists in orbit only , and differs of those of energylevels. since frequency 𝐟 𝐍 = 𝐟𝐑 = 2, 839845.𝟏𝟎𝟏𝟎 s−1 and exists in all atoms , due to the hydrogen first cave ,then this 𝐟𝐍 is the resonance frequency 𝐟𝐑 between all atoms and molecules. it was proved that → since fr = σ 𝛷 2πr and related to hydrogen cave r and stress σ only ← it is the unique common and neutral-monad , in all chemistry structures and composites . d-2..energy from equation e= h.𝐟 𝐍 = 6.62607.10−34 . 2,8398447.1010 = 18,817009.10−24 j / (1,6022. 10−19 ) = 1,174463 .10−4 ev , and conserved as thermal-energy 𝐄 𝐓 in kilo cal and which is continually increasing as , e t = 18,817009. 10−24 j / [(4,19. 103 ) = kcal] = 4,49093. 10−27 kcal . since 1ev = 96,363636 kj/mol thermal energy e t = 4,49093. 10−27 kcal / [96,363636.kcal/mol=1ev] = e t = 4,660399.10−29 ev. this happens because of the closed energy orbit-rims r , constant light velocity c , and from spin equation s = r m c . taking into consideration the thermal energy of a photon when it is pressing the surface of 1 m2 for 60s then , 𝐄 𝐏 = 20 kcal = 20.(4,19. 103 ) = 8,3777.104 j , and the ratio , [ e t / e p] = 4,49093. 10−27/20 = 2,24546.10−28 , which is a quantity not detected . the hydrogen caves created in the sun 1 million years-ago =106.365.24.3600 =3,1536.1013 sec ,is accumulated a thermal-energy of a magnitude e therm= 3,1536.1013.4,49093.10−27 = 1,41626.𝟏𝟎−𝟏𝟑 kcal , i.e. the stationary hydrogen-wave–cave needs 1quadrillion years to conserve 1-kcal thermal-energy . for half-frequency fr / 2 =1,4199223.1010 s−1, the kinetic energy is zero and the potential-energy is u = e = h f = 6,62606957.10−34 . 1,4199223.1010 = 9,4085039.10−24 j / (1,6022.10−19 ev) = 5,8722405.10−6 ev , which agree with the bohr-magneton . remarks : 1.. the real shear stress that exists in the force – carriers . from the static theory of equivalent tension (moore's cycle) is (1) = σ = √(𝛔𝟏 − 𝛔𝟐)² + 𝟒 𝛕𝟏𝟐 , and σ1,2 = (σ1+σ2 )/2 ± (½)√(σ1 − σ2)² + 4 τyz² , instead of the classical case which is σ = 2.τ max = √σ1 2 + 4. τ ² , . (2) . because nature always follows the uniform tension so it follows the (1) case. where exist the equation σ = √(𝛔𝟏 − 𝛔𝟐)² + 𝟒𝛕𝟏𝟐 , and σ 1,2 = (σ1+σ2) /2 ± (½)√(σ1 − σ2)² + 4τyz² = σ 2 [√5+1 ] = σ.[ √𝟓+𝟏 𝟐 ] , which is the golden ratio for stresses . 2.. the bernoulli astute about stresses-linear-distribution is equallytrue for bending too. 3.. in the case of concentrated stresses (such as closed or open conductors) ,the doubling of the main stresses observation is true , and this because of the golden ratio pattern. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 45 the fundamental particles origination mechanism in mfmf pns caves . 1d.. the spaces and the energy states : in articles [87-89 ] is elucidated the duality-photon as particle and as wave . the photon`s charge �̅� photon = v.σφ 2πr , transfers information`s and energy-storage in nature . in articles [87-88-89-90 ] are elucidated the gravitational constant g , the stresses σ , the light-velocity �̅� , the gravity �̅� , the hydrogen cave h , the electron �̅� , the electron-charge �̅� 𝐞 , the weak-forces �̅� 𝐅 , the strong – forces 𝐐𝐓 , and the uniform-resonance frequencies 𝐟 𝐑 , for atoms structure . in articles [91] is shown how ,the three circularly-placed physical-charged breakages on the-three spaces [⊕  ⊝] , or on the 3-extreme -triangles of stplmechanism which is the -thrust for the-origination of all the cosmic particles in linear & rotational conductors frequencies , either into the atoms or to their compounds [91] . 9a.. the flow plan of the → space energy ← universe . 1.. it was shown in [9-18] what is primary-neutral-space [pns] as well as what is infinity [15] , and rotational energy ≡ λ ≡ torque ≡ momentum [22] so [pns] → [a𝐏𝐀 ↔ 𝐏𝐁 b] ≡ work w =|ep |= [ |λ|. + λ x ] → w = ∫ p.ds = 0 → and so exists time t = ds / v = 0 . the cause is , because primary point , a , is nothing and then is quantized as → an point b (which is following the principle of virtual displacements w = ∫ p.ds = 0 = force x displacement = energy x space , and according to the ancient greek philosopher anaximander [ the non-existent ( i.e. the point a ) , exist when is done , it occurs as ( point b ) i.e. ≡ [ το μη ον , ον γίγνεσθαι ] = [ το τίποτα υπάρχει όταν αυτό γίνεται ] time is not existing because ds = ∞ . 0 = constant [21-22] . the relative range is the displacement ab → which is the monad-space as ab = 0 → k → the infinity anti-space. 2.. [pns] the energy-quaternion [ |λ|. + λ x ] ≡ �̅� ≡ [ λ ± λ x ] ≡ |�⃡�𝐨| 𝐰. 𝐞^{ [ �⃗⃖� i / √ λ' �⃗⃖� ] . [arc cos (w|λ|/2 . |√ �⃡� `𝐨. �⃡�𝐨|] } which is the beyond-gravity forced field . [25] . as in case (1) → there is no change of ds → so time t = ds / v = 0 . the cause is the lever arm moment of the primary forces and is quantised as the torque , < the torque modelling of microscopic description > , and later the momentum [19] . the relative range of the infinite points is the displacement ab = �⃡� to infinity ∞ . 3.. [pns] → the temperature-quaternion[ λ , ± λ x ] = z o = λ = n.r.t / v( λ = c) → the gas equation → no change of ds → so time t = 0 . the cause is the heat causing vibration on molecules and is quantised as → intensity ( pressure σ = 𝟐𝛑𝐫.𝐟 𝚽 = 𝟐𝛑𝐫.𝐜 𝚽.𝛌 = ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 46 the fundamental particles origination mechanism in mfmf pns caves . 𝟐𝛑𝐫.𝛔 𝛌 ) between them . the relative range of the infinite points in displacement is ab = λ = 𝟐𝛑𝐫 which is the wavelength λ. 3.a → k1 ≡ z = |𝑧 0| . e i. ( 9.π /2 ).10 = the energy under planck length , the tank cavity of gravity where �̅� 𝐄 = 0 and et = λ.v b + λ x v.�̅� and is the accelerating removing , rotating energy λ to v�̅� , m = 0 and et = λ.v.b + λ x v.�̅� and it is the linearly removing , energy λ towards v b , so no change of ds → therefore time t = 0 . the cause is the high heat conservational balanced tank of gravity and is quantised as the fundamental particles ( bosons and fermions) . [30] the relative range is of the infinite points in displacement |ab| to infinity . 3.b → k2 ≡ z = |𝑧 0|.𝐞− 𝐢( 𝟓𝛑 𝟐 )𝟏𝟎 = the energy in planck length → ∞ changes of ds → so time t = t . cause are the infinite changes of space and is quantised as → matter , energy and the existent as caves . the material-points occupy a relative range in planck`s length as in fig-1 3.c. → k3 ≡ z = |𝑧 0 =λ|. e i. ( π / 2 ) . 10 = the black hole temperature , balanced tank energy length , where pv = n.r.t and ( pa = wd = σ.t 4 ) → no change of ds → so time is t = constant . cause is the very high heat causing vibration on molecules and is quantized as → intensity = ( pressure = fd = c. �̅� = ± co .w.[√ a² x²] ) i.e. cause → (constant co) → quantized as new monad and relative range is infinity , or the meter of space-energychanges is time t , which exists in k2 quantized-region only. from quaternion equation �̅� = [s  �̅�. i] = [s² ±�̅�.s²] = [ �̅�↔�̅� ] = [ +|�̅�|² ↔ |�̅�|² ] is seen that motion , +|�̅�|² at point a , is transferred as motion |�̅�|² at point b . this issues from quaternion equation �̅� = [s  �̅�. i] = [s² ±�̅�.s²] = [�̅�↔�̅�] = [ +|�̅�|² ↔ |�̅�|² ]. fig -1 i.e. the vectors-sum of , n , vectors , either on the same linear-direction or , on non linear-directions results on the – resultant vector �⃗⃖� of the linear-vibration and between the quaternion edge-poinds [ �̅�↔�̅� ] . conclusions –1: energy in primary-neutral-space [pns] and , in under planck1s length ≡ the tank cavity of gravity is an standing wave with no-time existing as t = ds / v = 0 . conclusions –2: since in primary-neutral-space [pns] and , in under planck1s length ≡ the tank cavity of gravity existing as t = ds / v = 0 . the only meter of changes is the period of rotation of the [⊝ ↻ ⊕] as in the next figure-11 . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 47 the fundamental particles origination mechanism in mfmf pns caves . figure –11-the rotation of opposites [⊝ ↻ ⊕] and the how and why spin is equal to 1 , 𝟏 𝟐 in (1). the glue-bond-pair of opposites [⊝ σ ⊕] of the rectilinear motion for both → great circles and the small circles , creates on the stress-common-curve rotation of circle of radius r , the velocity v̅ = w.r = 2π t .r = 2πr. f = [ σ 2 ].(1+√5) = ± σ.φ , and the frequency f = (1+ √5 ] ) . σ 4πr = σ φ 2πr , of a period t = 4πr σ(1+√5) = 2πr σ.φ and , ± σ , where then exist two equal and opposite forces , → the centripetal , fp , and the centrifugal , ff force . the produced work as energy is → e = h f = �̅� = momentum = r m v = r [ 𝜋 𝑟² 2 ] w r = [ 𝜋 r4 2 ]w = [ 𝜋 r4 2 ] 2πf = π². r4.f = �̅� , where f = (1+ √5 ).σ 4πr = σ.φ 2πr , meaning that the stress pointy rotation creates the angular – momentum and momentum as , �̅� , �̅� , in the zero wave-note . [70] the glue-bond pair of opposites [⊝ ⊕] in the → left direction of small circles creates a rotation on circle of radius , r , with velocity v = w.2r = 2π t .2r = 4πr. f = [ σ 2 ].(1+√5) , where frequency f = (1+ √5 ] ).σ 8πr = σ φ 2πr , period t = 8πr σ(1+√5) = 2π(2r) σ.φ , where ± σ are the two equal and opposite centripetal ,fp , centrifugal ,ff force , with energy → e = h.f = h.σφ 2πr , in one wavenote. the glue-bond pair of opposites [⊝ ⊕] in the → right direction of small circles creates a rotation on circle of radius , r = 2 r , with velocity v = w.2r = 2π t .2r = 4πr .f = [ σ 2 ].(1+√5) , where frequency f = (1+ √5 ] ) . σ 8πr , period t = 8πr σ(1+√5) = 4πr σ.φ and the ± σ are the two equal and opposite stresses , or centripetal , fp, centrifugal , ff force . energy is → e = h f = (1+ √5 ] ) .σh 8πr = σφh 4πr , in the one wave note . in (2) the hollow-cone of 90ᵒ , between angular-momentum-vector �̅� and angular-velocity vector �̅� when illuminated by a circularly polarized light beam , then any changes in spin ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 48 the fundamental particles origination mechanism in mfmf pns caves . ≡ �̅� of the polhode cone pot and the exchange of linear and angular momentum between electromagnetic fields and the material media are shown as the profiles of the phase and the angular-velocity-vector in the potcone cross-sectional plane , i.e. the space can be eternally twisted but cannot disappear . [70] measurements of physical-properties such as position , momentum , spin and polarization , performed on entanglement particles gives rise to seemingly paradoxical effects considering systems as a whole and the erp paradox . the answer is given in figure 1-(the right-2) where is shown each property of the material-point . the spins [⊝ ↻ ⊕] presuppose [⊝ , ⊕] material-points which are the caves in pns space . conclusions the energy in spins , presuppose that the caves to be in [pns] space and in the under planck1s length . the ± spin is created from the opposite centripetal , 𝐅𝐩 , centrifugal , 𝐅𝐟 , forces . the energy e is initially absorbed in [pns] and in all quantized regions , either as momentum (mv) or as rotational momentum (λ = r. mv) . from momentum relation �̅� = m r v = m r² w = m r² (2πf) = j.w 2 = [ πr⁴ 4 ].[2πf] = π ²𝐫⁴f 2 , then the mass of the elementary particles is → m = 2π³𝐫⁴,f ² r²2πf = 𝛑²𝐫² 𝟏 f = [ 𝛑𝐫.𝐯 𝟐 ]v , or 𝐦 𝐄𝐏 = 𝛑.𝐫² 𝟒 = π.(1,07.10−7)² 4 = 8,992023.10−15 kg , in the min planck`s cave 𝐝 𝐏𝐜 = 1,07. 10−7 m → which agree with the diffraction energy mechanism for all space levels of quantization and which are the energy particles only ,where for base e = 2,71828 and k = 0 → 𝐋 𝐯= e^i.( ± π/2).b the space is e^ (-15,7079) = 1,78118 .10 ̄ 7 m since velocity v = s / t = distance / time then for constant s , then v is also constant and t = t , the period of vibration and for any constant momentum λ = λ2 m2 v²2 / 2 = λ2 m2 [wλ2 ] ² = 4π². m2 . λ³2 .f ² = [4π². m2 . λ³2 ] f ² = co. f ² = λ , i.e. in all rotational systems time t is constant and equal to the period of rotation which is the meter of changes . for monads in k2 ≡ pns region et = [λ +λ x ] = [λ.m+λxm ] = √ m. v²e + [λ. vb + λ 𝐱 vb ]² = = √ m. v²e + [λ. vb + s ² ] ² = h . f = h / t = h .v / λ , i.e. in every monad issues λ.mv/2 = h f = λ λ and kinetic-energy λ = h f / λ = co.f where the constant 𝐂𝐨 = e/ f ² a gauge magnitude depended on the angular velocity �̅� , the velocity �̅� , and then wavelength λ is conserved as momentum (mv) or angular momentum ( λ = r. mv̅ = m. wλ²/4) or both . considering oscillatory motion as the simplest case of energy dissipation , and this because any other motion is not differently conserved , then the work embodied in dipole is an < spring-mass system with viscous dumping > with co variants , energy e , mass m , velocity �̅� = �̅�e = ẋ wavelength λ , and then energy dissipation is the damping force equal f d = c ẋ where c is a constant and ẋ the velocity. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 49 the fundamental particles origination mechanism in mfmf pns caves . following the steady-state of displacement and velocity then → x = a. sin(wt θ) ← and ẋ = w.a .cos(wt-θ) = w x , where the energy dissipated per cycle is the executed work which is , w d = f d .dx = ∲ c x̅ dx = ∲ c x̅ ²dt = π c w a² = (π/4) c wλ² = [π²/2] c. f . λ² → the energy dissipated per cycle wd by the damping force 𝐅 𝐝 , is mapped by writing velocity ẋ in the form ẋ = wa.cos(wt θ) = ±wa.[√1 − sin²(wt − θ) ] =  w . [ √a² − x² ] where then the dumping force f d = c. ẋ = ± co .w. [ √a² − x²] and by rearranging [ cos²(wt-θ) +sin²(wt-θ) =1] then is fd ² / [ co .w. a ] ² + x² / a² = 1 , i.e. energy = motion is mapped as an ellipse with 𝐅 𝐝 ,x mapped along the vertical and horizontal axes respectively and the energy dissipated per cycle is given by the area enclosed by the above ellipse . the total energy e = λ2 . λ which is embodied in monad ab̅̅ ̅̅ is moving as an ellipsoid in the configuration of co variants λ , m , v = �̅�e , as kinetic ( energy of motion ω̅ ) and potential energy (the stored energy in λ , m , v) by rotation and displacement , on the principal axis (through the centre of monad ) , which is mapped out , as in solid material configuration by the nib of the rotational-vector (ω̅ ) = δ r̅ c) = [ v̅ c + w̅ . r̅ n] . δt , as the inertia-ellipsoid [ the poinsot's ellipsoid of construction ] in an absolute frame which instantaneously rotates around vector axis w̅ , θ , with the constant polar distance w̅.fd/|fd| and the constant angles ϑs , ϑb , traced on reference cone [body frame] and on [ absolute frame ] cone , which are rolling around the common axis of , w̅ vector , without slipping , and if (ω̅ ) = fe , is the diagonal of the energy cuboid with the three dimensions a̅ , b̅ , c̅ then follow pythagoras conservation law , with the three magnitudes [j,e,b ] of energy state following → cuboidal , plane , or the linear diagonal direction . [ 25 26 ]. the transportation of energy on edge points happens in material geometry , while the transportation of the edge-points as , positions , exists in vector geometry only. since in vibrations the deformation of the spring is an equilibrium position in ∆ , and the spring-forse k ∆ = w = m g = gravitational force w acting on mass m , then exists → m �̈� = σf = w = k [ ∆ + x] ….(1) , and because k ∆ = w we obtain w ²n = k/m.. (2) and (1) becomes → �̈� + 𝐰 ²𝐧 = 0 ,…(3) ← i.e. the transportation of energy is a harmonic vibration in cave ∆ , means the wave pattern . it is shown [92] that in pns-space the two equal and opposite stresses ± σ , are the two centripetal , 𝐅𝐩 , centrifugal , 𝐅𝐟 force where energy is → e = h f = (1+ √5 ] ) .σh 8πr = σφh 4πr , in the one wave note .and the frequency f = (1+ √5 ] ) . σ 8πr , period t = 8πr σ(1+√5) = 4πr σ.φ . 10.a.. the e-spaces and the energy regular polygons . a. the geometrical solution : a.. the even and odd n-polygons [ 80 ] : ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 50 the fundamental particles origination mechanism in mfmf pns caves . figure –12aan even and an odd n-polygon in circle o ,oa with diameters , ak a2k , passing from point a2k , as vertex (apex) of the polygone , and diameters , ak+2 m1 to be perpendicular to side a1a2 . the projections of the heights 𝒉 𝒏 . let be the n-polygon a1 , a2 , a3 , ak , ak+1 , ak+2 , a2k , in circle (o , oa1) , (e) a straight line not intersecting the circle d1 , d2 , d2k , the heights of the vertices to (e) line , h1 , h2 , h2k+1 , the heights of the midpoints mk mk+1 of the sides to (e) line φ h−λ , , the angle of the vertices and the ( ⏊ ) semi -sides , and ok = h , the height from the center o to (e) line . to proof : in any n polygon , the sum , σ = σ (h) , of the heights , d1 , d2 , d2k , of the vertices a1 , a2 , a3 , ak , ak+1 , ak+2 , a2k , where n = 2k , from any straight line (e) is equal to σ = σ (h) = n . ok = n . h . the geometrical proof in [ 80] remarks : 1.. in the system of regular polygons the , interior angles ( w ) and gradient ( φ ) , heights ( h ) and their differences , δh , – summation and subtraction of heights are interconnected and intertwined at the common circle [ a , δh = 𝐡 𝐀𝐡 𝐁 ] producing the common ( n+1 ) , odd – regular polygon . this property allows to refer the regular polygons to the tangents of any triangle a,b,c, as it is the 8..[a]  { stpl } mechanisms : the six triple points line mechanism . i.e. the fundamental particles origination happens from the electric and magnetic field of the 3 conductors on 2 anti-parallel-strands , fig– 9a 2.. 𝐬𝐢𝐧 𝛗 = ( 𝐡 𝛗 𝛌 𝛗 ) , is the harmonic mean between [ 𝐡 𝐀 𝐡 𝐁 ] , [ 𝟏 𝟐 (𝐫 𝐚 − 𝐫 𝐛) ] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 51 the fundamental particles origination mechanism in mfmf pns caves . the force harmonic vibration & the unbalance vibration system : figure –12bthe complex frequency response from the vector-force-polygon is seen that the terms of the forced harmonic vibration , ( m ẍ + cẋ + k x ) = 𝐅 𝟎 .sin(w t) , are perpendicular to the y-y axis .when the force changes to = 𝐅 𝟎 .cos(w t) , then the terms are perpendicular to the x-x axis , i.e. vector-force is 𝐅 𝟎 .[ cos(w t) + sin(w t) ] = 𝐅 𝟎 .e i.[w t] the rotation unbalance vibration system : for an unbalance-rotational system , the source of vibration excitation is considered an spring mass system constrained to move in the vertical dirtection and is excited by an unbalanced rotation. the unbalance is represented by an eccentric mass m ,with eccentricity e , and rotates with angular velocity w , on the nonrotating mass m m , with displacement x + | 𝐞 √𝟐(𝐧) | , the equation of motion is , [ m-m] ẍ + m d² dt² [x +| e √2(n) |] + c ẋ + kx = 0 , which can be rearranged to → m ẍ + c ẋ + k x = m | 𝐞 √𝟐(𝐧) | ← …... (c-2) . as in fig-12b-2. by tuning the system to the frequency of the exciting force fo= 1 such that w ² = k 2 m 2 , then system acts as an vibrator-absorber and reduces the motion of the main mass m ≡ m16 ≡ m1 and m 8 ,4 ,2 ≡ m2 ≡ m , to zero . making the substitution , w11² = k 1 m1 , w22² = k 2 m 2 , and assuming the motion to be harmonic , then the steady-state solution is , xs = m | e √2(n) | w ² √(k−mw2)2+ [cw]2 , and tanφ = c w k−mw ² and so the reduced to non-dimensional form equation for amplitudes is , x s = | m e m√2(n) |.| [ 𝐰 w n. ] ² √(1−[ 𝐰 w n. ] 2) 2 + [2𝜁 𝐰 w n. ] 2| , and tanφ = 2 ζ .[ 𝑤 w n ] 1−( 𝐰 wn )² …... (c-3) which is identical with , m ẍ + c ẋ + k x = fo.sin wt ….. (b-4) for steadystate solution . the resonance frequency 𝐟 𝟏 = w1 2π. = 1 2π. √ 𝑘 𝑚 − 𝑥²1 2m². , from (c-3) , a). choosing the damping factor-values ζ and the eccentric weight exciter e , is possible to define the resonant amplitude x = 1 2 ζ. [ m e m ] , and by altering the eccentric mass-m the prober x = m e m . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 52 the fundamental particles origination mechanism in mfmf pns caves . b). above un balance happens in rotating machines or wheels , which can be repaired by adding or detaching ± masses , or the opposite ∓ detaching – adding masses. 1). for small values of , 𝐰 wn << 1 , both the inertia m.w².x , and damping forces c w x , are small which results in a small phase angle φ . the magnitude of the impressed force f0 , is then nearly equal to the spring-force , k x , as in fig.12.b-2 2). for value , 𝐰 wn = 1 , the phase angle φ = 90 °, as in diagram .the inertia force mw²x , is larger and balanced by the spring force k x . whereas the impressed force f0 is >> the damping force . the amplitude at resonance is x = f 0 𝒄.wn = f 0 𝟐𝛇 𝐤 , or from diagram. 3). for large values , 𝐰 wn >> 1 , the phase angle approaches φ = 180 ° , and the impressed force f0 , is expended almost in overcoming the large inertia force , mw²x . a. the natural analogous of the energy motion through the paths . the propagation path of vibrations & the vibration-absorber . figure –12c the electromagnetic-wave {�̅�. fn̅ + �̅�.𝐟𝐧 } into the atoms-bracket-axes-hook. the process of the photo elastic fringe – pattern [96a] in-1 the unloaded beam ( f = 0 ) is the length ab̅̅ ̅̅ . in-2 the loaded beam ( f = f ) is the deflection-curve ab⏟ dh , the strain , where exist , a.. the shape of the beam is the space , the strain , the figure of the system , b.. the shape of the line-stresses is the figure-fringe the path of energy . c.. the colure of the figure-fringe . is the motion = the energy the forced vector. in-3 the loaded and deflected-beam are the spaces = the figures of system. the loaded and deflected-beam strain-lines-fringes , are the path of motion , the loaded and deflected-beam colure fringes , are the kind of forced vector, ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 53 the fundamental particles origination mechanism in mfmf pns caves . the complex force – vector harmonic vibration & the vibration-absorber . figure –12dthe complex -force -vector is 𝐅 𝟎 .[ cos(w t) + sin(w t) ] = 𝐅 𝟎 .e i.[w t] . in-1 the terms = ( m ẍ , cẋ , k x ) , of vector force polygon are perpendicular to y-y axis. in case of complex -force -vector the polygon becomes an absorber which reduces the motion from kinetic energy to potential energy (a magnetic field) . in-2 the 2-dof system with vector force such that , w ²11 = k 1 m1 , w ²22 = k 2 m2 , acts as an vibrator absorber , and reduces the motion of the mass m 1 to zero .the two natural frequencies , f 1 , f 2 , are perpendicular [ ⏊ ] each other as the diagrams. this happens for every action on the 8.[ a , b , c , d ] physical mechanisms . in-3 the{ stpl } six triple points line mechanism is formed from three tangents from a ,b ,c spaces on 2r circle . since pascal-desargues lines meet on markos stpl mechanism lines ,then transfer the properties & information ( f 1 , f 2 , ,, f n ) from the complex-vector-absorber = 𝟐𝐑̅̅ ̅̅ , to the vector-force-polygons remarks : 1.. the n-knots figures is any n-dof system , which characteristic equation results to an algebraic equation of 3-n degree , with main equation of motion [ λ.m + k ] x = 0 2.. the n-knots figures exist in figures where the vector -force polygons have the same number n , as the in figures . in (3) the circularlycharged breakages are the three3 elements [σ = n . h]  [s² = ⊕ , 2s²= , s² = ⊝] which formulate the 3-knots figures. this {stpl}-mechanism of 3 elements ( n=3 ) determinates the mould which denotes the number n , of the spaces for elementary-particles vector. 