id	sid	eid	entity	type
ejpam-5887	1	1	european journal	ORG
ejpam-5887	1	2	2025	DATE
ejpam-5887	1	3	18	CARDINAL
ejpam-5887	1	4	2	CARDINAL
ejpam-5887	1	5	5887	DATE
ejpam-5887	1	6	1307-5543	DATE
ejpam-5887	1	7	new york	GPE
ejpam-5887	1	8	n. m. noureldeen1,6	PERSON
ejpam-5887	1	9	j. jenisha2	PERSON
ejpam-5887	1	10	k. jeya daisy3	PERSON
ejpam-5887	1	11	jeyanthi4,∗	GPE
ejpam-5887	1	12	m. e. abdel-aal5	PERSON
ejpam-5887	1	13	1	CARDINAL
ejpam-5887	1	14	saudi arabia 2	GPE
ejpam-5887	1	15	reg.no	GPE
ejpam-5887	1	16	23213042092003	CARDINAL
ejpam-5887	1	17	tamilnadu	NORP
ejpam-5887	1	18	india	GPE
ejpam-5887	1	19	manonmaniam sundaranar university	ORG
ejpam-5887	1	20	tamilnadu	NORP
ejpam-5887	1	21	india	GPE
ejpam-5887	1	22	3	CARDINAL
ejpam-5887	1	23	tamilnadu	NORP
ejpam-5887	1	24	india 4	ORG
ejpam-5887	1	25	628215	DATE
ejpam-5887	1	26	tamilnadu	NORP
ejpam-5887	1	27	india	GPE
ejpam-5887	2	1	5	CARDINAL
ejpam-5887	2	2	13518	DATE
ejpam-5887	2	3	egypt 6	ORG
ejpam-5887	2	4	ain shams university	ORG
ejpam-5887	2	5	egypt	GPE
ejpam-5887	5	1	0	CARDINAL
ejpam-5887	5	2	1	DATE
ejpam-5887	6	1	1	CARDINAL
ejpam-5887	6	2	0	CARDINAL
ejpam-5887	6	3	1	DATE
ejpam-5887	7	1	f⋆(v	ORG
ejpam-5887	7	2	∏ uv∈e(g	PERCENT
ejpam-5887	7	3	i)−	NORP
ejpam-5887	7	4	j)| ≤ 1	DATE
ejpam-5887	7	5	j ∈	PERSON
ejpam-5887	7	6	0	CARDINAL
ejpam-5887	7	7	1	DATE
ejpam-5887	8	1	0	CARDINAL
ejpam-5887	8	2	1	DATE
ejpam-5887	8	3	1	CARDINAL
ejpam-5887	9	1	bistar	GPE
ejpam-5887	9	2	bistar	GPE
ejpam-5887	10	1	2020	DATE
ejpam-5887	10	2	05c78	CARDINAL
ejpam-5887	10	3	1	CARDINAL
ejpam-5887	11	1	the past six decades	DATE
ejpam-5887	12	1	n. m. noureldeen	PERSON
ejpam-5887	12	2	j. jenisha	ORG
ejpam-5887	12	3	jeyadaisy@holycrossngl.edu.in	PERSON
ejpam-5887	12	4	k. jeya daisy	PERSON
ejpam-5887	13	1	mohamed.abdelghani@fsc.bu.edu.eg	PERSON
ejpam-5887	13	2	m. e. abdel-aal	PERSON
ejpam-5887	14	1	1	CARDINAL
ejpam-5887	15	1	2025	CARDINAL
ejpam-5887	16	1	n. m. noureldeen et al	PERSON
ejpam-5887	17	1	j. pure	PERSON
ejpam-5887	17	2	18	CARDINAL
ejpam-5887	17	3	2	CARDINAL
ejpam-5887	17	4	2025	CARDINAL
ejpam-5887	17	5	5887 2	DATE
ejpam-5887	17	6	21 one	CARDINAL
ejpam-5887	19	1	1967	DATE
ejpam-5887	19	2	rosa	ORG
ejpam-5887	19	3	10	CARDINAL
ejpam-5887	20	1	4	CARDINAL
ejpam-5887	21	1	3	CARDINAL
ejpam-5887	24	1	two	CARDINAL
ejpam-5887	25	1	the subsequent years	DATE
ejpam-5887	27	1	11	CARDINAL
ejpam-5887	27	2	6	CARDINAL
ejpam-5887	29	1	2	CARDINAL
ejpam-5887	29	2	5	DATE
ejpam-5887	29	3	9	DATE
ejpam-5887	29	4	12	CARDINAL
ejpam-5887	30	1	1	CARDINAL
ejpam-5887	30	2	3	CARDINAL
ejpam-5887	33	1	0	CARDINAL
ejpam-5887	33	2	1	DATE
ejpam-5887	34	1	1	CARDINAL
ejpam-5887	34	2	0	CARDINAL
ejpam-5887	34	3	1	DATE
ejpam-5887	35	1	f⋆(v	ORG
ejpam-5887	35	2	∏ uv∈e(g	PERCENT
ejpam-5887	35	3	i)−	NORP
ejpam-5887	35	4	j)| ≤ 1	DATE
ejpam-5887	35	5	j ∈	PERSON
ejpam-5887	35	6	0	CARDINAL
ejpam-5887	35	7	1	DATE
ejpam-5887	36	1	0	CARDINAL
ejpam-5887	36	2	1	DATE
ejpam-5887	36	3	k−1	GPE
ejpam-5887	39	1	bistar	GPE
ejpam-5887	39	2	bipartite	NORP
ejpam-5887	39	3	second	ORDINAL
ejpam-5887	40	1	third	ORDINAL
ejpam-5887	41	1	fourth	ORDINAL
ejpam-5887	43	1	1	CARDINAL
ejpam-5887	43	2	4	CARDINAL
ejpam-5887	44	1	g′	PERSON
ejpam-5887	44	2	g′	PERSON
ejpam-5887	44	3	2	CARDINAL
ejpam-5887	44	4	4	CARDINAL
ejpam-5887	45	1	s′(g	GPE
ejpam-5887	45	2	∈	ORG
ejpam-5887	45	3	3	CARDINAL
ejpam-5887	45	4	8	CARDINAL
ejpam-5887	46	1	g1	ORG
ejpam-5887	46	2	gn	PERSON
ejpam-5887	46	3	2	CARDINAL
ejpam-5887	47	1	vi ∈ n. m. noureldeen	PERSON
ejpam-5887	48	1	j. pure	PERSON
ejpam-5887	48	2	18	CARDINAL
ejpam-5887	48	3	2	CARDINAL
ejpam-5887	48	4	2025	CARDINAL
ejpam-5887	48	5	5887 3	DATE
ejpam-5887	48	6	1	CARDINAL
ejpam-5887	48	7	2	DATE
ejpam-5887	48	8	∈	ORG
ejpam-5887	49	1	1 ≤	QUANTITY
ejpam-5887	49	2	1	CARDINAL
ejpam-5887	49	3	2	CARDINAL
ejpam-5887	49	4	bistar	GPE
ejpam-5887	49	5	bipartite	ORG
ejpam-5887	49	6	k1,n	ORG
ejpam-5887	49	7	the bistar graph bn	ORG
ejpam-5887	50	1	two	CARDINAL
ejpam-5887	50	2	bipartite	NORP
ejpam-5887	50	3	graph km	PERSON
ejpam-5887	51	1	1	CARDINAL
ejpam-5887	51	2	k1,n	ORG
ejpam-5887	53	1	k1,n	ORG
ejpam-5887	53	2	k1,n	ORG
ejpam-5887	54	1	1 ≤ i ≤ n	QUANTITY
ejpam-5887	54	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	55	1	n ≡	ORG
ejpam-5887	55	2	mod k	PERSON
ejpam-5887	55	3	0 ≤	MONEY
ejpam-5887	56	1	1	CARDINAL
ejpam-5887	56	2	0	CARDINAL
ejpam-5887	56	3	1	DATE
ejpam-5887	56	4	2	DATE
ejpam-5887	57	1	0	CARDINAL
ejpam-5887	57	2	1 ≤	QUANTITY
ejpam-5887	57	3	f(uu⌊n k ⌋+i	PERSON
ejpam-5887	58	1	1 ≤	QUANTITY
ejpam-5887	60	1	2	CARDINAL
ejpam-5887	61	1	1	CARDINAL
ejpam-5887	62	1	1 ≤	QUANTITY
ejpam-5887	62	2	0	CARDINAL
ejpam-5887	63	1	1	CARDINAL
ejpam-5887	63	2	1	CARDINAL
ejpam-5887	63	3	1 ≤	QUANTITY
ejpam-5887	64	1	1	CARDINAL
ejpam-5887	64	2	1	CARDINAL
ejpam-5887	64	3	0 ≤	MONEY
ejpam-5887	64	4	r.	NORP
ejpam-5887	65	1	j ∈	PERSON
ejpam-5887	65	2	0	CARDINAL
ejpam-5887	65	3	1	DATE
ejpam-5887	65	4	2	DATE
ejpam-5887	66	1	k1,n	PERSON
ejpam-5887	67	1	1	CARDINAL
ejpam-5887	67	2	4	CARDINAL
ejpam-5887	67	3	k1,9	PERSON
ejpam-5887	67	4	1	CARDINAL
ejpam-5887	67	5	0	CARDINAL
ejpam-5887	67	6	3	CARDINAL
ejpam-5887	67	7	0	CARDINAL
ejpam-5887	67	8	1	CARDINAL
ejpam-5887	67	9	4	CARDINAL
ejpam-5887	67	10	k1,9 n. m. noureldeen et al	PERSON
ejpam-5887	68	1	j. pure	PERSON
ejpam-5887	68	2	18	CARDINAL
ejpam-5887	68	3	2	CARDINAL
ejpam-5887	68	4	2025	CARDINAL
ejpam-5887	68	5	5887 4	DATE
ejpam-5887	68	6	21	CARDINAL
ejpam-5887	68	7	2	CARDINAL
ejpam-5887	70	1	vi	PERSON
ejpam-5887	70	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	70	3	e(bn	PERSON
ejpam-5887	71	1	uui	GPE
ejpam-5887	71	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	72	1	n ≡	ORG
ejpam-5887	72	2	mod k	PERSON
ejpam-5887	72	3	0 ≤	MONEY
ejpam-5887	73	1	1	CARDINAL
ejpam-5887	73	2	e(bn	PERSON
ejpam-5887	73	3	0	CARDINAL
ejpam-5887	73	4	1	DATE
ejpam-5887	73	5	2	CARDINAL
ejpam-5887	74	1	f(uv	PRODUCT
ejpam-5887	74	2	0	CARDINAL
ejpam-5887	74	3	0	CARDINAL
ejpam-5887	74	4	1 ≤	QUANTITY
ejpam-5887	74	5	f(uu⌊n k ⌋+i	PERSON
ejpam-5887	76	1	1 ≤	QUANTITY
ejpam-5887	77	1	2	CARDINAL
ejpam-5887	77	2	1	CARDINAL
ejpam-5887	78	1	1 ≤	QUANTITY
ejpam-5887	78	2	f(vvi	CARDINAL
ejpam-5887	78	3	0	CARDINAL
ejpam-5887	78	4	1 ≤	QUANTITY
ejpam-5887	78	5	1	CARDINAL
ejpam-5887	78	6	f(vv⌊n k	PERSON
ejpam-5887	79	1	0	CARDINAL
ejpam-5887	79	2	n ≡ 1	ORG
ejpam-5887	79	3	2	CARDINAL
ejpam-5887	79	4	3	CARDINAL
ejpam-5887	79	5	1	CARDINAL
ejpam-5887	79	6	mod k	PERSON
ejpam-5887	79	7	3 1	DATE
ejpam-5887	79	8	n ≡ 0	ORG
ejpam-5887	79	9	mod k	PERSON
