id	sid	eid	entity	type
cana-1164	1	1	1074	CARDINAL
cana-1164	2	1	6s	CARDINAL
cana-1164	2	2	2024	CARDINAL
cana-1164	2	3	97	CARDINAL
cana-1164	2	4	kumar thakur	PERSON
cana-1164	2	5	gautam chandra ray 𝟐	PERSON
cana-1164	2	6	pinkimani	NORP
cana-1164	2	7	goswami 𝟑 1department	PERSON
cana-1164	2	8	india	GPE
cana-1164	3	1	2department	PERSON
cana-1164	3	2	cit	ORG
cana-1164	3	3	india	GPE
cana-1164	3	4	3department	CARDINAL
cana-1164	3	5	university of science and technology	ORG
cana-1164	4	1	∗corresponding	PERSON
cana-1164	4	2	23-05-2024	CARDINAL
cana-1164	4	3	10-07-2024	CARDINAL
cana-1164	4	4	22-07-2024	CARDINAL
cana-1164	4	5	γ(𝑛	ORG
cana-1164	5	1	≡ 𝑦(𝑚𝑜𝑑 𝑛	ORG
cana-1164	5	2	∈	ORG
cana-1164	5	3	γ(𝑛	LOC
cana-1164	6	1	γ(𝑛	LOC
cana-1164	7	1	1	CARDINAL
cana-1164	7	2	γ(𝑛	LOC
cana-1164	9	1	11a07	CARDINAL
cana-1164	9	2	1	CARDINAL
cana-1164	10	1	recent years	DATE
cana-1164	10	2	3	CARDINAL
cana-1164	10	3	4	DATE
cana-1164	10	4	6	DATE
cana-1164	10	5	8	DATE
cana-1164	10	6	10	DATE
cana-1164	10	7	11	DATE
cana-1164	10	8	13	DATE
cana-1164	10	9	14	DATE
cana-1164	10	10	17	DATE
cana-1164	10	11	18	DATE
cana-1164	10	12	20	CARDINAL
cana-1164	12	1	γ(𝑛	ORG
cana-1164	12	2	γ(𝑛	LOC
cana-1164	12	3	∈	ORG
cana-1164	12	4	3	CARDINAL
cana-1164	12	5	6	DATE
cana-1164	12	6	9	DATE
cana-1164	12	7	12	DATE
cana-1164	12	8	14	DATE
cana-1164	12	9	17	DATE
cana-1164	12	10	18	DATE
cana-1164	12	11	γ(𝑛	LOC
cana-1164	13	1	1	CARDINAL
cana-1164	13	2	2	DATE
cana-1164	14	1	1	CARDINAL
cana-1164	20	1	γ(𝑛	LOC
cana-1164	21	1	γ(𝑛	LOC
cana-1164	22	1	1074	CARDINAL
cana-1164	23	1	6s	CARDINAL
cana-1164	23	2	2024	CARDINAL
cana-1164	23	3	98	CARDINAL
cana-1164	24	1	section 2	LAW
cana-1164	25	1	section 3	LAW
cana-1164	25	2	γ(𝑛	LOC
cana-1164	26	1	section 4	LAW
cana-1164	26	2	γ(𝑛	LOC
cana-1164	27	1	1	CARDINAL
cana-1164	27	2	γ(𝑛	LOC
cana-1164	28	1	2	CARDINAL
cana-1164	29	1	γ(𝑛	ORG
cana-1164	29	2	γ(𝑛	LOC
cana-1164	29	3	γ(𝑛	LOC
cana-1164	30	1	two	CARDINAL
cana-1164	32	1	≡ 𝑦(𝑚𝑜𝑑 𝑛	ORG
cana-1164	33	1	γ(𝑛	LOC
cana-1164	33	2	𝑉(γ	ORG
cana-1164	34	1	𝐴(γ(𝑛	ORG
cana-1164	35	1	𝑉(γ	ORG
cana-1164	35	2	𝑣1 𝑘 ≡	ORG
cana-1164	35	3	1	CARDINAL
cana-1164	36	1	γ(𝑛	LOC
cana-1164	37	1	γ(𝑛	LOC
cana-1164	37	2	5	CARDINAL
cana-1164	37	3	15	DATE
cana-1164	37	4	16	DATE