3.. the case of {tcic}≡ [ σ = n . h ]  8 x ① which is the physical mechanism , gives the n-masses of atom`s &compounds determining the atoms complex-vector-response. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 54 the fundamental particles origination mechanism in mfmf pns caves . b… general : 1.b. the spaces and the energy states . 1…the objective reality is composed of two elements , that of distance . the space ds , and that of motion , or else the called energy , and it is the content of all sciences [32] 2…the euclidean-geometry describes the space only but not the energy as is motion [48] 3...the solution of all the unsolved-ancient-greek-problems [50] opened the way to the material-geometry [54] which incorporates the motion → in-space as (⊕↔⊝) ≡ 0,[61] the space exists in energy caves as energy-quantum-quantities [39] , while motion or energy exist in caves as an confined stationary-wave which is either static or an moving energy-storage or an energy-box [68] . the any-two opposites (+) , (-) exist in nature and are found everywhere from → zero-point (0) ≡ [+ = -] or as (+ a) + (a) = 0 in e-geometry and → [0] = [⊕↔⊝] ≡ f n , [⊕←ds→⊝] ≡ ds in material-geometry , to the aperon , ± ∞ where → the space ≡ ds , is the nodes distance , and motion exist as the vibration type ← [52] 4..the electron-nutation in hydrogen cave produces energy due to g effect , as an minimum frequency fn ≡ 2,8398447. 1010 h and thus exists in all atoms . this energy in hydrogen cave is an electrical -magnetic conductor, which is the pin of atom-plug and enters into the other atom-sockets , which are the orbit-bracket–hooks . these are the hands of atoms , i.e. → the atoms-plug , placing their pins into the other atoms drains ≡ holes , and so is done that what we say the bonding of atoms . 5.. because of the resonance frequency between all atoms , and because of the common hydrogen , so atoms bond by pinning and form molecules and all compounds in this cosmos . this is possible because the periodic-system of atoms is now modulated into 4 new categories with only-one newcriterion which is the number of (±) energy conductors in atoms , related to their stability . 6..the first solid (4 points not coinciding ) is the tetrahedron , and the first material space for the two elements (+) , (-) , are (4 x 2 = 8 positions for electrons ) the-cube , i.e. the tetrahedron is into the cube and into its ex sphere , and this because all the circles on tetrahedron exist on sphere . 7.. when 1-8 electrons ⊝ are placed on the cube's vertices and into this onion system , then automatically are formed the 14 , energy-conductors , where issues [⊕ ↔ ⊝] = [ p ↔ e ] = [atom-plug = + electric vector ↔ magnetic vector ≡ drain ≡ hole ] , and are the actions between opposite . with this new modulation [ the ,1 , 2 ,3 , 4 , energyconductors ≡ atoms-four fundamental frequencies] , conductors exist on system and are active + as → the-electric-vectors ← and simultaneously passive ≡ drains (-) as → the-magnetic vectors ≡ holes ← . the zero-state conductors are those whose 8 cube positions are all filled with electrons . the multiple – plug-pins atoms can plug all the less number electrons atoms and can be-plugged from all the greater number electronsatoms .with these four modulations are formed all atoms and their compounds these are the mechanisms of , water , the cells , rna , dna , cancer ,and everything in this energy space universe . 8..in the same way , is that where an 110 volt refrigerator can’t-plug into 220 volt socket and this because are different in construction , the same to atoms-plug which differ to the ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 55 the fundamental particles origination mechanism in mfmf pns caves . number and the length of their pins . hydrogen is the multiple-plug-pins atom which can plug all atoms and itself. 9..the new-atom-structure is consisted of → a) the nucleus in-sphere b) the dynamic electro-magnetic -tetrahedron , c) the n-outercube , d) the orbitals in ex-sphere. 10..in the new-atom-structure , the master-keys resonances are the communications 11.. article [110] is the completion of [72-82] and of physical interpretation of the three constants of nature , that of planck constant 𝐋 𝐏 , newton`s gravitational constant g , and the gravity constant g ,with a rigorous geometrical and mechanical logic . it was shown [33-36] that un-clashed fragments through the center , o , consist the mediumfield material-fragment → [ ± s² ] = [mfmf] ≡ the chaos , the base for all motions ← and gravity which is a force [ i] while the clashed with the constant velocity , c̅ , consist the dark matter [ ± .c̅.s ] and the dark energy [c̅. i] , so declaring that → antimatter-galaxies and antimatter-asteroids can exist only as dark-matter or and dark-energy and not as antimatter light , c . from the velocity breakages [ ± s² = ± (w r)²] , [ i = 2(w r)² ] and the between instress σ , become the waves {the distance ds = r =|aae| which is the work embedded in monads and it is what is vibrated as frequency f } of the material-point with their vibrating equations of motion and after to become the waves . a → the particles with inherent vibration occupying the distances r = ds =positions |aae| the {monad = 𝐎𝐊𝟎 = r} ≡ energy in vacuum spaces are the fundamental particles , as [ ± v̅.s²] → fermions , quarks and leptons , and → [  v̅. i] → the bosons , b → the gravity-antigravity field-energy g without vibration being a stationary-rotating spinning material point m-p. [ ± s² ] → the [mfmf] neutral field ≡ the energy chaos , and their negative energy binder field [ i ] → as ± positions , for �̅� = ± �̅� , then + �̅� = + gravity – spin = + 𝛑.𝐫𝟑𝛔.𝚽 𝟐  the gravity spin �̅� = gravity – spin = 𝛑𝐫𝟑𝛔.𝚽 𝟐  the antigravity spin c → the dark-matter and the dark-energy constituents are as below , [ ±c̅.s² ] → the dark-matter , and the binder gravity-force g̅ ≡ [ i] , [ c̅. i ] → the expanding dark -energy , positive-energy , which both are moving with light velocity , c̅ , causing the universe to grow. from above in , a , and ,c, case → energy as velocity ,v̅, and , c̅ , exists in the discrete monads ,± v̅.s² , and , ± c̅.s² only . 2 , the case of transportation of energy , from chaos to stationary-material-points . dark energy de ≡ [ �̅� . i] (©) , acting  positive-energy , on the five constituents { [ ( i) , (+s²) , (-s²) , (+cs²) , (-cs²) ] } then produces , [ ± s² ] → the [mfmf] field [ ± c̅.s² ] → dm-de field of , dm = dark matter , and df = anti matter = dark energy . [ ±v̅.s² ] → the pns space , particles fermions , and [ i ] → gf ≡ gravity-force in dm-de stationary rotating = field . [ v̅. i ] → the bosons , [ c̅. i ] ≡ the de → dark energy → the travelling-energy with , �̅� , velocity. 𝐜 x (©) [ i ] ≡ gf → the gravity force . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 56 the fundamental particles origination mechanism in mfmf pns caves . in all above issue kepler laws , denoting that macrocosm and microcosm obey the newton`s laws of motion , in all scales , as was proofed . figure 13 vector-geometry follows the geometry rules as the left figure. euclideanvector carry point a to point b as , ab , following the geometry rules , while the material – points ≡ quaternions , carry the motion ≡ energy , from point a , to the point , b , linearly . in both cases the points are nothing possessing zero-magnitude and infinite directions . the two points in e-geometry consist the straight-line-segment possessing |ab| magnitude and the |a→b| directions while the two-points in material-geometry denote two-elements as one (+) and one (-) performing the space ≡ |ab| of motion = energy ≡ [a ↔ b] in |ab| space .the above comparison can bee seen in figure-1also . conclusions – 3 : from above analysis is seen the path which energy as motion follows to propagate . this path is the resultant of the vector-sum , or the quaternion as material–point . the vector-sum-length as a geometrical-resultant-vector is the quaternion`s edge points energy-vibration which transfers the motion from the vibrating-edge-points . thus stress → σ = [ 2πr φ ² ] . f n ← is the way of information and storage in nature . extending above to all planetary systems and to universe is seen that big bang . has never been existed , inversely energy ≡ motion ≡ matter followed the (+) stress-storage , ± σ = [ 2πr φ ² ] . f n , and anti-matter followed the (-) stress-storage and material-geometry. 2.b. the complex numbers & quaternion . 1… de moivre´s formula for complex numbers states that the multiplication of any two complex numbers say z1 , z2 , or [z1 = x1 + i. y1 , z2 = x2 + i. y2] where x = real [z] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 57 the fundamental particles origination mechanism in mfmf pns caves . the real-part and y = imaginary[z] . the imaginary part of , z , is the multiplication of their moduli r1 , r2 , where the moduli , r , is the magnitude [ r = | r | = √x² + y² ] and the addition of their angles , φ1 , φ2 , where φ = arg.(z) = atan2.(y , x) and so , z1 * z2 = ( x1 + i. y1) . ( x2 + i. y2) = r1. r2 [cos(φ1+ φ2) + i . sin(φ1+ φ2) and when z1 = z2 = z , and φ1 = φ2 = φ , then z = x + i y and z . z = z² = r².[cos2φ+i.sin2φ) and for , w , complex numbers becomes the euler`s , zw = rw. [ cos(wφ) + i. sin(wφ)] and so for r = 1 then → zw = 1w. [ cos φ+i. sin φ ] w = [ cos.(wφ) + i sin.(wφ) ] . the n.th root of any number z , is a number b [ √z n = b] such that bn = z and when z is a point on the unit circle r =1 , the first vertex of the polygon where φ = 0 is [ b = (cosφ + i sinφ)] ⁿ = b ⁿ = z = cos(nφ) + i .sin(nφ)= [cos(360/n) + i. sin(360/n)] ⁿ = cos 360 + i.sin360 = 1+ 0.i = 1 i.e. the w-spaces which are the repetition of any unit complex number z (multiplication by itself ) is equivalent to the addition of their angle and the mapping of the regular polygons on circles with unit sides , while the n spaces which are the different roots of unit 1 and are represented by the unit circle and have points z = 1 as one of their vertices , are mapped as these regular polygons inscribed the unit circle , figure-1(1) since also 𝐳𝐰 = 𝐳− 𝐧 and 𝐳− 𝐧 = 𝐳−𝟏/ 𝐧 = 𝐳+ 𝐧 therefore complex numbers are even and odd functions , i.e. are symmetrical about y , axis ( mirror ) and also about the origin . euler`s rotation in 3d space is represented by an axis (vector) and an angle of rotation which is a property of complex number and defined as �̅� = [ s ± �̅�.i ] where s , �̅� are real numbers and i the imaginary part such that i ² = 1 . extending imaginary part to three dimensions , 𝐯𝟏 , i , 𝐯𝟐 , j , 𝐯𝟑 , k → �̅�. i becomes quaternion which has 1 + 3 = 4 degrees of freedom i.e. [ dx – dy – dz – df ] ≡ the space energymode . 2… de moivre´s formula for the n-th roots of quaternion ,where q = k.[ cos.φ+ [ i].sin.φ] is for w = 1/n , 𝐪𝐰 = 𝐤𝐰. cos.wφ + ε . sin.wφ] where , q = z = ± ( x+y.i ) decomposed into its scalar (x) and vector part [y.i] and this because all the inscribed regular polygons in the unit circle have this first vertex at points , 1 , or at ,1 , (for real part φ = 0 , φ = 2π ) and all others at imaginary part where , k = 𝐓 𝐳 = tensor (the length) of vector , z , in euclidean coordinates which is k = 𝐓 𝐳=√𝐱 ² + 𝐲²𝟏 + 𝐲²𝟐 + 𝐲²𝐧 and for imaginary unit vector �̅� (𝐚𝟏, 𝐚𝟐, 𝐚𝟑, 𝐚𝐧,w) the unit vector ,ε, of imaginary part is → ε = (y.i / 𝐓𝐲) = [y. i ] / [𝐓𝐲] = ± ( 𝐲𝟏 . 𝐚𝟏+ 𝐲𝟐 . 𝐚𝟐) / √ 𝐲²𝟏 + 𝐲²𝟐 + 𝐲²𝐧 , the rotation angle 0 < = φ < 2π , φ = ± sin 1 ( 𝐓𝐲 / 𝐓𝐳 ) , cosφ = x / 𝐓 𝐳 , which follow pythagoras theorem for them and for their reciprocal quaternions �̅�` [�̅� . �̅� `= 1] . since also the directional derivative of the scalar field relation y.( 𝐲𝟏 , 𝐲𝟐 , …. 𝐲𝐧 ,,,) in the direction , i , is → i ( 𝐲𝟏 , 𝐲𝟐 …. 𝐲𝐧 ) = 𝐢 𝟏. 𝐲𝟏 + 𝐢 𝟐. 𝐲𝟐 + i.n.𝐲𝐧 and is defined as → i. grad y = 𝐢 𝟏.( 𝝏𝒚 𝝏𝟏 ) + 𝐢 𝟐.( 𝝏𝒚 𝝏𝟐 ) +…= [ i. ].y , which gives the change of field y , in the direction → i , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 58 the fundamental particles origination mechanism in mfmf pns caves . and [i. ] is the single coherent unit , therefore coexistance between spaces ,antispaces and sub-spaces of any monad → 𝐳 = x + y .i = 𝐀𝐁̅̅ ̅̅ , is happening through the general equation which follows 𝐦 [𝐬+𝐯 𝐢] = 𝐪𝐰 = (𝐓𝐪) . [cos.wφ + ε .sin.wφ] , where , m = 𝐥𝐢𝐦(𝟏 + 𝟏/𝐰)𝐰 for w = 1 → ∞ , q = z = ± ( x+y.i ) sinφ = y / √𝐱² + 𝐲² 𝟐 , cos.φ = x / √𝐱 ² + 𝐲 ² 𝟐 , |z| = √𝐱² + 𝐲² 𝟐 𝐓𝐪 = √𝐱 ² + 𝐲²𝟏 + 𝐲²𝟐 + ⋯ . . 𝐲²𝐧 , ty = √ 𝐲²𝟏 + 𝐲²𝟐 +. . … 𝐲²𝐧 , ε = (y.i / 𝐓𝐲) = [y. i ] / [𝐓𝐲] = ± ( 𝐲𝟏.𝐚𝟏+ 𝐲𝟐.𝐚𝟐) / √ 𝐲²𝟏 + 𝐲²𝟐 + ⋯ 𝐲²𝐧 i.e. the spaces , antispaces and sub-spaces of any monad → 𝐳 = x + y.i = 𝐀𝐁̅̅ ̅̅ follows the euclidean geometry (2a) and the energy flow (2b) , as in figure -1-3-21. the vectors-sum of vectors , either on points on the same linear-direction of vectors , or on non-linear-directions , results onto the-resultant-vector �⃗⃖� , of all system-vectors and becomes a linear-vibration between quaternion-edge-points. this remark denotes that quaternions which are composed of real (+charge) and of imaginary part (charge) act only on the resultant-direction end-points carry energy from one edge to anotheredge as in fig – 6 the above property of imaginary-part ≡ energy shows the how elementary particles are originated from the three charges (of the spaces [⊕  ⊝]) , circularly-placed on the-three-triangles of stplmechanism . [93] from planck`s energy e = h.fp = [ 6,6262.10−34 j.s ] x [ [ 3,28393.1015 / s] = 2 ,176.10−18 j , and in ev becomes → [ 2 ,176.10−18] / [1,6.10−19] = 13,6 ev or , energy in gravity-layer g  13,6 ev = h f = h / t = h /√𝐠. 𝐚³. \figure 14 : the action of quaternion �̅� . �̅� = [s + �̅�. i] ² ≡ s² |�̅�| ² + [2.�̅�].|s r|. i ≡ ≡ s² s² + 2.𝐬² and follows pythagoras theorem of the euclidean geometry. 3b…: the stability of space anti-space , in the neutral-space the number of moulds of stationary-states are as , 2n=1−4 points ≡ 2 ,4 ,8 ,16 ≡ 4–types i.e. monads are formed from the first 4moulds as → the 1-point which is everywhere, ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 59 the fundamental particles origination mechanism in mfmf pns caves . the 2points which form vectors , the 3points which form the planes , the 4points which form the volumes . for n > 4 as this is the atom structure, follows the sustainability of the spaces and anti-spaces . since space-energy structures follow the m-geometry stability these follow the mould of spaces 2 n = 3 = 8 elements which is the-cube as was shown before for the atoms structure fig-5-28 . the atoms-bonding follows → space ≡ electric-field & anti-space ≡ magnetic-field ← 4b… the quantum interference of the duality-photon : quantum interference is the phenomenon when , a photon interferes with itself .[92e]. this proposition is right when photon is considered having a twin-existance , [101]-3c.p-21. the problem is to determine the expression of the power developed by gravitational force. from dual-photon-equation , v̅.[ fn̅ + fn ] ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. b̅n + | c ̅| fn ,when c = 0 or c = c⃖ , then photon motion becomes → | 𝐯 𝛑².𝐫⁴𝐧 | . b̅n ← ≡ spin . i.e. the stationary photons exist as their spin only. this rest-space is the part | 𝐯 𝛑².𝐫⁴𝐧 |. b̅n and its communication code to be the term | 𝐯 𝛑².𝐫⁴𝐧 |=| 𝐰 𝐫 𝛑𝟐.𝐫⁴ | = | 𝟐 𝐯 𝟐𝛑 𝟑𝐫 ⁴ | ,the rest space is a stationary wave with fundamental frequency f 1= 1 t 1 = λ̅ c , with an electric wave , that is acting on vector c ̅ = ( i. cx + j. cy ) . it was proved [78] that the newton`s gravitational constant g does not effect on plane surfaces , but on volume surfaces only . this property of , g , means that the only one force which effects on the rest-space in torsional motion , and onto the light velocity vector c ̅, is the gravity force g only. figure 15-the stability of {⊕space [electric wave},{⊝-anti-space [magnetic wave] of sub-space { [⊕↔⊝] ≡ ds ≡ [⊕←ds→⊝] – or –the neutral-space [a e , b e , c e]} : ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 60 the fundamental particles origination mechanism in mfmf pns caves . since c is constant , it is analyzed to , ( cx , cy ) in any two directions x , y where x ⏊ y and y to be the gravity direction on where gravity is acting . the produced work is in y range , s y = cy t 1 2 g 𝑡2 = c sin 𝑎𝑡 1 2 g 𝑡2 , where maximum range is s max−y = cy 2/ 2g = (c sin𝑎𝑡) ² 2 g = [ c ² 2g ] sin² 𝑎 , happening at angle a = 90° and where sin² 𝑎 = 1 . since at point (1) the velocity v y = 0 , the motion in 0 -1 , may be considered as falling in linear motion and so issues v y = g t or time t = v y g = (c sin𝑎𝑡) g , as halve period with total period → t = 2 (c sin𝑎𝑡) g . in figure are seen the 4-point positions , at where change the phase and the work done in each phase of photon`s inner motion . the 4 phases correspond to the wavelength . λ , as λ / 4 . distance with 90° = 𝜋 2 to be the change of phase . this means that the produced work in prior phase area as the golden – ratio pattern is stored in the next and perpendicular phase-area . the total distance for doing work is the wavelength λ , in total period t as λ = cx t = c .cos 𝑎 .t or λ = c . cos 𝑎 . 2 (c sin𝑎𝑡) g = [ 2 c ² g ] sin 𝑎 , cos 𝑎 = [ 𝐜 ² 𝐠 ] 𝐬𝐢𝐧 𝟐𝒂 , ……(1) since from trigonometry exists , sin 2𝑎 = 2 sin 𝑎 . cos 𝑎 . the maximum wavelength λ happens when 𝐬𝐢𝐧𝟐𝒂 =1 or 2a = 90 ° and a = 45 ° , i.e. →the total motion in the wavelength , λ , corresponds to the angle a = 45 °, meaning that the motion of space , the electric-field , and anti-space , the magnetic-field , equilibrium as an bellow . the two wings in 90° type simultaneously oscillate to its 45 ° bisector -axis , and the produced work from the electric-field-area is stored into the magnetic-field area . the common axis of the two perpendicular fields , carry the g-ratio surfaces , runs with the same velocity c ̅ to the direction of the inner motion or, from 90°→ 45° , the created work w = c y g , is symmetrically stored from 0°→ 45°. the created work in [ 0-1 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 1`-2 ] = λ / 4 . surface [ 0 -1 -1` ] = ⏊ surface [ 1` -2` -2 ] . the created work in [ 1-2 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 2-3 ` ] = λ / 4 . surface [ 1 -2 -1` ] = ⏊ surface [ 2` -3` -2 ] . the created work in [ 2-3 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 3 `-4 ] = λ / 4 . surface [ 2 -3 -3` ] = ⏊ surface [ 3` -4` -4` ] . the created work in [ 3-4 ] is in → λ / 4 of transverse surface and is stored in the horizontal and perpendicular surface [ 3 `-4 ] = λ / 4 . surface [ 3 -4 -2 ] = ⏊ surface [ 4` -5` -4 ] . since from trigonometry issues sin 2𝜋= sin(180 − 2𝑎) = sιn 2(90 − 𝑎) then (1) becomes . λ = c cos 𝑎 2 (c sin𝑎𝑡) g = [ 2 c ² g ] sin 𝑎 , cos 𝑎 = [ 𝐜 ² 𝐠 ] 𝐬𝐢𝐧 𝟐𝒂 = [ 𝐜 ² 𝐠 ].sin(𝜋 − 2𝑎) = [ 𝐜 ² 𝐠 ].sin( 𝜋 2 − 𝑎) …(2) since 𝐬𝐢𝐧 𝟐𝒂 = sin(180 − 2𝑎) = sin(𝜋 − 2𝑎) , and also 𝐬𝐢𝐧 𝟐𝒂 = sin(180 − 2𝑎) = sin( 𝜋 2 − 𝑎) ……(1a) ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 61 the fundamental particles origination mechanism in mfmf pns caves . from dual-photon-equation , v̅.[ fn̅ + fn ] ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. b̅n + | c ̅| fn ,when c = 0 or c = c⃖ , then photon motion becomes → | 𝐯 𝛑².𝐫⁴𝐧 | . b̅n ← ≡ spin . i.e. the stationary photons exist as their spin only . the rest-space is the part | 𝐯 𝛑².𝐫⁴𝐧 |. b̅n where v = n.π.c , and communication code to be the term | 𝐯 𝛑².𝐫⁴𝐧 |= | n.𝜋𝑐 𝛑².𝐫⁴𝐧 | = | 𝑛.𝑐 𝝅.𝒓⁴ |=| 𝐰 𝐫 𝛑𝟐.𝐫⁴ | = | 𝟐 𝛑.𝐟 𝛑 𝟐𝐫³ | = | 𝟐 .𝐟 (𝛑 𝐫 ³) | with electromagnetic radiation the in monads frequency-f , which is satisfied by the relation | 2rw 2 v |= π /2 , π , 3π /2 .,,, n.[π/2] , and w = n.[ �̅� 2r ]x[ π 2 ] , where n = 1 → ∞ . when two photons strike each other c = c⃖ , or reflect on a wall , then c = 0 , and from spin �̅� = b̅ = π ². r ⁴. f1 its angular velocity w = n.[ �̅� 2r ] x [ π 2 ] = 2π.𝐟𝟏 , where then its fundamental frequency 𝐟𝟏 = n.[ �̅� 𝟐𝐫 ]x[ 𝟏 𝟒 ] = n.[ �̅� 𝟖.𝐫 ] which for n = [ 16.𝑟² v.c ] increases to λ c . r3 and photon-motion becomes → | 𝐯 𝛑².𝐫⁴𝐧 | . b̅n ← ≡ spin , i.e. the stationary photons are spin only and consist an stationary harmonic vibration , (1) – (1a) , with the same period t of the two waves [92-e}. the effective-range of the wavelength λ , for the dual-photon is related to the consistence v x , v y , velocities to any two perpendicular directions of electromagnetic fields e x , e y . the maximum displacement in y , direction is the centrifugal λ x = v² y 2 e y and sisin(φ) λ y = = c2. sin ²(φ) 2 e y , where φ = the phase difference between the two vibrations . since time t , is the same between the two vibrations where then t = v y e y = c.sin(φ). e x and since the period = 2.t then t = 2c.sin(φ). e x …... (2) from the in x → direction velocity v x = c . cos(φ) , and displacements , λ x , λ y , become , λ x = v x t = c. cos(φ){ 2c.sin(φ). e x } = λ x = { 2.𝑐² 𝐸 𝑥 }. sin(φ).cos(φ) = { c ² e x }. sin(2. φ) (3) since in trigonometry issues , sin(2φ) = sin(𝜋 − 2φ) = sin( 𝜋 2 − φ) and in physics e x = g , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 62 the fundamental particles origination mechanism in mfmf pns caves . then → 𝛌 𝐱 = { 𝐜 ² 𝐠 } . 𝐬𝐢𝐧(𝝅 − 𝟐𝛗) , 𝛌 𝐲 = { 𝐜 ² 𝐠 } . 𝐬𝐢𝐧( 𝝅 𝟐 − 𝛗) ← …….(4) since monads are complex quantities as ab ≡ �̅� ≡ x + i. y , then are quaternions where material-points ≡ quaternions , which carry the → motion ≡ energy ← from the two edge-poinds [ �̅�↔�̅� ] , as this is from point a , to point b circularly . since λ x ⏊ λ y , then issues λ y = [ i = √−1 ] * λ x , or equivalent to multiplying the quantities along the x – x axis , by i = √−1 . i.e. photons occupy 1.. the dual-property of particles and waves , 2.. the dual-property of stationary and spin , 3.. the dual-property of simultaneous existence , 4.. as electric and magnetic fields in , 𝛌 𝐱 , 𝛌 𝐲 , 5.. the bellow motion in transverse e-mwaves as is the way to fly , 6.. the transverse e-m waves are ⏊ to the propagation-direction , 7.. the electric effective range is thrusted through g only. [ fig-5 ] . 