id sid eid entity type cana-826 1 1 1074 CARDINAL cana-826 1 2 31 CARDINAL cana-826 2 1 4s CARDINAL cana-826 2 2 2024 CARDINAL cana-826 2 3 9 CARDINAL cana-826 2 4 2 1associate CARDINAL cana-826 2 5 mathematics(ca NORP cana-826 2 6 tamilnadu NORP cana-826 2 7 india GPE cana-826 3 1 2research CARDINAL cana-826 3 2 tamilnadu NORP cana-826 3 3 india GPE cana-826 3 4 c.nandhini@vcw.ac.in article GPE cana-826 3 5 15-04-2024 CARDINAL cana-826 3 6 03-06-2024 CARDINAL cana-826 3 7 22 CARDINAL cana-826 8 1 2000 CARDINAL cana-826 8 2 05c65 1 CARDINAL cana-826 9 1 5 CARDINAL cana-826 10 1 first ORDINAL cana-826 10 2 6 CARDINAL cana-826 13 1 3 CARDINAL cana-826 14 1 2 CARDINAL cana-826 14 2 4 CARDINAL cana-826 15 1 9 CARDINAL cana-826 15 2 10 DATE cana-826 15 3 11 CARDINAL cana-826 16 1 8 CARDINAL cana-826 17 1 7 CARDINAL cana-826 18 1 lanzhen yang PERSON cana-826 18 2 hua mao PERSON cana-826 19 1 12 CARDINAL cana-826 20 1 section 2 LAW cana-826 20 2 section 3 LAW cana-826 21 1 section 4 LAW cana-826 23 1 mailto:c.nandhini@vcw.ac.in PERSON cana-826 23 2 1074 CARDINAL cana-826 23 3 31 CARDINAL cana-826 24 1 4s CARDINAL cana-826 24 2 2024 CARDINAL cana-826 24 3 10 CARDINAL cana-826 25 1 2.1 CARDINAL cana-826 26 1 3 CARDINAL cana-826 28 1 0 ≀ QUANTITY cana-826 29 1 2.2 CARDINAL cana-826 30 1 3 CARDINAL cana-826 30 2 ifs ORG cana-826 31 1 six CARDINAL cana-826 31 2 cartesian NORP cana-826 31 3 π‘ˆ2 DATE cana-826 31 4 𝑖=1 GPE cana-826 31 5 ∈ ORG cana-826 31 6 π‘ˆ1 PRODUCT cana-826 31 7 Γ—2 DATE cana-826 31 8 Γ—2 GPE cana-826 31 9 Γ—2 GPE cana-826 31 10 ≠𝑛 + ORG cana-826 31 11 𝑖=1 GPE cana-826 31 12 ∈ ORG cana-826 31 13 π‘ˆ1 PRODUCT cana-826 31 14 π‘ˆπ‘–2 Γ—3 π‘ˆπ‘–3 Γ—3 …×3 ORG cana-826 31 15 𝑖=1 GPE cana-826 31 16 𝑖=1 GPE cana-826 31 17 ≠𝑛 + ORG cana-826 31 18 𝑖=1 ORG cana-826 31 19 ∈ ORG cana-826 31 20 π‘ˆ1 PRODUCT cana-826 31 21 π‘šπ‘Žπ‘₯⁑(𝜈1 ORG cana-826 31 22 𝜈2 CARDINAL cana-826 31 23 ∈ ORG cana-826 31 24 π‘ˆ1 PRODUCT cana-826 31 25 𝜈2 CARDINAL cana-826 31 26 ∈ ORG cana-826 31 27 π‘ˆ1 PRODUCT cana-826 31 28 ∈ ORG cana-826 31 29 1074 CARDINAL cana-826 31 30 31 CARDINAL cana-826 32 1 4s CARDINAL cana-826 32 2 2024 CARDINAL cana-826 32 3 11 CARDINAL cana-826 33 1 𝑖=1 GPE cana-826 33 2 𝑖=1 GPE cana-826 33 3 ∈ ORG cana-826 33 4 π‘ˆ1 PRODUCT cana-826 33 5 1,2,3,4,5,6 CARDINAL cana-826 34 1 2.3 CARDINAL cana-826 35 1 9 CARDINAL cana-826 35 2 0 ≀ QUANTITY cana-826 35 3 ≀ 1 CARDINAL cana-826 35 4 1,2 CARDINAL cana-826 35 5 βŠ† CARDINAL cana-826 35 6 π‘ˆ Γ— π‘ˆ ORG cana-826 35 7 π‘ˆ Γ— π‘ˆ ORG cana-826 35 8 0 ≀ QUANTITY cana-826 36 1 cartesian NORP