cana-1164	37	5	19	DATE
cana-1164	37	6	20	CARDINAL
cana-1164	38	1	0	CARDINAL
cana-1164	38	2	γ(𝑛	LOC
cana-1164	38	3	0	CARDINAL
cana-1164	39	1	1	CARDINAL
cana-1164	40	1	∈	ORG
cana-1164	41	1	1	CARDINAL
cana-1164	44	1	γ(𝑛	LOC
cana-1164	44	2	1	CARDINAL
cana-1164	44	3	γ(𝑛	LOC
cana-1164	46	1	1	CARDINAL
cana-1164	48	1	0	CARDINAL
cana-1164	48	2	1074	CARDINAL
cana-1164	49	1	6s	CARDINAL
cana-1164	49	2	2024	CARDINAL
cana-1164	49	3	99	CARDINAL
cana-1164	49	4	1	CARDINAL
cana-1164	49	5	γ(𝑛	LOC
cana-1164	49	6	two	CARDINAL
cana-1164	50	1	1	CARDINAL
cana-1164	52	1	ℤ𝑛 ∗	ORG
cana-1164	52	2	𝜙(𝑛	PRODUCT
cana-1164	52	3	euler	PERSON
cana-1164	53	1	1	CARDINAL
cana-1164	53	2	0	CARDINAL
cana-1164	55	1	∩	PERSON
cana-1164	55	2	γ(𝑛	LOC
cana-1164	55	3	2.1	CARDINAL
cana-1164	56	1	22	CARDINAL
cana-1164	56	2	γ(𝑛	LOC
cana-1164	60	1	two	CARDINAL
cana-1164	61	1	two	CARDINAL
cana-1164	62	1	two	CARDINAL
cana-1164	66	1	2.1	CARDINAL
cana-1164	67	1	γ(𝑛	LOC
cana-1164	67	2	γ(𝑛	LOC
cana-1164	67	3	γ(𝑛	LOC
cana-1164	69	1	one	CARDINAL
cana-1164	74	1	0	CARDINAL
cana-1164	75	1	0	CARDINAL
cana-1164	75	2	0	CARDINAL
cana-1164	75	3	0	CARDINAL
cana-1164	76	1	𝐴𝑚𝑚	PERSON
cana-1164	76	2	0	CARDINAL
cana-1164	76	3	zero	CARDINAL
cana-1164	77	1	1074	CARDINAL
cana-1164	78	1	6s	CARDINAL
cana-1164	78	2	2024	CARDINAL
cana-1164	78	3	100	CARDINAL
cana-1164	80	1	3.	CARDINAL
cana-1164	80	2	γ(𝑛	LOC
cana-1164	81	1	3.1	CARDINAL
cana-1164	82	1	γ(𝑛	LOC
cana-1164	82	2	1	CARDINAL
cana-1164	82	3	vi	PERSON
cana-1164	82	4	0	DATE
cana-1164	82	5	3.2	CARDINAL
cana-1164	83	1	γ(𝑛	LOC
cana-1164	83	2	1	CARDINAL
cana-1164	83	3	vi	PERSON
cana-1164	83	4	0	DATE
cana-1164	83	5	𝐴−(γ(𝑛	CARDINAL
cana-1164	83	6	γ(𝑛	LOC
cana-1164	83	7	γ(𝑛	LOC
cana-1164	85	1	𝐴𝑖𝑗	GPE
cana-1164	85	2	[𝑎𝑢𝑣]𝑞×𝑞	PERSON
cana-1164	85	3	i ≠ j. where	PERSON
cana-1164	85	4	1	CARDINAL
cana-1164	85	5	uth vertex	ORG
cana-1164	89	1	𝐴𝑖𝑗	GPE
cana-1164	89	2	[𝑎𝑢𝑣]𝑞×𝑞	PERSON
cana-1164	89	3	i ≠ j. where	PERSON
cana-1164	89	4	1	CARDINAL
cana-1164	89	5	uth vertex	ORG
cana-1164	90	1	γ(𝑛	LOC
cana-1164	90	2	i.	PERSON
cana-1164	92	1	γ(𝑛	LOC
cana-1164	93	1	1074	CARDINAL
cana-1164	94	1	6s	CARDINAL
cana-1164	94	2	2024	CARDINAL
cana-1164	94	3	101	CARDINAL
cana-1164	95	1	γ(𝑛	LOC