8.. with the same initial velocity is succeeded the same length-range with two complementary angles . since v y = c. , c.. : the origination of the physical loops ≡ caves 1c.. the gravitation force g , and the kick start : [78-79] figure – 16. the newton`s universal laws in primary-material-point , above , is consisted of the two-primary-opposite-spaces , {+} {-} , poles , with infinite points and parallel-lines such that g is a uniform-pointy–force . from where g , is produced ? → it was prior referred that , periodic excitation between primary space (+) and anti-space (-) may exist only as collision of opposite , so energy captured in box-𝐁𝐏 , planck`s loop , contains these ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 63 the fundamental particles origination mechanism in mfmf pns caves . three constitute elements [(+) ,[↔], (-)] without the inner acceleration but the material– extreme case of periodic acceleration [→ ←] = 0 and which is the reciprocating motion so , above property of primary space exists also and for all primary complexspaces. complex spaces either at , [(+)] or [(-)] , constitutes may be realized , exist , separately because of the → primary-opposites which pull each other ← therefore the initial property of attraction , g , between primary-opposites continue issues and for all the other infinite composite and complex structures as well as in all groups of them . since force g is a uniform-pointy–force , therefore needs a conductor , a layer , a kind of mass , to be spread and act on it as surface-force , a layer which is the base or the conductor spin s , or the stress g , as this is the growing-golden-ratio-pattern φ. because of this stress-layer g , all the energy-structures present → reaction to any motion ← i.e. mass . the gravity acceleration g , produced from the in material-points acceleration is equal to the pressure σ , and is the conductor to all waves as , force g over-layer a → [a ≡ mass * g ] → gσ ≡ g , and stress = g = σ² r².φ² 8 from figure -15(3) is seen that gravitational force is attractive following newton inverse square law, and it is an attractive-force [↔] , on any two opposite-elements [ (+) , (-)] realized through a layer of unit-stresses [σ = g ] , and becomes from an huge magnet in planck`s loop containing the whole universe [89] . g kick-start , is planck`s frequency → fp = 1 / tp = w 2π = √ 1 g.a ³ 2 and in this world the how this frequency can → enter , format and cohesive , to the first or any other energy-cave in planck`s length , e-unit monad , 1 = g .𝐟 ²𝐧 . 𝐚 𝟑 = [ 𝟒𝛑² 𝐆𝐌 ] . 𝐟 ²𝐧 . 𝐚𝟑 , where g = [ 𝟒𝛑² 𝐆𝐌 ] as this was shown in [79] for unit-monad-relation . an energy-rim = cave is a plane surface , an orbit , representing an constant energy becoming from the squared frequency 𝐟𝐧², representing the imaginary part , and r p³ , representing the real – space part of monad as equation 1 = k.𝐟𝐧².𝐫𝐏³ . the stationary energy is spread in one plane as this happens for the stationary-waves in caves , while in propagating energy in two , as are the electromagnetic transverse waves . all these energy-rims consist the quantized-plane-curves as fig-15 . the two different motions of ⊕ space , ⊝ anti-space , in any cave r and in a finite space , the revolving and periodic excitation create an eternal frequency which influence all other spaces , the caves . the minimum quantized-energy is stored in the gravity-layer , g , becoming from the unit energy of the sinus orbit . from equation , g = t ² / a³ , and minimum a then period is t = √g. a³ = √9,8076925. (2,1145016. 10−11)³ = 3,04513.10−16 s , since the unit-work = sine integral =∫ sin t t 𝑡 0 dt =1 , the semi-major axis , a , is a = 2,1145016.10−16 m , where frequency t −1 = fp = 3,28393..1015 /s , corresponds to a sphere-loop in planck`s scale as (4 / 3).π.r³ = 3,96 . 10−32 m³ , and then the energy in this planck`s loop is the minimum quantized . from planck`s energy e = h.fp = [ 6,6262.10−34 j.s ] x [ [ 3,28393.1015 / s] = ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 64 the fundamental particles origination mechanism in mfmf pns caves . 2 ,176.10−18 j , and in ev becomes → [ 2 ,176.10−18] / [1,6.10−19] = 13,6 ev or , energy in gravity-layer g  13,6 ev = h f = h / t = h /√𝐠. 𝐚³. the above quantity of energy consist the hydrogen minimum energy-rim , which becomes from loop-equation a = √𝐓𝟐/𝐠 𝟑 = √ [3,04513.10−16]² 9,80769251 3 = 2,1145016.10−16 m , and it is the energy-cave for the unit energy quantity . from newtonian constant of gravitation relation issues , g = e = h . fn = [ 𝐜.𝐫³ 𝐚³ ].[ gl kl ] = g .k e = g . [ gl kl ] , and since for the first chemical-neutral-material-cave , r , constants gl ,kl are equal to unity i.e. 𝐠𝐋 = 𝐤𝐋 = 1 , then above energy of e = 13,6 ev , in hydrogen-plane-orbit corresponds to the minimum-energy-cave , or → the phys-quantized-energy-structure ← since g pushes → g , on the earth-unit-coefficient , k e , and because is the starting for the first time beginning , of this mechanism then from g = g.[gl kl] ≡ g.[1*1] ≡ → g ← or g = g , meaning that in earth system of gravity , the newton`s gravitational constant g , and gravity g are equal , while in all other relative systems are equal to the proportionality of their local-constant 𝐤 𝐋 . gravity , g , is the minimum energy becoming from the in-storages angular velocity acceleration a = v²/ r ≡ 9, 8076941 . in the material-point . [72] photon is a material-point , box br , with fix-ends inward-cave r , and which is the energy storage 𝐁𝐑, outward-cave-r is an electromagnetic-radiation on wavelength λ = c t = c / fp which em-radiation , carries the box br . universal gravitational constant g = g k related to g , kr , is a force and through the static g becomes the principal stress  σ , or frequency f r which exists in nature as motion in the minimum resonance golden-ratio-frequencies fr = fn = 1 , and this because of the periodic motion , in excitation , where issues the coulomb-dipole law and which coulomb inverse law is as , f = kc [q1.q2 / r²] = kc [⊕→←⊝] /r² = 8 πr(1+√5) [ b r² ] , and coulomb constant kc = 9.109 nm2/c² , i.e. because of the periodic excitation between the , space (+) and anti-space (-) , [⊕→←⊝] exists only collision of opposite where the inner acceleration is equal to zero gravity g , and this because of the material extreme-case of the periodic acceleration {[→←]=0} which is zero , the net force vanishes , issues the coulomb dipole-law where static stress g , becomes from the stationary constant dipole moment �̅� ≡ �̅� ≡ as momentum which is the analogous in the revolving motion where then issues → g = g k = 𝐤𝐄 g = g.𝐤𝐑 𝐠𝐑 = �̅� , 𝐠𝐜 = σ = 𝐅𝐨𝐫𝐜𝐞 𝐀𝐫𝐞𝐚 = 𝐠.𝐌𝐚𝐬𝐬 𝐀𝐫𝐞𝐚 for earth-system mass me = 5,9723.1024 kg and radius re = 6378,137 km = 6,378.106 m , then earth-constant 𝐤𝐄 = r²e me = [6,378.106]² 5,98.1024 = 6,811551810−12 and g = g 𝐤𝐄 = [ 9,8076941] . 6,8116.10−12 = 6 , 68056.𝟏𝟎−𝟏𝟏 m3/ n.s² becoming from g , 𝐤𝐄 only , i.e. g as force pushes → g as stress energy in → 𝐤𝐄 constant . now is proved that , the gravitational constant g is the mechanism or the mould , for the first-kick-start upon this unit-granular-energy-stress-layer , g , to ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 65 the fundamental particles origination mechanism in mfmf pns caves . formulate in that orbit a , into planck`s cave the lightest and the less-energy mass particle of this universe , which is the hydrogen with the minimum quantized-energy of 13,6 ev. the centripetal acceleration in photon cave due to pressure σ ,where issues only rotational energy e r = b̅ . w̅ = 2l= j.w² , and 2e = b f , or → 2.(0,5.m.c²) = m.c² = 0,5.π.𝑟4.𝑐2/4 , then 4. �̅� . f = 𝐜𝟐. 𝐫𝟒 , where the golden ratio frequencies → fn = [ n.σ.φ 2.π.r ] ≡ n.(1+√5 ) .σ 2.𝜋.𝑟 = = 𝐜𝟐. 𝐫𝟒 / 4.�̅� = e h ← i.e. the box 𝐁 𝐏 , or the planck`s cave , carried with light velocity c and pressed into the planck`s loop , as n 𝐜𝟐 = n. c c , becomes the gravity g , which is a stress by which gravitational-force g is communicating as , g g = [ π r v4 2 ].𝑒3 = 9,808238 m/s . gravity is the minimum unit-stress-monad in planck`s scale becoming from a m-point and which is under -planck`s length region . since also acceleration in a material-point ( centrifugal-centripetal ) becomes from the principal stresses  σ , therefore constant g g ≅ ± σ = force area = g.mass area = g k , and for unit-g constant k becomes , k = system area system mass g where → 1 𝐠 = k g = system area system mass and g = k g g , and for the earth g ≡ g.ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[ gl kl ] ≡ 9,808238* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² so g , is the pulling and cohesive force on all the quantized-energy-structures which communicates with everything due to periodic excitation on all spaces . from equation g = g / [gl kl ] , is seen that g becomes from g under local-laws , and also , gravity g , is the unit-energy-path , quanta of gravitation , due to m-point frequency fg ≅ ± 𝜎 from the revolving motion , and also the unit-energy-stress per surface and for the force g , is the permitted path , or the communication conductor on all energy wave structures , in and out planck`s length . remarks : gravitational force ,g, is the pressure which every object in the universe exerts on every other , whether small or big and is equal to fgrav = g.m.m d² = g.𝐑²𝐄 m.m md² = g.𝐑²𝐄 .m d² = mg.𝐑²𝐄 d² , and g = g.𝐑²𝐄 m.m.d² md² = g.[ 𝐑²𝐄 ] m = g . ke , where ke = [ 𝐑²𝐄 ] m , and g = the minimum quantized work as acceleration ≡ 9, 8076941 m/s² ke = the earth local-coefficient between the two constants g , g . for earth (e) → k e = r ²e / m e , for bodies (b) → k b = r ²b / m b , for any body → local gravity g l = ke / kl from relation g = g 𝐤𝐄 = 9,8076941.6,8116.10−12 = 6,68056.𝟏𝟎−𝟏𝟏 m3/ n.s² and from relation 1 = c. r³. 𝐟𝐏² → fp = √1/cr³ and from f = e/h then 𝐄 𝐡 = √1/cr³ or → e = h . √1/c. r³ , e² h² = 1 cr³ → and , e ² = 𝐡² 𝐜 𝐫³ , is an energy relation between c , r . instances : [77] 1.. for earth-system mass me = 5,9723.1024 kg and for area → radius 6378,137 km = 6,378.106 m then the earth-constant 𝐤𝐄 = [6,378.106]² 5,9723.1024 = 6,811551810−12 and g ≡ σ.𝚽³ = g 𝐤𝐄 = 9,8076941* 6,8115518.10−12 = 6,6805616*𝟏𝟎−𝟏𝟏 m³/kg .s² i.e. gravitational-constant g becomes from g , 𝐤𝐄 , and issues g = 6,6805616.10−11 m3 / n.s² ……..….(7) for black-holes-system mass m bh = 4,0.10 52 kg and for the area ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 66 the fundamental particles origination mechanism in mfmf pns caves . radius 3,08.1025 m , then black-hole-constant kbh = [3,08.1025]² 4,0.1052 = 2,3716.10−2 𝐠𝐁𝐇 = k e k bh = 6,81155.10−12 / 2,3716.10−2 = 2, 872.10−10. ge → i.e. for earth-unit-coefficient ke = 6,81155.10−12 → black-hole –unit-coefficient kbh = 2,3716.10−2 m2/kg , and gravity acceleration of a black-hole is , 𝐠𝐁𝐇 = 2,872.10−10. ge , an expected explanation . for all planes issues g = 𝐤𝐄 g = 𝐠 . 𝐤𝐋 𝐠𝐋 , and for black-holes also where , g = g . 𝐤𝐁𝐇 . 𝐠𝐁𝐇 = 9,8076941* 2,3716.10−2 * 2,872.10−10 = 6,6805616*𝟏𝟎−𝟏𝟏 meaning that gravitational constant is the same for all systems . the gravitational force , g , is the pressure which every object in the universe exerts on every other , whether small or big and from newton are equal to → 𝐅𝐠𝐫𝐚𝐯 = 𝐆.𝐌.𝐦 𝐝² = 𝐠.𝐑²𝐄 𝐌.𝐦 𝐌𝐝² = 𝐠.𝐑²𝐄 .𝐦 𝐝² = 𝐦𝐠.𝐑²𝐄 𝐝² , g = 𝐠.𝐑²𝐄 𝐌.𝐦.𝐝² 𝐌𝐝² = 𝐠.[ 𝐑²𝐄 ] 𝐌 = g . 𝐤𝐄 where 𝐤𝐄 = 𝐠.[ 𝐑²𝐄 ] 𝐌 = 𝐠.𝐌 𝐆 g =the minimum quantized work as change of velocity → acceleration ≡ 9,807694 m/s² 𝐤𝐄 = the earth local-coefficient between the two constants , g , g , and 𝐤𝐄.g = g .m for earth (e) → 𝐤 𝐄 = 𝐫 ²𝐄 / 𝐦 𝐄 , for bodies (b) → 𝐤 𝐁 = 𝐫 ²𝐁 / 𝐦 𝐁 , for any body → local gravity 𝐠 𝐋 = 𝐤𝐄 / 𝐤𝐋 2.. coulomb electrical force , 𝐅𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐧 = 𝐤𝐜 𝐐𝟏.𝐐𝟐 𝐝² = [⊕→←⊝] 𝐝² = 𝟖 𝛑𝐫(𝟏+√𝟓) [ 𝐁 𝐫² ] , where coulomb constant 𝐤𝐂 = 9.𝟏𝟎𝟗 nm²/c² 3.. the work done by the electric field to rotate the dipole is w = 𝐅𝐞𝐥𝐞𝐜𝐭𝐫𝐨𝐧 . 𝐄 𝐅𝐢𝐞𝐥𝐝 . 4.. the work done in material point needs a path to exit from the box 𝐟𝐑 = [ 𝐁𝐏𝐇 ] ≡ [ 𝐟𝟏=𝐍 , 𝐟𝟐 , 𝐟𝟑 , 𝐟 𝐑 = 𝐰²𝐍 ] ≡ [e² + h²] , and from where is propagated . resonance-path happens as the force , em-radiation in two directions , which can travel in any closed system , and for solids through cauchy-stress-tensor where the two conveyers e ⊥ b ⊥ r ≡ 𝛔𝟏 ⊥ 𝛔𝟐 ⊥ 𝛔𝟑 , can carry the energy storage r in system [ + 0 ] and change the inner-structure of this system to another or destroy it. in [70] , the work produced in material-point 𝐀𝐁⃖⃗⃗⃗ ⃗ is equal to → w = 2l = �̅� . �̅� = j.w² ← consisting the first-energy-store which is a stationary wave with , n lobes as , 𝐖𝐧(𝐧+𝟏) = [ 𝟒𝝅𝐫²𝐟𝟏 𝟑 ].n.(n+1) and wavelength 𝛌𝐍 = 𝛔.(𝟏+√𝟓) 𝟒𝛑𝐫 = 𝒏 .�̅� 𝟒𝛑𝐫² = 𝟐𝒏 .�̅� 𝐰 of r = 𝒎 .�̅� 𝐪.𝐁 related to m-field b in [75-b] ,the permeable-resonance-path for m-point is → the resonance-frequencies 𝐟𝐑 [s≡ 𝐟𝟏=𝐧 ,𝐟𝟐, 𝐟𝐑 =𝐰²] with 𝐟𝐧= [n (𝟏+√𝟓)𝛔 𝟒𝛑𝐫 = 𝐧𝛔.�̅� 𝟖 𝐫² ] related to spin �̅� , stress 𝛔 and cave r . since the frequency in material-point is → ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 67 the fundamental particles origination mechanism in mfmf pns caves . 𝐟𝐧 = ( 𝐧𝛔 𝟖 𝐫 ² ) .�̅� ≡ [ 1+√5 2 ] 𝛔 𝟐𝛑𝐫 ≡ [ nσ 8 r ² ].b̅ ≡ [ 1+√5 2 ] stress perimeter ≡ [ 1+√5 2 ] force [n] area∗l[m3] ≡ 𝐄 𝐡 , then it occupies the property of the golden-ratio pattern φ , and equation defines that the material point of frequency 𝐟𝐧 , when collide with another material point , or with any other particle or particles then produces another monad as → 1 ≡ a new quaternion and the first continuous to be of the same identity , frequency 𝐟𝐧 , as before and from euler`s , rigid body dynamics work → w = 2l = b̅ . w̅ = j . w² ≡ h. 𝐟𝐧 ← i.e. the frequency of photon , is embodied with the → growing-golden-ratio-pattern φ and it uses the vibrating physical structures , or the same the granular material-instruments , to kick start all of them and everything in this world . the how in [75-b] page 34 . because of the periodic excitation between , space (+) and anti-space (-) , [⊕→←⊝] exists only collision of opposite , where the inner acceleration is equal to zero gravity g and this because of material extreme-case of the periodic acceleration {[→←] = 0} , and which is zero and the net force vanishes , and in where issues the coulomb dipole-law and where acceleration g becomes from the stationary constant dipole moment �̅� ≡ �̅� ≡ momentum and which is the analogous in the revolving motion in where then issues → gravitational force  g = g k = 𝐤𝐄 g = g . 𝐤𝐑 𝐠𝐑 = �̅� . 𝐠𝐜 = σ = 𝐅𝐨𝐫𝐜𝐞 𝐀𝐫𝐞𝐚 = 𝐌𝐚𝐬𝐬 𝐀𝐫𝐞𝐚 g in page-33 [77] , the earth-constant 𝐤𝐄 = r²e me = [6,378.106]² 5,98.1024 = 6,811551810−12 and the gravitational force g = g 𝐤𝐄 = [ 9,8076941] . 6,8116.10−12 = 6,68056.𝟏𝟎−𝟏𝟏 m3/n.s² becoming from g , 𝐤𝐄 only, i.e. g is a force which pushes → g as energy in → 𝐤𝐄 from gravity relation g = g 𝐤𝐄 =9,8076941.6,8116.10−12= 6,68056.𝟏𝟎−𝟏𝟏 m3/ n.s² and from energy-loop relation 1= c. r³. 𝐟𝐏² → fp=√1/cr³ , from f = e / h then f = 𝐄 𝐡 = √1/cr³ or → e = h.√1/c. r³ , and e² h² = 1 cr³ → e ² = h² c r³ , an energy relation between the , c , r . this relation is used for blackholes energy where r , result to zero , [71] the permissible permeable resonance paths for the means , is as below . 1) solids → the normal-mode-vibration system { w² [m] + [k] (x) } = 0 2) liquids → the cauchy stress-tensor as momentum equation { ∇.σ = -∇p +∇.τ } 3) gazes → the combined avogadro`s pressure law { pv = n rt = n.mv²/3 } 4) crystals → the cauchy ellipsoid-stress-tensor where { e⊥b⊥r ≡ σ1⊥ σ2⊥σ3 } 5) molecules → the lattice-crystal-arrangement with their chemical bonds relation 6) atoms → the energy-rims with chemical bonds , ionic , covalent , relation 7) particles → the resultance of the one-dimensional-collision v̅ij= v̅j v̅i = w̅ij. r̅ij ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 68 the fundamental particles origination mechanism in mfmf pns caves . 8) m-points → the resonance-frequencies 𝐟𝐑 [s ≡ f1= n ,f2, f3,fr=w² ] = 𝐟𝐧 = [⊕ ↔ ⊝] = n (1+√5)σ 4πr = n.σ.b̅ 8 r² → i.e. the golden-ratio-pattern , which is related to spin. 9) cave-orbit→ the relation c.l s = lv , light-velocity 3.108 m/s*1.10−42 m cave =3.10−34 m²/s are the cave-energy-plane-rims in atom`s , planet orbits. in vibrations of systems , for orbits issues → w = 4π² . 𝐫³ 𝐓² . �̅� = 4π².r ³. 𝐟²𝐩 . �̅� ← which is the energy in the conic-sections of kepler`s unit orbits . the energy-storages r = n.[ λ 2 ] ≡ wn(n+1)= [ 4𝜋r²f1 3 ].n.(n+1) , are travelling through bodies and follow, lame stress ellipsoid n1² + n2² + n3² = t1² σ1² + t2² σ2² + t3² σ3² =1 on principal stresses σ1, σ2, σ3 , which is the passage through which forces (the em-radiation) travel in any solid either it is in motion or is at rest . laplace`s orbital angular-momentum ei.2πn = 1 and for n = 0 ,1,2 ,3,. n , consist the eigenvalues operator lz which agree with prior resonance-frequencies 𝐟𝐑 [s≡ f1=n , f2, f3,f r = w² ] as wavelengths λ ≡ [f1 , f2…fn= w²] ≡ the n lobes , or → fn = n (1+√5)σ 4πr = nσ.b̅ 8 r² , as the principal-stresses σ , and resonance frequencies 𝐟𝐑 relation , which is energy stored in the mp-lobes. [70] type particles at sites type of bounding-force properties em-radiation ionic : ⊕ , ⊝ ions electrostatic ⊕ ↔⊝ non-conductors infrared molecular : atoms or molecules dipole attraction repulsion non-conductors chemical-bonds . covalent : atoms networkbonds between atoms -non-conductors em-spectrum metallic : atoms ions and electrons attraction conductors e-conduction . the kinetic-energy e k of a moving material-point , as this is the photon , is stored as motion in its storage , r = [ n. λ/2] , with the n frequencies f n = n.f 1 , with the n lobes , and with an fundamental frequency f 1 . from above is seen the passage and the-how emradiation can travel in crystals and which are the cauchy-stress-tensor , where exists e ⊥ b ⊥ r ≡ σ1 ⊥ σ2 ⊥ σ3 , in-where energy propagates along directions without any birefringence , and carries the energy-storage r , which radiation is the conveyer. above procedure can be used in cells , where cells are cases of an birefringence material and the resonance-passage happens as the force , an em-radiation in two directions , can travel in cell through cauchy-stress-tensor where the two conveyers e ⊥ b ⊥ r ≡ σ1 ⊥ σ2 ⊥ σ3 , and can carry the energy-storage , r , in cell , and alter the inner-structure of cell to another desirable property. from inner-velocity equation v = w r = (2π/t).r =2π.f1 r , wavelength λ = ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 69 the fundamental particles origination mechanism in mfmf pns caves . ct=c /f 1, cave r = n.[λ/2], then r = n.(c/2f1) and v =2π.f 1[n.c/2f1] = n.π.c or v= n.π.c ....(4) showing that velocities in lobes are , n.π ,times that of light and for n = 1 then v = π.c , more than three times faster of light velocity. because of the above velocity v , an e field is produced , which produces the 𝛛𝐃/𝛛𝐭 field , which in turn produces the h field which produces the 𝛛𝐁/𝛛𝐭 field and which again produces the e field , i.e. the total em-field regenerates itself as it rotates , and is a phenomenon happening in a propagating plane-wave . its permeable resonance path is impossible in an three-times stronger em-field because issues … (4) 2c.. the united-coulomb-newton law for interactions : for a stationary interaction the coulomb-law-force is 𝐅𝐂 = 𝐤[𝐪𝟏𝐪𝟐] 𝐫² , and for charges 𝐪𝟏, 𝐪𝟐 where constant k = 9,0.109 n.m²/c² , and r , is the distance between the charges. for a stationary interaction newton law for masses m1, m2 is 𝐅𝐍 = 𝐆[𝐦𝟏𝐦𝟐] 𝐫² , and for v̅=r̅ conjugating interaction , energy e = 𝐯² 𝟐 [m1 + m2] = 𝐫² 𝟐 [ f1 g + f2 g ] = 𝐅.𝐫² 𝟐 [ 1 g + 1 g ] = 𝐅.𝐫² 𝟐 = = 𝐅.𝐫² 𝟐 [ 2.g g ² ] = 𝐅𝐍 .𝐫² 𝐠 , where 𝐅𝐍 is newton force, and e is the energy of the field-system.[ g+g g∗g } equating the two forces then k [q1q2] r² = ge r ² , or  k [𝐪𝟏, 𝐪𝟐] = g . e ……..(1) equation (1) relates any two charges q1, q2, with field-energy e , under vector �̅� = �̅� = �̅� 3c.. the origination of light velocity c : gravity g ≡ 𝐓² 𝐚³ ≡ 𝟏 f ²n.𝐚³ ≡ 9, 8076941 → the minimum granular quantized energy , the forces beyond-planck-length are → f = φ³. [⊕ 𝛔 ⊝] = σ x 𝚽 ³ ← (1)-[87] force g acts → on spinning m-points on �̅� through g̅ → on planck`s b̅ → on 𝐠𝐆 , through the resonance frequency 𝐟 𝐑. it was proved that g = σ φ³, and σ = 𝐆 φ³ = 𝐆.𝛔³ c ³ , or → σ ² g = c ³…(2) , where σ is a stress between the frequencies . from force-relation f = σ a = (2πf r) a φ = w r a φ = �̅� a φ , is seen that force g becomes a velocity �̅� . in the case of planck`s length this velocity is �̅� = w r = 𝛔.𝚽 𝐫 𝐫𝐩 = σ x φ = | 𝐆 𝐋𝐏 . r φ³ |φ = [ 𝐆.𝐋𝐏 𝐫.𝚽² ] and from gravity relation g ≡ σ x 𝚽³ , then issues → [ �̅� = 𝐆 𝐋 𝐏 𝐫.𝚽² ] = [ �̅� = 𝟐.𝐆 𝐋 𝐏 𝝀.𝚽² ] which is the light-velocity vector . since r , becomes from the beyond-planck`s-region then  �̅� = f.φ a = [ g φ a ] …...(f) equation (f) may be written as force g = �̅� 𝐀 𝚽 , which is a viscusdampingforce where a φ is a constant of proportionality and the equation of motion is 𝐅𝐆 = [ 𝐀 𝚽 ] . �̇� force g is applied in all quantized-universe a as velocity �̅� and as stress σ on spaces anti-spaces as spaces-relation 𝐋𝐩 = ei.( π 2 +2kπ).b ,where from material-geometry [24-58] , space ≡ 1 and anti-space ≡ √1 = + 1 , 1 , and √−1 = i = the imaginary part into the → anti-space + space ≡ motion ≡ i ≡ √−1 2 ≡ √−1 4 ≡ e−i.( π 4 ).b = 0,707106781 . b . the base b , which for natural logarithms issues << the natural logarithm ln(x) of a magnitude x , is the power to which , e , would have to be raised to equal x >> and is defining that → ln(x) is the period-needed to grow x , as this is in integration ∫ x , so ex ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 70 the fundamental particles origination mechanism in mfmf pns caves . is the amount of growth after period x , and the possible-repetitive-permutations for moulds and elements in which issues 𝐌𝐨𝐮𝐥𝐝 𝐄𝐥𝐞𝐦𝐞𝐧𝐭𝐬 = the growth-periods i.e. when mould = elements then is succeeded maximum and for e is ee, and any logx x . for logx x and base x = 10 then log10 10 =1010 and for the two elements [⊕ , ⊝] is 10[10]² = 1020 positions ≡ distances ≡ r and since 10−x = 1 10x then b = 10−20 , and the anti-space + space-positions are , 0,707106781.10−20 = 1, 0707106781.10−19. in this way the non-dimensional-number [ 0,707106781*10−1 = 0, 0707106781] , is quantized in cave , r , as distance and becomes the-dimensional-space in the decimal system b =10 as cave r , and which cave is monad r = 1, 0707106781 or , r = 1, 0707106781 m , having unit-area π.r² =3.601588 m2 , and the qua-universe is a = 3.601588 . 𝟏𝟎−𝟏𝟗 m2/s ,which are the space +anti-space positions in universe ...(i) from (h) , light-velocity �̅� = [ 𝐆 𝚽 𝐀 ] = 𝟔,𝟔𝟕𝟑𝟔𝟗𝟐.𝟏𝟎−𝟏𝟏𝟏,𝟔𝟏𝟖𝟎𝟑𝟑𝟗𝟖𝟖𝟕 𝟑,𝟔𝟎𝟏𝟓𝟖𝟖 .𝟏𝟎−𝟏𝟗 . = 2.99819938 .