cana-826 36 2 1,2,3,4,5,6 CARDINAL cana-826 36 3 2.2 CARDINAL cana-826 37 1 fourth ORDINAL cana-826 37 2 π‘šπ‘Žπ‘₯⁑(𝜈1 ORG cana-826 37 3 𝜈2 CARDINAL cana-826 37 4 2.4 CARDINAL cana-826 38 1 10 CARDINAL cana-826 38 2 π‘ˆ,𝐸 NORP cana-826 38 3 𝐸1 DATE cana-826 38 4 πΈπ‘š PERSON cana-826 38 5 πœˆπ‘—(𝑒𝑖)|0 ≀ FAC cana-826 39 1 ≀ 1 CARDINAL cana-826 39 2 1,2 CARDINAL cana-826 39 3 1,2 CARDINAL cana-826 39 4 ⋃ supp(𝐸𝑗 PERSON cana-826 39 5 1,2 CARDINAL cana-826 39 6 𝐸𝑗 ORG cana-826 39 7 membership & nonmembership ORG cana-826 39 8 0 ≀ MONEY cana-826 39 9 membership & nonmembership ORG cana-826 39 10 0 ≀ QUANTITY cana-826 39 11 β€’ CARDINAL cana-826 40 1 β€’ CARDINAL cana-826 40 2 0 CARDINAL cana-826 41 1 1074 CARDINAL cana-826 41 2 31 CARDINAL cana-826 42 1 4s CARDINAL cana-826 42 2 2024 CARDINAL cana-826 42 3 12 CARDINAL cana-826 42 4 3 CARDINAL cana-826 42 5 3.1 CARDINAL cana-826 43 1 ℍ𝔾 ORG cana-826 43 2 β„°1 CARDINAL cana-826 43 3 β„°2 ORG cana-826 43 4 πœˆπ‘—(𝑒𝑖)|0 ≀ FAC cana-826 44 1 ≀ 1 CARDINAL cana-826 44 2 1,2 CARDINAL cana-826 44 3 ℰ𝑗 β‰  βˆ… PERSON cana-826 44 4 1,2 CARDINAL cana-826 44 5 ⋃ DATE cana-826 44 6 1,2 CARDINAL cana-826 44 7 𝑉 βŠ† π‘ˆ ORG cana-826 44 8 ℍ𝔾 ORG cana-826 45 1 1 CARDINAL cana-826 45 2 𝑠2π‘’π‘–βˆˆπ‘‰ ORG cana-826 46 1 0.4,0.6 ORG cana-826 46 2 β„°1 CARDINAL cana-826 46 3 β„°2 ORG cana-826 46 4 β„°3 CARDINAL cana-826 46 5 β„°4 CARDINAL cana-826 47 1 3.1 CARDINAL cana-826 48 1 1074 CARDINAL cana-826 48 2 31 CARDINAL cana-826 49 1 4s CARDINAL cana-826 49 2 2024 CARDINAL cana-826 49 3 13 CARDINAL cana-826 49 4 3.2 CARDINAL cana-826 50 1 ℍ𝔾 ORG cana-826 51 1 two CARDINAL cana-826 52 1 max PERSON cana-826 52 2 πœ‡π‘—(𝑒𝑖+1))),min ORG cana-826 52 3 max PERSON cana-826 52 4 1,2 CARDINAL cana-826 52 5 1,2 CARDINAL cana-826 52 6 1 CARDINAL cana-826 53 1 3.3 CARDINAL cana-826 54 1 ℍ𝔾 ORG cana-826 55 1 ℰ𝑗 LOC cana-826 56 1 max PERSON cana-826 56 2 πœ‡π‘˜(𝑒𝑖)),min 𝑗 PERSON cana-826 56 3 max PERSON cana-826 56 4 1,2 CARDINAL cana-826 56 5 1,2 CARDINAL cana-826 56 6 1 CARDINAL cana-826 57 1 3.4 CARDINAL cana-826 58 1 ℍ𝔾 ORG cana-826 60 1 πœ‡π‘—(𝑒𝑖)),max PRODUCT cana-826 60 2 0 CARDINAL cana-826 60 3 0 CARDINAL cana-826 61 1 1 CARDINAL cana-826 62 1 3.5 CARDINAL cana-826 63 1 ℍ𝔾 ORG cana-826 64 1 at least one CARDINAL cana-826 64 2 zero CARDINAL cana-826 64 3 1 CARDINAL cana-826 65 1 𝑠2π‘’π‘–βˆˆπ‘‰ ORG cana-826 66 1 3.6 CARDINAL cana-826 67 1 ℍ𝔾 ORG cana-826 68 1 𝑒1⁑ ORG cana-826 68 2 β„°1⁑⁑ GPE cana-826 68 3 β„°2⁑ GPE cana-826 68 4 0 CARDINAL cana-826 69 1 1 CARDINAL cana-826 69 2 𝑠2π‘’π‘–βˆˆπ‘‰ ORG cana-826 70 1 3.7 