ejpam-5887	79	10	3	CARDINAL
ejpam-5887	79	11	2	CARDINAL
ejpam-5887	79	12	2	CARDINAL
ejpam-5887	79	13	n ≡ 1	ORG
ejpam-5887	79	14	mod k	PERSON
ejpam-5887	79	15	3	CARDINAL
ejpam-5887	79	16	f(vv⌊n k ⌋+i	PERSON
ejpam-5887	80	1	k −	PERSON
ejpam-5887	83	1	1 ≤	QUANTITY
ejpam-5887	84	1	2 1	DATE
ejpam-5887	86	1	1 ≤	QUANTITY
ejpam-5887	86	2	2n	CARDINAL
ejpam-5887	87	1	2n	CARDINAL
ejpam-5887	87	2	k + 1	FAC
ejpam-5887	87	3	1	CARDINAL
ejpam-5887	87	4	n ≡ 0	ORG
ejpam-5887	87	5	mod k	PERSON
ejpam-5887	87	6	2⌊nk	CARDINAL
ejpam-5887	87	7	3	CARDINAL
ejpam-5887	87	8	2 ≤	QUANTITY
ejpam-5887	89	1	3 2⌊nk	CARDINAL
ejpam-5887	89	2	1	CARDINAL
ejpam-5887	89	3	1	CARDINAL
ejpam-5887	90	1	2	CARDINAL
ejpam-5887	90	2	3	CARDINAL
ejpam-5887	90	3	n ≡ 1	ORG
ejpam-5887	90	4	mod k	PERSON
ejpam-5887	90	5	k ≥ 3	ORG
ejpam-5887	90	6	2⌊nk	CARDINAL
ejpam-5887	90	7	1	CARDINAL
ejpam-5887	91	1	0 2⌊nk ⌋+ 2	CARDINAL
ejpam-5887	91	2	1 ≤	QUANTITY
ejpam-5887	93	1	1	CARDINAL
ejpam-5887	93	2	n ≡ k	ORG
ejpam-5887	94	1	mod k	PERSON
ejpam-5887	95	1	2⌊nk	CARDINAL
ejpam-5887	95	2	1	CARDINAL
ejpam-5887	95	3	0	CARDINAL
ejpam-5887	95	4	1	CARDINAL
ejpam-5887	95	5	2 ≤	QUANTITY
ejpam-5887	97	1	2⌊nk	CARDINAL
ejpam-5887	97	2	2	CARDINAL
ejpam-5887	98	1	1	DATE
ejpam-5887	98	2	mod k	PERSON
ejpam-5887	98	3	k −	PERSON
ejpam-5887	101	1	+ 1	DATE
ejpam-5887	101	2	1 ≤	QUANTITY
ejpam-5887	103	1	1	CARDINAL
ejpam-5887	104	1	i)−	NORP
ejpam-5887	104	2	j)| ≤ 1	DATE
ejpam-5887	104	3	j ∈	PERSON
ejpam-5887	104	4	0	CARDINAL
ejpam-5887	104	5	1	DATE
ejpam-5887	107	1	2	CARDINAL
ejpam-5887	107	2	4	CARDINAL
ejpam-5887	107	3	2	CARDINAL
ejpam-5887	107	4	n. m.	PERSON
ejpam-5887	109	1	j. pure	PERSON
ejpam-5887	109	2	18	CARDINAL
ejpam-5887	109	3	2	CARDINAL
ejpam-5887	109	4	2025	CARDINAL
ejpam-5887	109	5	5887 5	DATE
ejpam-5887	109	6	21	CARDINAL
ejpam-5887	109	7	2	CARDINAL
ejpam-5887	109	8	4	CARDINAL
ejpam-5887	109	9	b9,9 theorem 3	ORG
ejpam-5887	110	1	3 ≤ k ≤	QUANTITY
ejpam-5887	110	2	2	CARDINAL
ejpam-5887	112	1	3 ≤ k ≤	QUANTITY
ejpam-5887	112	2	2	CARDINAL
ejpam-5887	112	3	⌊n(n−1	CARDINAL
ejpam-5887	112	4	2k	CARDINAL
ejpam-5887	112	5	1	CARDINAL
ejpam-5887	114	1	2k	CARDINAL
ejpam-5887	114	2	⌊n(n−1	CARDINAL
ejpam-5887	114	3	2k	CARDINAL
ejpam-5887	115	1	1	CARDINAL
ejpam-5887	115	2	two	CARDINAL
ejpam-5887	116	1	n ≤	FAC
ejpam-5887	117	1	0	CARDINAL
ejpam-5887	117	2	1	CARDINAL
ejpam-5887	118	1	2k	CARDINAL
ejpam-5887	118	2	2k	CARDINAL
ejpam-5887	118	3	1	CARDINAL
ejpam-5887	118	4	1	CARDINAL
ejpam-5887	118	5	1	CARDINAL
ejpam-5887	118	6	2	DATE
ejpam-5887	119	1	1	CARDINAL
ejpam-5887	120	1	vf∗(i	PERSON
ejpam-5887	123	1	2k	CARDINAL
ejpam-5887	123	2	⌊n(n−1	CARDINAL
ejpam-5887	123	3	2k ⌋+1	CARDINAL
ejpam-5887	123	4	⌋+1	LAW
ejpam-5887	124	1	|vf∗(0)−vf∗(j)|	CARDINAL
ejpam-5887	124	2	1	CARDINAL
ejpam-5887	124	3	1	CARDINAL
ejpam-5887	124	4	2	DATE
ejpam-5887	124	5	k−1	GPE
ejpam-5887	125	1	kn	PERSON
ejpam-5887	125	2	3 ≤ k ≤	QUANTITY
ejpam-5887	125	3	2	CARDINAL
ejpam-5887	126	1	4	CARDINAL
ejpam-5887	127	1	bipartite	ORG
ejpam-5887	127	2	graph km	PERSON
ejpam-5887	127	3	mod k	PERSON
ejpam-5887	127	4	n ≡ r2	ORG
ejpam-5887	127	5	mod k	PERSON
ejpam-5887	129	1	mod k	PERSON
ejpam-5887	129	2	n ≡ r2	ORG
ejpam-5887	129	3	mod k	PERSON
ejpam-5887	130	1	n)| = k(⌊mk	PERSON
ejpam-5887	131	1	n)| = k(k⌊mk	PERSON
ejpam-5887	131	2	⌋⌊ n k ⌋+	PERSON
ejpam-5887	135	1	⌋⌊ n k	PERSON
ejpam-5887	136	1	k⌊mk	PERSON
ejpam-5887	137	1	⌋+1	LAW
ejpam-5887	138	1	r1+r2 k	PERSON
ejpam-5887	139	1	r1+r2 k ⌋+	PERSON
ejpam-5887	140	1	1	CARDINAL
ejpam-5887	140	2	r1 + r2	ORG
ejpam-5887	140	3	r1+r2 k	PERSON
ejpam-5887	141	1	⌋⌊ n k	PERSON
ejpam-5887	143	1	+ 1	CARDINAL
ejpam-5887	143	2	r1+r2 k ⌋+1	PERSON
ejpam-5887	145	1	n. m. noureldeen et al	PERSON
ejpam-5887	146	1	j. pure	PERSON
ejpam-5887	146	2	18	CARDINAL
ejpam-5887	146	3	2	CARDINAL
ejpam-5887	146	4	2025	CARDINAL
ejpam-5887	146	5	5887 6 of 21 3	DATE
ejpam-5887	148	1	5	CARDINAL
ejpam-5887	149	1	0(mod k	PERSON
ejpam-5887	151	1	k ≤	ORG
ejpam-5887	151	2	t ≥ 1	DATE
ejpam-5887	152	1	0 ≤ j ≤	QUANTITY
ejpam-5887	153	1	j k⌋	PERSON
ejpam-5887	154	1	0	CARDINAL
ejpam-5887	154	2	1	DATE
ejpam-5887	157	1	j k⌋	PERSON
ejpam-5887	157	2	j k⌋+ 1	PERSON
ejpam-5887	158	1	j k⌋+ 1	PERSON
ejpam-5887	158	2	1	CARDINAL
ejpam-5887	158	3	1	CARDINAL
ejpam-5887	158	4	2	DATE
ejpam-5887	159	1	1	CARDINAL
ejpam-5887	161	1	3.1	CARDINAL
ejpam-5887	162	1	3	CARDINAL
ejpam-5887	162	2	4	DATE
ejpam-5887	162	3	5	DATE
ejpam-5887	162	4	6	CARDINAL
ejpam-5887	164	1	two	CARDINAL
ejpam-5887	167	1	1	CARDINAL
ejpam-5887	169	1	4tk	ORDINAL
ejpam-5887	170	1	1	CARDINAL
ejpam-5887	172	1	mod k	PERSON
ejpam-5887	174	1	5	CARDINAL
ejpam-5887	176	1	t ≥ 1	DATE
ejpam-5887	176	2	0 ≤	MONEY
ejpam-5887	177	1	2	CARDINAL
ejpam-5887	177	2	2tk	ORDINAL
ejpam-5887	178	1	4tk + 4r	DATE
ejpam-5887	179	1	4	CARDINAL
ejpam-5887	179	2	⌊4rk	GPE
ejpam-5887	179	3	4t+	CARDINAL
ejpam-5887	179	4	1	CARDINAL
ejpam-5887	179	5	1 ≤	QUANTITY
ejpam-5887	179	6	2	CARDINAL
ejpam-5887	180	1	1	CARDINAL
ejpam-5887	180	2	2 2	DATE
ejpam-5887	180	3	3 2t+	CARDINAL
ejpam-5887	181	1	2t+	CARDINAL
ejpam-5887	182	1	1	CARDINAL
ejpam-5887	182	2	1 ≤	QUANTITY
ejpam-5887	182	3	2	CARDINAL
ejpam-5887	182	4	4	CARDINAL
ejpam-5887	182	5	4t+1	CARDINAL
ejpam-5887	183	1	2t+1	GPE
ejpam-5887	183	2	t ≥ 1	DATE
ejpam-5887	184	1	|vf∗(0)−vf∗(j)|	CARDINAL
ejpam-5887	184	2	1	CARDINAL
ejpam-5887	184	3	1	CARDINAL
ejpam-5887	184	4	2	DATE
ejpam-5887	184	5	k−1	GPE
ejpam-5887	185	1	4	CARDINAL
ejpam-5887	185	2	⌊4rk	GPE
ejpam-5887	185	3	4	CARDINAL
ejpam-5887	185	4	+ 1	DATE
ejpam-5887	186	1	⌊4rk	GPE
ejpam-5887	187	1	1 ≤	QUANTITY
ejpam-5887	188	1	2	CARDINAL
ejpam-5887	188	2	2	CARDINAL
ejpam-5887	190	1	+ 1	DATE
ejpam-5887	190	2	t ≥ 1	DATE
ejpam-5887	191	1	1	CARDINAL
ejpam-5887	191	2	1	CARDINAL
ejpam-5887	191	3	2	DATE
ejpam-5887	193	1	1	CARDINAL
ejpam-5887	194	1	k ≤ n. n. m. noureldeen et al	ORG
ejpam-5887	195	1	j. pure	PERSON
ejpam-5887	195	2	18	CARDINAL
ejpam-5887	195	3	2	CARDINAL
ejpam-5887	195	4	2025	CARDINAL
ejpam-5887	195	5	5887 7	DATE
ejpam-5887	195	6	21	CARDINAL
ejpam-5887	195	7	7	CARDINAL
ejpam-5887	196	1	3	CARDINAL
ejpam-5887	199	1	vi	PERSON
ejpam-5887	199	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	200	1	uui	GPE
ejpam-5887	200	2	uvi	GPE
ejpam-5887	200	3	1 ≤ i ≤ n	QUANTITY
ejpam-5887	201	1	0	CARDINAL
ejpam-5887	201	2	1	DATE
ejpam-5887	201	3	2	CARDINAL
ejpam-5887	201	4	0	CARDINAL
ejpam-5887	201	5	1	CARDINAL
ejpam-5887	201	6	2	CARDINAL
ejpam-5887	201	7	1	CARDINAL
ejpam-5887	201	8	1	CARDINAL
ejpam-5887	201	9	2	CARDINAL