𝟏𝟎𝟖 m/s i.e. gravity force g ≡ motion , exists without mass and is quantized → spread in {space and anti-space} or ≡ qua ≡ {is the quantized-units in-area of space and anti-space }, and as a mould 𝚽 golden-ratio , exists in the impedance b . impedance in mechanics is the friction coefficient , where for force g to be proportional to the light-velocity c = �̅� , then the harmonic – oscillation is an dumped-oscillator , and can oscillate due to the excitation as , a.. with a frequency lower than in the undamped case and an amplitude decreasing with time , the under-damped-oscillator which originates the quantum of motion which is a limiting case between the oscillatory and non-oscillatory motion the critical damping , and which originates the light-velocity �̅� , [49] b.. with undamped case frequency and decay to the equilibrium-position without oscillation and the over-damped-oscillator originating the n-times velocity c . what is quantization of motion and impedance is analytically referred in [83] a parallel solution becomes also from the attendant geometry logic , the three elements ≡ digits of material-geometry are →{⊕ , [⊕↔⊝] , ⊝} ≡ [+ ,0, -]← the permutation , arrangement of the two-elements p1 2 = 2 , i.e. are→ [⊕,⊝] [⊝,⊕]← the three-elements in space need p 1 3 = 3.(3-1).(3-2) = 6 positions and the same for three-elements in anti-space need p 1 3 = 3.(3-1).(3-2) = 6 positions , and total places → p1 3 . p1 3 = 6 x 6 = 36 positions = a , for spaces and anti-spaces as impedance and as before for logx x and base x = 10 then log10 10 =1010 and for the two elements [⊕ , ⊝] the growth is 10 [10] ² = 1020 positions ≡ distances ≡ r , and since issues 10−x = 1 10x then b = 36.10−20 , and  �̅� = f φ a = [ g φ a ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9795163. 108 m/s i.e. the ubiquity of material geometry in electromagnetism is everywhere . the gravity constant g , is the permeable path for inner stress σ , to pass the material`s point a surface 4πr³/3 and to expenditure its energy .the same exists also to the electromagnetic force which is associated with a fundamental property of matter which is the electric charge and which is a clue to the ubiquity of electromagnetism . from equation of gravitation g = ke g = g. [ kr gr] seems that the two constants are related i.e. act each other through local-coefficients or through field-lines , called the medium or permissible path which are as σ = 𝐅 𝐀 = 2πrf φ = w r φ = 𝐯 𝚽 , velocity vector in a unit-space 𝚽. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 71 the fundamental particles origination mechanism in mfmf pns caves . it was shown that the first path is gravity , g , and original field-lines of force , g , are distorted by these charges , local-coefficients , the layers following newton`s laws . the original field-lines terminate at the surface on one side of the medium , and new field lines originate from the other side of it . it was shown that the momentum vector ,�̅� , is equal to spin s , because it is following the stationary wave nodes principle in the material-points creating the minimum quantized energy which is conserved in lobes . this property is extended also to the number of lobes as well as to , π , number as velocity { v = n.π.c } which is the minimum number relating lines and surfaces . analogous happens in equation when v = c , and → r = c . from inner-velocity equation v = w.r = (2π / t).r = 2π.f1. r , of fundamental frequency f1 , of wavelength λ = c.t = c / f1 , and cave r = n.[λ/2] , then r = n.(c /2f1) and v = 2π.f1 . [n. c / 2f1] = n.π.c or v = n . π . c ……..(π) equation (π) shows that velocities in lobes are , n.π times that of light , following , π , number in circle , i.e. in material-points exist velocities multi-times that of light and the minimum surface-constant, unit π , or the growth of the velocity-vectors occurs in lobes by following the logarithm laws of energy-constant c which acts on space constant π. from velocity , v = n.π.c, is seen that light-velocity is the quantum of unit-velocity in planck`s length . the why velocity c and π , is such in [42-51-63] kepler`s laws → explain how the planets move around the sun but not the why. newton`s laws → explain the why by filling this gap by a force f = g.m 1.m 2/ r². acting instantly between the bodies that are moving around each other but not their nature and not the how force is acting . markos-spaces → explain the why by filling the gap with a double ocean of the pointy-spinningmaterial-points , becoming from the stationarymaterial-points , photons or electrons , and from the moving-energystorages , the photons , which orientate and re-orientate the stationary-spins , and explain the how and the why all motions follow the ubiquity of the electromagnetism , starting from the primary material-points, photons ,which through the golden-ratio-frequency-growth effect and conserve the whole natural-world from microcosm to macrocosm . a) in vacuum ≡ [10−62-10−35m] , the {energy-space-monad |�̅�| ≡ |ab|},vanishes. b) in vacuum exists only pointy vibrating ( the circular motion only ) between the opposite dipole [⊕↻↺⊝] generating the , w , b , momentum. c) as unit-energy vectors �̅� = σ / b̅ → exists on {unit-space-monad |ab| ≡ |r̅|← and then → {the energy–space monad |�̅�| ≡ |ab| ≡ the existing universe} d) stpl-line-machine is the physical rotor for the cosmic-particles origination . the gravitational-force g , acting in the beyond-planckcave , and on the light velocity vector �̅� , creates electron-charge �̅� electron = g c √2 = 1,574.10−19 c . which agrees with that of coulomb`s charge q̅ , and the material-points �̅� photon= g √2.f , while by effecting on the whole-planck-cave 𝐋 𝐏 ≡ e i . (5π/2) .10 , creates the pointy-gravity force as this is the ocean of spins g̅ , and which oriented-spins , originate gravity �̅� = [↓= ↻] , and antigravity g = [↑= ↺] as electron e̅ [72] . the light-velocity c̅ of photons becoming from the action of g on planck`s length , when act on a-cave , r = 𝐡 𝐜.𝐙𝐜 ≠ 𝐋 𝐏 then originates the hydrogen-cave e = h fm , because angular-momentum �̅� = 2l w̅ = 2l 2πf = [ 2l 2π ].[ 1 f ] = constant 2π [ 1 fm ] = spin [70]. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 72 the fundamental particles origination mechanism in mfmf pns caves . the four energy-space constants in nature are , [99] 𝐋 𝐏 ≡ 𝐞−i.( π 2 +2kπ).b ≡ ei .(5π/2) .10 ≡ ei .(-5π/2) .10 ≡ { √3.π. 1,616199.10−35 m } → and is the planck`s length cave , g ≡ 𝐓² 𝐚³ ≡ 𝟏 f ²n.𝐚³ ≡ 9, 8076941 → the minimum granular quantized energy , from the gravity frequency fg , representing the bedding-quantum-raw-material of force g , on all energy structures . g ≡ σ.𝚽³ ≡ g. ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[gl kl ] ≡ 9,8076941* 6,8116.10−12 ≡ 6,68056.𝟏𝟎−𝟏𝟏 m3/ n.s² the pulling and cohesive bond , of all the quantized-energy-structures in all spaces . 𝐟𝐧 ≡ { [s ≡ bp ≡ em-r ≡ f1=n , f2 , f3, fd ,,f n] ≡ n (1+√5)σ 4πr = nσ.b̅ 2 r² and from cave 𝛌𝐍 = 4𝜋.r c nσ.(1+√5) = 𝜋2.𝑟 4 c n.b̅ ] } , the amount of energy , the unit meter of motion , in the torsional and resonance-augmented-golden-ratio-pattern φ , and �̅� = �̅� = σ . φ = f a φ = [ g φ a ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9985163. 108 m . 4c.. the origination of electron-charge �̅� ≡ �̅� : from relation g ≡ g.ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[gl kl ] = g k , is seen that gravitational-force g , is the spring-like central-force from a fix point , the source, on an attached , probe, mass as → f = k r = k r.r̅ , as the 1-dof equation , r̈ + w ² r = 0 , where k = the unit-spring-force ≡ [meter of area].[meter of force ≡ stress] ≡ π g ≡ with a general solution r = a sinwnt + b coswnt , where a , b are constants and evaluated from the initial velocity-conditions which become r = [x́(0) / wn] . sinwnt + x(0).coswnt . above equation represents the natural-frequency of the celestial-structures , and for planck`s length the only one primary-particle occupying the less frequency and negative-charge , which is the electron equation and with the solution as , 𝐰𝐧 𝟐𝛑 = 𝐟𝐞 = 1 2𝜋 √ k m , or 4 π² f ²e . me = k = π g and or → 𝐦𝐞 = 𝐠 𝟒 𝛑 𝐟 ²𝐞 from planck`s min-equation f e = e / h = [-13,6 x 1,6.10−19 = 2,176.10−16 joule] / [ 6,6262.10−34 j.s ] = 3, 283998.𝟏𝟎𝟏𝟔 /s ,where in hydrogen atom energy e = h f = -13,6 ev substituting all the minimum-meters of planck`s scale then , electron mass is , 𝐦𝐞 = g 4 π f ²e = 9,808238 4.π.[3,28399.1016 ]² = 7 , 2373149. 𝟏𝟎−𝟑𝟎 kg ……… (4b) 𝐟 𝐞 = 3, 283998.𝟏𝟎𝟏𝟔 /s , and loop 𝐋 𝐞 = 2 , 3762992.𝟏𝟎−𝟏𝟔 m …..(4c) relation 4π²f²e .me = k = π g , denotes that electric-field generated by an electron has both , a component oscillating at the electron`s compton frequency 𝐟 𝐞 , which is ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 73 the fundamental particles origination mechanism in mfmf pns caves . the energy-part , and the non-oscillating component , the electric-field-lines as the energy-conductor , the space-part , which is connecting gravity with electric-field. so , electron , is the minimum-energy massive particle in hydrogen-cave {-13,6 ev} , becoming from material-point in planck`s-length , minimum-cave, occupying the less unit frequency in planck`s – length ,with negative-charge { 𝐟𝐞= 3,283998.1016 /s }. because of the negative-energy in hydrogen-cave , and the less-negative energy of rims which consist the plane-traces-conductors or the paths in-where electrons roll so energy rims consist the energy-minimum-paths in hydrogen-cave-potentials , as en en−1 = [me.v²] .[2n-1] , where the electrons , the electric–charges , roll . electron charge �̅� becomes from magnetic field m which creates the electric-field e , which is acting on charge q , and the acting force per second creates work which is conserved and coincide with the planck`s constant h . this is because h → j s = n m s = power ,where from, energy = power x time , issues the beyond planck`s length 𝐋𝐏 , and the voltage v is → q̅ ≡ ke vp =1 = me 𝑐 ² 2 = g c ² 8πf ².1 , and v̅ ≡ vp ≡ c .charge total−energy= h = c .q̅ h ...(1) using the two energy-equations for linear-motion → fn = 1 2π √ k m 2 and orbital-motion a =√ 1 k .f ² 3 , for unit-energy-space-frequency k = g , a = π , then → g .f ². π ³ = 1 ...(2) frequency fn =√ 1 g.π ³ 2 =√ 1 9,808238.π ³ 2 =1,8133418.10−3 , i.e. the unit-charge-cave , q̅ into hydrogen cave [a =1,82043047.10−12m].[1,813342.10−3/s] = 3,3010625.𝟏𝟎−𝟏𝟓 c from equations charge and voltage is the self-growing property of frequency 𝐟𝐧 in material-point , therefore and for hydrogen-cave is equal to → q̅.φ , because gravitational force is equal to → the geometric-resultant of light-velocity c , acting on electron-unit-charge �̅� ← or , g = c √𝟐 �̅� , then electron-charge is , q̅ = g c √2 = 6,680561 .10−11 1,41429.[2,9979346.10 8] = 1,58.10−19 coulomb , the known coulomb`s charge quantity, from gl ,kl universal constants the newton`s gravitational constant force is , g ≡ g.ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[gl kl ] ≡ 9,808238* 6,8116.10−12 ≡ 6,68056.10−11 𝐦³ 𝐍𝐬² the in depth position of electrons allows to the work produced in prior phase-areas as golden ratio pattern , to be stored in the next . energy = motion / t ≡ ( v 2πr ) . [σ + σ φ] = �̅� . [ 𝛔 𝟐𝛑𝐫 + 𝛔𝚽 𝟐𝛑𝐫 ] ≡ �̅� . [. fn̅ + 𝐟𝐧 ] ≡ moving storage →[ �̅� . fn̅ ] ← + moving-frequency → [ �̅�.𝐟𝐧 ] ← ≡ material-point i.e. the energy produced in photon-cave is consisted of two-moving-storages , that travel as wave [�̅�.𝐟𝐧] →[ �̅� . fn̅ ] → [ �̅� = �̅� = λ f φ ] → [ s ≡ em-r ≡ f1=n , f2 , f3, fd ,,f n= w² ] , and as particle →[ f1= (e²+h²) = n (1+√5)σ 2πr = nb̅ π² r⁴ ] →{w≡em-r≡ [εe² + μb²] = 2.λc.sin.2φ} and is the duality of an energy-storage s ≡ { [⊕← 𝐫 →⊝] + motion m . [87-89] 5c.. the origination of hydrogen cave h : [99] from kepler third law , the energy-closed-space equation and newton`s laws of motion constant k = v². r = ( w r ) ².r = [ 𝟐𝛑 𝐓 𝐫 ] ². r = 𝟒𝛑² 𝐫² 𝐓² . r = 𝟒𝛑² 𝐫 ³ 𝐓² = 4π². 𝐫³ 𝐓² = 4π².r³.𝐟²𝐩 ……..(k) ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 74 the fundamental particles origination mechanism in mfmf pns caves . because (k) is constant , r ³.f ²p , is also a constant multiplication of cave , r , and the frequency f , and the work ≡ motion are conserved in cave , r , as the , n , frequencies fn = n (1+√5)σ 2πr = nb 𝜋2r4 , and for a damping-cave → r(t) = r(t + w) ← as planck`s scale is with min-damping = 1 and unit-energy-quantity 𝐖𝐮 , ( the critical-energy-unit in min , r ) , is this unit-stress-gravity g , as k = e = t² a³ = g = 1 f ².a³ , i.e. → stress �̅� , when is entering into the minimum cave , a , of a minimum surface , then from the period of rotation t on the perimeter , is created in surface the minimum quantity of energy-cave and is the hydrogen-atom , where gf ² ≡ the energy-part embodied with stress �̅� , and the cave a³ , is the space-part , in 3-dof space as → t ² = g a³ = 9,808238.[ 2 ,1145016.10−11 ] ³ = 9 , 2728158.10−32 s , and period t = 3,04513.10−16 s , or frequency f = 3,2839982.1015 / s . from equation e = h f = 6.62607.10−34 . 3,2839982.1015 = 2,175999. 10−18 j / (1,6022. 10−19 ) = 13,59999 ev above quantized energy of 13,6 ev , corresponds to the hydrogen-atom-cave . it was shown that in conservative systems of a central-force , the total energy e is conserved and at periapsis , energy e = 𝐆.𝐌𝐦 𝟐 𝐚 and e = √1 + 2el2/g²m²m³ and for e = 0 then → e = g²m² m³ 2 l² , i.e. energy is always negative . hydrogen cave constant is k = e = t² a³ = g = [ 4π² gm ] therefore gm = 4π²a g and from newton total-energy e = gm m 2 a , rotational-momentum l = √(1 − e2). gmm². a eccentricity e = 0 , g mm = 2a.e , and then → l² = gmm².a = 2ae[a] = 2a²e . i.e. equation l² = 2a².e , denotes that angular -momentum l , in orbit rims is always negative and equal to , l = a √𝟐. 𝐚𝐄 . so , hydrogen-atom , is the minimum-energy-cave in planck`s-length , occupying the minimum negative-energy in rims ≡ caves , the space , where quantum-energy-rims are the plane traces-conductors in where can roll the ⊝ charged electrons . in hydrogen-cave , the resonance frequency of an electron ⊝ , and a proton ⊕ , happens from the stationary unit-cave of a system of kepler second planetary-law as equation , 4 π² m f ²o = k , and from constant law of areas 1 = k .𝐟 ²𝐨 𝐚 𝟑 .their common constant-energy k , is → k = 4 π² m f ²p = 1 𝐟 ²𝐩 𝐚𝟑 or , f ⁴p = 1 4π²ma3 and fp = √ 1 4π²m.a3 4 . a loop in planck`s length 2a = l p = π.1,616199.10−35 m with frequency f = c 𝜆 , where then issues f ⁴p = 1 4π²ma3 = [ c 𝜆 ] ⁴ = [ 1 𝜆 ] ⁴.{𝑐4 = 1 [4πa3]mπa } , and for the unit intensity i = energy area = e = hf a = 4πa3 = 1kw a = 1 [4πa3] = 1 𝜆⁴ {mc4. πa}= 1 𝜆⁴ {1}.80,760865.1032. (3,14).0,8081.10−35 =0,5865.{ 1 𝜆⁴ }, and e-intension is → i = 0,5865 . { 𝟏 𝝀⁴ } ← or , is the rayleigh scattering , i.e. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 75 the fundamental particles origination mechanism in mfmf pns caves . the quanta of energy , passing through a unit-surface or volume , which becomes from the kepler`s second and third law mechanism , is inversely proportional to the fourth power of the wavelength 𝝀 . in outer electromagnetic fields , frequency , w , is that of inner motion → w = 2π t = 2π.fn ≡ [ 1+√5 2 ] σ 2πr = 𝛔.𝚽 𝐫 , which continuous to follow the growth-golden-ratio-pattern . photon occupies such the property of material point as the growth-pattern .with this way photon`s frequency → fp = 1 / tp , kicks-start everything found on its-way . from the resonance constant k = 1 f ²n.a ³ → or 1 = k .f ²n. a3 ← and from kepler`s laws and newton`s lagrange laws for orbits , the cave-semi-major-axis is as , a = √t2/ k 3 = √ 1 g.f ² 3 =√ 4.π².r² g.σ².φ² 3 = √ [4π2].x r² [g.φ2].x σ² 3 , and 𝐟 𝐑 = wr 2π = √ k=1 g.a ³ 2 = √ 𝛔 g .φ2.a³ 2 ≡ √ k md ev …(a) i.e. it is the resonance-frequency fr , in photons loop , and which is the energy-space quanta , following the growth-golden ratio-pattern , which travels in all cosmos. a)... it was shown that when two photons strike each other , c = c⃖ , or reflect on a wall , then c = 0 and so their fundamental frequency f1 = n.[ �̅� 2r ]x[ 1 4 ] = n.[ �̅� 8.r ] which for n = [ 8.𝜋𝑟𝑐 v.λ ] increases to λ c , therefore for n = n.[ 8.𝜋𝑟𝑐 v.λ ] = n.| c 𝜆 | and for n → ∞ then f1 = e / h =∞ ...(b) in dual photon v̅ [ σφ 2πr + σ 2πr ] ≡ v̅ . [ fn̅ + fn ] , colours are the still-sub-units of storage [ v̅ . fn̅ ] , and exist as frequencies , violet , blue , green , which stabilize with their complementary colors yellow , orange , red , of wavelength d = λ → 𝝀 𝐦𝐢𝐧  𝝀 𝐦𝐚𝐱 = 3.10−7m  3.√3.10−7m . b).. the quanta of energy , passing through a unit-surface or volume is inversely proportional to the fourth power of the wavelength 𝜆 as i = 0,5865 . { 1 𝜆⁴ } . when fbe = e / h = ∞ then the energy-intension increases and the wavelength 𝝀 decreases and for fbe = ∞ then 𝝀 = 0 . this is what happens in black-holes where d = 𝜆 = 0 , and in where e = fbe = ∞ . hydrogen-energy e = h.fp = [ 6,6262.10−34 j.s ] . [ 3,28393.1015 / s] = 2,176.10−18 j and in ev → [2 ,176.10−18] / [1,6.10−19] = 13,6 ev , above quantity of energy consist the hydrogen minimum energy-rim , becoming from equation a = √𝐓𝟐/𝐠 𝟑 = √ [3,04513.10−16]² 9,80769411 3 = 2,1145016.10−11 m , for unit energy cave . from angular-momentum b = r.mv = r [ πrv 2 ] v = 𝛑 𝐫² 𝟐 v² = 𝛑 𝐫² 𝟐 v² = πr³ 2 [n.π.c]² = 𝛑³ 𝐫³ 𝟐 c² , or black-hole-energy → 𝐁𝐄 = 2.π⁵.r³.f ² = (𝛑𝐫)³.w² ← i.e. velocity in black-holes is related to cave , r ³, and energy w² times of light velocity. c).. since the primary motion ≡ [pm] exist from the three breakages only , and stpl is the n =3 mechanism creating elementary particles  [n-knots=3] in 𝐕 gra = 7,593.1035 v. explanations : ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 76 the fundamental particles origination mechanism in mfmf pns caves . the action-spaces ≡ the infinite quaternion monads → �̅� ≡ ( s + v̅.i ) ≡ 𝐀𝐁̅̅ ̅̅ ≡ ≡ √s2 + [ v̅. i ]² ] , or �̅� ≡ x + i . y ≡ 𝐀𝐁̅̅ ̅̅ ≡ moduli , r , is their magnitude [ r = | r | = √ x2 + y2 ] , on unit-diameter ab̅̅ ̅̅ . the [qmas] . for quaternion |v̅| = s breakages 𝐳 * 𝐳 = (s + v̅.i )² = s² +[v̅]² +2.sv̅.i = s² – v̅ ² ± 2.s v̅ = |s|² |�̅�|² ± 2.|s|.|s̅| , i.e. in figure -3 the vector of amplitude is quaternion �̅� ≡ x + i . y , with vector �̅� , to be from definition of the complex vector space , the set of vectors of length (the n vertices which form the n polygon sides , or the norm of the complex vectors ) a...the 4-geometrical spaces , [ space , anti-space , neutral-space , sub-space ] consist → the geometrical mould of 𝐀𝐁̅̅ ̅̅ monad ≡ [pm-2r-m] ← b...the vectors of amplitude z = 2r , on 𝐀𝐁̅̅ ̅̅ monad , consist the energy-mould . of 2r-form .the physical -mechanisms → [ physical mould of 2rmonad ] ≡ [gm-ab-m] ← are such and dependent on the n number of accessories they use . a… the gravity force g ≡ motion ≡ σ.𝚽³ , exists without mass and is quantized spread in{space and anti-space or ≡ qua ≡ {is the quantized-units in-area of space and anti-space }, and as a mould 𝚽 golden-ratio , exists as impedance b. placing [ physical mould of 2rmonad ] on [ geometrical mould of 𝐀𝐁̅̅ ̅̅ monad ] [ pm-2r-m ] → ⇅ ← [ gm-ab-m ] then → gravitational-force g ≡ [ σ = n . h ] , and n-na = 1  creates & constructs, for n =1 abm = [σ = n h]  g / c = charge �̅� = 𝐆 𝐜 √𝟐 = 1,58.10−19 coulomb . 1)..→ g ← g ≡ σ.𝚽³ ≡ g. ke ≡ g .[gl kl ] ≡ [ t²p 𝐚³ ].[gl kl ] , and g = 𝐆 ke = [ 6,673692 .10−11 6,8116 .10 −12 . ] ≡ 9,8076941 . gravity-acceleration in black-holes g g = [ π r v4 2 ].𝑒3 = 9,808238 m/s . for b = 36.10−20 , �̅� = f φ a = [ g φ a ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9895163.108 m/s 2).. → c̅ ← �̅� = �̅� = σ . φ = f a φ = [ g φ a ] = [ 6,673692 .10−11.1,6180339887 36 .10 −20 . ] = 2.9985163.108 m . 3).. → q̅ ← from g = c √𝟐 �̅� , electron-charge is , q̅ = g c √2 = 1,58.𝟏𝟎−𝟏𝟗 coulomb , b… the{tvpm} mechanisms , from charges on two-perpendicular-vectors-edges ≡ markos two-vectors three-poles mechanism is case {qua} ≡ [ σ = n . h ]  ∞ x 𝚽 which is the physical mechanism , and gives the ∞ impedance b , determining the objective reality which is the existing universe . the two ⏊ vectors produce the ∞ squares ∞ , 𝐀𝐁̅̅ ̅̅ monad ≡ ∞ [ gm-ab-m ] with ∞ , 2r-form [ pm-2r-m ] and n ≡ ∞ number of accessories , as from monad r = σ j.w² = 𝛔 �̅� and velocity �̅� ≡  σ φ . c…primary motion ≡ [pm] exist from the three breakages only [s² = ⊕ ,2s² = , s² = ⊝] , on triangle 3-knots with n ≡ 3 number of accessories , therefore [stpl] mechanism is the n = 3 accessories-mechanism creating the elementary particles , [n-knots=3] in the gravity voltage  𝐕 gra = 7,593.1035 v. this physical mechanism creates the elementary particles their n = 1 ∞ , masses of atom`s & compounds determining their atoms complex-vector-response .these sparse particles exist in all of the ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 77 the fundamental particles origination mechanism in mfmf pns caves . universe in the sense that they are uniformly distributed . d…the { tcic } mechanisms ≡ [ σ = n . h ]  8 x ① is the physical mechanism , from charges in three-vectors-conductors-edges ≡ markos tetrahedron cube inscribed – circumscribed circles .the tetrahedron conductors -vectors da̅̅ ̅̅ ⏊ db̅̅ ̅̅ ⏊ dc̅̅ ̅̅ curry energy as electromagnetic-fields (forces-frequencies-stresses-any motion) to the inscribed sphere (nucleus of atoms), to cube vertices (electrons-position) ,to the circumscribed sphere (the electrons orbits) for equilibrium and for transferring information and signals . a.. the protons and the nucleus structure : from above [gm-ab-m] ≡ 8 x [(+)[↔](-)] ≡ neutral = n , then a... for 1 proton [(+)[↔](-)]↔(+) ≡ neutral → n = 1 p = 1 b... for 2 proton (+)↔[(-)↔(+)][(+)↔(-)]↔(+) ≡ neutral → p n n p c... for 3 proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)] ≡ neutral → n p n p n p d... for 4 proton (+)↔[(-)(+)](+)[(-)(+)](+)[(-)(+)](+)[(-)(+)] → p n p n 0 n p n p h... for k proton (k1)↔[(-)(+)](k2)[(-)(+)](+)[(-)(+)](+)[(-)(𝐡𝐤)] → kp kn kp kn 0 kn kp kn kp where n-na ≡ k ≡ the number of accessories of {tcic} mechanisms ≡ [ σ = n . h ] on → kp kn kp kn 0 kn kp kn kp k space [ gm-ab-m ] . from above is seen the way of atoms construction. (geometrical-mould of 𝐀𝐁̅̅ ̅̅ -monad). e…from energy = motion / t ≡ ( v 2πr ).[σ+σ φ] = �̅�.[ 𝛔 𝟐𝛑𝐫 + 𝛔𝚽 𝟐𝛑𝐫 ] ≡ �̅� . [. fn̅ + 𝐟𝐧 ] ≡ moving storage →[ �̅� . fn̅ ] ≡ [gm-ab-m] ← + moving-frequency → [�̅�.𝐟𝐧] ≡ [pm-2r-m] ← ≡ [ σ = n . h ] , and n-na = 1  material-point i.e. the energy produced in photon-cave is consisted of two-moving-storages that travel as wave [�̅�.𝐟𝐧] →[ �̅� . fn̅ ] → [ �̅� = �̅� = λ f φ ] → [s≡ em-r ≡ f1=n] & as particle→[ f1= (e²+h²) = n (1+√5)σ 2πr = nb̅ π² r⁴ ]→{w≡em-r≡ [εe²+μb²] = 2.λc.sin.2φ} and the duality of an energy-storage ≡{[gm-ab-m] ≡ s = [⊕← 𝐫 →⊝] + motion ≡ [pm-2r-m] ≡m≡ �̅� − 𝐕𝐞𝐜𝐭𝐨𝐫 } , issuing [pm-2r-m] = √−1 .[gm-ab-m] , therefore → photon-mechanism is for both travelling as particle and as wave , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 78 the fundamental particles origination mechanism in mfmf pns caves . 