cana-1164	99	1	γ(𝑛	LOC
cana-1164	99	2	γ(𝑛	LOC
cana-1164	100	1	3.1	CARDINAL
cana-1164	101	1	2	CARDINAL
cana-1164	102	1	1	CARDINAL
cana-1164	102	2	2	CARDINAL
cana-1164	103	1	0	CARDINAL
cana-1164	103	2	1	CARDINAL
cana-1164	103	3	1	CARDINAL
cana-1164	103	4	1	CARDINAL
cana-1164	103	5	0	CARDINAL
cana-1164	103	6	0	CARDINAL
cana-1164	103	7	3	CARDINAL
cana-1164	103	8	0	CARDINAL
cana-1164	103	9	0	CARDINAL
cana-1164	103	10	5	CARDINAL
cana-1164	103	11	1	CARDINAL
cana-1164	103	12		ORG
cana-1164	104	1	1	CARDINAL
cana-1164	104	2	1	CARDINAL
cana-1164	104	3	1	CARDINAL
cana-1164	104	4	0	CARDINAL
cana-1164	104	5	3	CARDINAL
cana-1164	104	6	0	CARDINAL
cana-1164	104	7	1	CARDINAL
cana-1164	104	8	1	CARDINAL
cana-1164	104	9	5	CARDINAL
cana-1164	104	10		ORG
cana-1164	104	11	1	CARDINAL
cana-1164	104	12	1	CARDINAL
cana-1164	104	13	1	CARDINAL
cana-1164	104	14	0	CARDINAL
cana-1164	104	15	3	CARDINAL
cana-1164	104	16	0	CARDINAL
cana-1164	104	17	1	CARDINAL
cana-1164	104	18	1	CARDINAL
cana-1164	104	19	5	CARDINAL
cana-1164	104	20		ORG
cana-1164	104	21	𝑅2	CARDINAL
cana-1164	104	22	↔	ORG
cana-1164	104	23	𝐶2 ↔ 𝐶6	MONEY
cana-1164	104	24	𝑅3	CARDINAL
cana-1164	104	25	𝐶3 ↔ 𝐶6	ORG
cana-1164	105	1	𝐶5 ↔ 𝐶6	ORG
cana-1164	105	2	1	CARDINAL
cana-1164	105	3	1	CARDINAL
cana-1164	105	4	0	CARDINAL
cana-1164	105	5	1	CARDINAL
cana-1164	105	6	3	CARDINAL
cana-1164	105	7	0	CARDINAL
cana-1164	105	8	2	CARDINAL
cana-1164	105	9		ORG
cana-1164	105	10	2	CARDINAL
cana-1164	107	1	1074	CARDINAL
cana-1164	108	1	6s	CARDINAL
cana-1164	108	2	2024	CARDINAL
cana-1164	108	3	102	CARDINAL
cana-1164	109	1	0	CARDINAL
cana-1164	109	2	5 1 3 2 4	TIME
cana-1164	109	3	1	CARDINAL
cana-1164	109	4	1	CARDINAL
cana-1164	109	5	0	CARDINAL
cana-1164	109	6	1	CARDINAL
cana-1164	109	7	3	CARDINAL
cana-1164	109	8	0	CARDINAL
cana-1164	109	9	2	CARDINAL
cana-1164	109	10		ORG
cana-1164	109	11	0	CARDINAL
cana-1164	109	12	0	CARDINAL
cana-1164	113	1	1	CARDINAL
cana-1164	114	1	0	CARDINAL
cana-1164	116	1	1	CARDINAL
cana-1164	117	1	0	CARDINAL
cana-1164	119	1	2	CARDINAL
cana-1164	119	2	2	CARDINAL
cana-1164	120	1	2	CARDINAL
cana-1164	120	2	2	CARDINAL
cana-1164	122	1	5 1 3 2 4	TIME
cana-1164	122	2	1	CARDINAL
cana-1164	122	3	3	CARDINAL
cana-1164	122	4	0	CARDINAL