CARDINAL cana-826 71 1 ℍ𝔾 ORG cana-826 71 2 two CARDINAL cana-826 71 3 ℍ𝔾 ORG cana-826 71 4 1 CARDINAL cana-826 72 1 𝑠2π‘’π‘–βˆˆπ‘‰ ORG cana-826 73 1 ℍ𝔾 ORG cana-826 74 1 3.8 CARDINAL cana-826 75 1 ℍ𝔾 ORG cana-826 78 1 2βˆ’πœ‡π‘–(𝑒𝑖)βˆ’πœˆπ‘–(𝑒𝑖 CARDINAL cana-826 78 2 1βˆ’πœˆπ‘—(ℰ𝑗) 2βˆ’πœ‡π‘—(ℰ𝑗)βˆ’πœˆπ‘—(ℰ𝑗 CARDINAL cana-826 79 1 1 CARDINAL cana-826 79 2 𝑠2π‘’π‘–βˆˆπ‘‰ ORG cana-826 80 1 1074 CARDINAL cana-826 80 2 31 CARDINAL cana-826 81 1 4s CARDINAL cana-826 81 2 2024 CARDINAL cana-826 81 3 14 CARDINAL cana-826 81 4 3.1 CARDINAL cana-826 82 1 1 CARDINAL cana-826 84 1 ℍ𝔾 ORG cana-826 84 2 β„°2 ORG cana-826 84 3 two CARDINAL cana-826 85 1 two CARDINAL cana-826 86 1 βˆƒ ORG cana-826 86 2 2 CARDINAL cana-826 86 3 u & v ORG cana-826 86 4 u & v ORG cana-826 86 5 1 CARDINAL cana-826 88 1 3.2 CARDINAL cana-826 89 1 ℍ𝔾 ORG cana-826 90 1 ℍ𝔾 ORG cana-826 90 2 π‘ˆ1 PRODUCT cana-826 90 3 π‘ˆ2 CARDINAL cana-826 90 4 π‘ˆ,π‘ˆ1 CARDINAL cana-826 92 1 one CARDINAL cana-826 92 2 π‘ˆ1 DATE cana-826 92 3 π‘ˆ2 DATE cana-826 94 1 ℍ𝔾 ORG cana-826 95 1 βˆƒ ORG cana-826 95 2 ∈ ORG cana-826 96 1 π‘ˆπ‘’ β‰  PERSON cana-826 96 2 π‘ˆπ‘’ β‰  PERSON cana-826 96 3 π‘ˆπ‘’ PERSON cana-826 97 1 ∈ ORG cana-826 100 1 βˆƒ ORG cana-826 101 1 π‘ˆ,π‘ˆ1 ∩ π‘ˆ2 PRODUCT cana-826 101 2 one CARDINAL cana-826 101 3 π‘ˆ1 DATE cana-826 101 4 π‘ˆ2 DATE cana-826 102 1 ℍ𝔾 ORG cana-826 103 1 ℍ𝔾 ORG cana-826 104 1 π‘’πœ–π‘ˆ1 CARDINAL cana-826 105 1 ∈ ORG cana-826 105 2 ℍ𝔾 ORG cana-826 106 1 βˆƒ ORG cana-826 106 2 π‘’πœ–π‘ˆ1 DATE cana-826 106 3 ∈ ORG cana-826 106 4 π‘ˆ1 PRODUCT cana-826 107 1 one CARDINAL cana-826 107 2 π‘ˆ1 DATE cana-826 107 3 π‘ˆ2 CARDINAL cana-826 110 1 ℍ𝔾 ORG cana-826 111 1 1074 CARDINAL cana-826 111 2 31 CARDINAL cana-826 112 1 4s CARDINAL cana-826 112 2 2024 CARDINAL cana-826 112 3 15 CARDINAL cana-826 112 4 3.3 CARDINAL cana-826 113 1 ℍ𝔾 ORG cana-826 113 2 ∈ ORG cana-826 113 3 ℍ𝔾 ORG cana-826 115 1 ℍ𝔾 ORG cana-826 115 2 w GPE cana-826 115 3 ℍ𝔾. CARDINAL cana-826 115 4 w CARDINAL cana-826 116 1 w ORG cana-826 116 2 1 CARDINAL cana-826 116 3 2 CARDINAL cana-826 116 4 w ORG cana-826 116 5 w ORG cana-826 116 6 w PERSON cana-826 116 7 π‘’π‘˜βˆ’1 PERSON cana-826 117 1 w GPE cana-826 118 1 j PERSON cana-826 118 2 ℍ𝔾 GPE cana-826 118 3 w ORG cana-826 120 1 4. CARDINAL cana-826 122 1 two CARDINAL cana-826 122 2 11 CARDINAL cana-826 122 3 10 CARDINAL cana-826 123 1 β€’ CARDINAL cana-826 125 1 β„°1 CARDINAL cana-826 125 2 β„°2 ORG cana-826 127 1 β„°10 ORG cana-826 128 1 β€’ CARDINAL cana-826 128 2 two CARDINAL cana-826 129 1 two CARDINAL cana-826 129 2 first ORDINAL cana-826 