6c.. the epilogues of prior . figure – 17 : the periodic motion in caves follows material geometry rules . in (1). is shown the geometrical expose of , dynamic-space-energy relation g .r ³.𝐟²𝐩= 1 in (2). is shown the mechanical impress of the , orbit-space-energy relation g .r ³.𝐟²𝐩= 1 in (3). is shown the extreme design of the , dynamic-space-energy relation g .r ³.𝐟²𝐩= 1 in (4). is shown the geometrical creation of the energy-monads-motion & the physical monads motion from their unit relation g .r ³.𝐟²𝐩= 1 the origination of anti – gravity stress g : 6c1.. the origination & the nature of gravity g . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 79 the fundamental particles origination mechanism in mfmf pns caves . figure -17a-:the stability of ±spin as an electromagnetic wave of two frequencies ,0, c from relation angular velocity c = w r = 2πfn.r = 2πr .fn = σ φ . since light-velocity is created from g beyond the planck`s length r p =1,61626.10−35 m , therefore issues c = w r = w r p = 2π rp .fn = σ φ , and from the light-velocity properties becomes , a).. the light velocity c is constant . since c is constant then is analyzed to , ( cx , cy ) in any two directions x , y where x ⏊ y and y to be the gravity direction on where gravity is acting . in figure (8) are seen the 4-point positions , at where velocity phase changes and the work done in each phase of photon`s inner motion . the 4-phases correspond to the wavelength . λ , as λ / 4 . in distance with 90° = 𝜋 2 to be the change of phase . this means that the produced work in the prior phase-area is as the golden-ratio pattern which is stored in the next and perpendicular phase-area . the light velocity �̅� , changes phase every 90° and the created work { w = c y x [quarter circle 1-o-4] } is symmetrically stored from 0° → 90° alternately + , , , + , in [quarter circle 1-o-4] in the [quarter circle 4-o-2] into [quarter circle 2-o-3] into [quarter circle 3-o-1] the momentum ≡ spin ≡�̅�≡ r m v = [ 𝛑.𝐫𝟐 𝟒 ].�̅� x �̅� where ± [ 𝛑.𝐫𝟐 𝟒 ] = the quarter circle. the quarter circle , + + , consist the + spin ≡ �̅� which is the gravity force , and the quarter circle , , consist the -spin ≡ �̅� which is the gravity force from total ,+, -,-,+, when r → 0 or when �̅� doesn`t act on the quarter circle then happens black-hole. in [108] is determined the when newton’s gravitational force 𝐆 , stops acting thrusting , on the light velocity vector �̅� , and happen the black-holes . b).. the dual property of the simultaneously existing spin as electric and as magnetic ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 80 the fundamental particles origination mechanism in mfmf pns caves . field in wavelengths , λ x , λ y , as effective range , is thrusted through the light velocity �̅� , and finally is from the newton’s gravitational force 𝐆 . in planck`s length r p =1,616255.10−35 m the created work from �̅�, on 1-4 position is stored in the [1-o-4] quarter-circle surface and at phase-4 where c = 0 is then stored in the [4-o-2 ] quarter circle surface . at phase-2 where �̅� = �̅� motion is stored in the [2-o-3 ] quarter circle surface , and at phase-3 where �̅� = 0 motion is stored in the [3-o-1] quarter circle surface , i.e. is alternately repeated the + [ 𝛑.𝐫𝟐 𝟒 ] =the +quarter circle [1-o-4] , and the [ 𝛑.𝐫𝟐 𝟒 ] =the quarter circle [4-o-2] both in the same plane-surface [1-4-2-3] of motion . the surface [ 1 – 4 – 2 – 3 ] is ⏊ on the spin axis → u – d . from fn = σ 2πr φ and c̅ = w̅.r = 2πfn.r = 2πr .fn = σ φ , are frequencies , 0 , c , relation. at c = 0 exist minima or maxima where is the change of two energy-semicircles. c).. the origination of spin in cave h . from orbit vibration fn = (1+√5)σ 4πr = σ 2πr φ and from relation g.a³.f ²n = 1 , f ²n = σ² 4π²r² φ² = = 1 g .a³ then g.a³.φ² = 4𝜋²𝑎² 𝜎² , or 1). a = 1 g [ 2𝜋 σ φ ] ² , and from work w = 2e = b w = j.w² , or 2e = 2π.f and �̅�f = e π then 2). the energy in planck-scale-cave 2e = b̅fn = [ (1+√5)σ 4πr ].b̅ = [ σ 2πr ].b̅ φ , or e = [𝚽 𝛔 𝟒𝛑𝐫 ].�̅� i.e. the energy in planck-cave-particles is dependent on their spin only , and for electron r = a e → e = [φ σ 4π.r ].b̅ =1,6180339[ 1 4𝜋.1,6840307.10−17 ]x x 2,845976. 10−17= 21,76.10−19 1,6.10 19 . = 13,6 ev , which is the min-voltage in hydrogen cave . d).. the angular momentum in cave c̅ r , in cave , r , the energy-vector �̅� is positive for o3⃗⃗⃗⃗ ⃗ , o1⃗⃗⃗⃗ ⃗ , o4⃗⃗⃗⃗ ⃗ , directions while , the energy-vector �̅� is negative for o3⃐⃗⃗⃗⃗⃗ , o2⃐⃗⃗⃗⃗⃗ , o4⃐⃗⃗⃗⃗⃗ , directions . since total-energy l = b w = j.w 2 w = j.w² 2 then 2l = j.w² , then angular momentum relation b̅ = r.mv = r [ π.r⁴ 2 ]. 2πf r = π ².𝐫𝟔.f . and b = j w i.e. angular momentum �̅� = r m v = j.w 2 = [ π.𝐫𝟒 4 ] c r = [ 𝛑.𝐫𝟑 𝟒 ].�̅� = [ 𝛑.𝐫 𝟐 𝟒 ].�̅� x �̅� ......(a) ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 81 the fundamental particles origination mechanism in mfmf pns caves . for �̅� = ± �̅� , then + �̅� = + gravity – spin = + [ 𝛑.𝐫𝟐 𝟒 ].�̅� x �̅� , the + gravity force �̅� = gravity – spin = [ 𝛑.𝐫𝟐 𝟒 ].�̅� x �̅� , the -antigravity force from momentum relation b̅ = m r v = m r² w = m r² (2πf) = j.w 2 = [ πr⁴ 2 ].[2πf] then spin s ≡ angular-momentum �̅� ≡ �̅� = j.w 2 = πr⁴ 4 [2πf]= π².r⁴.[f] = 𝛑.𝐫³𝚽.𝛔 𝟐 ≡ [𝟏+√𝟓 ].𝛔.𝛑.𝐫³ 𝟒 = 𝛑𝐫𝟑𝛔.𝚽 𝟐 ..(b) for �̅� = ± �̅� , then + �̅� = + gravity – spin = + 𝛑.𝐫𝟑𝛔.𝚽 𝟐 , the gravity spin �̅� = gravity – spin = 𝛑𝐫𝟑𝛔.𝚽 𝟐 , the antigravity spin from spin-momentum relation s = �̅� = 𝐄 𝐰 = 𝐄 𝟐𝛑𝐟 = 𝐄𝐫 𝛔𝚽 = 𝐫𝐆 𝛔𝚽 , and �̅�² = 𝐉²𝛔² 𝐫² φ², so → [�̅�]² = [ 𝚽𝐉𝛔 𝐫 ]²← while from gravity-force 𝐅𝐆 ≡ 𝐦𝐆g= g. [σ c].r = 𝐦𝐆 𝐜² 𝐫 = jw².𝐠𝐆 = 𝐯² 𝐫 . [ 𝛑𝐫𝟒 𝟐 ] 𝐯² 𝐫² = 𝛑𝐫𝐯𝟒 𝟐 , and gravity-acceleration ± 𝐠𝐆 = [ 𝛑𝐫𝐯𝟒 𝟐 ]. 𝒆𝟑 , geometry growth [ 𝛑𝐫𝐯𝟒 𝟐 ]=[ 𝟑,𝟏𝟒𝟏𝟓𝟗𝟐𝟔([√𝟓+𝟏]. √𝟐 𝟒 .𝟏𝟎−𝟑𝟓).(𝟐,𝟗𝟗𝟕𝟗𝟑𝟒𝟓𝟖.𝟏𝟎𝟖 )𝟒 𝟐 ].𝒆𝟑 = = 6,044981.𝟏𝟎−𝟑𝟓. 80,773378.𝟏𝟎𝟑𝟐. 20,085536 = 𝐠𝐆 = ± 9,8076925 . i.e. gravity + 𝐠𝐆 = + 9,8076925 = 𝛝𝐫c̅ 𝛝𝐭 = + �̅� , and antigravity= 𝐠𝐆 = 9,8076925 = 𝛝𝐫c̅ 𝛝𝐭 = �̅� are the two opposite forces as spin and for the stability of the spin �̅� , equilibrium. for c = 0 exist minima or maxima and thus spin is equal to the negative gradient of the potential energy in the two interchange energy semi circles . centripetal force originates the two-momentum ± �̅� , �̅� = r m v = [ 𝛑𝐫𝟑𝛔.𝚽 𝟐 ] ≡ gravity – antigravity – spin. 6c2.. the stability of equilibrium : in the conservative system , of cave �̅� r , the total energy 𝐄 𝐓 remains constant and the total-energy is 𝐄 𝐓 = 𝐄 𝐊 + 𝐄 𝐔 = 𝟏 𝟐 [�̇�] ² + u(x) = constant …..(1) where 𝐄 𝐊= the kinetic energy and 𝐄 𝐔 = the potential energy . for y = �̇� then , y = �̇� =± √𝟐. [𝐄 − 𝐔(𝐱)] …(2) the differential equation of motion in cave �̅� r , for conservative systems is ẍ = f(x) and because ẍ = ẋ (d�̇� /dẋ ) then ẋ. dẋ = f(x)dx =0 ..(3) by integrating then ẋ² 2 ∫ 𝑓(𝑥)𝑑𝑥 𝑥 0 = ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 82 the fundamental particles origination mechanism in mfmf pns caves . e , and by comparison to priors then → u(x) = ∫ 𝑓(𝑥)𝑑𝑥 𝑥 0 or f(x) = 𝐝𝐔 𝐝𝐱 = �̈� …(4) i.e. for the conservative system , of cave �̅� r , the force = �̅� = r m �̅� = j.w² 2 = [ 𝛑.𝐫𝟐 𝟐 ].�̅� ² is equal to the negative gradient of the potential energy into 2-energy semi-circles. for y = �̇� then (3) becomes dy dx = f (x) y = φ (x , y ) , and the singular points correspond to f (x) = 0 , and y = ẋ = 0 and hence are the equilibrium points . for every point x , y in the phase-plane for which φ (x y) is not indeterminate exists a unique slope of the trajectory. when the velocity vector is at points 1,2 then y = 0 and f(x , y) = c ≠ 0 then the slope of the trajectory is infinite and angle φ = 90ᶱ . since motion in cave �̅� r , is a single-dof oscillator , then motion equation is ẍ + w² x = 0 .equation indicates that at the equilibrium singular points , 3 , 4 , the slope of the potential energy curve u(x) must be zero, meaning minima or maxima , and which is the ± change of direction for the four energyquarter circles [1-o-4] , [4-o-2] , [2-o-3] , [3-o-1] in ( cx , cy ) where 𝐜𝐲 ̅̅̅̅ = �̅� = spin direction , conclusions : 1… from a-momentum �̅� = r m �̅� = j.w² 2 = 𝟏 𝟐 [�̇�]² +u(x) = 𝟏 𝟐 [ y]² + 𝟏 𝟐 k x² = constant , then spin s ≡ �̅� = 1 2 [y]² + 1 2 k x² = [y] ² + w ² x² = c = constant , or → y² + w² x² = c ←….(a) equation (a) is a series of ellipses determined by c , and for w = 1 y/w = y then reduce to circles y² + x² = c , in �̅� r plane and perpendicular to s since , [ �̅� r ] ⊥ �̅� . 2… in the conservative system of cave �̅� r , where �̅� = light-velocity = 2.9985163. 108 m/s , r = planck`s -length = √2 4 . 10−35 m , motion in cave is an single-dof oscillator as equation ẍ + w² x = 0 . equation of motion is a circle on where exist the 2-singular points 3 , 4 , meaning minima or maxima , where issues dy dx = f (x) y = φ (x , y ) , and are in the mode-shapes at the points of equilibrium . at singular points 3 , 4 , velocity 𝐜𝐲 ̅̅̅̅ = �̅� = 0 and spin �̅� = gravity = 0 , and since dy dx < 0 , then spin �̅� changes direction becoming �̅� = antigravity = 0 , meaning that at singular points gravity & antigravity are zero . 3…at point 1 , velocity 𝐜𝐲 ̅̅̅̅ = + �̅� because of 𝐜𝐲 ̅̅̅̅ , +↓ direction and corresponds the spin �̅� = gravity = �̅� = r m �̅� which is constant and equal to work = 𝐜𝐲 ̅̅̅̅ x [quarter circle 1-o-4] at point 2 , velocity 𝐜𝐲 ̅̅̅̅ = �̅� because of 𝐜𝐲 ̅̅̅̅ , -↑ direction and corresponds the spin �̅� = gravity = �̅� = r m �̅� which is constant and equal to work = 𝐜𝐲 ̅̅̅̅ x [quarter circle 2-o-3] where the [semicircle 4-1-3] = + �̅� ≡the gravity , and [semicircle 4-2-3] ≡ �̅� = the ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 83 the fundamental particles origination mechanism in mfmf pns caves . antigravity .the produced work from the eternal rotation is store in the semicircles as , (+) total work = [semicircle 4-1-3] = + �̅� and (-) total work =[semicircle 4-2-3] = �̅� 4…from newton`s law of universal gravitation m r. �̅� ² = g 𝐦.𝐌 𝐫 ² , and when m = 1 , issues �̅� ² = 𝐆.𝐌 𝐫 ³ , and b = j w , therefore angular velocity �̅� = ± √ gm r³ 2 = √ gπ.r4 2.r³ 2 = √ π.gr 2 2 . this angular-momentum is identical with spin which is trapped in caves`s loops which are in phase with each other. the amplitude of oscillation varies from zero at nodes to maxima at antinodes .since frequency ≡ 𝐟𝐧 = ( nσ 8 r ² ).�̅� , so energy-caves are stationary wave-fringes 5… the total-energy , e t , in caves , [70] a… → in any moving system e t = e r + e k = πc² 8 r³ + 1 2 m.v² = 3,535.1016. r ³ + 1 2 m.v² , b… → light-velocity c̅ of photons becomes from the action of g on plancks length lp = r p =1,6162.10−35 m cave r, and when cave r = 𝐡 𝐜.𝐙𝐜 ≠ 𝐋 𝐏 originates hydrogen-cave. it is shown later that when the g action area resistance ≡ gravitational-impedance z g−i = 6,344907.10−35 kg differs for any reason then black-holes happen. c… → euler`s rotating rigid body .the d-work is w = 2l = b̅ . w̅ = j .w² ≡ h. 𝐟𝐧 , and for m mass w = h fm ,angular-momentum �̅� = 2l w̅ = 2l 2πf = [ 2l 2π ].[ 1 f ] = constant 2π [ 1 fm ]= spin d… → in central-motion ,kepler constant k = 4π². r³. fp², or 1 = [ 4𝜋² 𝑘 ] r³. fp², 1 = c . r³. 𝐟𝐏² where for → r → 0 then 𝐟 𝐏 → ∞ ← , a case of a black-hole in any central motion . e… →the vibrations in orbit-systems are as → w = 4π² .[ 𝐫³ 𝐓² ] . �̅� = 4π². r ³. 𝐟²𝐩 . �̅� ← f… → in [pns] neutral space , dumping force is fd = c . ṙ = ± co .w. [√a2 − r ²] and fd ² / [ co .w. a ] ² + r ² / a² = 1 , i.e. energy ≡ motion is mapped as an ellipse with axis , 𝐅𝐝 , x , and energy dissipated per cycle , the area enclosed by ellipse. g… → the energy in [pns] , is either as momentum (mv) or as rotational momentum (λ = r.mv) . velocity �̇� , in form ẋ = w a cos.(wt θ) = ± wa.[ √1 − sin²(wt − θ) ] =  w . [ √a² − x² ] where momentum relation �̅� = m r v = m r² w = m r² (2πf) = j.w 2 = [ πr⁴ 4 ].[2πf] = π2r4f 2 ,and mass of elementary-particles is m = π2r4f 2w.r² = π2r3 2v f = π1r2 4 , or the repeated quarter circle [ 𝛑.𝐫𝟐 𝟒 ] surface of the ± spin dual property in planck`s length and area , 𝐀 𝐐𝐂 = [ 𝛑.𝐫² 𝟒 ] = 𝟑,𝟏𝟒𝟏𝟓𝟗𝟐𝟔.( √𝟐 𝟒 .𝟏𝟎−𝟑𝟓)² 𝟒 = 1,1107206.10−70 m² . since issues the monad of mass to be [ ev c² ] = 1,782662.10−36 kg , then energy dissipated per 𝐴 𝑄𝐶 = 𝐄 𝐐𝐂 = ( 1,1107206.10−70 m²/ 1,782662. 10−36 = 6,230715. 10−35 kg , or from the g action area resistance ≡ gravitational-impedance z iqc , exists ziqc =√2𝑐3a qc / g , and 𝐦 𝐏𝐍𝐒 ≡ ziqc = √2.𝑐3a qc / g = 1,4143.(27.10"24)1,1107206.10−70 𝟔,680056.10−11 = 6,344907.10−35 kg . the result agree with prior because in [pns] dipole of space exists the impedance ziqc , which doesn`t exist in [mfmf] space . [22-26] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 84 the fundamental particles origination mechanism in mfmf pns caves . h… → in [mfmf] space , work = motion = energy is zero as w = ∫ p. ds a b = 0 , and this because ds = v⃗ = 0 and time is not existing and v⃗ = 0 as relation , s = �⃗� .t . each unit ab⃗⃗⃗⃗ ⃗ = ds > 0 exists by this inner impulse ( p ) and so p a+p b = 0 . i.e. the position and dimension of all points which are connected across universe and that of spaces exists because of this static inner impulse , on the contrary should be one point only or , primary point a = black hole → ds = 0 and p = ∞ . since impulse is ∞ then may be → vacuum , momentum or potential or induced potential and it is a type of effect , to push the nature → (impulse ≡ motion ≡ energy) ← 6c3.. the nature of antigravity (g ) . from mechanics in case that , in an axisymmetric rotating body , with constant angular-velocity , w , the moment of inertia , 𝐉𝐱 = 𝐉𝐲 , is equal about the two of the three principal axis , and then , 1.. the angular – velocity vector , �̅� , describes the ellipsoid of angular velocity and its nib describes a cone of which plane-base is fixed simultaneously , 2.. the angular-momentum , �̅� , describes the ellipsoid of angular-momentum and its nib describes a cone also of which plane-base is also fixed . 3.. the nib of angular velocity vector , �̅� , describes on the tangential-plane of the angular-momentum-ellipsoid , the herpolhode , while , 4.. the nib of angular-momentum , �̅� , describes on the tangential-plane , of the angular-velocity-ellipsoid , the polhode . 5.. the fixed-tangential-planes on , �̅� , and , �̅� , nib are alternately perpendicular to , b̅ , and , w̅ , common central axes of rotation . 6.. the kinetic rotational – energy of monad , which is the work in monad , is the scalar quantity , l , the vector angular momentum quantity is , �̅� , and the vector of angular-velocity quantity is , �̅� , which three monads are related as �̅�.�̅� = 2l = j.w² where j = the moment of inertia around the axis of rotation = ⇅ 7..all above happen in , material-point , where the positive ⊕ constituent the + �̅� , is eternally self rolling on the negative ⊝ constituent ,the �̅� , with angular-velocity w , in infinite spherical traces , either at great-circles [+ or ] , or small-circles , [+] where is the clockwise left direction and [-] where is anti-clockwise right direction or any other close spherical-curve , and applying all laws of mechanics into this energy chaos , is thus created the → first–discrete-energy–monad �̅� , the material point ← i.e. the angular momentum �̅� , is the quantum of physics and is , for all the energy – space-universe .the energy = e dissipated per cycle is equal to the work wd = e = h f = h(1+ √5 ] ) 4π .[ 𝜎 𝑟 ] = 8.kζ(πr²) and is stored in monad , which is a stationary wave with and in , n , loops as 𝐖𝐧 = [ 𝟒𝝅𝒓² 𝟑 ].𝐟𝐧 = [ 𝟒𝝅𝒓² 𝟑 ].𝐧𝐟𝟏 = n (𝟏+√𝟓)𝛔𝐫 𝟑 = n [𝛔𝚽.𝐫] 𝟏,𝟓 . 8.. geometry has as monad ,the discrete continuity 𝐀𝐁⃖⃗⃗⃗ ⃗ -length becoming from the zero ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 85 the fundamental particles origination mechanism in mfmf pns caves . point ≡ 0 , and mechanics-physics the recent-acquisition of the material geometry, where zero-point , 0 =  = {⊕+⊝} = the material-point = the quantum of space , as , the positive space and the negative anti-space . [58] in fig-8 , the light velocity vector �̅� = 2,9979. 10 8 m/s rotates in the planck`s length perimeter 𝐫 𝐏 = 1,616255.10−35 m , and is thrusted through the newton’s – gravitational force 𝐆 , during rotation is created work which is stored in the circle surface as angular momentum �̅� = r m v = j.w 2 = [ π.𝐫𝟒 4 ] c r = [ 𝛑.𝐫𝟑 𝟒 ].�̅� = [ 𝛑.𝐫 𝟐 𝟒 ].�̅� x �̅� , in semi circle {+[ 4 -1-3]} , and semi circle { [ 4-2-3] }, for the stability of equilibrium . i.e. gravity + g ≡ the vector of angular-momentum �̅� ≡ + spin ≡ ↓↻ ≡ the work stored in the [4-1-3] semi circle of the , �̅� �̅� vectors circle surface produced from the gravitational-force , g , as acting into the – planckcave , and on the light velocity vector in , + ↓ �̅� direction . antigravity g ≡ the vector of angular -momentum �̅� ≡ spin ≡ ↑↺ ≡ the work stored in the complementary [4-2-3] semi circle of the , �̅� �̅� vectors circle-surface which is produced from the gravitational-force , g , as acting into the planckcave , and on the light velocity vector in the , ↑ �̅� direction . figure – 17b the material , lrc circuit on orbit , on focus-planet-sector |f↔p| in (1). force g , as wave , is directed to the center of rotation f , and is proportional to the distance pf ≡ focus planet .the gravitational potential-energy 𝐠𝐆 = 9, 8076925 is stored in → focus-planet-sector ≡ fp ← which is the material capacitor stores charge as that of material-lrc-circuit , and inductors . because of the chains of spins and of their periodic excitation [↔] , is thus created a magnetic field due to the lrc-circuit and which is tuning to the critical quantum and critical-state 𝐠𝐆 . the chains of spins are pointy vibrating with their characteristic ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 86 the fundamental particles origination mechanism in mfmf pns caves . frequencies f = (1+ √5 ] ) .σ 4πr b̅ , and filling up the entire universe . since inner-stresses 𝛔𝟏,𝟐 = σ1/2 ± (½)√σ1² + 4. σ1² = σ1/2 [1±√5] follow the golden ratio on stresses then this quantum-energy gg produced , is the state causing them to magneticallyresonate . in material-point-chain of caves r = λ/2 , and since ẋ = w.r → w = ẋ/r then golden ratio damping-force is the work produced with unit damping-ratio ζ=1 .since also ẋ = v then exists , fd = c. ẋ = 1m.wn .ẋ = 1.[ (π r²/2) v2]. ẋ ² 𝑟 = [ 𝛑𝐫𝐯𝟒 𝟐 ] = g = 𝟐 𝒓 [ σ (1+√5) ] ² = 𝟒𝝈² 𝒓 [ 3+√5 ] . this dissipation of energy is determined under conditions of cyclic oscillations , and is dependent on glue-bond σ , and r , cave . since r is in denominator then for the very small caves , the under planck`s caves, damping-force becomes infinite independently of glue-bond . this may be considered as a type which happens in black holes as this happens in algebra inverse fractions also . in (2) is presented the back-up electromagnetic current flowing in opposite direction fp , by changing the spin direction of the sector-material-points such that work w = g . from kepler`s 2nd law the , area s , swept by any focus-planet-sector ≡ fp is the constant k , and equal to, s² = l² t² 4m² = π²a².[ b = πa ( l² 2me ) ] , or t ² a ² = 4π²m 2e = 4π² 2e/m = 4π² a gm and → 𝐓 ² 𝐚 ³ = 𝟒𝛑² 𝐆𝐌 = k and k = 𝟏 𝐟 ²𝐧 .𝐚 ³ , from which issues → 1 = k . 𝐟²𝐧 . 𝐚 𝟑 and , k = 4π² gm where is seen that , resonance between focus-planet exists through space a and energy f . from the second-order differential equation excited by a harmonic external force , ft .sin wt , and is as , m d²x dt² + c dx dt + k . x = ft . sin w t , corresponds physically to the free damped vibration , where x = the displacement , dx / dt = the velocity and , d²x / dt² = the acceleration of monad , m ,c ,k constants , with the general solution given by the equation x = a . 𝑒𝑠1.𝑡+ b.𝑒𝑠2.𝑡 + x sin(wt-φ) ……(1) the equation → l d²q dt² + r dq dt + 1 𝐶 q = e t . sinwt .corresponds physically to the free damped vibration , where charge q = is the physical property of matter that causes it to experience a force which can be positive or negative , dq / dt = the least quantized amount of charge , and d²q / dt² = the space distribution of charge , and l , r , c inductance , resistance , elasticity constants with general solution given by the equation q = a . 𝑒𝑠1.𝑡+ b. 𝑒𝑠2.𝑡 + x sin(wt-φ) …...(2) equations (1) , (2) give the analogic relation of the classical mechanics [space position x] and the electromagnetism [quanta of energy , q ] of storing and removing of energy in energy-space cosmos .distributed forces l1l2 = l (di/dt) , r1r2 = r . i , c1c2 = q / c respectively , showing the identification of the mechanical and physical laws . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 87 the fundamental particles origination mechanism in mfmf pns caves . the way that potential-energy is stored , is that of material lrccircuit , which is for the gravitational-potential-energy the material-capacitor or the → focus-planet–sector – stores-change ← which develop a voltage in response to that charge .the coil of wire is the infinite stationary-dipole-spinning material-points of this → focus – planet sector ← which develops the back-electromagnetic-force ,when current through them changes . in-cave-box exist two motions ,revolving and periodic, the acceleration of gravity g ≡  σ exists for the first box-𝐁𝐑 , while local-extreme-case for the second box-𝐁𝐏 . in this gravity g ≡  σ , is locally altering s , by changing the principal-stress σ with local uniform pressure 𝐠𝐋 ≡ g ke = g * [force/area] = g , i.e. is proved that the minimum local-energy acceleration is the known universal gravitational-constant g = g k = 𝐤𝐄 g = 𝐤𝐋 σ = g gl kl such for macrocosm as for microcosm , and both cases obeying newton`s laws of motion . 4.. since electron is rotating in orbits so electromagnetic wave is also rotating , therefore is needed a proper-stationary-magnet on which rotation becomes oscillation in order to succeed a clear 3-dimensioned , φ ³, magnetic-resonance-imaging [the mri ] and the other systems use this property of bottom-information . this property of spin in all depths of energy caves allows the granularity of energy in energyloops , and after to be bonded .this property of electron may be used for cancer chromosomes . 6c4.. the conservation of energy in → velocity �̅� and in gravity �̅� caves ← since gravity�̅� is the energy ≡the angular-momentum {�̅� =r m v}in planck`s length r p , and it is a stationary-wave or an electromagnetic wave-signal which transmits informations then follows the waves-frequency-signal-spectrum .the gravitational constant g is the active force on light velocity �̅� and consequently on g̅ also . from waves theory → position x = a.sinwt and acceleration �̈� = a w ² ← where the signal-spectrum is consisted of the amplitude 𝐀 𝐂 and of the angular frequency �̅� , from where all other properties follow. since �̈� ts the acceleration force of position x , then for planck`s length issues ẍ = g = a²g.w ², or �̅̅̅� = √g/a² = √6,68. 