ejpam-5887	201	10	1	CARDINAL
ejpam-5887	201	11	1	CARDINAL
ejpam-5887	202	1	1	CARDINAL
ejpam-5887	203	1	3	CARDINAL
ejpam-5887	204	1	2	CARDINAL
ejpam-5887	204	2	6	CARDINAL
ejpam-5887	204	3	8	CARDINAL
ejpam-5887	204	4	5	CARDINAL
ejpam-5887	204	5	3	CARDINAL
ejpam-5887	205	1	6	CARDINAL
ejpam-5887	205	2	n ≥ 3	DATE
ejpam-5887	205	3	3	CARDINAL
ejpam-5887	206	1	8	CARDINAL
ejpam-5887	207	1	4	CARDINAL
ejpam-5887	210	1	vi	PERSON
ejpam-5887	210	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	211	1	uui	GPE
ejpam-5887	211	2	uvi	GPE
ejpam-5887	211	3	1 ≤ i ≤ n	QUANTITY
ejpam-5887	212	1	4	CARDINAL
ejpam-5887	212	2	4	CARDINAL
ejpam-5887	212	3	8	CARDINAL
ejpam-5887	212	4	12	CARDINAL
ejpam-5887	213	1	5	CARDINAL
ejpam-5887	213	2	4	CARDINAL
ejpam-5887	214	1	4	CARDINAL
ejpam-5887	215	1	2	CARDINAL
ejpam-5887	215	2	1 or	CARDINAL
ejpam-5887	215	3	2	CARDINAL
ejpam-5887	216	1	2	CARDINAL
ejpam-5887	216	2	3	CARDINAL
ejpam-5887	216	3	1	CARDINAL
ejpam-5887	216	4	1	CARDINAL
ejpam-5887	216	5	2	DATE
ejpam-5887	216	6	3	DATE
ejpam-5887	217	1	4	CARDINAL
ejpam-5887	218	1	6	CARDINAL
ejpam-5887	218	2	4	CARDINAL
ejpam-5887	218	3	n ≥ 4	DATE
ejpam-5887	218	4	9	CARDINAL
ejpam-5887	219	1	5	CARDINAL
ejpam-5887	219	2	2	CARDINAL
ejpam-5887	222	1	vi	PERSON
ejpam-5887	222	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	223	1	uui	GPE
ejpam-5887	223	2	uvi	GPE
ejpam-5887	223	3	1 ≤ i ≤ n	QUANTITY
ejpam-5887	223	4	0	CARDINAL
ejpam-5887	223	5	1	DATE
ejpam-5887	223	6	2	DATE
ejpam-5887	223	7	3	DATE
ejpam-5887	223	8	4	CARDINAL
ejpam-5887	223	9	4	CARDINAL
ejpam-5887	223	10	1	CARDINAL
ejpam-5887	223	11	3	CARDINAL
ejpam-5887	223	12	2	CARDINAL
ejpam-5887	223	13	1	CARDINAL
ejpam-5887	223	14	4	CARDINAL
ejpam-5887	223	15	f(vv2	GPE
ejpam-5887	224	1	3	CARDINAL
ejpam-5887	224	2	1	CARDINAL
ejpam-5887	224	3	2	CARDINAL
ejpam-5887	224	4	3	CARDINAL
ejpam-5887	224	5	4	CARDINAL
ejpam-5887	224	6	2	CARDINAL
ejpam-5887	225	1	1	CARDINAL
ejpam-5887	225	2	5	CARDINAL
ejpam-5887	227	1	4	CARDINAL
ejpam-5887	228	1	5	CARDINAL
ejpam-5887	228	2	vf∗(i	PERSON
ejpam-5887	228	3	0	CARDINAL
ejpam-5887	228	4	1	CARDINAL
ejpam-5887	228	5	0	CARDINAL
ejpam-5887	228	6	1	DATE
ejpam-5887	228	7	2	DATE
ejpam-5887	228	8	3	DATE
ejpam-5887	228	9	4	DATE
ejpam-5887	228	10	2	CARDINAL
ejpam-5887	229	1	1	CARDINAL
ejpam-5887	229	2	1	CARDINAL
ejpam-5887	229	3	2	DATE
ejpam-5887	229	4	3	DATE
ejpam-5887	229	5	4	CARDINAL
ejpam-5887	230	1	1	CARDINAL
ejpam-5887	230	2	two	CARDINAL
ejpam-5887	230	3	2	CARDINAL
ejpam-5887	230	4	3	CARDINAL
ejpam-5887	231	1	2	CARDINAL
ejpam-5887	231	2	1	CARDINAL
ejpam-5887	231	3	4	CARDINAL
ejpam-5887	232	1	2	CARDINAL
ejpam-5887	233	1	5	CARDINAL
ejpam-5887	234	1	8	CARDINAL
ejpam-5887	234	2	12	CARDINAL
ejpam-5887	235	1	3	CARDINAL
ejpam-5887	236	1	2 or	CARDINAL
ejpam-5887	236	2	0	CARDINAL
ejpam-5887	236	3	1	DATE
ejpam-5887	236	4	2	DATE
ejpam-5887	236	5	3	DATE
ejpam-5887	236	6	4	DATE
ejpam-5887	236	7	1 or	CARDINAL
ejpam-5887	236	8	0	CARDINAL
ejpam-5887	236	9	1	DATE
ejpam-5887	236	10	2	DATE
ejpam-5887	236	11	3	DATE
ejpam-5887	236	12	4	CARDINAL
ejpam-5887	237	1	2	CARDINAL
ejpam-5887	237	2	n. m. noureldeen et al	PERSON
ejpam-5887	238	1	j. pure	PERSON
ejpam-5887	238	2	18	CARDINAL
ejpam-5887	238	3	2	CARDINAL
ejpam-5887	238	4	2025	CARDINAL
ejpam-5887	238	5	5887 8	DATE
ejpam-5887	238	6	21	CARDINAL
ejpam-5887	238	7	3	CARDINAL
ejpam-5887	238	8	2	CARDINAL
ejpam-5887	239	1	1	CARDINAL
ejpam-5887	239	2	1	CARDINAL
ejpam-5887	239	3	2	DATE
ejpam-5887	239	4	3	DATE
ejpam-5887	239	5	4	DATE
ejpam-5887	240	1	5	CARDINAL
ejpam-5887	241	1	n = 4	ORG
ejpam-5887	241	2	10	CARDINAL
ejpam-5887	241	3	16	CARDINAL
ejpam-5887	242	1	5	CARDINAL
ejpam-5887	242	2	5	CARDINAL
ejpam-5887	243	1	6	CARDINAL
ejpam-5887	243	2	n ≥ 5	DATE
ejpam-5887	243	3	5	CARDINAL
ejpam-5887	244	1	3.2	CARDINAL
ejpam-5887	245	1	k ≤ n.	ORG
ejpam-5887	245	2	3	CARDINAL
ejpam-5887	245	3	4	DATE
ejpam-5887	245	4	5	DATE
ejpam-5887	245	5	10	CARDINAL
ejpam-5887	247	1	two	CARDINAL
ejpam-5887	250	1	1	CARDINAL
ejpam-5887	255	1	mod k	PERSON
ejpam-5887	257	1	5	CARDINAL
ejpam-5887	259	1	t ≥ 1	DATE
ejpam-5887	259	2	0 ≤	QUANTITY
ejpam-5887	260	1	2	CARDINAL
ejpam-5887	260	2	2tk	ORDINAL
ejpam-5887	263	1		ORG
ejpam-5887	263	2	3	CARDINAL
ejpam-5887	263	3	0 3t+ 1	CARDINAL
ejpam-5887	263	4	1	CARDINAL
ejpam-5887	263	5	3 3	DATE
ejpam-5887	263	6	1	CARDINAL
ejpam-5887	263	7	1	CARDINAL
ejpam-5887	263	8	2 ≤	QUANTITY
ejpam-5887	264	1	2	CARDINAL
ejpam-5887	264	2	∈	ORG
ejpam-5887	264	3	0	CARDINAL
ejpam-5887	264	4	1	DATE
ejpam-5887	265	1	1	CARDINAL
ejpam-5887	265	2	 2t+ 1	DATE
ejpam-5887	265	3	2 2t+ 1 or 2t+ 2	DATE
ejpam-5887	265	4	1	CARDINAL
ejpam-5887	265	5	3 2t+ 1	CARDINAL
ejpam-5887	265	6	1	CARDINAL
ejpam-5887	265	7	4 2t or 2t+ 1	DATE
ejpam-5887	265	8	3	CARDINAL
ejpam-5887	265	9	1	CARDINAL
ejpam-5887	265	10	5 2t+	CARDINAL
ejpam-5887	265	11	2t+	CARDINAL
ejpam-5887	266	1	1	CARDINAL
ejpam-5887	266	2	2 ≤	QUANTITY
ejpam-5887	266	3	2	CARDINAL
ejpam-5887	266	4	∈	ORG
ejpam-5887	266	5	0	CARDINAL
ejpam-5887	266	6	1	CARDINAL
ejpam-5887	268	1	0 ≤	MONEY
ejpam-5887	268	2	3	CARDINAL
ejpam-5887	268	3	≥ 3t+ 1	CARDINAL
ejpam-5887	268	4	t ≥ 1	DATE
ejpam-5887	269	1	1	CARDINAL
ejpam-5887	269	2	2	CARDINAL
ejpam-5887	269	3	2	CARDINAL
ejpam-5887	269	4	t ≥ 1	DATE
ejpam-5887	270	1	|vf∗(0)− vf∗(j)|	PERSON
ejpam-5887	270	2	1	CARDINAL
ejpam-5887	270	3	1	CARDINAL
ejpam-5887	270	4	2	DATE
ejpam-5887	270	5	k−1	GPE
ejpam-5887	271	1	3t+⌊3rk	CARDINAL
ejpam-5887	271	2	3	CARDINAL
ejpam-5887	272	1	+ 1	DATE
ejpam-5887	272	2	2 ≤	QUANTITY
ejpam-5887	273	1	2	CARDINAL
ejpam-5887	273	2	2t+	CARDINAL
ejpam-5887	273	3	t ≥ 1	DATE
ejpam-5887	274	1	|vf∗(0)− vf∗(j)|	PERSON
ejpam-5887	274	2	1	CARDINAL
ejpam-5887	274	3	1	CARDINAL
ejpam-5887	274	4	2	DATE
ejpam-5887	274	5	1	CARDINAL
ejpam-5887	275	1	k ≤ n. theorem	PERSON
ejpam-5887	275	2	11	CARDINAL
ejpam-5887	276	1	3	CARDINAL
ejpam-5887	279	1	vi	PERSON
ejpam-5887	279	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	280	1	uui	GPE
ejpam-5887	280	2	vui	GPE
ejpam-5887	280	3	1 ≤ i ≤ n	QUANTITY
ejpam-5887	282	1	0	CARDINAL
ejpam-5887	282	2	1	DATE
ejpam-5887	282	3	2	CARDINAL
ejpam-5887	282	4	n. m. noureldeen et al	PERSON
ejpam-5887	283	1	j. pure	PERSON
ejpam-5887	283	2	18	CARDINAL
ejpam-5887	283	3	2	CARDINAL
ejpam-5887	283	4	2025	CARDINAL
ejpam-5887	283	5	5887 9 of 21	DATE
ejpam-5887	283	6	1	CARDINAL
ejpam-5887	284	1	2	CARDINAL
ejpam-5887	284	2	1	CARDINAL
ejpam-5887	284	3	2	CARDINAL
ejpam-5887	284	4	1	CARDINAL
ejpam-5887	285	1	1	CARDINAL
ejpam-5887	285	2	3	CARDINAL
ejpam-5887	286	1	2	CARDINAL