cana-1164	122	5	2	CARDINAL
cana-1164	122	6	0	CARDINAL
cana-1164	122	7	0 1 1	QUANTITY
cana-1164	122	8		ORG
cana-1164	122	9	0	CARDINAL
cana-1164	122	10	0	CARDINAL
cana-1164	122	11	0	CARDINAL
cana-1164	122	12	0	CARDINAL
cana-1164	122	13	0	CARDINAL
cana-1164	124	1	1	CARDINAL
cana-1164	126	1	0	CARDINAL
cana-1164	126	2	0 1 1	QUANTITY
cana-1164	128	1	1	CARDINAL
cana-1164	130	1	0	CARDINAL
cana-1164	130	2	0 1 1	QUANTITY
cana-1164	131	1	3.1	CARDINAL
cana-1164	136	1	3.2	CARDINAL
cana-1164	137	1	γ(𝑛	LOC
cana-1164	143	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	143	2	1074	CARDINAL
cana-1164	144	1	6s	CARDINAL
cana-1164	144	2	2024	CARDINAL
cana-1164	144	3	103	CARDINAL
cana-1164	144	4	1	CARDINAL
cana-1164	148	1	𝑏𝑖𝑗]𝑛×𝑛	ORG
cana-1164	148	2	1	CARDINAL
cana-1164	150	1	2	CARDINAL
cana-1164	150	2	1	CARDINAL
cana-1164	156	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	156	2	𝑏𝑖𝑗]𝑛×𝑛	ORG
cana-1164	157	1	𝑑𝑖𝑗]𝑛×𝑛	ORG
cana-1164	157	2	2	CARDINAL
cana-1164	157	3	1	CARDINAL
cana-1164	161	1	3.1	CARDINAL
cana-1164	162	1	i. 𝐴γ	PERSON
cana-1164	168	1	𝐴γ −)𝑡	ORG
cana-1164	168	2	3.3	CARDINAL
cana-1164	169	1	1	CARDINAL
cana-1164	173	1	γ(𝑛	LOC
cana-1164	174	1	𝑎𝑖1	WORK_OF_ART
cana-1164	176	1	1	CARDINAL
cana-1164	176	2	𝑅𝑖	ORG
cana-1164	176	3	0	CARDINAL
cana-1164	176	4	1	CARDINAL
cana-1164	176	5	1074	CARDINAL
cana-1164	177	1	6s	CARDINAL
cana-1164	177	2	2024	CARDINAL
cana-1164	177	3	104	CARDINAL
cana-1164	178	1	3.1	CARDINAL
cana-1164	181	1	1	CARDINAL
cana-1164	181	2	1	CARDINAL
cana-1164	182	1	1	CARDINAL
cana-1164	182	2	3.3	CARDINAL
cana-1164	184	1	3.4	CARDINAL
cana-1164	190	1	γ(𝑛	LOC
cana-1164	191	1	𝑎𝑖1	WORK_OF_ART
cana-1164	193	1	1	CARDINAL
cana-1164	194	1	3.5	CARDINAL
cana-1164	199	1	γ(𝑛	LOC
cana-1164	201	1	1	CARDINAL
cana-1164	202	1	3.6	CARDINAL
cana-1164	203	1	1	CARDINAL
cana-1164	208	1	γ(𝑛	LOC
cana-1164	210	1	1	CARDINAL
cana-1164	210	2	1	CARDINAL
cana-1164	210	3	3.2	CARDINAL
cana-1164	212	1	3.7	CARDINAL
cana-1164	215	1	1074	CARDINAL
cana-1164	216	1	6s	CARDINAL
cana-1164	216	2	2024	CARDINAL
cana-1164	216	3	105	CARDINAL
cana-1164	219	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	219	2	γ(𝑛	LOC
cana-1164	220	1	𝑅1	GPE
cana-1164	220	2	𝑅2	DATE
cana-1164	220	3	𝑅𝑛	PERSON