10−11/[2,6262. 10−35]² so gravity-frequency→ �̅̅̅� = 0,5057.1030 h and gravity-wave-frequency 𝐟 𝑮𝑾=8,05.1028 h ← the gravitational force action : [[g]] → {[�̅� ] ± 𝐄𝒏}≡ �̅� ≡ 𝐄𝒏+{[g] ± [�̅� ]} and → 𝐄 𝐑 = 𝐄 𝐆 + 𝐄 �̅� ← where , [[g]] = [gravitational constant g] ≡ the signalling.(w c) → the carrier wave [[�̅�]] = any action of [g on�̅�] ≡ any signalling.(�̅�) modulating velocity-wave [[�̅�]] = the new modulated [g ↔ �̅�] signal ≡ f �̅� m+wc 2 + f �̅� m−wc 2 ≡ [mas] compound = the demodulated [mas] ≡ f w+2wc 2 + f w−2wc 2 ≡ [dmas] . �̅� , since [ g = the carrier ] →[ �̅� = the modulator ]→[ g+ �̅� = the modulated ] and [ g+ �̅� ] + �̅� ↔ [ dm = antidotes ] = the demodulation ] i.e. the communications–frequencies in planck`s cave , are the wave resonances which consist the master-keys of these wave caves . the conservation of energy in caves and in atoms compounds . [106] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 88 the fundamental particles origination mechanism in mfmf pns caves . the atoms – actions : [[a]] → {[b]±𝐄𝒏} ≡ c ≡ 𝐄𝒏+ {[a] ± [b]},→ 𝐄𝒏= 𝐄𝐂 + 𝐄𝐀 ← and where [[a]] = [atom or compound] ≡ the signalling.(wc) → the carrier wave [[b]] = any[a-or-c] ≡ any signalling.(w) wave = the modulating wave [[c]] = the new modulated wave [a] signal ≡ f w+wc 2 + f w−wc 2 ≡ [mas] antidote = the demodulated [mas] ≡ f w+2wc 2 + f w−2wc 2 ≡ [dmas] , �̅� , as the atoms action (1) ... [a] → {[b]±𝐄𝐍} ≡ [c] ≡ ± 𝐄𝐍 + {[b] + [a]} , , [a] ≡ [c] +{ 𝐗 𝐍} [a] = the initial , [c] = the final , [ x n ] = {[b] ± [en]} ≡ the antidote . the atoms action (2) …. [a] → [ ± e n ] ≡ [c] ≡≡≡ ± 𝐄𝒏 + [a] , , , , [a] ≡ [c ] + [ e n ] [a] = the initial , [c} = the final , [ e n ] ≡ the antidote , the circular-signal-action is as , i → f = [ i + f ] and [ i + f ] ↔ [ dm ] ≡ antidote , or still-actions [ i = carrier ] →[ f = modulator ]→[ i+f = modulated ] → → [ i+f ] + m ↔ [ dm = antidotes] = the demodulation ] the communications –frequencies in atoms caves , are their wave resonances which consist the master-keys of all the wave caves . 6c5.. the energy superflow in mfmf-pns space-levels for black-holes origination. 1)..it was prior proved that angular-momentum of the primary-motion in pns space is a quaternion of dimension power ( s + �̅� i) ¹/ d = √( s + �̅� i) d , which follows that of spaces origination . 2)it was prior proved [70] that in pns space the inner forces of a system , are the two equilibrium centripetal and centrifugal forces due to the eternal , ± σ , stresses of opposites [⊕↔⊝] originate motion u , as 𝐉𝟏. 𝐮² 𝐜𝐨𝐬𝛝 𝐉𝟑 𝐰𝟑 .u + s . q = 0 . this quadratic equation has 4 solutions where the angular velocity complex number u = |�̅�| = s q b = s q j3w = |�̅�| = dφ dt = [ j3. w3 2 j1.cos ϑ ] ± [ k pns 2 j1.cos ϑ ].i = s ± v i , or → angular-velocity |�̅�| = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ± √− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ) ² 𝟐 = s ± {√�̅� }.i ← where s = [ j3. w3 2 j1.cos ϑ ] , �̅� = [ k pns 2 j1.cosϑ ] , 𝐤 𝐏𝐍𝐒 = √ [ j3. w3]² − 4. sq. j1 cos ϑ 2 the constant work exists on the common central axes of rotation which is fixed to the tangential-planes of �̅� & �̅� nib , being alternately perpendicular to , b̅ , and , w̅ , and found to be as �̅� . �̅� = 2l = j.w² . the frequency monad of rotation in the pns space is as → 𝐟 𝐏𝐍𝐒 = 𝐬 𝐐 𝟐𝛑.𝐉𝟐.𝐰 ← with the following 4 solutions for |�̅�| , solution-1  | 𝑤 1 | = [ s ] + [ √�̅� ] → real root ≡ huge minerals at edge of universe. solution-2  | 𝑤 2 | = [ s ] [ √�̅� ] → real root ≡ atoms & compounds into structures. solution-3  | 𝑤 3 | = [ s ] + [ √�̅� ].i → imaginary root ≡ huge bh at centres of galaxies. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 89 the fundamental particles origination mechanism in mfmf pns caves . solution-4  | 𝑤 4 | = [ s ] [ √�̅� ].i → imaginary root ≡ micro bh at centre of molecules. where bh = black hole . from 4-above issues , a-velocity |𝑤1 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 +√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 a-velocity |𝑤2 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 -√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 a-velocity |𝑤3 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 +√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬 𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 . i a-velocity |𝑤4 ̅̅ ̅̅ | = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 ≡ 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 -√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 )² 𝟐 . i angular-velocity �̅� is a vector , and rotates in one axis k with respect to variable �̅� momentum and , the ellipsoid of |�̅�| ≡ quaternion of dimension power (s ± �̅� i)¹/d the inertia j-ellipsoid of |�̅�| gives two anti-symmetrical inertial ellipsoids , a real and an imaginary in their polhode and herpolhode cones as below, the-real �̅� ≡ { polhode ≡ cone ops , herpolhode ≡ cone opt , [ ↻ rolling ] on the the ellipsoid axis , ↓ normal on the unmovable & attitude-plane of the [ pspt ] truncated-polar-cone }. the-imaginary �̅� ≡ {polhode = cone ops , herpolhode = cone opt [↺ rolling ] on the anti-symmetrically-ellipsoid-axis, ↑ normal on the unmovable & attitude-plane of the [ pspt ] truncated-polar-cone } . it has been proved that �̅� , �̅� vectors exist as , e = energy =motion , s = space =length --------------------------------------------------------------------------------------------------------------- both magnitudes define the unit-energy vectors → { unit-e vector r = σ j.w² = 𝛔 �̅� } ← when the unit-e.vector r is placed on [ab≡2r s.diameter of spaces-neutral-circle] then energy-monad |�̅�| , exists on , space-monad |ab| ≡ |�̅�| and is {e+s -monad |�̅�|} or is the { energy – space monad |�̅�| ≡ |ab| } , where the magnitudes, spaces sides 𝛌𝐧 = 2,√r2 − a²n , & the quantum critical-side 𝐚𝐧 = 1 2 √4r2 − λ²n , every=monad occupies ab = 2|r| diameter = space & unit-e-vector r= σ j.w² = σ b̅ = motion the quantization of energy in spaces happens by following the archimedes formula for duplication of sides as 𝛌𝟐𝐧=√2r2 − r√4r2 − λ²n & critical-sides 𝐚𝟐𝐧= 1 2 √4r2 − λ²2n the energy-monad |�̅�| = σ j.w² = 𝛔 �̅� remains the same in pns space while the spaces sides change as 𝛌𝟐𝐧 , 𝐚𝟐𝐧 , when the monad |�̅�| enters the 2n . ------------------------------------------------------------------------------------------------------------------------- the 4 solution for real and imaginary root is as below , the |w1 ̅̅ ̅̅ | solution is the real -root giving the strongest in amplitude energy-structions. the |w2 ̅̅ ̅̅ | solution is the real -root giving the weakest in amplitude energy-structions. the |𝑤3 ̅̅ ̅̅ | solution is the imaginary-root giving the mega tapered-amplitude e-structions. the |𝑤4 ̅̅ ̅̅ | solution is the imaginary-root giving the micro black-holes structions . from gravity relation g = g 𝐤𝐄 =9,8076941.6,8116.10−12= 6,68056.𝟏𝟎−𝟏𝟏 m3/ n.s² from ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 90 the fundamental particles origination mechanism in mfmf pns caves . energy-loop relation 1 = c. r³. 𝐟𝐏² → fp=√1/cr³ , and from f = e / h then f = 𝐄 𝐡 = √1/cr³ or → e = h .√1/c. r³ , and e² h² = 1 c r³ or , e ² = h² c r³ , an energy relation between the , c , r . this relation is used for blackholes energy where r , result to zero , the permissible-permeable-resonance-paths for their means are for , 7) particles → the resultance of the one-dimensional-collision v̅ij= v̅j v̅i = w̅ij. r̅ij 8) particles → the cauchy ellipsoid-stress-tensor where { w̅ ⊥ b̅ ⊥ r̅ ≡ σ1⊥ σ2⊥σ3 } 9) m-points → the resonance-frequencies 𝐟𝐑 [s ≡ f1= n ,f2, f3,fr=w²]= 𝐟𝐧 = [⊕↔⊝] = n (1+√5)σ 4πr = n.σ.b̅ 8 r² → i.e. the golden-ratio-pattern , which is related to spin. ------------------------------------------------------------------------------------------------------------- from solution-3 the magnitude d becomes maximum when 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 ≡ maximum or ,when |θr̅| axis is ⊥ b̅ and cone-angle 𝛉  90 . from energy relation e = w = 2l = b̅ .w̅ = j.w² angular-momentum �̅� = [3 ,3770539.10−96j].[5,613308.1045hz] = 1,895644.𝟏𝟎−𝟓𝟎 j/s, i.e. the planck`s cave ,10−35 m , is the safety-valve ≡ the critical state for the stress ellipsoids ,b̅ w̅, vectors in order that this total-energy to superflow and be still into vacuum ≡ [10−62-10−35m]. this vacuum between the |gravity cave planck`s cave| consists the black holes chaos , as the recycling–energy into which → the { energy – space monad |�̅�| ≡ |ab| } , vanishes ≡ decomposes ← ------------------------------------------------------------------------------------------------------------- in physics exist ,two different potential-energies between planck spaces and gravity spaces as uplanc and ugravity .the difference ∆u = ugraupla is the flow-energy between the two energy levels l pla = 6,6262.10−34 m , and l gra= 1,777.10−62 m , from cave l pla = 6,6262.10−34m and fp = √ g a³ 2 = √ 9,8078 (6,6262.10−34)³ 2 = 1,83585.10 50 h then → e pla = h.fp = 6.62607.10−34 . 1,83585.10 50 = 1,216447. 1017 j the minimum charge in planck`s cave → q pla = 1cb = 1,6 .10−19 ev , from voltage relation ∆u = energy charge issues , 1volt = 1joule 1culomb or 1v = 𝟏𝐉 𝟏𝐂 ….(v) in order that 1cb may overflow and flow from planck`s cave to the gravity cave , then must exist a voltage ∆u > = 7,593.10 35 ev , and be as ....(v) 𝐕 𝐆𝐑𝐀 = 𝐡.𝐟𝐏 . 𝟏𝐂𝐛 = 1 ,216447.1017 j 1,6022.10−19ev = 7,593.𝟏𝟎 𝟑𝟓 v , i.e. → when the energy in planck`s cave lpla , increases to an voltage >= 7,593.10 35 v then the motion≡ energy between the two opposite elements of mp-dipole [⊕↔⊝] becomes a stationary flow [⊕,⊝] ≡ charges [⊕--⊝ ] in gravity cave l gra. so energy ≡ motion , exists in l gra as an → pointy rotational centripetal force ← ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 91 the fundamental particles origination mechanism in mfmf pns caves . → when the energy in planck`s cave lpla , decreases to an voltage =< 7,593.10 35 v then motion ≡ energy between the two opposite elements of m-p-dipole [⊕↻↺⊝] creates the a-velocity-vector �̅� which rotates in the axis k with respect to variable �̅�. so when unit-energy vectors �̅� = σ / �̅� exists on the unit-space-monad |ab| ≡ |�̅�| , then is → {the energy –space monad |�̅�| ≡ |ab| ≡ the existing universe} ← for the chaos motion → angular – momentum ≡ the rotational energy �̅� ← for the pns – spaces → angular – momentum ≡ the rotational energy �̅� ← from fig-13 and solution-3 , magnitude d becomes maximum when 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 ≡ maximum or , when |θr̅| axis is ⊥ b̅ and cone-angle θ  90 , where energy e =w= 2l= b̅ .w̅ = j.w² angular-momentum �̅� = [3 ,3770539.10−96j].[1,15346.1051hz] = 3,895312.𝟏𝟎−𝟒𝟓 j/s, i.e. the planck`s cave , 10−35 m , is the safety-valve ≡ the critical state for the stress ellipsoids ,b̅ w̅, vectors in order that this total-energy to superflow and be still into vacuum ≡ [10−62-10−35m] . this vacuum between the |gravity cave planck`s cave| consists the black holes chaos , as the recycling–energy into which → the { energy – space monad |�̅�| ≡ |ab| } , decomposes ← {fig-13} in figure , the herpolhodecone plane is a circle where vector b̅ rotates , and when 𝛉  90 . then cone becomes a circle , where energy follows [ ⊕  ⊝ ] mode as → + when 𝛉 = 0 = < 90 , and when 𝛉 = 90 – 180 ← it was shown that , the energy from chaos [⊕  ⊝ becomes momentum �̅� , �̅� , and is composed ≡ quantized through the physical-[stpl]-line-mechanism , in-circle  triangles  ex-circle  ex-triangles consisted with the three spaces . on this pm exist the three breakages [s² = ⊕ , 2s ² =  , s ² = ⊝] which circularly-charge on the three – extreme triangles {a b c} , {ka kb kc}, {ae be ce} and thus launch the energy-quantity ± 𝐐𝐩−𝐩= [ 𝒄.𝑺 𝒓 ] ≡ centripetal-force from chaos to pns space , on the pascal`s -desargues points-line through the stpl circuit caves r , and from the ptolemy`s & ceba`s still-mechanism . {fig-24} remark . 1.. electron , e , me = 0 , 511mev = 0, 511.10−6 ev . [1,80. 10−27] = 9,198.10−34 kg 𝐚𝐞 = 5,0.10−18 m ,charge 𝐂 𝐞 =1,602.10−19 c , spin 𝐒𝒆 = 𝐒 2 =2,845976.10−34{kg/m/s}, using 𝐄 𝐞𝐊 = 𝐤 𝐫 + 𝒄𝐒 𝟐 𝟐𝐦.𝐫𝟐 = 36.10−20 7,10−18 + 3.108(5,691952.10−34)² 2.9,198.10−34[5.10−18]² = 51,428 ev + 2,111843.1010j = 51,43 ev+ 1,32 .1029 ev , i.e. the energy increases inverse analogous to the cave . 2…since primary motion ≡ [pm] exist from the three breakages only stpl is the n =3 mechanism creating elementary particles as [n-knots=3] in 𝐕 gra = 7,593.1035 v . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 92 the fundamental particles origination mechanism in mfmf pns caves . the unit-space -vectors , �̅�.�̅� , & the unit-energy-vectors , �̅�.�̅� , ellipsoids . figure.17c .. shown the geometrical-meaning of ao̅̅ ̅ ri̅ , √ri² − (ao̅̅ ̅ ri̅)² terms of radius oa i in (1) is shown the inertial ellipsoid ( o, a̅) of radii , a̅, and the interchangable momentum–ellipsoid (o , ρ̅) of radii , ρ̅. (o , ρ̅) . the same for {b̅ , (o, b̅)} in (2) is shown the momentum energy ellipsoid ( o, b̅) of radii , b̅, and the interchangable angular velocity – ellipsoid (o , w̅) of radii , w̅ . s = space in (3) is shown in mohr-method , the geometrical construction from the two interchangable ellipsoids→the energy-rotational momentum–ellipsoid ( o , ρ̅) , where { work } = the angular-velocity-vector -unit-sphere , s =the inertialellipsoid (o , a̅) , { force }= the reaction to the velocity-change-motion , s = the mass ellipsoid { mass } = ( gd = ja ≡ m̅ ) , and between the types of ellipsoids issues ρ̅ . a̅ = j a ..(12) the above property of the two interchangable-ellipsoids defines the deep relation between , the angular-velocity-ellipsoid → j1w1² + j2w2² + j3w3² = 2l = c = ja w̅ ² ….(13) and the momentum-energy-ellipsoid → 1 j1 b1² + 1 j2 b2² + 1 j3 b3² = 2l = c = ja w̅ ² .....(13a) conclusions : 1..from the generated radius �̅� =the inertial-ellipsoid of a solid fig-17-1 corresponds another radius b̅ , fig-17-2 momentum –energy– ellipsoid , to the common-point o of rotation .the radius �̅� is perpendicular to the tangential-plane at the nib of the �̅� angular velocity-ellipsoid . similarly the radius �̅� is perpendicular to the tangential-plane at the nib of the �̅� of the angular-momentum-ellipsoid . relation ρ̅ . a̅ = j a= a constant , means ρ̅ ⊥ a̅ ≡ space + anti-space ≡ motion  �̅� ≡ [ i ≡ √−1 2 ] * ρ̅ ≡ the argand-diagram. from relation (12) , the vertical-projection of the radius , ρ̅ , at the corresponding conjugate radius a̅ , of the unit sphere , denotes the meter to the solid last angular momentum . because the unit-energy-vectors , �̅�.�̅� , are related as �̅� = ab = σ / b̅ = σ / j.w² , i.e. the common velocity vector c = w r = 2.b / π r ³ , therefore issues and for all the 5-spaces . 2..inversely the unit-energy-vectors , �̅�.�̅� , decompose when bcrit=j w = 𝜋r3 2 [c/r] = [ 𝜋r2 2 ]c or when �̅� becomes maximum , i.e. for gravity and antigravity happens when |θr̅| axis is perpendicular (⊥) to b̅ and the cone-angle θ  90 , and energy e =w= 2l= b̅ .w̅ = j.w². from the �̅� physical -magnet , �̅� is quantized to the 2-constitutes ⊕ ⇉ ⊝ as the prior . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 93 the fundamental particles origination mechanism in mfmf pns caves . 6c6.. the de-quantization of energy from , pns  mfmf , ≡ empty space . figure.17d.. the angular-velocity |�̅�| = 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ±√− 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 + ( 𝐉𝟑. 𝐰𝟑 𝟐 𝐉𝟏.𝐜𝐨𝐬𝛝 ) ² 𝟐 = s ± {√�̅� }.i  |𝑤3 |= [s]+[√�̅� ].i , s = [ j3. w3 2 j1.cos ϑ ] , �̅� = [ k pns 2 j1.cosϑ ] , 𝐤 𝐏𝐍𝐒 = √ [ j3. w3]² − 4. sq. j1 cos ϑ 2 from 10.c, the unit-energy-vectors , �̅�.�̅� , fp = 1,836.1045hz in pns–space become 1).. from relation w̅ =2π.f pns , then 𝐰 pns = 56,13308 .1044hz = 5,613308 .1045hz , 2).. from centripetal force in mfmf space fc = σ = m.v³ r =j w² r =b̅ w r , then b̅ =j w ,or b̅ = j w = [ π.r4 2 ].w = π[1,6162.10−35]4 (5,613308 .1045) 2 = 3 ,3770539.10−96 j and for ev , b̅ = 3 ,3770539.10−96 j / 1,6022 .10−19 c = 3,754935.10−76 ev. 3).. from sol -3 the magnitude d becomes maximum when 𝐬𝐐 𝐉𝟏.𝐜𝐨𝐬𝛝 ≡ maximum or , when |θr̅| axis ⊥ b̅ and cone-angle 𝛉  90 . from energy relation e = w = 2l = b̅ .w̅ = j.w² angular-momentum �̅� = [3 ,3770539.10−96j].[5,613308.1045hz] =1,895644.𝟏𝟎−𝟓𝟎 j/s, i.e. the planck`s cave ,10−35 m , is the safety-valve ≡ the critical state for the stress ellipsoids ,b̅ w̅, vectors in order that this total-energy to superflow and be still into vacuum ≡ [10−62-10−35m]. this vacuum between the |gravity cave planck`s cave| ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 94 the fundamental particles origination mechanism in mfmf pns caves . consists the black holes chaos , as the recycling–energy , into which → the { energy – space monad |�̅�| ≡ |ab| } , vanishes ≡ decomposes ← → in physics exist , two different potential-energies between planck spaces and gravity spaces as uplanc and ugravity .the difference ∆u = ugraupla is the flow-energy between the two energy levels l pla = 6,6262.10−34 m , and l gra= 1,777.10−62 m , the total energy in planck`s cave is → e pla = 1,216447. 1017 j , and the minimum charge in planck`s cave → q pla = 1cb = 1,6 .10−19 ev , from voltage relation ∆u = energy charge issues , 1volt = 1−joule 1−culomb or 1v = 𝟏𝐉 𝟏𝐂 ….(v) in order that 1cb may overflow and flow from planck`s cave to the gravity cave , then must exist a voltage ∆u > = 7,592354 .10 35 ev , and be as ..….(v) 𝐕 𝐆𝐑𝐀 = 𝐡.𝐟𝐏 . 𝟏𝐂𝐛 = 1 ,216447.1017 j 1,6022.10−19ev = 7,593.𝟏𝟎 𝟑𝟓 v , i.e.  when the energy in planck`s cave = lpla , increases to an voltage = 7,593.10 35 v then the motion≡ energy between the two opposite elements of mp-dipole [⊕↔⊝] becomes a stationary flow [⊕,⊝] ≡ charges [⊕  ⊝] in gravity cave l gra . so energy ≡ motion , exists in l gra as an → pointy rotational centripetal force ←  when the energy in planck`s cave lpla , increases to an voltage 𝐕𝐂𝐑 > 7,593.10 35 v then motion ≡ ⇄ ≡ energy between the two opposite elements of m-p-dipole [⊕↻↺⊝] creates the negative-angular -velocity-vector �̅� ↓ , which rotates in the symmetrical to neutral circle axis k = ↓ with respect to the interchangable variable �̅� ↓ . this is the case of the black-holes origination .  when the energy in planck`s cave lpla , decreases to an voltage = < 3,695.10 30 v then motion ≡ ⇄ ≡ energy between the two opposite elements of m-p-dipole [⊕↻↺⊝] creates the positive-angular -velocity-vector �̅� ↑ , which rotates in the symmetrical to neutral circle axis k = ↑ with respect to the interchangable variable �̅� ↑ . this is the case of the universe origination . so thus , are defined the unit-energy vectors , �̅� = σ / �̅� = σ / j.w² , which exist on the unit – space – monad |ab| ≡ |�̅�| , and which is { the energy – space – monad |�̅�| ≡ |ab| ≡ the existing universe }. in fig-14 are shown the 5-states for angle → θ  0ᶱ , 45ᶱ , 90ᶱ , 135ᶱ , 180ᶱ . 1.. for angle → θ  00ᶱ , �̅� ≡ �̅� , voltage = 7,593.10 35 v , ↑ on neutral – circle 2.. for angle → θ  45ᶱ , �̅� / �̅� , voltage < 7,593.10 35 v , < ≠ neutral – circle 3.. for angle → θ  90ᶱ , �̅� ⏊ �̅� , voltage ≡ 7,593.10 35 v , ⇉ in neutral circle 4.. for angle → θ  135ᶱ , �̅� \ �̅� , voltage > 7,593.10 35 v , ≠ > neutral – circle 5.. for angle → θ  180ᶱ , �̅� ⏊ �̅� , voltage >=7,593.10 35 v , ⇅ off neutral-circle 4).. from solution-4 , the {energy-space-monad |r̅| ≡ |ab|} vanishes in bh-chaos , where energy increases to an voltage >> 7,593.10 35 v and then motion ≡ energy between the two opposite elements collides them to the mp-dipole [⊕↔⊝]. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 95 the fundamental particles origination mechanism in mfmf pns caves . 5).. from solution-5 , the {energy-space-monad |r̅| ≡ |ab|} vanishes in bh-chaos , & energy is quantized to the 2-constitutes from the �̅� , physical-magnet ⊕ ⇉ ⊝. figure.17e..the stability of photon as electromagnetic wave with the two frequencies . the dual property of the simultaneous existence as electric and as magnetic fields in wavelengths , λ x , λ y , as effective range .which is thrusted through the newton’s gravitational force g only , becoming from the trigonometry relations [the space ] , sin(2φ) = sin(𝜋 − 2φ) = sin( 𝜋 2 − φ ) , and from physics [energy] e x = g , maxwel electric displacement current 𝝑d / 𝝑t is created from the ( 𝐜𝐱 , 𝐜𝐲 ) vectors motion happening from gravity continuous action , and producing the work w = 𝐜 𝐲 g for photon { markos rotating method of{ 2-vectors 3-poles mechanism }which circles and squares as areas pass from the point m determined by the position 𝐀 𝐞 , and where is such that issues π r ² = cmnh square then .the squaring of the circle becomes at the equilibrium positions which are from relation → π [ 𝐂𝐀 √𝟐 ]² = cm² = |cm`|² ← namely the square | cmnh | = square | cm`n`h`| . because of the bellow motion then , the total motion in the wavelength , λ , corresponds to the angle a = 45 °, meaning that the motion of space ≡ the electric-field , and anti-space ≡ the magnetic-field equilibrium as an bellow . the two wings in 90° type simultaneously oscillate to its 45 ° bisector -axis , and the produced work from the electric-field-area is stored into the conjugate magnetic-field-area . the common axis of the two perpendicular fields , carry the golden-ratio energy-surfaces , and runs with the same velocity c ̅ to the direction of the inner motion . the epilogus in [89 96] it was proved and elucidated that , a.. monad is quaternion [x + i y] = λ̅ type , and its energy is vector as q⃗⃗ = {λ}.x , where x = the displacement . since 1 = c. r³. fp² = constant , so → r³. fp² = constant . since also ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 96 the fundamental particles origination mechanism in mfmf pns caves . λ = 2r then r³. fp² = λ ³. fp² = c , and energy becomes →vector q⃗⃗ = {λ}.√c/ fp² 3 ← [𝐄 𝟏] b.. in mechanics an energy-vector is an conductor aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = |q⃗⃗ |  , and is the quaternion in material-geometry where zero-point 0 ≡  ≡ {⊕+⊝}on where motion |q⃗⃗ | ≡ �̅� 𝐱 as an electric field on aka axis and as q⃗⃗ |  ≡ �̅� 𝐲 ≡ magnetic field ⊥ aka , is transferred from one edge point a , to the other edge point k a as → �̅� , �̅� 𝐱 , �̅� 𝐲 ← [𝐄 𝟐] c.. energy is the work , the motion , produced in energy-monads and for photon is equal to w = 2l = b̅ . w̅ = r m v = [r. p] = j w² ≡ [ �̅� 𝐱 ⊥ �̅� 𝐲 ] where �̅� 𝐱 = �̅� 𝐲 , in the conductor aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = 2.r ≡ λ where , λ is -the -wavelength and p = m c = the momentum . [𝐄 𝟑] d.. it was prooved that → any diameter d of a circle in the stpl – circuit , the three – poles –squares-rotational system is an conductor where d = aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ = |q⃗⃗ |  = | 𝐜.