ejpam-5887	287	1	6	CARDINAL
ejpam-5887	287	2	5	CARDINAL
ejpam-5887	287	3	3	CARDINAL
ejpam-5887	288	1	10	CARDINAL
ejpam-5887	288	2	n ≥ 3	DATE
ejpam-5887	288	3	3	CARDINAL
ejpam-5887	289	1	12	CARDINAL
ejpam-5887	290	1	4	CARDINAL
ejpam-5887	290	2	2	CARDINAL
ejpam-5887	293	1	vi	PERSON
ejpam-5887	293	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	294	1	uui	GPE
ejpam-5887	294	2	vui	GPE
ejpam-5887	294	3	1 ≤ i ≤ n	QUANTITY
ejpam-5887	295	1	e(s′(k1,2	PERSON
ejpam-5887	296	1	0	CARDINAL
ejpam-5887	296	2	1	DATE
ejpam-5887	296	3	2	DATE
ejpam-5887	296	4	3	CARDINAL
ejpam-5887	296	5	1	CARDINAL
ejpam-5887	296	6	3	CARDINAL
ejpam-5887	296	7	1	CARDINAL
ejpam-5887	296	8	2	CARDINAL
ejpam-5887	297	1	2	CARDINAL
ejpam-5887	297	2	1	CARDINAL
ejpam-5887	297	3	1	CARDINAL
ejpam-5887	297	4	2	CARDINAL
ejpam-5887	297	5	1	CARDINAL
ejpam-5887	297	6	3	CARDINAL
ejpam-5887	297	7	1	CARDINAL
ejpam-5887	297	8	1	CARDINAL
ejpam-5887	298	1	1	CARDINAL
ejpam-5887	299	1	1	CARDINAL
ejpam-5887	299	2	4	CARDINAL
ejpam-5887	300	1	4	CARDINAL
ejpam-5887	300	2	3	CARDINAL
ejpam-5887	301	1	4	CARDINAL
ejpam-5887	301	2	0	CARDINAL
ejpam-5887	301	3	1	CARDINAL
ejpam-5887	301	4	0	CARDINAL
ejpam-5887	301	5	1	DATE
ejpam-5887	301	6	2	DATE
ejpam-5887	301	7	3	CARDINAL
ejpam-5887	301	8	1	CARDINAL
ejpam-5887	301	9	0	CARDINAL
ejpam-5887	301	10	1	DATE
ejpam-5887	301	11	2	DATE
ejpam-5887	301	12	3	CARDINAL
ejpam-5887	302	1	2	CARDINAL
ejpam-5887	303	1	1	CARDINAL
ejpam-5887	303	2	1	CARDINAL
ejpam-5887	303	3	2	DATE
ejpam-5887	303	4	3	CARDINAL
ejpam-5887	303	5	2	CARDINAL
ejpam-5887	303	6	1	CARDINAL
ejpam-5887	303	7	2	CARDINAL
ejpam-5887	303	8	1	CARDINAL
ejpam-5887	304	1	4	CARDINAL
ejpam-5887	305	1	8	CARDINAL
ejpam-5887	305	2	9	CARDINAL
ejpam-5887	305	3	5	CARDINAL
ejpam-5887	305	4	4	CARDINAL
ejpam-5887	306	1	10	CARDINAL
ejpam-5887	306	2	4	CARDINAL
ejpam-5887	306	3	n ≥ 4	DATE
ejpam-5887	306	4	13	CARDINAL
ejpam-5887	307	1	5	CARDINAL
ejpam-5887	310	1	vi	PERSON
ejpam-5887	310	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	311	1	uui	GPE
ejpam-5887	311	2	vui	GPE
ejpam-5887	311	3	1 ≤ i ≤ n	QUANTITY
ejpam-5887	313	1	0	CARDINAL
ejpam-5887	313	2	1	DATE
ejpam-5887	313	3	n ≤ 3	ORG
ejpam-5887	314	1	1	CARDINAL
ejpam-5887	314	2	2	CARDINAL
ejpam-5887	314	3	3 1	CARDINAL
ejpam-5887	314	4	2	CARDINAL
ejpam-5887	314	5	2	CARDINAL
ejpam-5887	314	6	3 2	CARDINAL
ejpam-5887	314	7	1	CARDINAL
ejpam-5887	314	8	3	DATE
ejpam-5887	315	1	1	CARDINAL
ejpam-5887	315	2	1	CARDINAL
ejpam-5887	315	3	3	CARDINAL
ejpam-5887	315	4	2 3	DATE
ejpam-5887	315	5	3 4	DATE
ejpam-5887	315	6	1	CARDINAL
ejpam-5887	315	7	1	CARDINAL
ejpam-5887	315	8	2	CARDINAL
ejpam-5887	315	9	2	CARDINAL
ejpam-5887	315	10	3	CARDINAL
ejpam-5887	315	11	1 2	CARDINAL
ejpam-5887	315	12	1	CARDINAL
ejpam-5887	315	13	2	CARDINAL
ejpam-5887	315	14	3 3	DATE
ejpam-5887	315	15	2	CARDINAL
ejpam-5887	315	16	2	CARDINAL
ejpam-5887	315	17	3 4	DATE
ejpam-5887	315	18	3	CARDINAL
ejpam-5887	315	19	|ef	ORG
ejpam-5887	315	20	j ∈	PERSON
ejpam-5887	315	21	0	CARDINAL
ejpam-5887	315	22	1	DATE
ejpam-5887	315	23	2	DATE
ejpam-5887	315	24	3	DATE
ejpam-5887	315	25	4	CARDINAL
ejpam-5887	316	1	5	CARDINAL
ejpam-5887	316	2	n ≤ 3	PRODUCT
ejpam-5887	316	3	n. m. noureldeen et al	PERSON
ejpam-5887	317	1	j. pure	PERSON
ejpam-5887	317	2	18	CARDINAL
ejpam-5887	317	3	2	CARDINAL
ejpam-5887	317	4	2025	CARDINAL
ejpam-5887	317	5	5887 10 of 21	DATE
ejpam-5887	317	6	10	CARDINAL
ejpam-5887	317	7	12	CARDINAL
ejpam-5887	318	1	5	DATE
ejpam-5887	318	2	5	CARDINAL
ejpam-5887	319	1	10	CARDINAL
ejpam-5887	319	2	5	CARDINAL
ejpam-5887	319	3	n ≥ 5	DATE
ejpam-5887	319	4	4	CARDINAL
ejpam-5887	319	5	bistar	GPE
ejpam-5887	321	1	14	DATE
ejpam-5887	323	1	at least one	CARDINAL
ejpam-5887	323	2	0	CARDINAL
ejpam-5887	326	1	⋃	PERSON
ejpam-5887	326	2	1≤i≤n	CARDINAL
ejpam-5887	328	1	⋃	PERSON
ejpam-5887	328	2	vi ∈	PERSON
ejpam-5887	328	3	1 ≤	QUANTITY
ejpam-5887	329	1	kt	PERSON
ejpam-5887	330	1	0	CARDINAL
ejpam-5887	330	2	1	DATE
ejpam-5887	331	1	1	CARDINAL
ejpam-5887	331	2	j ∈	PERSON
ejpam-5887	331	3	0	CARDINAL
ejpam-5887	331	4	1	DATE
ejpam-5887	333	1	0	CARDINAL
ejpam-5887	333	2	1	DATE
ejpam-5887	333	3	k−1	GPE
ejpam-5887	333	4	n ≡	ORG
ejpam-5887	333	5	mod k	PERSON
ejpam-5887	333	6	0 ≤	MONEY
ejpam-5887	334	1	∈	ORG
ejpam-5887	334	2	1 ≤	QUANTITY
ejpam-5887	334	3	0	CARDINAL
ejpam-5887	334	4	1 ≤	QUANTITY
ejpam-5887	334	5	1	CARDINAL
ejpam-5887	334	6	1 ≤	QUANTITY
ejpam-5887	334	7	2⌊n−1 k	PERSON
ejpam-5887	334	8	2	CARDINAL
ejpam-5887	334	9	2⌊n−1 k	QUANTITY
ejpam-5887	334	10	1 ≤	QUANTITY
ejpam-5887	335	1	k − 1	PERSON
ejpam-5887	335	2	k − 1)⌊n−1	ORG
ejpam-5887	335	3	1 ≤	QUANTITY
ejpam-5887	335	4	f(ek⌊n−1 k ⌋+i	PERSON
ejpam-5887	336	1	mod k	PERSON
ejpam-5887	336	2	1 ≤	QUANTITY
ejpam-5887	336	3	1− k⌊n−1 k ⌋.	PRODUCT
ejpam-5887	336	4	1	CARDINAL
ejpam-5887	336	5	vf∗(i	PERSON
ejpam-5887	337	1	j ∈	PERSON
ejpam-5887	337	2	0	CARDINAL
ejpam-5887	337	3	1	CARDINAL
ejpam-5887	337	4	2	CARDINAL
ejpam-5887	340	1	4.1	CARDINAL
ejpam-5887	340	2	path union of star	ORG
ejpam-5887	340	3	k1,m	PERSON
ejpam-5887	340	4	n ≡ 0	ORG
ejpam-5887	340	5	1	CARDINAL
ejpam-5887	340	6	mod k	PERSON
ejpam-5887	341	1	n ≡ k	ORG
ejpam-5887	342	1	mod k	PERSON
ejpam-5887	343	1	mod k	PERSON
ejpam-5887	344	1	n. m. noureldeen	PERSON
ejpam-5887	345	1	j. pure	PERSON
ejpam-5887	345	2	18	CARDINAL
ejpam-5887	345	3	2	CARDINAL
ejpam-5887	345	4	2025	CARDINAL
ejpam-5887	345	5	5887 11	DATE
ejpam-5887	345	6	21	CARDINAL
ejpam-5887	345	7	15	CARDINAL
ejpam-5887	346	1	k1,m	PERSON
ejpam-5887	346	2	n ≡ 0	ORG
ejpam-5887	346	3	1	CARDINAL
ejpam-5887	346	4	mod k	PERSON
ejpam-5887	349	1	{vi	PERSON
ejpam-5887	349	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	349	3	1 ≤	QUANTITY
ejpam-5887	352	1	vnv	ORG
ejpam-5887	352	2	j n	PERSON
ejpam-5887	352	3	1 ≤	QUANTITY
ejpam-5887	352	4	1 ≤	QUANTITY
ejpam-5887	354	1	mod k	PERSON
ejpam-5887	354	2	14	DATE
ejpam-5887	356	1	0	CARDINAL
ejpam-5887	356	2	1	DATE
ejpam-5887	356	3	2	DATE
ejpam-5887	356	4	k−1	GPE
ejpam-5887	356	5	n ≡ 0	ORG
ejpam-5887	356	6	1	CARDINAL
ejpam-5887	356	7	mod k	PERSON
ejpam-5887	356	8	andm ≡	PERSON
ejpam-5887	356	9	mod k	PERSON
ejpam-5887	356	10	1 ≤	QUANTITY
ejpam-5887	356	11	1	CARDINAL
ejpam-5887	356	12	0	CARDINAL
ejpam-5887	356	13	1 ≤	QUANTITY
ejpam-5887	357	1	1	CARDINAL
ejpam-5887	357	2	two	CARDINAL
ejpam-5887	358	1	case(i	PERSON
ejpam-5887	358	2	n ≡ 0	ORG
ejpam-5887	359	1	mod k	PERSON
ejpam-5887	362	1	0	CARDINAL
ejpam-5887	362	2	1 ≤	QUANTITY
ejpam-5887	363	1	1 ≤	QUANTITY
ejpam-5887	364	1	1	CARDINAL
ejpam-5887	365	1	1	CARDINAL