129 3 11 CARDINAL cana-826 131 1 β„°2 ORG cana-826 133 1 β„°5 CARDINAL cana-826 133 2 second ORDINAL cana-826 133 3 10 CARDINAL cana-826 135 1 β„°6 CARDINAL cana-826 136 1 β„°9}⁑. β€’ DATE cana-826 138 1 at least one CARDINAL cana-826 141 1 1074 CARDINAL cana-826 141 2 31 CARDINAL cana-826 142 1 4s CARDINAL cana-826 142 2 2024 CARDINAL cana-826 142 3 16 CARDINAL cana-826 143 1 4.1 CARDINAL cana-826 143 2 β€’ CARDINAL cana-826 145 1 β€’ 𝑣𝑗(𝑒𝑛 CARDINAL cana-826 147 1 β€’ CARDINAL cana-826 147 2 πœ‡π‘–π‘—(β„°π‘š ORG cana-826 150 1 4.1 CARDINAL cana-826 150 2 4.2 CARDINAL cana-826 150 3 1074 CARDINAL cana-826 150 4 31 CARDINAL cana-826 151 1 4s CARDINAL cana-826 151 2 2024 CARDINAL cana-826 151 3 17 CARDINAL cana-826 151 4 1 CARDINAL cana-826 151 5 1 CARDINAL cana-826 151 6 0.7139 2 u2 MONEY cana-826 151 7 0.6663 3 u3 PERCENT cana-826 151 8 0.6892 4 PERCENT cana-826 151 9 0.6894 5 u5 MONEY cana-826 151 10 0.6248 6 u6 MONEY cana-826 151 11 u7 ORG cana-826 151 12 0.6,0.0015 CARDINAL cana-826 151 13 0.7140 8 u8 MONEY cana-826 151 14 0.5875 9 PERCENT cana-826 151 15 u9 ORG cana-826 151 16 0.5,0.005 CARDINAL cana-826 151 17 0.3,0.05 CARDINAL cana-826 151 18 0.5758 11 MONEY cana-826 151 19 0.5442 12 u12 MONEY cana-826 151 20 0.6664 13 CARDINAL cana-826 151 21 14 MONEY cana-826 151 22 0.4,0.0015 CARDINAL cana-826 151 23 0.7137 16 MONEY cana-826 151 24 0.6664 18 MONEY cana-826 151 25 0.7,0.002 CARDINAL cana-826 151 26 0.7689 21 CARDINAL cana-826 151 27 u21 ORG cana-826 151 28 0.92,0.04 CARDINAL cana-826 151 29 0.9423 24 β„°1 MONEY cana-826 151 30 0.5,0.002 CARDINAL cana-826 151 31 0.6662 25 β„°2 MONEY cana-826 151 32 0.4,0.01 CARDINAL cana-826 151 33 0.6226 26 β„°3 MONEY cana-826 151 34 0.6633 27 β„°4 MONEY cana-826 151 35 0.3,0.005 CARDINAL cana-826 151 36 0.5870 28 CARDINAL cana-826 151 37 0.2,0.05 CARDINAL cana-826 151 38 0.5429 29 β„°6 QUANTITY cana-826 151 39 0.5878 30 β„°7 MONEY cana-826 151 40 β„°8 CARDINAL cana-826 151 41 0.6663 32 β„°9 MONEY cana-826 151 42 0.6,0.002 CARDINAL cana-826 151 43 0.7139 33 CARDINAL cana-826 152 1 0.7131 CARDINAL cana-826 152 2 1074 CARDINAL cana-826 152 3 31 CARDINAL cana-826 153 1 4s CARDINAL cana-826 153 2 2024 CARDINAL cana-826 153 3 18 CARDINAL cana-826 153 4 4.3 CARDINAL cana-826 155 1 1074 CARDINAL cana-826 155 2 31 CARDINAL cana-826 156 1 4s CARDINAL cana-826 156 2 2024 CARDINAL cana-826 156 3 fig:4.4 iftshg PERSON cana-826 156 4 2 CARDINAL cana-826 156 5 3 CARDINAL cana-826 157 1 4.4 CARDINAL cana-826 159 1 1074 CARDINAL cana-826 159 2 31 CARDINAL cana-826 160 1 4s CARDINAL cana-826 160 2 2024 