�̅� 𝐫 ² | = 𝐈𝐄 the electric-wave-intensity , and a ↑ ka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ≡ 𝐈𝐌 is the magnetic–wave intensity. their square -vectors – area = [ aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ]x[a ↑ ka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ] = [�̅� 𝐱�̅� 𝐲] = the energy square ≡ cm x cm = cmnh square = on circle ( e , ea ) , and from relation �̅� = σ . φ = f a φ area a = f / σ = |cm|² / σ , i.e. plane-force f = σ . |cm|² , and in space volume-force f = σ . |cm| ³ ≡ [𝐄 𝟒] or [ aka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ]x[a ↑ ka ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ]x[d] = cmnh x 𝐐 ⤱ ⇉⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = cmnh x �̅� ⤱ ⇉ , where |𝐐 ⤱⇉⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | ≡ �⃗⃗� , which is the energy-quantum in the propagating electromagnetic wave where | 𝐈𝐄 ⊥ 𝐈𝐌 | . since motions b̅xw̅ , are constants from (b) and from poinsot`s solution w1² + j2 w2² = a² , then angular velocity-ellipsoid polar-paths are the parallel circles and perpendicular to the angular-momentum ellipsoid nibs.. [𝐄 𝟓] , [70] e.. when is found an obstacle at point 𝐊𝐀 , then the particles of conductor medium oscillate perpendicular . this obstacle at ka point , is the presentation of the , ⊕ constituent and the absent of ⊝ constituent and so  , transversepropagating-wave , or moving energy-vector |⊕ 𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ |=|↕| at node ka is equal to e-vector| ⊕ 𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ |≡ |↔| at node a , and energy volume is doubled as , 2l = b̅ .w̅ , in kaa⃐⃗⃗⃗ ⃗⃗ ⃗⃗⃗ conductor meaning that the meter of volume is the duplication of total energy l .the reaction to the motion of squares {|qx =co ⇉| x|qy= cb ⇈|} ≡ |𝐐 ⤱⇉⃗⃗⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ | ≡ [ 𝐜𝐒 𝐫² ].|𝐐 ⇉⃗⃗⃗⃗⃗⃗ ⃗⃗ | is equal to quantum �⃗⃗� . [𝐄 𝟔] f.. any two perpendicular energy-vectors , |𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↔ | , |𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ = ↕ | of this propagating energy , form the energy-square , |↔|.|↕|≡|𝐐 ⇉ |.| 𝐐 ⇈| ≡ q0 , on conductor aka and then energy-circular-prism = |aka| .{𝐐 ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ }.{ 𝐐 ⇈ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ } ≡ ≡ [�̅� 𝐱�̅� 𝐲].|aka| = | ↔ |.|↕|.|aka| , i.e. an energy-circular-cone which is a moving energy -volume and because from cauchy equations of stresses in three dimensions , the energy stress remains flat onlywhen the plane-section is a circle [𝐄𝟐 ] and the square becomes circle ,anti-quadrature. because from mechanics , the sphere occupies the least-resistance to motion where the geometrical square–cone [cm x ch x[co⃗⃗⃗⃗ ⃗ ↑] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 97 the fundamental particles origination mechanism in mfmf pns caves . is altered to equivalent energycircular cone [π.[ 𝐂𝐌 𝟐 ]² (x) [co⃗⃗⃗⃗ ⃗ ↑] and to the energy unit -volume co³.[𝐐 ⇉⃗⃗⃗⃗⃗⃗ ⃗⃗ ] ≡ 2 .sphere 𝟒𝛑 𝟑 r ³ and to energy-sphere cone ⸿ = k π r³ , where the geometrycube [|↔|.|↕|.|𝐀𝐊𝐀|] = x ³ so becomes the energy -sphere – cone as → k π r³ = x ³ ← [𝐄 𝟕] g.. since frequencies fn are the quantum of energy , and electromagnetic vectors are always perpendicular between them as | 𝐅 𝐄 ⊥ 𝐅 𝐌 | , so are everywhere in nature from microcosm to macrocosm . because gravitational force exists beyond planck`s length therefore the propagating-plane-force 𝐅𝐏 = σ . |cm|² , is continually perpendicular to the two perpendicular constituents of electromagnetic vectors as |𝐅𝐏 ⊥ 𝐅 𝐄 ⊥ 𝐅 𝐌 |. distance is the quantum of e-geometry , while material-point is the quantum in physics and in material – geometry which is the composition of opposites and are the elements in chemistry and physics . as in algebra zero ,0, is the master-key number for all positive and negative numbers because their sum and multiplication is zero , and the same on coordinate system , ± , axes pass from zero . [𝐄 𝟖] the rolling of positive ⊕ , constituent on the negative ⊝ , constituent , creates the neutral material point which equilibrium . spin �̅� , is its angular momentum and also the first -discrete-energy-monad which occupies discrete-value and direction , in contradiction to the point which is nothing , dimensionless and without any direction . point-caves are the energy-magnets from which the quantum of energy are generated . h.. the space is quantized as energy-caves under the effect of gravitational-force g , as well in conductors and this energy or motion is quantized as frequency in energy caves , or in conductors , following the kepler`s first-law of equal areas where in equal intervals and in conductors , is the electricity or electromagnetic waves . [𝐄 𝟗] quaternion [(+)↻↺(-)] is a quantum-mould for space [(+)(-)] and energy ≡ motion ≡ force x displacement as [↻↺ or ↑↔↓] ≡ standing box 𝐁𝐐 ≡ an material point which carries the principal stress σ between positions a (+) , b (-) , and σ , is the centripetal acceleration of minimum energy becoming from the in-storage ab acceleration and which is equal to the gravity g . from quaternion quantum mould [r + v̅ i ]𝟏/𝐰 = 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐰 is created the min-space ≡ cave r = 1, 07.𝟏𝟎−𝟕 m and the min-energy ≡ v̅ = w r = 2πr f , as frequency 𝐟𝐦= 2,839844 . 𝟏𝟎𝟏𝟎 h . g pushes the → min-quantum-energy ≡ motion 𝛔𝐄 , into the inscribed-cave π.[ 𝐂𝐌 𝟐 ] ² ≡ π.ea² . gravitational-force g effecting on light velocity �̅� creates the electron-charge �̅� and electron 𝐞 ̅, while acting on planck`s-cave on the gravity g ≡  σ , as g = g k = 𝐤𝐄 g = 𝐤𝐋 σ = g.gl kl , and �̅� effecting on the min-planck-cave in lp formulates hydrogen-cave h with its electron as he . [𝐄 𝟏𝟎] , [109] i.. the physical-notion of the duplication of the cube , in the quadrature physical energy mould shows that nature follows the geometry-rules in order to exist as space and as energy , which is motion of the ,+, constituent to the , , constituent . this motion is conserved by propagating in caves either for standing ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. b̅n or for moving caves as this is the duality-photon → v̅ . [ fn̅ + fn ] ≡ | 𝐯 𝛑².𝐫⁴𝐧 |. b̅n + | c̅ | fn .material-geometry meters for quantum-length is the quaternion ≡ dipole ≡ the two-points joint with motion between them ,while for the quantum-surface is the unit circle π.r ² = π , and is r =unit 1. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 98 the fundamental particles origination mechanism in mfmf pns caves . for the quantum-volumes , is the. unit volume sphere 4 3 π.r ³ = [ 4.r 3 ] (π.r ²) = 4 π 3 , for r = unit 1 . the unit-volume is formulated from the inscribed-square of unit circle-side co = cb = 1 /√2 , and when placed the-unit-energy-magnitude oc on square co x co then is unit-volume [co x co x co⃗⃗⃗⃗ ⃗ ↑] . this unit-volume when placed in , the three -poles squares-rotating-system ,then at point kn , unit volume [co x co x [co⃗⃗⃗⃗ ⃗ ↑] becomes [cm x ch ] x ch⃗⃗⃗⃗ ⃗ ↑] = circular-cone [π.( 𝐂𝐁 √𝟐 )²] . [co⃗⃗⃗⃗ ⃗ ↑] = unit sphere 4π 3 r ³ , equal to quantum of energy . because physical-magnet ⊕ ⇉ ⊝ compels force �̅� ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ , between a and ka points and then unit-sphere 4π 3 r ³ becomes the double unit-sphere 2.{ 4π 3 r³} and because the energy square [cakap] is double that the unit square |cb x co| , then force �̅� ⇉ aka⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗ is the pivot of spinning indicating ± �̅� ≡ �̅� , in all energy monads .[𝐄 𝟏𝟏] j.. gravity and antigravity are the two halve spins from the angular momentum �̅� ≡ ± work created from the velocity vector �̅� by rotating ↻ on planck`s cave diameter.[𝐄𝟏𝟐] this property , ± �̅� ≡ �̅� , is in all energy monads and when applied on initial , ⊕ ⇉ ⊝ , answers to the 3d-stability of all elementary-particles and cosmology equilibrium . gravity + g ≡ the vector of angular-momentum �̅� ≡ + spin ≡ ↓↻ ≡ the work stored in the [4-1-3] semi circle of the , �̅� �̅� vectors circle-surface ,produced from the gravitational-force , g , as acting into the -planckcave , and on the light velocity vector of , + ↓ �̅� direction . [107] antigravity g ≡ the vector of angular -momentum �̅� ≡ spin ≡ ↑↺ ≡ the work stored in the complementary [4-2-3] semi circle of the , �̅� �̅� vectors circle surface which is produced from the gravitational-force g , as acting into the-planck-cave , and on the light velocity vector of the , ↑ �̅� direction . k.. the black-holes exist as parallel origination either as  planar black-holes or  atoms black – holes , when g force is acting on the -planck – cave . [𝐄 𝟏𝟑] momentum b̅ & w̅ , as real ≡ the existing universe as is the real objective reality |�̅�| ≡ the numeric vector-value of the material-point [⊕, r ,⊝] in [pns] ≡ the work stored in polhode cone of the |𝐎𝐒̅̅ ̅̅ | rolling nib -vector produced from the eternal centripetal force , 𝐅 𝐂 as the newton`s action in the planck`s cave, and from the �̅� angular momentum in the , + ↓ b̅ , direction . the real type is applied on all different monads and their compounds monads . [109] momentum b̅ & w̅ , as imaginary ≡ the black holes as is their real objective reality ± �̅�.i ≡ s  the numeric amplitude-vector-value of material-point [⊕,r,⊝] in [pns] and the imaginary �̅�.i ≡ centripetal force 𝐅 𝐂 ≡ the wave amplitude -value ≡ the work stored in the herpolhode cone of the |𝐎𝐁̅̅ ̅̅ | rotating �̅� -vector and also produced from the eternal centripetal force 𝐅 𝐂 , as the newton`s reaction into the planck`s cave and from the �̅� angular momentum in , ↑ �̅� direction. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 99 the fundamental particles origination mechanism in mfmf pns caves . l.. in case of points a , b , c being on a sphere then [stpl] becomes a cylinder as this is the angular-momentum �̅� = 2l w̅ = 2l 2πf = [ 2l 2π ].[ 1 f ] = constant 2π [ 1 fm ] = spin [89] d... the origination of the fundamental particles 1d… the balancing of the space and anti-space . [64] ↺ ↻ ↻↺ → ↻↺ + ↺ + ↻ figure -18: the work w, of two-quaternion-monads 𝐳 = (s + v̅. i) ≡ 𝐀𝐁̅̅ ̅̅ ≡ �⃗⃗� is the action of any two tangential & opposite equilibrium-dipole [⊕↔⊝] with v = w⃗⃗⃗ .r. for quaternion actions exists the square law for both real and imaginary part , as 𝐳 * 𝐳 = (s + v̅. i )² = s² +[v̅. ]² +2.sv̅. i = s² – v̅ ² ± 2.s v̅ = |s|² -|s̅|² ± 2.|s|.|s̅| , for |v̅| = s vorticities ± λ (rotating energy) , becomes from the collision on common-circle and from the thrust on the velocity-breakages .the work is , w = [ n . p ] = [ λ.λ] , where λ = displacement of a to b and it is a scalar magnitude called wavelength of dipole ab. λ = the amount of rotation on dipole ab ( this is angular momentum and it is a vector ) . momentum ± λ̅ = r.m.v = r m wr = mr².w̅ , where w̅ is the angular velocity (spin) which maps velocity vector v̅ on the perpendicular to ± λ̅ plane with two components v̅ e ⊥ v̅ b . the tangential velocity v̅ e = wr , is the quaternion v̅ e = w r = z̅ = [ s + v̅. i ] where s = v̅ e = |r.w̅| and v̅. i = |w̅ x r̅ |. in a spherical cave the biaxial ellipsoid (σx = σy) exists as momentum +λ on caves of diameter 2r with parallel circles → 0 .the biaxial anti-ellipsoid (σx = σy) exists as equal and opposite momentum λ on the same diameter 2r with anti parallel circles → 0 . equilibrium of the two ellipsoids ± λ , presupposes a stabilizersystem attached to ellipsoids such that opposite momentum is distributed to the center of mass of the total system , recover equilibrium , which is the center of the spherical cave. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 100 the fundamental particles origination mechanism in mfmf pns caves . the biaxial ellipsoid and anti-ellipsoid are inversely directed and rotated in the same circle , so the two velocity vectors , v̅ ,− v̅ , collide .this collision of the two opposite velocity-vectors is the physical-action ( thrust ) of the two-quaternion and their action is , ( s + v̅. i )(©)( s + v̅. i ) = [s + v̅ . i] ² = s² + | v̅| ². i ² + 2|s|.|v̅ |. i = s² |v̅| ² + 2|s|.| w̅ r|. i = s² |v̅| ² + [2w̅].|s| |r|. i , where , s² = (r w) ² → is the real-part of the new-quaternion, |v̅|² = |w̅ x r̅|² = (w.r) ² → the always negative-anti-space ( a vector ⊥ to w r plane ) , [2 �̅�].|s|.|�̅�. i = 2w.(sr). i → the double angular-velocity-term , and for the pns space where |s| = |v̅|² then s² = (w.r) ² ≡ ⊕ ≡ s ² = + charge |v̅|² = |w̅xr̅|² = (w.r) ² ≡ ⊝ ≡ s ² = charge [2w̅].|v̅||.r̅. i = 2 |v̅|² = 2 (w̅.r) ≡ 2[⊕↔⊝] ≡ 2s ² = 0 charge i.e. for the equilibrium-recovery (maybe a surface cylinder with 2r diameter) , fig-16-17 2d.. the vibration of particles in all levels [54] . figure 19 : the glue-bond electrons of nucleus rotor , form rhodonea-like curves curves created by rotor`s electron are such that they keep a continuous pressure on protons. 1.. the continuous action of negative ⊝ , on one positive ⊕ , is equivalent to a circular motion of the ⊝ , ⊕ , with velocity v = √f. r/m , and this because exists l = r = constant . 2.. the continuous action of negative ⊝ , on two positive ⊕⊕ of glued bond , is equal to a circular and other of one-step longer motion of ⊝ to ⊕⊕ , or circular motion on ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 101 the fundamental particles origination mechanism in mfmf pns caves . the common–length , l , in polar equations is , l = a . cos(c.θ) or l = a . sin(c.θ) where a = the length of the base =2r+2r = 4r , c = rim`s number which produce a c-petaled rose , if c is odd or 2c-petaled rose and if c is even number. the period of both cos(cθ) , sin(cθ) is 2π/c , and the number of petals for the period is [ 0 ,2π/c] will be c or 2c , according to as c is odd or c is even , so the degree of the cartesian equation of the curve is , c +1 for odd-c , and 2(c+1) for even-c . 3.. the continuous action of negative ⊝ , on three positive ⊕⊕⊕ of glued bond , is equivalent to a circular-and other of two-step longer motion of ⊝ to ⊕⊕⊕ , or circular motion on the common–length . length , l , in polar equations is , l = a . cos(c.θ) or l = a . sin(c.θ) , where a is the length of the base = 2r + 2r + 2r = 6r , c = rim`s number which produce a c-petaled rose , where c is odd or 2c petaled rose number . this is seen and from equations issuing in figure.21-2 → r = rd.sin φ and r = rd.sinφ where a , are the equal radius of point a of polar path . rhodonea curves follow the odd or and even number of protons in nucleus by changing their rolling length a = (2r) , and the polar – length l = a . cos(c.θ) depending on ,θ, as above . the common bonding point of the negative circle and the positive line of the rolling circle obliges that point to follow the cycloid motion in maximum degree and thus executing hypocycloid motion . particles vibration in all levels of material-geometry . 4.. the n-knots figures exist in figures where the vector -force polygons have the same number n , as the in figures . for the circularlycharged breakages the 3-elements [σ = n . h]  [s² = ⊕ , 2s²= , s² = ⊝] formulate the primary-particles 3-k,figures. since n =3 , the only shape in spaces is the regular triangle , and from this n < 4→ n there are vast regions of space that are very nearly empty of them , 5.. the case of {tcic}≡ [ σ = n . h ]  8 x ① which is the physical mechanism , gives the n-masses of atom`s &compounds determining the atoms complex-vector-response. these particles exist in all of the universe in the sense that they are uniformly distributed throughout space and consist the objective reality which is the existing universe ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 102 the fundamental particles origination mechanism in mfmf pns caves . figure – 20: the carbon , nitrogen , oxygen , sulfur , atoms-conductors sustains into the first 4-dimentional spinning , tetrahedron cube – sphere mould . figure– 20a:the hydrogen , carbon , oxygen , atoms-conductors into the sphere-cube mould . deka hexahedron , octahedron ≡ cube , tetrahedron ≡ 4-points , triangle ≡ 3-points , vector ≡ 2-points ≡ the closed-conductor , 1-point ≡ the opened -conductor ≡ the bracket-hook . oxygen is the element which fills 8 2 = 6 vertices of cube as el → { 1𝑒 0𝑒 1𝑒 1 𝑒 2⃡ 0 𝑒 1𝑒 1𝑒 1𝑒 } ≡ (± 2 2 ) , where ⓪ is the first , zero-acting cube filled with 1e , therefore is the only steady element which occupies the first energy-level ① with 2 electron-positions as ① = 2e = 2⃡ positions at 1⃗ , 1⃗⃖ the electron being in the hydrogen-cave precesses because of the different axis of rotation and nutation`s , from the continuous and immense-communication to the effect of gravity g . since electron is continually oscillating , with the nutation-frequency 𝐟 𝐍 , so produces a continuous and oscillating magnetic-field –mwhich in turn is the source of an oscillating electric-field – e which implies the regeneration of each other , i.e. it is a propagating electromagnetic-wave where e = b c , and with a quantum-energy e = h fn or e = 2μ.b , this energy e consists the hydrogen bracket hook hbh , or [bh⃗⃗⃗⃗ ⃗] and it is the only free monad which can bond to all energy structures . 5d.. the natural electromagnetic energy-tetrahedron and , the 3conductors into the energy cube and spheres : [102] ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 103 the fundamental particles origination mechanism in mfmf pns caves . figure – 20.athe dynamic loading of → tetrahedron – cube sphere ← atom –structure. material geometry uses the only-three elements for the origination of primary particles , as weak , w−zo and strong w+z+0,− forces , spin b̅ = s̅ , hydrogen cave , r h , and electron e−. where [0] ≡  ≡ the rest-energy ≡ [⊕ ↔ ⊝] → [ ⊕ ∷ 2 +⊝ ∷ 2 ] , split into two . and , [0] ≡ zd̅̅̅̅ ≡ the rest space ≡ conductor − dm⃗⃗⃗⃖ ← + z e⃗⃗ ⃗ → , from relation [⊕←ds→⊝] , where their properties and corelations are all motions into their spaces , from energy. in fig-20b the electric magnetic field of 3-conductors which lie on the dabc tetrahedron .the stability of the → { ⊕ 4-space [d a b c]} regular – tetrahedron , occurs into the {⊝ 8-anti-space cube , [da1a d1 , b b1cc1]}, and the positive → { ⊕ 8 space – cube , [da1a d1 , b b1cc1]} , into the {⊝ 16-anti-space [ 1 ,2 ,3 ,4 ,5 , 6, 7, 8 9 ,10 ,11, 12 , 13 ,14 , 15, 16] } sphere cube on vertices.= point [the point 16 ≡ |⇉|z|] consists the navel-string gate into the triangle-circle-system {z-(∆d,abc) , ∆[t1t2 t3]} of the stationary bases which results on the 2 -anti-parallel-strands . the dual photon v̅ [ σφ 2πr + σ 2πr ] ≡ v̅.[ fn̅ + fn ] , occupies stresses = σ and velocities v̅ , in the tiny-caves r . the colours in light , are the still-sub-units in storage →[v̅ . fn̅ ]← and exist as frequencies of → violet , blue , green , with their complementary yellow , orange , red ← every 8-electrons are vibrations on atoms-cube-structure {a tetrahedron in cube in a sphere } whether it be sound or light . above structure is followed by all compounds . molecules are systems consisted of the periferal ≡ skeleton 𝐌 𝟏𝟔 and the central ≡ fittings 𝐌 𝟖 , and the accessories ≡ 𝐌 𝟒 or 𝐌 𝟐 , 𝐌 𝟏 . this property of atoms and of the compounds is the critical-valve of switching the motion as this happens in the electro-magnetic solenoid valves with high or low pressure and flow rates directly or not . figure – 20.b:the electric magnetic field of 3, conductors results on 2 -anti-parallel strands .the stability of →{ ⊕ 4-space [d a b c]} regular tetrahedron , in {⊝ 8-antispace cube , [ da1b d1 , c c1ab1]}, and the →{ ⊕ 8space [ da1bd1 , cc1ab1]} cube, into the {⊝ 16-anti-space [ 1 ,2 ,3 ,4 ,5 , 6, 7, 8 9 ,10 ,11, 12 , 13 ,14 , 15, 16]} sphere ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 104 the fundamental particles origination mechanism in mfmf pns caves . cube vertices . point [16 ≡ |⇉|z|] consists the navel-string gate in {z-(∆d,abc),∆[t1 t2 t3]} stationary bases . stability analysis in [88] data : a.. the atoms cube coordinate-system : the relation of a cube of side a ,inscribed in a sphere of diameter d =2r , is in their volumes as v sphere = 4.𝜋 r ³. 3 = 𝜋 d ³. 6 , v cube = a³ and since d² = 2a² + a² = 3,a² , then a = d. √3 = d.,√3 3 = √3,d 3 = 2√3,r 3 ,and the coordinates of the 8-cube-vertices are for center o → 𝐗 𝐎 = 0 , 𝐘 𝟎 = 0 , 𝐙 𝟎 = 0 ← 𝐗 𝟏 = − a. 2 , 𝐘 𝟏 = − a. 2 , 𝐙 𝟏 = − a. 2 , 𝐗 𝟐 = + a. 2 , 𝐘 𝟐 = − a. 2 , 𝐙 𝟐 = − a. 2 , 𝐗 𝟑 = + a. 2 , 𝐘 𝟑 = + a. 2 , 𝐙 𝟑 = − a. 2 , 𝐗 𝟒 = − a. 2 , 𝐘 𝟒 = + a. 2 , 𝐙 𝟒 = − a. 2 , 𝐗 𝟓 = − a. 2 , 𝐘 𝟓 = + a. 2 , 𝐙 𝟓 = + a. 2 , 𝐗 𝟔 = − a. 2 , 𝐘 𝟔 = − a. 2 , 𝐙 𝟔 = + a. 2 , 𝐗 𝟕 = + a. 2 , 𝐘 𝟕 = − a. 2 , 𝐙 𝟕 = + a. 2 , 𝐗 𝟖 = + a. 2 , 𝐘 𝟖 = + a. 2 , 𝐙 𝟖 = + a. 2 , since tetrahedron in cube is formed in three equal-diagonals 𝐝 𝟏−𝟑 = a.√2 , 𝐝 𝟏−𝟓 = a.√2 , 𝐝 𝟏−𝟕 = a.√2 of the perpendicular 3-planes , p x−y = p y−z = p z−x = a² , and since issues 𝐝 𝟏−𝟑 ⏊ 𝐝 𝟏−𝟓 ⏊ 𝐝 𝟏−𝟕 , these consist the-scanning vectors-plotting for all systems . since the cube’s outer-sphere diameter d , is equal to the cube-diagonals [1-8] , [2-5] ,[3-6] , [4-7] , therefore consist 4-big circles on sphere and perpendicular between them , in contradiction to the 3 circles in the 3-planes which consist the small circles . the geometrical matric of tetrahedron-cube-sphere mould = ( stcs ) where , 𝐏𝟏 𝐏𝟐 𝐏𝟑 𝐏𝟓 𝐏𝐨 𝐏𝟒 𝐏𝟔 𝐏𝟕 𝐏𝟖 p o = the coordinates x o , y o , z o of the common center of 2-spheres . p 1 = the coordinates x 1 , y 1 , z 1 of the tetrahedroncube point -1 p 2 = the coordinates x 2 , y 2 , z 2 of the sphere cube point 2 p 3 = the coordinates x 3 , y 3 , z 3 of the tetrahedroncube point -3 p 4 = the coordinates x 4 , y 4 , z 4 of the sphere cube point 4 p 5 = the coordinates x 5 , y 5 , z 5 of the tetrahedroncube point -5 p 6 = the coordinates x 6 , y 6 , z 6 of the sphere cube point 6 p 7 = the coordinates x 7 , y 7 , z 7 of the tetrahedroncube point -7 p 8 = the coordinates x 8 , y 8 , z 8 of the sphere cube point 8 on the above geometrical coordinates system are placed the 4-spaces from the oz = od extended conductor and equal to monads . for the cave-spin , �̅� = r m v which is a monad , exist the 4-spaces as fig-1 → (( the n spaces of spin �̅� are the polygons 𝐒𝐧 , the n anti-spaces of spin �̅� are the polygons 𝐒𝐧 )) ← the sub-spaces ± √ �̅� 𝒏=𝟏−∞ are the polygons with n = 1 ≈ ∞ knots since knots are the degrees of the polynomials of each monad then , the oz monad determines the number of knots beginning from number 3 , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 105 the fundamental particles origination mechanism in mfmf pns caves . 