ejpam-5887	365	2	1 ≤	QUANTITY
ejpam-5887	367	1	1	CARDINAL
ejpam-5887	367	2	+ 1	DATE
ejpam-5887	368	1	2n	CARDINAL
ejpam-5887	368	2	k 2	ORG
ejpam-5887	368	3	1 ≤	QUANTITY
ejpam-5887	370	1	1	CARDINAL
ejpam-5887	370	2	+ 2	DATE
ejpam-5887	370	3	2n	CARDINAL
ejpam-5887	372	1	k − 1	PERSON
ejpam-5887	372	2	1 ≤	QUANTITY
ejpam-5887	372	3	k−1)n	GPE
ejpam-5887	373	1	n k	PERSON
ejpam-5887	373	2	1 +	DATE
ejpam-5887	373	3	1	CARDINAL
ejpam-5887	374	1	1 ≤	QUANTITY
ejpam-5887	375	1	1	CARDINAL
ejpam-5887	377	1	n k	PERSON
ejpam-5887	377	2	0 ≤	MONEY
ejpam-5887	380	1	1	CARDINAL
ejpam-5887	380	2	n ≡ 1	ORG
ejpam-5887	380	3	mod k	PERSON
ejpam-5887	380	4	1 ≤	QUANTITY
ejpam-5887	380	5	1	CARDINAL
ejpam-5887	380	6	1 ≤	QUANTITY
ejpam-5887	380	7	j n	PERSON
ejpam-5887	381	1	0	CARDINAL
ejpam-5887	381	2	1 ≤ j ≤ ⌊mk ⌋ 1	QUANTITY
ejpam-5887	381	3	1 ≤	QUANTITY
ejpam-5887	381	4	2	CARDINAL
ejpam-5887	381	5	2⌊mk	CARDINAL
ejpam-5887	381	6	1 ≤	QUANTITY
ejpam-5887	383	1	k − 1	PERSON
ejpam-5887	383	2	1 ≤	QUANTITY
ejpam-5887	383	3	k ⌋+j n	ORG
ejpam-5887	383	4	1 ≤ j ≤ r.	QUANTITY
ejpam-5887	383	5	1	CARDINAL
ejpam-5887	383	6	1 ≤	QUANTITY
ejpam-5887	383	7	1	CARDINAL
ejpam-5887	383	8	0 ≤	MONEY
ejpam-5887	383	9	r.	NORP
ejpam-5887	383	10	j ∈	PERSON
ejpam-5887	383	11	0	CARDINAL
ejpam-5887	383	12	1	DATE
ejpam-5887	384	1	n ≡ 0	ORG
ejpam-5887	384	2	1	CARDINAL
ejpam-5887	384	3	mod k	PERSON
ejpam-5887	386	1	3	CARDINAL
ejpam-5887	386	2	4	CARDINAL
ejpam-5887	386	3	4.kv	CARDINAL
ejpam-5887	386	4	1,5	CARDINAL
ejpam-5887	386	5	3	CARDINAL
ejpam-5887	386	6	n. m.	PERSON
ejpam-5887	388	1	j. pure	PERSON
ejpam-5887	388	2	18	CARDINAL
ejpam-5887	388	3	2	CARDINAL
ejpam-5887	388	4	2025	CARDINAL
ejpam-5887	388	5	5887 12	DATE
ejpam-5887	388	6	21	CARDINAL
ejpam-5887	388	7	3	CARDINAL
ejpam-5887	388	8	4	CARDINAL
ejpam-5887	388	9	4.kv	CARDINAL
ejpam-5887	388	10	1,5	CARDINAL
ejpam-5887	388	11	16	CARDINAL
ejpam-5887	389	1	k1,m	PERSON
ejpam-5887	389	2	n ≡ k− 1	ORG
ejpam-5887	389	3	mod k	PERSON
ejpam-5887	389	4	0	CARDINAL
ejpam-5887	389	5	mod k	PERSON
ejpam-5887	392	1	{vi	PERSON
ejpam-5887	392	2	1 ≤ i ≤ n	QUANTITY
ejpam-5887	392	3	1 ≤	QUANTITY
ejpam-5887	395	1	vnv	ORG
ejpam-5887	395	2	j n	PERSON
ejpam-5887	395	3	1 ≤	QUANTITY
ejpam-5887	395	4	1 ≤	QUANTITY
ejpam-5887	397	1	mod k	PERSON
ejpam-5887	397	2	14	DATE
ejpam-5887	399	1	0	CARDINAL
ejpam-5887	399	2	1	DATE
ejpam-5887	399	3	2	DATE
ejpam-5887	399	4	k−1	GPE
ejpam-5887	399	5	n ≡ k−1	ORG
ejpam-5887	399	6	mod k	PERSON
ejpam-5887	399	7	≡ k−1	GPE
ejpam-5887	399	8	mod k	PERSON
ejpam-5887	399	9	1 ≤	QUANTITY
ejpam-5887	401	1	1 ≤	QUANTITY
ejpam-5887	401	2	15	CARDINAL
ejpam-5887	401	3	0	CARDINAL
ejpam-5887	401	4	1 ≤	QUANTITY
ejpam-5887	402	1	1	CARDINAL
ejpam-5887	404	1	n−	ORG
ejpam-5887	405	1	1 ≤	QUANTITY
ejpam-5887	405	2	1	CARDINAL
ejpam-5887	406	1	1 ≤	QUANTITY
ejpam-5887	406	2	2	CARDINAL
ejpam-5887	406	3	k + 2 ≤ i ≤ n	FAC
ejpam-5887	406	4	1 ≤	QUANTITY
ejpam-5887	407	1	k − 1	PERSON
ejpam-5887	407	2	k + 2 ≤ i ≤ n	FAC
ejpam-5887	407	3	1 ≤	QUANTITY
ejpam-5887	407	4	k + 2	ORG
ejpam-5887	407	5	m− k + 2	PERSON
ejpam-5887	407	6	m− k + 2	PERSON
ejpam-5887	407	7	m− k + 2	PERSON
ejpam-5887	407	8	k⌊nk ⌋⌊	PERSON
ejpam-5887	407	9	k − 1)(1	ORG
ejpam-5887	407	10	k⌊nk ⌋⌊	PERSON
ejpam-5887	407	11	k − 1)(1	ORG
ejpam-5887	407	12	1 ≤	QUANTITY
ejpam-5887	408	1	1	CARDINAL
ejpam-5887	408	2	k⌊nk ⌋⌊	PERSON
ejpam-5887	408	3	k − 1)(1	ORG
ejpam-5887	408	4	0 ≤	MONEY
ejpam-5887	410	1	1	CARDINAL
ejpam-5887	411	1	j ∈	PERSON
ejpam-5887	411	2	0	CARDINAL
ejpam-5887	411	3	1	DATE
ejpam-5887	412	1	n ≡ k−1	ORG
ejpam-5887	412	2	mod k	PERSON
ejpam-5887	412	3	0, k−1	PERCENT
ejpam-5887	412	4	mod k	PERSON
ejpam-5887	414	1	4	CARDINAL
ejpam-5887	414	2	5	CARDINAL
ejpam-5887	414	3	4.kv	CARDINAL
ejpam-5887	414	4	1,5	CARDINAL
ejpam-5887	414	5	4	CARDINAL
ejpam-5887	415	1	n. m. noureldeen et al	PERSON
ejpam-5887	416	1	j. pure	PERSON
ejpam-5887	416	2	18	CARDINAL
ejpam-5887	416	3	2	CARDINAL
ejpam-5887	416	4	2025	CARDINAL
ejpam-5887	416	5	5887 13	DATE
ejpam-5887	416	6	21	CARDINAL
ejpam-5887	416	7	4	CARDINAL
ejpam-5887	416	8	5	CARDINAL
ejpam-5887	416	9	4.kv	CARDINAL
ejpam-5887	416	10	1,5	CARDINAL
ejpam-5887	416	11	4.2	CARDINAL
ejpam-5887	417	1	n ≡ 0	ORG
ejpam-5887	417	2	1	CARDINAL
ejpam-5887	417	3	mod k	PERSON
ejpam-5887	418	1	n ≡ k	ORG
ejpam-5887	419	1	mod k	PERSON
ejpam-5887	420	1	mod k	PERSON
ejpam-5887	421	1	17	CARDINAL
ejpam-5887	422	1	bistar graph p	ORG
ejpam-5887	422	2	n ≡ 0	ORG
ejpam-5887	422	3	1	CARDINAL
ejpam-5887	422	4	mod k	PERSON
ejpam-5887	425	1	{vi	PERSON
ejpam-5887	426	1	1 ≤ i ≤ n	QUANTITY
ejpam-5887	426	2	1 ≤	QUANTITY
ejpam-5887	428	1	vnv	ORG
ejpam-5887	428	2	j n	PERSON
ejpam-5887	428	3	unu j n	PERSON
ejpam-5887	428	4	1 ≤	QUANTITY
ejpam-5887	428	5	1	CARDINAL
ejpam-5887	428	6	1 ≤	QUANTITY
ejpam-5887	430	1	0	CARDINAL
ejpam-5887	430	2	1	DATE
ejpam-5887	430	3	2	DATE
ejpam-5887	430	4	k−1	GPE
ejpam-5887	430	5	n ≡ 0	ORG
ejpam-5887	430	6	1	CARDINAL
ejpam-5887	430	7	mod k	PERSON
ejpam-5887	430	8	andm ≡	PERSON
ejpam-5887	430	9	mod k	PERSON
ejpam-5887	430	10	0 ≤	MONEY
ejpam-5887	430	11	1	CARDINAL
ejpam-5887	430	12	0	CARDINAL
ejpam-5887	430	13	1 ≤	QUANTITY
ejpam-5887	431	1	1	CARDINAL
ejpam-5887	431	2	0	CARDINAL
ejpam-5887	431	3	1 ≤	QUANTITY
ejpam-5887	431	4	two	CARDINAL
ejpam-5887	432	1	n ≡ 0	ORG
ejpam-5887	432	2	mod k	PERSON
ejpam-5887	434	1	0	CARDINAL
ejpam-5887	434	2	1 ≤	QUANTITY
ejpam-5887	435	1	1 ≤	QUANTITY
ejpam-5887	437	1	1 ≤	QUANTITY
ejpam-5887	437	2	k 1	FAC
ejpam-5887	438	1	k 2	ORG
ejpam-5887	440	1	k −	PERSON
ejpam-5887	440	2	1	CARDINAL
ejpam-5887	442	1	1	CARDINAL
ejpam-5887	442	2	mod k	PERSON
ejpam-5887	446	1	j ≡	PERSON
ejpam-5887	447	1	1	CARDINAL
ejpam-5887	447	2	1 ≤	QUANTITY
ejpam-5887	448	1	2	CARDINAL
ejpam-5887	448	2	1	CARDINAL
ejpam-5887	449	1	1	CARDINAL
ejpam-5887	449	2	1 ≤	QUANTITY
ejpam-5887	449	3	1 ≤	QUANTITY
ejpam-5887	449	4	1)⌊mk	CARDINAL
ejpam-5887	449	5	f(viv k⌊m	FAC
ejpam-5887	450	1	k − q	PERSON
ejpam-5887	450	2	j ≡	PERSON
ejpam-5887	451	1	1	CARDINAL
ejpam-5887	451	2	1 ≤	QUANTITY
ejpam-5887	452	1	2 1	DATE
ejpam-5887	453	1	1	CARDINAL
ejpam-5887	453	2	1 ≤ i ≤ n k	QUANTITY
ejpam-5887	453	3	1 ≤ j ≤	QUANTITY
ejpam-5887	454	1	k⌊m k ⌋+j n	FAC
ejpam-5887	457	1	1	CARDINAL
ejpam-5887	457	2	1 ≤	QUANTITY
ejpam-5887	458	1	2	CARDINAL
ejpam-5887	458	2	1	CARDINAL
ejpam-5887	459	1	1	CARDINAL
ejpam-5887	459	2	1 ≤	QUANTITY
ejpam-5887	459	3	1 ≤ j ≤	QUANTITY
ejpam-5887	459	4	n. m. noureldeen et al	PERSON
ejpam-5887	460	1	j. pure	PERSON
ejpam-5887	460	2	18	CARDINAL
ejpam-5887	460	3	2	CARDINAL