CARDINAL cana-826 160 3 20 CARDINAL cana-826 160 4 1 CARDINAL cana-826 160 5 2 CARDINAL cana-826 160 6 3 CARDINAL cana-826 160 7 4 CARDINAL cana-826 161 1 5 CARDINAL cana-826 161 2 6 CARDINAL cana-826 161 3 7 CARDINAL cana-826 161 4 8 CARDINAL cana-826 161 5 9 CARDINAL cana-826 167 1 two CARDINAL cana-826 168 1 1074 CARDINAL cana-826 168 2 31 CARDINAL cana-826 169 1 4s CARDINAL cana-826 169 2 2024 CARDINAL cana-826 169 3 4.1 CARDINAL cana-826 169 4 β€’ CARDINAL cana-826 171 1 β€’ CARDINAL cana-826 172 1 β€’ CARDINAL cana-826 173 1 β€’ CARDINAL cana-826 174 1 5 CARDINAL cana-826 178 1 1 CARDINAL cana-826 178 2 20 CARDINAL cana-826 178 3 1986 DATE cana-826 179 1 2 CARDINAL cana-826 179 2 6th ORDINAL cana-826 179 3 ifsa world congress ORG cana-826 179 4 sao paulo GPE cana-826 179 5 brazil GPE cana-826 179 6 551–554 DATE cana-826 180 1 3 CARDINAL cana-826 180 2 heidelberg GPE cana-826 180 3 new york GPE cana-826 180 4 2012 DATE cana-826 181 1 4 CARDINAL cana-826 181 2 korea GPE cana-826 181 3 2009 DATE cana-826 181 4 20-24 DATE cana-826 182 1 5 CARDINAL cana-826 182 2 north holland GPE cana-826 182 3 new york GPE cana-826 182 4 1976 DATE cana-826 183 1 6 CARDINAL cana-826 185 1 university of waterloo ORG cana-826 185 2 canada GPE cana-826 185 3 1973 DATE cana-826 186 1 muhammad akram PERSON cana-826 186 2 musavarah sarwar GPE cana-826 186 3 rajab ali borzooei PERSON cana-826 186 4 1074 CARDINAL cana-826 186 5 31 CARDINAL cana-826 187 1 4s CARDINAL cana-826 187 2 2024 CARDINAL cana-826 187 3 22 CARDINAL cana-826 187 4 intelligent & ORG cana-826 187 5 35 DATE cana-826 187 6 2018 DATE cana-826 187 7 1905 1922 DATE cana-826 188 1 8 CARDINAL cana-826 188 2 k k PERSON cana-826 188 3 akram m PERSON cana-826 188 4 springer-verlag berlin heidelberg ORG cana-826 188 5 7(2 CARDINAL cana-826 188 6 2016 CARDINAL cana-826 188 7 287 CARDINAL cana-826 189 1 9 CARDINAL cana-826 189 2 9th ORDINAL cana-826 189 3 new york GPE cana-826 189 4 20(2006 DATE cana-826 189 5 139 CARDINAL cana-826 189 6 150 CARDINAL cana-826 190 1 10 CARDINAL cana-826 190 2 bulgarian NORP cana-826 190 3 9(2 CARDINAL cana-826 190 4 2009 DATE cana-826 190 5 46–53 CARDINAL cana-826 191 1 11 CARDINAL cana-826 191 2 pushpa publishing house ORG cana-826 191 3 14(1 CARDINAL cana-826 191 4 2013 DATE cana-826 192 1 12 CARDINAL cana-826 192 2 lanzhen yang PERSON cana-826 192 3 hua mao PERSON cana-826 192 4 36 CARDINAL cana-826 192 5 2019 CARDINAL cana-826 192 6 6641–6651 CARDINAL cana-826 193 1 13 CARDINAL cana-826 193 2 8 CARDINAL cana-826 193 3 1965 DATE cana-826 193 4 338–353 CARDINAL