cana-1164	221	1	3.1	CARDINAL
cana-1164	221	2	1,2,⋯	CARDINAL
cana-1164	221	3	𝑖=1	GPE
cana-1164	221	4	𝑖=1	GPE
cana-1164	222	1	3.8	CARDINAL
cana-1164	226	1	3.4	CARDINAL
cana-1164	226	2	3.2	CARDINAL
cana-1164	228	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	228	2	γ(𝑛	LOC
cana-1164	228	3	i. ∑𝑛𝑖=1 ∑	PERSON
cana-1164	228	4	𝑖=1	GPE
cana-1164	228	5	𝑖=1	GPE
cana-1164	229	1	3.9	CARDINAL
cana-1164	233	1	3.10	CARDINAL
cana-1164	237	1	3.3	CARDINAL
cana-1164	240	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	240	2	γ(𝑛	LOC
cana-1164	240	3	i. ∑𝑛𝑖=1 ∑	PERSON
cana-1164	240	4	𝑖=1	GPE
cana-1164	240	5	𝑖=1	GPE
cana-1164	241	1	3.11	CARDINAL
cana-1164	244	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	244	2	γ(𝑛	LOC
cana-1164	244	3	γ(𝑛	LOC
cana-1164	246	1	0	CARDINAL
cana-1164	246	2	0	CARDINAL
cana-1164	246	3	0	CARDINAL
cana-1164	246	4	≠ 𝑗.	PERSON
cana-1164	246	5	0	CARDINAL
cana-1164	246	6	1	CARDINAL
cana-1164	246	7	1	CARDINAL
cana-1164	248	1	1	CARDINAL
cana-1164	248	2	1	CARDINAL
cana-1164	249	1	1	CARDINAL
cana-1164	251	1	1	CARDINAL
cana-1164	251	2	+ 1	DATE
cana-1164	252	1	𝑏𝑖𝑗 i.e.	CARDINAL
cana-1164	254	1	𝑏𝑖𝑗]𝑛×𝑛.	DATE
cana-1164	255	1	𝑏𝑖𝑗]𝑛×𝑛	ORG
cana-1164	256	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	256	2	1074	CARDINAL
cana-1164	257	1	6s	CARDINAL
cana-1164	257	2	2024	CARDINAL
cana-1164	257	3	106	CARDINAL
cana-1164	257	4	0	CARDINAL
cana-1164	257	5	1	CARDINAL
cana-1164	257	6	0	CARDINAL
cana-1164	257	7	𝑏𝑖𝑘.	ORG
cana-1164	257	8	+ 1	DATE
cana-1164	258	1	∑ 𝑛 𝑘=1	PERSON
cana-1164	258	2	+ 1	DATE
cana-1164	260	1	3.12	CARDINAL
cana-1164	264	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	264	2	γ(𝑛	LOC
cana-1164	264	3	𝑖. 𝑒. 𝑣𝑖 ←	PERSON
cana-1164	264	4	γ(𝑛	LOC
cana-1164	266	1	3.11	CARDINAL
cana-1164	266	2	3.13	CARDINAL
cana-1164	269	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	269	2	γ(𝑛	LOC
cana-1164	271	1	at least two	CARDINAL
cana-1164	271	2	zero	CARDINAL
cana-1164	273	1	3 +⋯+	DATE
cana-1164	273	2	1	CARDINAL
cana-1164	275	1	γ(𝑛	LOC
cana-1164	275	2	γ(𝑛	LOC
cana-1164	275	3	1	CARDINAL
cana-1164	276	1	two	CARDINAL
cana-1164	276	2	more than two	CARDINAL
cana-1164	276	3	γ(𝑛	LOC
cana-1164	278	1	γ(𝑛	LOC
cana-1164	279	1	0	CARDINAL
cana-1164	283	1	3 +⋯+	DATE