3  is the figure of regular triangle in poly-3rd degree . the 3 and ± √ oz̅̅̅̅ 𝑛=3 4  is the figure of regular square in poly-4rd degree . the 4 and ± √ oz̅̅̅̅ 𝑛=4 5  is the figure of regular pentagon in poly-5rd degree . the 5 and ± √ oz̅̅̅̅ 𝑛=5 6  is the figure of regular hexagon in poly-6rd degree . the 6 and ± √ oz̅̅̅̅ 𝑛=6 7  is the figure of regular heptagon in poly-7rd degree . the 7 and ± √ oz̅̅̅̅ 𝑛=7 8  is the figure of regular octagon in poly-8rd degree . the 8 and ± √ oz̅̅̅̅ 𝑛=8 9  is the figure of regular eniagon in poly-9rd degree . the 9 and ± √ oz̅̅̅̅ 𝑛=9 n  is the figure of regular n-gone in poly-nrd degree . the n and ± √ oz̅̅̅̅ 𝑛=𝑁 remarks : 1… gravity force g ≡ motion , exists without mass and is quantized → spread in {space and anti-space} or ≡ qua ≡ {is the quantized-units in-area of space and anti-space }, and as a mould 𝚽 golden-ratio , exists in the impedance b . 2…the case of {qua} ≡ [ σ = n . h ]  ∞ x 𝚽 which is the physical mechanism , gives the ∞ impedance b determining the objective reality which is the existing universe. for n =2 then [σ = n h]  2 x g /φ , is the light-velocity �̅� = [ 𝐆 𝚽 𝐀 ] =2.9982.108 m/s. for n =1 then [σ = n h]  g / c = charge q̅ = 𝐆 𝐜 √𝟐 = 1,58.10−19 coulomb . 3… for n = 3 is stpl mechanism which creates the elementary particles , [n-knots = 3] in gravity-voltage 𝐕 gra = 7,593.1035 v. this is because the primary motion ≡ [pm] exist from the three 3 breakages elements [σ = n . h]  [s² = ⊕ , 2s²= , s² = ⊝] b.. the atoms-elements mass , stiffness flexibility [101] 6d..the dimensioning of the [stpl] – mechanism defining the lengths , a = kb kc , b = kc ka , c = ka kb , d = [ a+b+c 2 ] =the semi-perimeter then the inscribe radius r = √d(d−a)(d−b)(d−c) d = |oa| , the coordinates for the point k=da are [ bc b+c−a ] : [ ca c+a−b ] : [ ab a+b−c ] , while for point o are [ b+c−a a ] : [ c+a−b b ] : [ a+b−c c ] . fig-17 the stpl mechanism is the mould consisted from any common circle o,oa=[oa`≡ oae], o,ob = [ob`≡ obe] , o,oc = [oc`≡ oce] , and the common lines da-pa , db-pb , dc-pc all on a line of stpl . on the infinite sectors ada-apa , bdb-bpb , cdc-cpc vibrate , the ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 106 the fundamental particles origination mechanism in mfmf pns caves . breakages [± s² = ± (w r)²] and [ i = 2(w r)²] forming all families of curves and the euler savary coupler curves of the cubic-of-stationary curvature mechanism of spaces , anti-spaces vibrating the end-curves . the dimensioning of the mechanism is possible by using the analytical geometry. [91] the dimensioning of proton – neutron in fig-26 , and others with compounds in fig-27 .the dimensioning of hydrogen and atoms in fig-28-30, the linear-vibrations [ ẍ + w ² x = 0 ] of three-masses , occur on two-line-vectors perpendicular each other , vibrating on straight-line , ÿ + w ² y = 0 , of x ⊥ y plane and follow the lissajous shapes , [83] , where for , a.. difference of phase dφ = 90⁰ emission is → the eight-shapes ꝏ . b.. difference of phase dφ = 0⁰ emission is → the ellipse-shapes ∝ . c.. difference of phase dφ = 45⁰ emission is → the double-saddle-shapes . ց , ѡ . for planck length [73] p-49, was shown that the rotated energy case , when s = 0 and cosφ = 0 , exists for angle φ = π /2 and quaternion ( s + v̅ i )𝟏/𝐰 = 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐰...(1) the dimension power → w = b ← and for k = 1 then (1) becomes , [84] -p.74 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐰 = 𝐞− 𝐢.(𝛑/𝟐+𝟐𝐤𝛑).𝐛 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝐛 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝟏𝟎 [86] …... (2) equation (2) fits , as minimum cave , in the planck length and is 𝐋 𝐩 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝟏𝟎 .....(3) equation (3) is the smallest energy-unit of space , and this because is for s = 0 and k = 1. it was shown [31] that space and energy is quantized and measured on the two constant and natural numbers e , π , where for base the natural logarithm , e , and exponent the decimal base , b = 10 . from → z¹/w = ( s + �̅� i ) ¹/ w = |zo|−w.[cos.(φ + kπ)/w + i.sin.(φ + kπ)/w] =|zo|−w . e−i.(φ+kπ).w for cos.(φ+kπ)/w = 0 then exists only the imaginary part of monad (�̅� i ) ≠ 0 , where φ = π /2 and then , z¹/w = |zo| ̄ w . e i.(φ+kπ)/w = e−i.( π 2 +kπ).10 which is the diffraction energy mechanism for all space levels of quantization which are the energy particles only i.e. the energy particles in stationary caves are z¹/w = |zo| ̄ w. lv = e-monad. extending quantization of energy according to exponential formula→ 𝐋 𝐯 = 𝐞− 𝐢.(𝟓𝛑/𝟐).𝟏𝟎 then 𝐋 𝐯 on the decimal base b = 10 and for k = ± 0 → ± ∞ , are the energy caves as , for base e = 2,71828 and base b = 10 then e^ (13,8155) = 1 .10 ̄ 6 m for base e = 2,71828 and k = 0 lv = e^i.( ± π/2)b then e^ (-15,7079) = 1,78118 .10 ̄ 7 m for base e = 2,71828 and base b = 10 then e^ (16,1181) = 1 .10 ̄ 7 or r = 1,07.𝟏𝟎−𝟕 m. placing r , in the dynamic-space-energy relation when g =1 then r ³.𝐟²𝐩= 1 and 𝐟²𝐩 = 1 r ³ = 8,0647139.1020 m and occurs the , minimum frequency 𝐟𝐦𝐢𝐧 = 2,839844. 𝟏𝟎𝟏𝟎 h …(4) 7d.. the cosmic particles in the cycloid , anti-cycloid evolute : the moving monads in cycloid equilibrium in the anti cycloid ( evolute ). the equilibrium of the resultant forces is obtained by the above cycloid , anti-cycloid evolute forces , or by the rotor and spin in one rim , as was shown. ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 107 the fundamental particles origination mechanism in mfmf pns caves . in rims is possible the forming velocities greater than that of light , and this because of different , r , and the tangential velocities are kept with the same angular velocity , the spinning , unless spinning changes in such way to keep light-velocity constant . figure 25 the equilibrium of the resultant forces , in the cycloid and for motion wavelength , and of any other removal monad . the intrinsic space is (wavelength = [λ] ) , as the energy quanta . the equilibrium of the three-spaces , the space anti-space , neutral-space : [60] the stability of elementary particles , conductors and the atoms -nucleus fig-24 the equations of motion become everywhere from any opposite space ≡ ⊕ , anti space ≡ ⊝ . during these motions as , ds . is done a constant-work , k , or and from their velocities �̅� and this work may be positive or negative for equilibrium . in f-23 cycloid anti-cycloid motion is as , d. the origination of gravity g , antigravity [g ] in fig-20.b , conductor d = d = aka where ⊕ constituent attacks ⊝ by a hook`s law force as stress σ = e u , and an coulomb force f [+ ↔ −] = [⊕↔⊝ ] r² = [2 σ] r² , creating work ≡ motion as was shown it was shown that , the stability of nucleus forces , is succeeded through the pascal`s ceba`s triangle launched caves r , while leptons from their anti leptons existence in the same launched pascal`s caves r . [58] the equilibrium of the three-spaces is done on the three cyclic-quadrilaterals (abeaece) (bcebeae) , (caecebe) of the 6 conductors each for , [a , b , c , d , p , q ] x [3-abc] = 18 conductors . linearly is through the products of the diagonal connectors as , → → → [aae].[bece] = [ace].[aebe]+[abe].[aece] , [bbe].[ceae] = [bae].[bece]+[bce].[beae] , [cce].[aebe] = [cae].[bece]+[cbe].[ceae] and , rotationally through the ratio of the diagonals ≡ the conductors | akb akc |x| bkc bka |x| cka ckb | = 1 or  rotational-equilibrium as , [akb].[cka].[bkc] = [akc].[bka].[ckb] , of all opposites  tangential-electromotive forces . from mechanics-physics , all systems possessing elasticity ≡ motion and reaction to motion , the called mass , are capable of free vibration or vibration taking place in the absence of external excitation [7] .this principle issues for both closed or open systems. i.e. either for the nucleus where issues the pascal`s ceba`s triangle stability , or for the orbitals where issues the tetrahedron cube – sphere – mould . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 108 the fundamental particles origination mechanism in mfmf pns caves . 8d.. the stability of the cosmic particles conductors and atoms – nucleus . figure -26: the stability of → { ⊕ space [a b c] },{⊝ anti-space [ka kb kc]} , of the sub-space [a e , b e , c e]}, and neutral-space {[⊕↔⊝] ≡ ds ≡ [⊕←ds→⊝]}. since , energy of the first wheel-rim or orbital is distributed to the couple of (2e) electrons of the orbital , then this is a cycloid anti-cycloid stability system . this still systems of space –anti space does not mean matter–antimatter but the state of equilibrium in caves ≡ the ceba`s energy triangle . a…the space anti-space and , neutral-space conductors : [60] the equations of motion become everywhere from any opposite , space ≡ ⊕ , and anti-space ≡ ⊝ . during these motions as , ds . is done a constant-work , k , or and from their velocities �̅� . in figure conductor d = d = aka where ⊕ constituent attacks ⊝ by a hook`s law force as stress σ = e u , and an coulomb force f [+ ↔ −] = [⊕↔⊝ ] r² = [2 σ] r² , creating work ≡ motion as was shown . [58] it was shown that , the stability of forces is succeeded in the pascal`s launched caves r while leptons , from their anti leptons existence in the same launched pascal`s-caves r . the equilibrium of the three-spaces is succeeded on the three cyclic quadrilaterals → (abeaece) , (bcebeae) , (caecebe) of 6 conductors each [a , b , c , d , p , q ] x [3-abc] = 18 conductors . linearly is through the products of the diagonal connectors as , → → → [aae].[bece] = [ace].[aebe]+[abe].[aece] , [bbe].[ceae] = [bae].[bece]+[bce].[beae] ,[cce].[aebe] = [cae].[bece]+[cbe].[ceae] and , rotationally through the ratio of diagonals ≡ the conductors | akb akc |x| bkc bka |x| cka ckb | = 1 or the  rotational-equilibrium [akb].[bkc].[cka] = [akc].[bka].[ckb] ,of all above opposites → tangential-electromotive-forces ← from mechanics ,all systems possessing elasticity ≡ motion and reaction to motion , called mass , are capable of free vibration or vibration taking place in the absence of external excitation . [7] from lorentz-force , f = �̅� �̅� x �̅� 𝐅 , where , q̅ ≡ charge , v̅ ≡ the velocity of charge , b̅ f ≡ the strength of magnetic field and centrifugal force , 𝐅 𝐂𝐅 = m.𝐰²𝐑.r . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 109 the fundamental particles origination mechanism in mfmf pns caves . from cave-spin , �̅� = r m v then mass m = ( s r.v ) .equating equations then , f ≡ fc f ≡ q̅ v̅ x b̅ f , and 𝐐  = m.w²r.r = ( s r.v ).w².r = s.w² v = s.v ² r².v ≡ | 𝐯.�̅� 𝐫² | ….(emf) where exists rmin= √πb−n 2 , [95] , [96bb] . electromotive-forces , f = �̅� �̅� x �̅� 𝐅 = 𝐐  = 𝐯.�̅� 𝐫² ≡ | 𝐜.�̅� 𝐫 ² |, are the elementary-particles {f, l ,q }. from electromotive-force in min-cave r = 1,7724538.10−3 m. exists electron force-charge q̅ e as 𝐐  = | 𝐜.�̅� 𝐫 ² | = [2,998.108].[1,0546.10−34]j (1,7724538.10−3) ² = 1,6022.10−19 ev , which is the electron-force-charge and also from the gravitational-stress-charge �̅� electron = g c √2 = 6,6736923 .10−11 1,41429.2,9979346.108 = 1,574.10−19 c . electromotive-force in the min-cave 𝑒−15,572707= r = 1,7252787.10−7 m , and the gravitational-force g becomes also as , 𝐐  = | 𝐜.�̅� 𝐫 ² | = [2,998.108].[6,62606957.10−34] (1,7252787.10−7) ² = 6,673692. 10−11 , which is the g physical-forcecharge. when the heap of the elementary particles on da-pa line , is compressed and gets out the line then is a reason for black hole creation . b…the relation of the n , degree of polynomials and electromotive-forces vibration . the transportation of energy on edge-points happens in material geometry , while the transportation of the edge-points as , positions , exists in vector geometry only. since in vibrations the deformation of the spring is an equilibrium position in ∆ , and the spring-forse k ∆ = w = m g = gravitational force w acting on mass m , then exists → m �̈� = σf = w = k [ ∆ + x] ….(1) , and because k ∆ = w we obtain w ²n = k/m.. (2) and (1) becomes → �̈� + 𝐰 ²𝐧 = 0 ,… (3) ← i.e. the transportation of energy is a harmonic vibration in cave ∆ , meaning the wave pattern . from before the cave-spin , �̅� = r m v is a monad , and occupies the 4-spaces as the fig–1 . i.e. → (( the n spaces of spin �̅� are the polygons 𝐒𝐧 , the n anti-spaces of spin �̅� are the polygons 𝐒𝐧 )) ← the sub-spaces ± √ �̅� 𝒏=𝟏−∞ are the polygons with n = 1 ≈ ∞ knots since the number of knots n , are the degrees n of the polynomials of each monad and since the n masses spring 𝐦 𝐧 is an equilibrium position in ∆ , of the equation of motion → �̈� + 𝐰 ²𝐧 = 0 ← then the 4-spaces of monad are the polygons of n-knots and equal to the n degrees roots of the polynomials . the process of actions in the 4=spaces 1…a heap of n-masses vibrates and equilibrium . 2…a heap of n-masses equilibrium at the n-mode shapes waveform . 3…a heap of n-masses of n-mode shapes occupies the resultant e-spectrum 4…each-one of the n-masses occupies n-mode shapes , n-element .spectrum , 4-p.spaces , 4.n.. regular-polygons of n-degree , with 4.n..knots and roots . 5…each-one of the n-masses is an e -monad occupying 4-primary-spaces , and ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 110 the fundamental particles origination mechanism in mfmf pns caves . forms 4.n.. regular-polygons of the n-degree , and with 4.n..knots as roots .. 6…each-one of the n-masses is an e-monad occupying all above properties (signals). 9d.. the origination of proton and neutron cosmic particles . the elements in the proton & neutral-caves → { masses – charges forces } [89] figure-27 : the sphere-structure of the hydrogen-nucleus , ⊕ proton ,[⊕↔⊝]=neutron proton is consisted of three-primary-opposite-spaces ⊕ , ⊝ , [⊕↔⊝] , having masses m p , charges q̅p , and caves a p = r . the nucleus`s cave r , exists in the inscribed to the tetrahedron sphere . the elements in proton are the two u-quarks and one d-quark . the hydrogen-nucleus is in the inscribed sphere from the { inscribed sphere -tetrahedron – cube circumscribed sphere system }. in order to shift the u-d-quarks of an anti-proton into a proton spin-pair require an extra input of energy-(mev) , so would proton paired with a neutron be stable . the total mass mt in proton follows , the parallel connections resistors inverse law , where the proton total-harmonic mass ≡ mt is → 1 mt = 1 mu + 1 mu + 1 md , the system totalharmonic-charge ≡ qt ≡ 2.qu + qd = 2.(2/3).e – (1/3) e = + 3 3 e = + 1,6022.10−19 c , and the system-resonance-charge qt = + 1, 6022.10−19 c ...(2) the proton totalharmoniccharge → qt ≡ 2.qu + qd ,and for the resonance frequency of the unit-cave is the stationary-system becoming from kepler second planetary-law as equation , 4 π² m f ²o = k , and constant law of areas 1 = k .𝐟 ²𝐨 𝐚 𝟑 . their common k , is the ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 111 the fundamental particles origination mechanism in mfmf pns caves . constant-energy → k = 4 π² m f ²p = 1 𝐟 ²𝐩 𝐚𝟑 or , f ⁴p = 1 4π²ma3 and f p = √ 1 4π²m.a3 4 . the measured magnitudes are as follows , a-proton⊕→ mass 𝐦𝐩 = 1,672.10−27 kg →charge 𝐂 𝐩 =1,602.10−19 c → a = 8,4.10−16 m b-electron⊝→mass 𝐦𝐞 = 9,11.10−31 kg →charge 𝐂 𝐞 =1,602.10−19 c → a = 5,0.10−17 m c-neutron[⊕↔⊝]→mass 𝐦𝐧 = 1,672.10−27 kg →charge 𝐂 𝐧 = 0 ,0 c → a = 1,7.10−15 m d-u-quark →from equal masses mp =2.mu + md =3.mu =1,672.10−27 kg and quark masses 𝐦𝐮 = 𝐦𝐝 = mp / 3 = 5,573.10−28 kg , and 1 mt = 1 mu + 1 mu + 1 md = 3 mp + 2.3 mp = 9 mp and proton resonance mass mt = m p 9 = 1,857777.10−28 kg ≈ 16,7267 m ev/c² …….(m) for proton issues qp = 2.qu + qd = 2. 2 3 e 1 3 e = e , and the stability of forces is axial . electron-charge 𝐂𝐞 =1,602. 10−19 c , while 𝐂𝐪𝐮 = 2 3 e = + 𝟐 𝟑 1,602.10−19 c 𝐂𝐪𝐝 = 1 3 e = 𝟏 𝟑 1,602.10−19 c , and from qt ≡ 2.qu + qd , then proton resonance charge qt = 3.q p 3 = 1,602.10−19 c ………..(c) proton-resonance frequency f p =√ 1 4π²m.a3 4 = √ 1 4π²5,573.10−28(8,4.10−16)³ 4 = 5,26241.1017 h using the united newton-coulomb electro-mechanical equation, q b̅l=2π.m f , the proton magnetic-field b̅f = |2π.m t| qt f = 2𝜋.1,85777.10−285,262409.1017 1,6022.10−19 (kg/cs) = 3,83389.109 tesla which is the strength of a magnetar , i.e. a type of neutron-star having an extremely powerful magnetic-field , and electric-forces to be over ten-thousands-newton . the electric force between the u-quarks and d-quarks in proton is from coulomb law 𝐅𝐮𝐝−𝐩 = c q1.q 2 r ² = 8,9875.109(nm2/c²). 2 9 [1,602.10−19 c]² 1 (10−16)² = 1,997222.10 6 n ...(f p) and the electric force between the u-quarks and d-quarks in neutron is 𝐅𝐮𝐝−𝐧 = c q1.q 2 r ² = 8,9875.109(nm2/c²). 2 9 [1,602.10−19 c]² 1 (10−16)² = -1,997222.10 6 n ..(f n) i.e. forces between the opposites equilibrium-linearly ←[d-u-d]→ or →[u-d-u]← for the neutral-cave issues qn = 2.qd + qu = 2. 1 3 e + 2 3 e = 0.e , and the stability of forces is axial as in proton and this because the dynamic-strip-polygon doesn`t close . for , ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 112 the fundamental particles origination mechanism in mfmf pns caves . 1.. neutrinos , v , mass m = (3,11 .10−6) mev/c² x 1,8.10−28 = 5 , 598 .𝟏𝟎−𝟑𝟒 kg the spin is , 𝐒𝒗 = 5,691952.10−34{kg/m/s}, 𝐚𝐯 = 7,0.10−21 m , 𝐟𝐯 = [𝐄 𝐯𝐊 = mc²] / h = 5,6.10−34 [1017].1,6022.10−19 = 8,96912.10−36 j/[6,626.10−34 js = 13,536244. 𝟏𝟎𝟏/s from cave r = 7.10−21 m then f = √ 1 g.a³ 2 = √ 1 9,81(7.10−21)³ 2 = 5,347345.𝟏𝟎𝟑𝟎 h and exists , 𝐄 𝐯𝐊 = h 𝐟𝐯 = 6,62607.10−34j s.[ 5,347.1030] /(1,6022.10−19 ev) = 2,2114518.𝟏𝟎 𝟏𝟒ev using 𝐄 𝐯𝐊 = 𝐤 𝐫 + 𝒄𝐒 𝟐 𝟐𝐦.𝐫𝟐 = 36.10−20 7,10−21 + 3.108(5,691952.10−34)² 2.3,922.10−36[7,0.10−21]² =51,428 ev+ 2,528774.1017ev = 252,8774, [1015] tev → the total energy of the sun striking earth-face per second. 2…electron , e , me = 0 , 511mev = 0, 511.10−5 ev . [1,80. 10−27] = 9,198.10−31 kg 𝐚𝐞 = 5,0.10−18 m ,charge 𝐂 𝐞 =1,602.10−19 c , spin 𝐒𝒆 = 𝐒 2 =2,845976.10−34{kg/m/s}, from cave r = 5.10−18 m then f = √ g a³ 2 = √ 9,8078 (5.10−18)³ 2 = 2,801114.10 26 h , and by using 𝐄 𝐞𝐊 = 𝐤 𝐫 + 𝒄𝐒 𝟐 𝟐𝐦.𝐫𝟐 = 36.10−20 7,10−18 + 3.108(5,691952.10−34)² 2.9,198.10−34[5.10−18]² = 51,428 ev + 2,111843.1010j = 51,43 ev+ 1,32 .1029 ev → the energy of 133 gr to fall 1 meter against gravity. remarks : 1.. gravitational force → g ≡ σ a ≡ [ 2πrf φ ] a ≡ v̅ [ 𝐀 𝚽 ] ≡ σ.φ³ ≡ φ ² . [{σ φ}] ≡ g ≡ φ². [{σ φ}≡ 2πfp r ≡ w r ≡ v̅ ≡ m g = c̅ = 2.b πr³ ] → i.e. g is related to → motion ≡ work w , spaces r , anti-spaces 1/r , stresses σ , areas a , caves a , → periods t , frequencies f , angular-waves w , angular-momentum b , → spin b ≡ s , velocities v , light-velocity c , impedances zn, masses m , → gravity g̅ , charges , q̅ , electromagnetic – fields , e̅ , m̅ , hydrogen h , → atoms , molecules , golden-ratio φ , i.e. all-universe . markos 9/4/2020. by sweeping the φ surface through stresses σ ≡ [ 2πrf φ ] , and scanning frequencies f , explanation : the action-spaces ≡ the infinite quaternion monads → �̅� ≡ ( s + v̅.i ) ≡ 𝐀𝐁̅̅ ̅̅ ≡ ≡ √s2 + [ v̅. i ]² ] , or �̅� ≡ x + i . y ≡ 𝐀𝐁̅̅ ̅̅ ≡ moduli , r , is their magnitude [ r = | r | = √ x2 + y2 ] , on unit-diameter ab̅̅ ̅̅ . the [qmas] . for quaternion |v̅| = s breakages 𝐳 * 𝐳 = (s + v̅.i )² = s² +[v̅]² +2.sv̅.i = s² – v̅ ² ± 2.s v̅ = |s|² |�̅�|² ± 2.|s|.|s̅| , i.e. in figure -3 the vector of amplitude is quaternion �̅� ≡ x + i . y , with vector �̅� , to be from definition of the complex vector space , the set of vectors of length (the n vertices which form the n polygon sides , or the norm of the complex vectors ) a...the 4-geometrical spaces , [ space , anti-space , neutral-space , sub-space ] consist → the geometrical mould of 𝐀𝐁̅̅ ̅̅ monad ≡ [pm-2r-m] ← b...the vectors of amplitude z = 2r , on 𝐀𝐁̅̅ ̅̅ monad , consist the energy-mould . of 2r-form .the physical -mechanisms → [ physical mould of 2rmonad ] ≡ [gm-ab-m] ← are such and dependent on the n number of accessories they use . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals 113 the fundamental particles origination mechanism in mfmf pns caves . a… the gravity force g ≡ motion ≡ σ.𝚽³ , exists without mass and is quantized spread in{space and anti-space or ≡ qua ≡ {is the quantized-units in-area of space and anti-space }, and as a mould 𝚽 golden-ratio , exists as impedance b. placing [ physical mould of 2rmonad ] on [ geometrical mould of 𝐀𝐁̅̅ ̅̅ monad ] [ pm-2r-m ] → ⇅ ← [ gm-ab-m ] then → gravitational-force g ≡ [ σ = n . h ] , and n-na = 1  creates & constructs, figure -28: on any energy monad a ↔ b , exist the 4-primary spaces which are the roots of the nth degree polynomial , becoming from the equation of motion . f... references : [ 1] matrix structure of analysis by j.l.meek library of congress catalog 1971. 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[106] [m] programming the atoms and compounds and the unification of physics and chemistry —["jmseat 2024"] atul@scientificadvances.co.in [106] [m] programming the atoms and compounds and the unification of physics and chemistry — asrp.hspublishing@gmail.con [107-a] [m] the planck`s << duality angular momentum >> as gravity and . antigravity <editor@granthaalayah.com> research international journal [107] [m] the planck`s << duality angular momentum >> as gravity and antigravity http://science mpg ≡ markos georgallides [108] [m] the planar and atoms black holes , under the critic of material-geometry & planck–dual spin as gravity-antigravity http://science mpg ≡ markos georgallides [109] [m] the dual quaternion momentum -b as { the existing universe } & the black -holes , http://science mpg ≡ markos georgallides [110] [m] the origination-mechanism of fundamental particles in the energy-planck`s confinement and their existence, http://science mpg ≡ markos georgallides [111] [m] the polynomials , n-knots figures , edge points vibration & m-geometry . by http://science mpg ≡ markos georgallides [112] [m] an way for deceptioning the normal or the dangerous cells by http://science mpg ≡ markos georgallides [113] [m] the origination of universe from energy ≡ motion into the e-geometry . markos georgallides comes from cyprus and currently resides in the city larnaca , cyprus after being expelled from his home town famagusta-varosha , by the barbaric turks in august 1974. he works as a consultant civil and architect engineer having his own business . he is also the author of numerous scholarly articles focusing on euclidean and material geometry , and mathematical to physics and mechanics related subjects . he obtained his degree from the athens , national technical , polytechnic university [natua] athensgreece , and subsequently studied in germany , math theory of photoelasticity . ijo international journal of mathematics (issn: 2992-4421 ) volume 08 | issue 08 | august 2025 | https://ijojournals.com/index.php/m/index ijo journals http://science/ http://science/ http://science/ mailto:atul@scientificadvances.co.in mailto:asrp.hspublishing@gmail.con mailto:editor@g http://science/ http://science/ http://science/ http://science/ http://science/ http://science/ 