ejpam-5887	460	4	2025	CARDINAL
ejpam-5887	460	5	5887 14	DATE
ejpam-5887	462	1	j i	PERSON
ejpam-5887	462	2	1 ≤ i ≤	QUANTITY
ejpam-5887	463	1	1 ≤	QUANTITY
ejpam-5887	463	2	j ≤ m.	PERSON
ejpam-5887	463	3	2n	CARDINAL
ejpam-5887	467	1	1	CARDINAL
ejpam-5887	467	2	2n	CARDINAL
ejpam-5887	469	1	1 ≤	QUANTITY
ejpam-5887	470	1	1	CARDINAL
ejpam-5887	471	1	2n	CARDINAL
ejpam-5887	473	1	0 ≤ i ≤	QUANTITY
ejpam-5887	475	1	1	CARDINAL
ejpam-5887	475	2	n ≡ 1	ORG
ejpam-5887	475	3	mod k	PERSON
ejpam-5887	475	4	1 ≤	QUANTITY
ejpam-5887	475	5	1	CARDINAL
ejpam-5887	475	6	1 ≤	QUANTITY
ejpam-5887	475	7	1 ≤	QUANTITY
ejpam-5887	475	8	1	CARDINAL
ejpam-5887	475	9	1 ≤	QUANTITY
ejpam-5887	475	10	j n	PERSON
ejpam-5887	475	11	0	CARDINAL
ejpam-5887	475	12	1 ≤ j ≤ ⌊mk ⌋	QUANTITY
ejpam-5887	475	13	j n	PERSON
ejpam-5887	475	14	0	CARDINAL
ejpam-5887	475	15	1 ≤ j ≤ ⌊mk ⌋ − 1	QUANTITY
ejpam-5887	476	1	k ⌋+j n	PRODUCT
ejpam-5887	476	2	j ≡	PERSON
ejpam-5887	477	1	1	CARDINAL
ejpam-5887	477	2	1 ≤	QUANTITY
ejpam-5887	478	1	2	CARDINAL
ejpam-5887	478	2	1	CARDINAL
ejpam-5887	479	1	1	CARDINAL
ejpam-5887	479	2	1 ≤ j ≤ m− ⌊mk	QUANTITY
ejpam-5887	479	3	k ⌋ n	ORG
ejpam-5887	479	4	0	CARDINAL
ejpam-5887	479	5	≡ 1	ORG
ejpam-5887	479	6	2	CARDINAL
ejpam-5887	480	1	3	CARDINAL
ejpam-5887	480	2	1	CARDINAL
ejpam-5887	480	3	mod k	PERSON
ejpam-5887	481	1	3 1	CARDINAL
ejpam-5887	482	1	mod k	PERSON
ejpam-5887	483	1	3	CARDINAL
ejpam-5887	483	2	2	CARDINAL
ejpam-5887	483	3	2	CARDINAL
ejpam-5887	483	4	≡ 1	FAC
ejpam-5887	483	5	mod k	PERSON
ejpam-5887	484	1	3	CARDINAL
ejpam-5887	484	2	k ⌋+j n	PRODUCT
ejpam-5887	485	1	k −	PERSON
ejpam-5887	485	2	j ≡	PERSON
ejpam-5887	485	3	1	CARDINAL
ejpam-5887	485	4	1 ≤	QUANTITY
ejpam-5887	487	1	2 1	DATE
ejpam-5887	488	1	1	CARDINAL
ejpam-5887	488	2	1 ≤ j ≤ m− ⌊mk	QUANTITY
ejpam-5887	488	3	2⌊nk	CARDINAL
ejpam-5887	489	1	2mn k	PERSON
ejpam-5887	489	2	0	CARDINAL
ejpam-5887	489	3	2 ≤	QUANTITY
ejpam-5887	491	1	1 2⌊nk	CARDINAL
ejpam-5887	492	1	2mn	ORG
ejpam-5887	493	1	1	CARDINAL
ejpam-5887	494	1	mod k	PERSON
ejpam-5887	495	1	2⌊nk ⌋+ 2n⌊mk	CARDINAL
ejpam-5887	495	2	2⌊nk	CARDINAL
ejpam-5887	496	1	3	CARDINAL
ejpam-5887	496	2	2 ≤	QUANTITY
ejpam-5887	498	1	3 2⌊nk	CARDINAL
ejpam-5887	498	2	2⌊nk	CARDINAL
ejpam-5887	498	3	1	CARDINAL
ejpam-5887	498	4	1	CARDINAL
ejpam-5887	499	1	2	CARDINAL
ejpam-5887	499	2	3	CARDINAL
ejpam-5887	499	3	1	CARDINAL
ejpam-5887	499	4	mod k	PERSON
ejpam-5887	499	5	k ≥ 3	ORG
ejpam-5887	499	6	2⌊nk ⌋+ 2n⌊mk	CARDINAL
ejpam-5887	499	7	2(k	CARDINAL
ejpam-5887	499	8	1)⌊nk	CARDINAL
ejpam-5887	499	9	1	CARDINAL
ejpam-5887	499	10	0 2⌊nk ⌋+ 2n⌊mk	CARDINAL
ejpam-5887	499	11	2(k	CARDINAL
ejpam-5887	499	12	1)⌊nk	CARDINAL
ejpam-5887	499	13	2	CARDINAL
ejpam-5887	499	14	1 ≤	QUANTITY
ejpam-5887	501	1	1	CARDINAL
ejpam-5887	503	1	mod k	PERSON
ejpam-5887	503	2	2⌊nk ⌋+ 2n⌊mk	CARDINAL
ejpam-5887	503	3	2r⌊nk	CARDINAL
ejpam-5887	503	4	1	CARDINAL
ejpam-5887	503	5	0	CARDINAL
ejpam-5887	503	6	1	CARDINAL
ejpam-5887	503	7	2 ≤	QUANTITY
ejpam-5887	506	1	2⌊nk	CARDINAL
ejpam-5887	506	2	2r⌊nk	CARDINAL
ejpam-5887	506	3	2	CARDINAL
ejpam-5887	507	1	1	DATE
ejpam-5887	507	2	3	CARDINAL
ejpam-5887	507	3	+ 1	DATE
ejpam-5887	507	4	1 ≤	QUANTITY
ejpam-5887	509	1	1	CARDINAL
ejpam-5887	510	1	j ∈	PERSON
ejpam-5887	510	2	0	CARDINAL
ejpam-5887	510	3	1	DATE
ejpam-5887	511	1	n ≡ 0	ORG
ejpam-5887	511	2	1	CARDINAL
ejpam-5887	511	3	mod k	PERSON
ejpam-5887	513	1	5	CARDINAL
ejpam-5887	513	2	4	CARDINAL
ejpam-5887	513	3	5	CARDINAL
ejpam-5887	514	1	n. m.	PERSON
ejpam-5887	516	1	j. pure	PERSON
ejpam-5887	516	2	18	CARDINAL
ejpam-5887	516	3	2	CARDINAL
ejpam-5887	516	4	2025	CARDINAL
ejpam-5887	516	5	5887 15	DATE
ejpam-5887	516	6	5	CARDINAL
ejpam-5887	516	7	4	CARDINAL
ejpam-5887	516	8	18	CARDINAL
ejpam-5887	517	1	bistar graph p	ORG
ejpam-5887	517	2	n ≡ k− 1	ORG
ejpam-5887	517	3	mod k	PERSON
ejpam-5887	517	4	0	CARDINAL
ejpam-5887	517	5	mod k	PERSON
ejpam-5887	520	1	{vi	PERSON
ejpam-5887	521	1	1 ≤ i ≤ n	QUANTITY
ejpam-5887	521	2	1 ≤	QUANTITY
ejpam-5887	523	1	vnv	ORG
ejpam-5887	523	2	j n	PERSON
ejpam-5887	523	3	unu j n	PERSON
ejpam-5887	523	4	1 ≤	QUANTITY
ejpam-5887	523	5	1	CARDINAL
ejpam-5887	523	6	1 ≤	QUANTITY
ejpam-5887	525	1	0	CARDINAL
ejpam-5887	525	2	1	DATE
ejpam-5887	525	3	2	DATE
ejpam-5887	526	1	n ≡ k	ORG
ejpam-5887	527	1	mod k	PERSON
ejpam-5887	527	2	mod k	PERSON
ejpam-5887	529	1	1 ≤	QUANTITY
ejpam-5887	532	1	1 ≤	QUANTITY
ejpam-5887	532	2	17	CARDINAL
ejpam-5887	532	3	1 ≤	QUANTITY
ejpam-5887	534	1	1 ≤	QUANTITY
ejpam-5887	534	2	17	CARDINAL
ejpam-5887	534	3	0	CARDINAL
ejpam-5887	534	4	1 ≤	QUANTITY
ejpam-5887	535	1	1	CARDINAL
ejpam-5887	535	2	0	CARDINAL
ejpam-5887	535	3	1 ≤	QUANTITY
ejpam-5887	535	4	two	CARDINAL
ejpam-5887	537	1	mod k	PERSON
ejpam-5887	539	1	0	CARDINAL
ejpam-5887	539	2	n−	ORG
ejpam-5887	539	3	k + 2 ≤ i ≤ n	FAC
ejpam-5887	539	4	1 ≤	QUANTITY
ejpam-5887	541	1	1	CARDINAL
ejpam-5887	541	2	f(vn−k+1+iv	ORG
ejpam-5887	541	3	k n−k+1+i	PERSON
ejpam-5887	543	1	1 ≤	QUANTITY
ejpam-5887	544	1	2	CARDINAL
ejpam-5887	544	2	k n	PERSON
ejpam-5887	545	1	k n	PERSON
ejpam-5887	546	1	0	CARDINAL
ejpam-5887	546	2	f(un−k+1+iu	ORG
ejpam-5887	546	3	k n−k+1+i	PERSON
ejpam-5887	549	1	1 ≤	QUANTITY
ejpam-5887	550	1	2	CARDINAL
ejpam-5887	552	1	1	CARDINAL
ejpam-5887	553	1	k 2	ORG
ejpam-5887	553	2	2	CARDINAL
ejpam-5887	553	3	k + 1 ≤ j ≤ 3m k : :	FAC
ejpam-5887	553	4	k − 1	PERSON
ejpam-5887	553	5	k−1)m k + 1 ≤ j ≤ m	PERSON
ejpam-5887	554	1	2⌊nk	CARDINAL
ejpam-5887	555	1	2mn	ORG
ejpam-5887	556	1	0	CARDINAL
ejpam-5887	556	2	1	DATE
ejpam-5887	557	1	1 2⌊nk	CARDINAL
ejpam-5887	558	1	2mn	ORG
ejpam-5887	559	1	2 ≤	QUANTITY
ejpam-5887	560	1	2	CARDINAL
ejpam-5887	560	2	n. m. noureldeen et al	PERSON
ejpam-5887	561	1	j. pure	PERSON
ejpam-5887	561	2	18	CARDINAL
ejpam-5887	561	3	2	CARDINAL
ejpam-5887	561	4	2025	CARDINAL
ejpam-5887	561	5	5887 16	DATE
ejpam-5887	561	6	2⌊nk	CARDINAL
ejpam-5887	562	1	2mn	ORG
ejpam-5887	563	1	1	CARDINAL
ejpam-5887	564	1	1 2⌊nk	CARDINAL
ejpam-5887	565	1	2mn	ORG
ejpam-5887	566	1	0	CARDINAL
ejpam-5887	566	2	2 ≤	QUANTITY
ejpam-5887	567	1	2	CARDINAL
ejpam-5887	569	1	mod k	PERSON
ejpam-5887	570	1	1 ≤	QUANTITY
ejpam-5887	571	1	k + 2 ≤ i ≤ n	ORG
ejpam-5887	571	2	1 ≤	QUANTITY
ejpam-5887	573	1	0	CARDINAL
ejpam-5887	573	2	1 ≤	QUANTITY
ejpam-5887	574	1	1	CARDINAL
ejpam-5887	574	2	0	CARDINAL
ejpam-5887	574	3	1 ≤	QUANTITY
ejpam-5887	576	1	0	CARDINAL
ejpam-5887	576	2	n−	ORG
ejpam-5887	577	1	m− k	PERSON
ejpam-5887	579	1	m− k	PERSON
ejpam-5887	580	1	f(vn−k+1+iv	ORG
ejpam-5887	580	2	j n−k+1+i	PERSON