cana-1164	285	1	γ(𝑛	LOC
cana-1164	285	2	one	CARDINAL
cana-1164	285	3	zero	CARDINAL
cana-1164	286	1	𝐵γ	PERSON
cana-1164	286	2	γ𝑠.	CARDINAL
cana-1164	288	1	zero	CARDINAL
cana-1164	289	1	at least two	CARDINAL
cana-1164	289	2	zero	CARDINAL
cana-1164	290	1	3.14	CARDINAL
cana-1164	294	1	𝑎𝑖𝑗]𝑛×𝑛	ORG
cana-1164	294	2	γ(𝑛	LOC
cana-1164	295	1	𝐶γ	PERSON
cana-1164	296	1	at least two	CARDINAL
cana-1164	296	2	zero	CARDINAL
cana-1164	298	1	1	CARDINAL
cana-1164	300	1	3.13	CARDINAL
cana-1164	301	1	1074	CARDINAL
cana-1164	302	1	6s	CARDINAL
cana-1164	302	2	2024	CARDINAL
cana-1164	302	3	107	CARDINAL
cana-1164	302	4	3.1	CARDINAL
cana-1164	303	1	γ(𝑛	LOC
cana-1164	303	2	at least one	CARDINAL
cana-1164	303	3	0	CARDINAL
cana-1164	304	1	γ(𝑛	LOC
cana-1164	304	2	at least one	CARDINAL
cana-1164	305	1	0 or 𝑘𝜔(𝑛),	PERCENT
cana-1164	305	2	∈	ORG
cana-1164	306	1	+ 1	DATE
cana-1164	307	1	𝜙(𝑛	PRODUCT
cana-1164	307	2	𝜙(𝑛	PRODUCT
cana-1164	308	1	∈	ORG
cana-1164	308	2	0 ≤ 𝑚 ≤	MONEY
cana-1164	309	1	1,𝑚𝑘	CARDINAL
cana-1164	309	2	1(𝑚𝑜𝑑 𝑛	DATE
cana-1164	313	1	γ(𝑛	LOC
cana-1164	315	1	γ(𝑛	LOC
cana-1164	315	2	γ(𝑛	LOC
cana-1164	316	1	1	CARDINAL
cana-1164	316	2	∀𝑣 ∈ γ(𝑛	PERSON
cana-1164	316	3	at least one	CARDINAL
cana-1164	317	1	γ(𝑛	LOC
cana-1164	317	2	at least one	CARDINAL
cana-1164	319	1	0 or 𝑘𝜔(𝑛),	PERCENT
cana-1164	319	2	∈	ORG
cana-1164	320	1	at least one	CARDINAL
cana-1164	321	1	0	CARDINAL
cana-1164	322	1	0	CARDINAL
cana-1164	323	1	at least one	CARDINAL
cana-1164	324	1	0	CARDINAL
cana-1164	325	1	3.2	CARDINAL
cana-1164	327	1	γ(𝑛	LOC
cana-1164	327	2	at least one	CARDINAL
cana-1164	328	1	γ(𝑛	LOC
cana-1164	328	2	γ(𝑛	LOC
cana-1164	329	1	2	CARDINAL
cana-1164	330	1	γ(𝑛	LOC
cana-1164	330	2	0	CARDINAL
cana-1164	333	1	1074	CARDINAL
cana-1164	334	1	6s	CARDINAL
cana-1164	334	2	2024	CARDINAL
cana-1164	334	3	108	CARDINAL
cana-1164	334	4	3.1	CARDINAL
cana-1164	334	5	γ(𝑛	LOC
cana-1164	334	6	at least one	CARDINAL
cana-1164	334	7	γ(𝑛	LOC
cana-1164	334	8	indeg(𝑣𝑡	GPE
cana-1164	335	1	0	CARDINAL
cana-1164	336	1	𝐶𝑣𝑡	WORK_OF_ART
cana-1164	336	2	zero	CARDINAL
cana-1164	337	1	𝐶𝑣𝑡	PRODUCT
cana-1164	337	2	one	CARDINAL
cana-1164	338	1	𝑠𝑎𝑦),1 ≤ 𝑖	MONEY
cana-1164	338	2	one	CARDINAL
cana-1164	338	3	zero	CARDINAL