ejpam-5887	580	3	0	CARDINAL
ejpam-5887	581	1	2 ≤	QUANTITY
ejpam-5887	583	1	2	CARDINAL
ejpam-5887	583	2	m− k	PERSON
ejpam-5887	584	1	2 ≤	QUANTITY
ejpam-5887	584	2	j ≤	PERSON
ejpam-5887	586	1	1	CARDINAL
ejpam-5887	586	2	f(vn−k+1+iv	ORG
ejpam-5887	586	3	j n−k+1+i	PERSON
ejpam-5887	588	1	1 ≤	QUANTITY
ejpam-5887	590	1	1	CARDINAL
ejpam-5887	590	2	m− k	PERSON
ejpam-5887	591	1	2 ≤ j ≤ m	QUANTITY
ejpam-5887	591	2	f(vn−k+1+iv	ORG
ejpam-5887	591	3	j n−k+1+i	PERSON
ejpam-5887	593	1	1	CARDINAL
ejpam-5887	594	1	m− k	PERSON
ejpam-5887	596	1	m− k	PERSON
ejpam-5887	597	1	1	CARDINAL
ejpam-5887	597	2	f(un−k+1+iu	ORG
ejpam-5887	597	3	j n−k+1+i	PERSON
ejpam-5887	599	1	1 ≤	CARDINAL
ejpam-5887	599	2	j ≤	PERSON
ejpam-5887	601	1	1	CARDINAL
ejpam-5887	601	2	2n⌊mk	CARDINAL
ejpam-5887	602	1	1	CARDINAL
ejpam-5887	602	2	0 2n⌊mk	CARDINAL
ejpam-5887	602	3	2n	CARDINAL
ejpam-5887	602	4	1 ≤	QUANTITY
ejpam-5887	603	1	1	CARDINAL
ejpam-5887	604	1	2n⌊mk	CARDINAL
ejpam-5887	604	2	2n	CARDINAL
ejpam-5887	604	3	0 ≤	MONEY
ejpam-5887	607	1	1	CARDINAL
ejpam-5887	608	1	j ∈	PERSON
ejpam-5887	608	2	0	CARDINAL
ejpam-5887	608	3	1	DATE
ejpam-5887	609	1	n ≡ k	ORG
ejpam-5887	610	1	mod k	PERSON
ejpam-5887	610	2	mod k	PERSON
ejpam-5887	612	1	6	CARDINAL
ejpam-5887	613	1	3	CARDINAL
ejpam-5887	613	2	5.bv	CARDINAL
ejpam-5887	613	3	6	CARDINAL
ejpam-5887	613	4	6	CARDINAL
ejpam-5887	613	5	3	CARDINAL
ejpam-5887	613	6	5.bv	CARDINAL
ejpam-5887	613	7	3,3	CARDINAL
ejpam-5887	613	8	4.3	CARDINAL
ejpam-5887	614	1	3	CARDINAL
ejpam-5887	614	2	4	CARDINAL
ejpam-5887	615	1	≥ 3	DATE
ejpam-5887	616	1	n. m. noureldeen et al	PERSON
ejpam-5887	617	1	j. pure	PERSON
ejpam-5887	617	2	18	CARDINAL
ejpam-5887	617	3	2	CARDINAL
ejpam-5887	617	4	2025	CARDINAL
ejpam-5887	617	5	5887 17	DATE
ejpam-5887	617	6	21	CARDINAL
ejpam-5887	620	1	19	CARDINAL
ejpam-5887	621	1	k	PERSON
ejpam-5887	623	1	k ⌋.	ORG
ejpam-5887	623	2	0	CARDINAL
ejpam-5887	623	3	1	DATE
ejpam-5887	623	4	k−1	GPE
ejpam-5887	624	1	1	CARDINAL
ejpam-5887	625	1	k ⌋+ 1	ORG
ejpam-5887	627	1	+ 1	DATE
ejpam-5887	629	1	+ 2	CARDINAL
ejpam-5887	630	1	k ⌋ + 1	FAC
ejpam-5887	632	1	20	CARDINAL
ejpam-5887	633	1	≥ k. proof	PERSON
ejpam-5887	637	1	two	CARDINAL
ejpam-5887	638	1	case(i	PERSON
ejpam-5887	638	2	0	CARDINAL
ejpam-5887	638	3	1	DATE
ejpam-5887	638	4	2	DATE
ejpam-5887	638	5	k−1	GPE
ejpam-5887	638	6	vf∗(i	PERSON
ejpam-5887	639	1	0	CARDINAL
ejpam-5887	639	2	1	DATE
ejpam-5887	639	3	2	DATE
ejpam-5887	643	1	19	CARDINAL
ejpam-5887	644	1	n ≥ k. theorem 21	ORG
ejpam-5887	645	1	k. proof	PERSON
ejpam-5887	646	1	= kt	PERSON
ejpam-5887	646	2	t ≥ 1	DATE
ejpam-5887	647	1	n.ckt	ORG
ejpam-5887	647	2	ktn+n−1	PERSON
ejpam-5887	648	1	vf∗(i	PERSON
ejpam-5887	649	1	0	CARDINAL
ejpam-5887	649	2	1	DATE
ejpam-5887	649	3	2	DATE
ejpam-5887	649	4	k−1	GPE
ejpam-5887	651	1	2	CARDINAL
ejpam-5887	651	2	1	CARDINAL
ejpam-5887	651	3	2	DATE
ejpam-5887	651	4	k−1	GPE
ejpam-5887	652	1	20	CARDINAL
ejpam-5887	653	1	n.ckt	ORG
ejpam-5887	654	1	22	CARDINAL
ejpam-5887	655	1	3	CARDINAL
ejpam-5887	655	2	2	CARDINAL
ejpam-5887	655	3	2	CARDINAL
ejpam-5887	658	1	1 ≤ i ≤ n	QUANTITY
ejpam-5887	658	2	1 ≤	QUANTITY
ejpam-5887	661	1	n , vmi v1i	ORG
ejpam-5887	661	2	1 ≤ i ≤ n − 1	QUANTITY
ejpam-5887	661	3	1 ≤	QUANTITY
ejpam-5887	661	4	1	CARDINAL
ejpam-5887	663	1	0	CARDINAL
ejpam-5887	663	2	1	DATE
ejpam-5887	663	3	2	CARDINAL
ejpam-5887	663	4	t ≥ 1	DATE
ejpam-5887	663	5	n. m. noureldeen et al	PERSON
ejpam-5887	664	1	j. pure	PERSON
ejpam-5887	664	2	18	CARDINAL
ejpam-5887	664	3	2	CARDINAL
ejpam-5887	664	4	2025	CARDINAL
ejpam-5887	664	5	5887 18	DATE
ejpam-5887	664	6	0	CARDINAL
ejpam-5887	665	1	1	CARDINAL
ejpam-5887	665	2	1 ≤	QUANTITY
ejpam-5887	666	1	1	CARDINAL
ejpam-5887	666	2	0	CARDINAL
ejpam-5887	667	1	1 ≤ j ≤ 2	QUANTITY
ejpam-5887	667	2	1	CARDINAL
ejpam-5887	667	3	1	CARDINAL
ejpam-5887	667	4	1	CARDINAL
ejpam-5887	667	5	3	CARDINAL
ejpam-5887	667	6	4	CARDINAL
ejpam-5887	667	7	2	CARDINAL
ejpam-5887	667	8	j ≡ 1	ORG
ejpam-5887	667	9	2	CARDINAL
ejpam-5887	667	10	1 ≤ j ≤	QUANTITY
ejpam-5887	667	11	1	CARDINAL
ejpam-5887	667	12	2	CARDINAL
ejpam-5887	667	13	1	CARDINAL
ejpam-5887	667	14	0	CARDINAL
ejpam-5887	667	15	1	CARDINAL
ejpam-5887	667	16	4	CARDINAL
ejpam-5887	667	17	2	CARDINAL
ejpam-5887	667	18	j ≡ 2,	ORG
ejpam-5887	667	19	3	DATE
ejpam-5887	667	20	1 ≤ j ≤ 3t+ 1	QUANTITY
ejpam-5887	667	21	2t+ 1	DATE
ejpam-5887	667	22	2	DATE
ejpam-5887	667	23	1	CARDINAL
ejpam-5887	667	24	2	DATE
ejpam-5887	667	25	2t+ 1	DATE
ejpam-5887	667	26	1	CARDINAL
ejpam-5887	667	27	2	CARDINAL
ejpam-5887	667	28	2	CARDINAL
ejpam-5887	668	1	3	CARDINAL
ejpam-5887	668	2	2.c3t+1	CARDINAL
ejpam-5887	669	1	t ≥ 1	DATE
ejpam-5887	670	1	2	CARDINAL
ejpam-5887	670	2	0	CARDINAL
ejpam-5887	670	3	1	CARDINAL
ejpam-5887	670	4	2	CARDINAL
ejpam-5887	670	5	2	CARDINAL
ejpam-5887	670	6	2	CARDINAL
ejpam-5887	670	7	0	CARDINAL
ejpam-5887	670	8	1	CARDINAL
ejpam-5887	670	9	2	CARDINAL
ejpam-5887	671	1	2	CARDINAL
ejpam-5887	671	2	2	CARDINAL
ejpam-5887	671	3	2	CARDINAL
ejpam-5887	672	1	2.c3t+1	CARDINAL
ejpam-5887	672	2	t ≥ 1	DATE
ejpam-5887	672	3	3	CARDINAL
ejpam-5887	673	1	21	CARDINAL
ejpam-5887	673	2	2.c3	CARDINAL
ejpam-5887	673	3	t ≥ 1	DATE
ejpam-5887	673	4	3	CARDINAL
ejpam-5887	674	1	20	CARDINAL
ejpam-5887	674	2	n ≥ 3	DATE
ejpam-5887	674	3	3	CARDINAL
ejpam-5887	675	1	7	CARDINAL
ejpam-5887	675	2	3	CARDINAL
ejpam-5887	675	3	7	CARDINAL
ejpam-5887	675	4	0	CARDINAL
ejpam-5887	675	5	0	CARDINAL
ejpam-5887	675	6	7	CARDINAL
ejpam-5887	675	7	3	CARDINAL
ejpam-5887	675	8	23	CARDINAL
ejpam-5887	676	1	4	CARDINAL
ejpam-5887	676	2	3	CARDINAL
ejpam-5887	679	1	1 ≤ i ≤ n	QUANTITY
ejpam-5887	679	2	1 ≤	QUANTITY
ejpam-5887	682	1	n , vmi v1i	ORG
ejpam-5887	682	2	1 ≤ i ≤ n − 1	QUANTITY
ejpam-5887	682	3	1 ≤	QUANTITY
ejpam-5887	682	4	1	CARDINAL
ejpam-5887	683	1	0	CARDINAL
ejpam-5887	683	2	1	DATE
ejpam-5887	683	3	2	DATE
ejpam-5887	683	4	3	CARDINAL
ejpam-5887	683	5	3	CARDINAL
ejpam-5887	683	6	5	DATE
ejpam-5887	683	7	2	CARDINAL
ejpam-5887	683	8	f(vji	NORP
ejpam-5887	684	1	1	CARDINAL
ejpam-5887	684	2	1 1	DATE
ejpam-5887	684	3	1	CARDINAL
ejpam-5887	684	4	2 3	DATE
ejpam-5887	684	5	2	CARDINAL
ejpam-5887	684	6	1 ≤	QUANTITY
ejpam-5887	684	7	3	CARDINAL
ejpam-5887	684	8	f(vji	NORP
ejpam-5887	686	1	1	CARDINAL
ejpam-5887	686	2	1 ≤	QUANTITY
ejpam-5887	686	3	1	CARDINAL
ejpam-5887	686	4	4	CARDINAL
ejpam-5887	686	5	2	CARDINAL
ejpam-5887	686	6	2 ≤ j ≤ 3 2	QUANTITY
ejpam-5887	686	7	1	CARDINAL
ejpam-5887	686	8	3 3	DATE
ejpam-5887	686	9	2	CARDINAL
ejpam-5887	686	10	1	CARDINAL
ejpam-5887	686	11	4	CARDINAL
ejpam-5887	686	12	5	CARDINAL
ejpam-5887	686	13	n. m. noureldeen et al	PERSON
ejpam-5887	687	1	j. pure	PERSON
ejpam-5887	687	2	18	CARDINAL
ejpam-5887	687	3	2	CARDINAL
ejpam-5887	687	4	2025	CARDINAL
ejpam-5887	687	5	5887 19 of 21	DATE
ejpam-5887	688	1	1	CARDINAL
ejpam-5887	688	2	2	CARDINAL
ejpam-5887	688	3	3 2	CARDINAL
ejpam-5887	688	4	1	CARDINAL
ejpam-5887	688	5	3	CARDINAL
ejpam-5887	688	6	2	CARDINAL
ejpam-5887	688	7	5	CARDINAL
ejpam-5887	688	8	2⌊m4	CARDINAL
ejpam-5887	689	1	0 2⌊m4	CARDINAL
ejpam-5887	689	2	1	CARDINAL
ejpam-5887	689	3	1 ≤	QUANTITY
ejpam-5887	689	4	2⌊m4	CARDINAL
ejpam-5887	689	5	1	CARDINAL
ejpam-5887	689	6	3 2⌊m4	CARDINAL
ejpam-5887	689	7	1	CARDINAL
ejpam-5887	689	8	0	CARDINAL
ejpam-5887	689	9	2	DATE
ejpam-5887	690	1	2.c5	CARDINAL
ejpam-5887	690	2	4	CARDINAL
ejpam-5887	691	1	two	CARDINAL
ejpam-5887	692	1	case(i	PERSON
ejpam-5887	692	2	2	CARDINAL
ejpam-5887	692	3	4	CARDINAL
ejpam-5887	692	4	1 ≤	QUANTITY
ejpam-5887	692	5	2	CARDINAL
ejpam-5887	692	6	1	CARDINAL
ejpam-5887	692	7	2 ≤	QUANTITY
ejpam-5887	692	8	2r	PRODUCT
ejpam-5887	692	9	2r + 1	DATE
ejpam-5887	692	10	4	CARDINAL
ejpam-5887	693	1	2	CARDINAL
ejpam-5887	693	2	1 2t+ 1 or 2t+ 2	CARDINAL
ejpam-5887	693	3	2	CARDINAL
ejpam-5887	693	4	3	DATE
ejpam-5887	693	5	2	CARDINAL
ejpam-5887	693	6	1 2t+ 1	CARDINAL
ejpam-5887	694	1	2 2t+ 1 or 2t+ 2	DATE
ejpam-5887	694	2	3	CARDINAL
ejpam-5887	694	3	1	CARDINAL
ejpam-5887	695	1	2	CARDINAL
ejpam-5887	695	2	two	CARDINAL
ejpam-5887	695	3	2	CARDINAL
ejpam-5887	697	1	2t+1	PERSON
ejpam-5887	697	2	1	CARDINAL
ejpam-5887	697	3	2	DATE
ejpam-5887	697	4	3	CARDINAL
ejpam-5887	697	5	2t+1	GPE
ejpam-5887	697	6	2	CARDINAL
ejpam-5887	698	1	2	CARDINAL
ejpam-5887	698	2	3	CARDINAL
ejpam-5887	698	3	2t+1	DATE
ejpam-5887	699	1	2t+2	PERSON
ejpam-5887	699	2	1	CARDINAL
ejpam-5887	699	3	2	DATE
ejpam-5887	699	4	3	CARDINAL
ejpam-5887	700	1	2	CARDINAL
ejpam-5887	700	2	2t+1	DATE
ejpam-5887	700	3	2	CARDINAL
ejpam-5887	701	1	case(ii	PERSON
ejpam-5887	701	2	4	CARDINAL
ejpam-5887	701	3	1 ≤	QUANTITY
ejpam-5887	701	4	1 ≤	QUANTITY
ejpam-5887	701	5	3	CARDINAL
ejpam-5887	701	6	3r	CARDINAL
ejpam-5887	701	7	3r + 2	DATE
ejpam-5887	701	8	4	CARDINAL
ejpam-5887	703	1	1 or 3t+ 2	CARDINAL
ejpam-5887	703	2	1 3t+ 2	CARDINAL
ejpam-5887	703	3	2 3t+ 2 or 3t+ 3	DATE
ejpam-5887	703	4	3	CARDINAL
ejpam-5887	703	5	3	CARDINAL
ejpam-5887	703	6	1 3t+ 1 or 3t+ 2	CARDINAL
ejpam-5887	703	7	2 3t+ 2 or 3t+ 3	DATE
ejpam-5887	703	8	3	CARDINAL
ejpam-5887	704	1	1 ≤	QUANTITY
ejpam-5887	704	2	+ 1	DATE
ejpam-5887	704	3	3	CARDINAL
ejpam-5887	704	4	3	CARDINAL
ejpam-5887	705	1	3	CARDINAL
ejpam-5887	705	2	1	CARDINAL
ejpam-5887	705	3	2	DATE
ejpam-5887	705	4	3	CARDINAL
ejpam-5887	706	1	3	CARDINAL
ejpam-5887	706	2	1	CARDINAL
ejpam-5887	706	3	2	DATE
ejpam-5887	706	4	3	CARDINAL
ejpam-5887	706	5	3	CARDINAL
ejpam-5887	706	6	2	CARDINAL
ejpam-5887	707	1	21	CARDINAL
ejpam-5887	707	2	t ≥ 1	DATE
ejpam-5887	707	3	4	CARDINAL
ejpam-5887	708	1	20	CARDINAL
ejpam-5887	709	1	n ≥ 4	DATE
ejpam-5887	709	2	4	CARDINAL
ejpam-5887	710	1	n. m. noureldeen et al	PERSON
ejpam-5887	711	1	j. pure	PERSON
ejpam-5887	711	2	18	CARDINAL
ejpam-5887	711	3	2	CARDINAL
ejpam-5887	711	4	2025	CARDINAL
ejpam-5887	711	5	5887 20 of 21	DATE
ejpam-5887	711	6	1	CARDINAL
ejpam-5887	711	7	a. anto cathrin aanisha	PERSON
ejpam-5887	713	1	31(7):453–458	CARDINAL
ejpam-5887	713	2	2024	DATE
ejpam-5887	714	1	2	CARDINAL
ejpam-5887	714	2	c. m. barasara	PERSON
ejpam-5887	716	1	7(2):263–273	DATE
ejpam-5887	716	2	2018	DATE
ejpam-5887	717	1	3	CARDINAL
ejpam-5887	717	2	i. cahit	PERSON
ejpam-5887	719	1	23:201–207	CARDINAL
ejpam-5887	719	2	1987	DATE
ejpam-5887	720	1	4	CARDINAL
ejpam-5887	720	2	j. a. gallian	PERSON
ejpam-5887	722	1	combinatorics	ORG
ejpam-5887	722	2	2023	CARDINAL
ejpam-5887	723	1	5	CARDINAL
ejpam-5887	723	2	e. roshdy m. aboshady	PERSON
ejpam-5887	723	3	r. elbarkouky	PERSON
ejpam-5887	723	4	m. abdel-azim seoud	PERSON
ejpam-5887	725	1	the pakistan academy of sciences	ORG
ejpam-5887	725	2	57(4):23–32	DATE
ejpam-5887	725	3	2020	DATE
ejpam-5887	726	1	6	CARDINAL
ejpam-5887	726	2	s. somasundaram m. sundaram	PERSON
ejpam-5887	726	3	r. ponraj	PERSON
ejpam-5887	728	1	23e(1):155–163	CARDINAL
ejpam-5887	728	2	2004	DATE
ejpam-5887	729	1	7	CARDINAL
ejpam-5887	729	2	andrea	PERSON
ejpam-5887	729	3	martin bača	PERSON
ejpam-5887	729	4	muhammad irfan	PERSON
ejpam-5887	730	1	3	CARDINAL
ejpam-5887	731	1	akce international journal	ORG
ejpam-5887	731	2	16:310–318	CARDINAL
ejpam-5887	731	3	2019	DATE
ejpam-5887	732	1	8	CARDINAL
ejpam-5887	732	2	andrea	PERSON
ejpam-5887	732	3	p. jeyanthi	PERSON
ejpam-5887	732	4	k. jeya daisy	PERSON
ejpam-5887	734	1	21:15–35	DATE
ejpam-5887	734	2	2019	DATE
ejpam-5887	735	1	9	CARDINAL
ejpam-5887	735	2	r. rajeswari r. thamizharasi	PERSON
ejpam-5887	736	1	9(2):1–4	CARDINAL
ejpam-5887	736	2	2015	DATE
ejpam-5887	737	1	10	CARDINAL
ejpam-5887	737	2	a. rosa	PERSON
ejpam-5887	739	1	internat	PERSON
ejpam-5887	740	1	349–355, 1966	DATE
ejpam-5887	741	1	11	CARDINAL
ejpam-5887	741	2	c. m. barasara s. k.	PERSON
ejpam-5887	743	1	2(5):1436–1450,	DATE
ejpam-5887	743	2	2012	DATE
ejpam-5887	744	1	12	CARDINAL
ejpam-5887	744	2	c. m. barasara s. k.	PERSON
ejpam-5887	746	1	3(1):4–7	CARDINAL
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ejpam-5887	747	1	13	CARDINAL
ejpam-5887	747	2	c. m. barasara s. k. vaidya	PERSON
ejpam-5887	749	1	3(3):49–53, 2013	DATE
ejpam-5887	750	1	14	CARDINAL
ejpam-5887	750	2	c. m. barasara s. k. vaidya	PERSON
ejpam-5887	752	1	15(1):103–111	ORG
ejpam-5887	752	2	2013	DATE
ejpam-5887	753	1	15	CARDINAL
ejpam-5887	753	2	c. m. barasara s. k. vaidya	PERSON
ejpam-5887	755	1	22, 2014	DATE
ejpam-5887	756	1	16	CARDINAL
ejpam-5887	756	2	c. m. barasara s. k. vaidya	PERSON
ejpam-5887	757	1	15(1):61–74	CARDINAL
ejpam-5887	757	2	2015	DATE
ejpam-5887	758	1	17	CARDINAL
ejpam-5887	758	2	n. b. patel u. m. prajapati	PERSON
ejpam-5887	760	1	6:268–278	CARDINAL
ejpam-5887	760	2	2016	DATE
ejpam-5887	761	1	n. m. noureldeen et al	PERSON
ejpam-5887	762	1	j. pure	PERSON
ejpam-5887	762	2	18	CARDINAL
ejpam-5887	762	3	2	CARDINAL
ejpam-5887	762	4	2025	CARDINAL
ejpam-5887	762	5	5887 21 of 21	DATE
ejpam-5887	762	6	18	CARDINAL
ejpam-5887	762	7	n. b. patel u. m. prajapati	PERSON
ejpam-5887	764	1	18(1):69–76	DATE
ejpam-5887	764	2	2019	DATE
ejpam-5887	765	1	19	CARDINAL
ejpam-5887	765	2	n. b. patel u. m. prajapati	PERSON
ejpam-5887	767	1	j. app	PERSON
ejpam-5887	768	1	eng	GPE
ejpam-5887	769	1	12(1):191–199	DATE
ejpam-5887	769	2	2022	CARDINAL
ejpam-5887	770	1	•	CARDINAL