cana-1164	339	1	0	CARDINAL
cana-1164	339	2	3.15	CARDINAL
cana-1164	341	1	γ(𝑛	LOC
cana-1164	344	1	0	CARDINAL
cana-1164	344	2	γ(𝑛	LOC
cana-1164	346	1	1	CARDINAL
cana-1164	346	2	3.2	CARDINAL
cana-1164	347	1	𝐴γ𝑚 +	FAC
cana-1164	348	1	1 ≤ 𝑚	QUANTITY
cana-1164	348	2	1	CARDINAL
cana-1164	349	1	0	CARDINAL
cana-1164	351	1	3.16	CARDINAL
cana-1164	352	1	γ(𝑛	LOC
cana-1164	355	1	𝑑𝑒𝑡((𝐴γ +	PRODUCT
cana-1164	356	1	3.1	CARDINAL
cana-1164	358	1	3.15	CARDINAL
cana-1164	360	1	4	CARDINAL
cana-1164	364	1	algebraic multiplicities	PERSON
cana-1164	365	1	1074	CARDINAL
cana-1164	366	1	6s	CARDINAL
cana-1164	366	2	2024	CARDINAL
cana-1164	366	3	109	CARDINAL
cana-1164	367	1	multiplicities 𝑚(𝜆1),𝑚(𝜆2),𝑚(𝜆3),⋯	PERSON
cana-1164	367	2	𝑚(𝜆𝑡	ORG
cana-1164	367	3	𝑚(𝜆𝑡	ORG
cana-1164	368	1	𝑑𝑒𝑡((𝐴γ +	PRODUCT
cana-1164	370	1	𝑑𝑒𝑡((𝐴γ +	PRODUCT
cana-1164	370	2	n. ⇒	PERSON
cana-1164	371	1	− 𝜆𝐼)𝑡	PERSON
cana-1164	371	2	3.1	CARDINAL
cana-1164	375	1	γ(𝑛	LOC
cana-1164	377	1	γ(𝑛	LOC
cana-1164	377	2	γ(𝑛	LOC
cana-1164	377	3	γ(𝑛	LOC
cana-1164	377	4	algebraic multiplicities	PERSON
cana-1164	377	5	4.1	CARDINAL
cana-1164	378	1	11	CARDINAL
cana-1164	379	1	2	CARDINAL
cana-1164	381	1	1	CARDINAL
cana-1164	381	2	6 1	DATE
cana-1164	381	3		PERSON
cana-1164	382	1	1 1 1	DATE
cana-1164	384	1	2 5	DATE
cana-1164	384	2		ORG
cana-1164	385	1	4 7 4	DATE
cana-1164	385	2		ORG
cana-1164	386	1	8 8 1	TIME
cana-1164	387	1	0,0,1	CARDINAL
cana-1164	387	2	1	CARDINAL
cana-1164	387	3	1	CARDINAL
cana-1164	388	1	1074	CARDINAL
cana-1164	389	1	6s	CARDINAL
cana-1164	389	2	2024	CARDINAL
cana-1164	389	3	110	CARDINAL
cana-1164	389	4	0	CARDINAL
cana-1164	389	5	0	CARDINAL
cana-1164	390	1	3 6 1 2 5 4 7 8	TIME
cana-1164	390	2	1	CARDINAL
cana-1164	390	3	0	CARDINAL
cana-1164	390	4	1	CARDINAL
cana-1164	390	5	0	CARDINAL
cana-1164	390	6	0	CARDINAL
cana-1164	390	7	7	CARDINAL
cana-1164	390	8	8	CARDINAL
cana-1164	390	9		PERSON
cana-1164	390	10	−𝜆9	ORG
cana-1164	392	1	−𝜆2(𝜆	EVENT
cana-1164	393	1	1)5(𝜆 + 1)2	DATE
cana-1164	393	2	0	CARDINAL
cana-1164	393	3	0	DATE
cana-1164	393	4	−1,−1	PERSON
cana-1164	393	5	1	CARDINAL
cana-1164	393	6	1	CARDINAL
cana-1164	393	7	1	CARDINAL
cana-1164	393	8	1	CARDINAL
cana-1164	393	9	1	CARDINAL
cana-1164	394	1	11	CARDINAL
cana-1164	395	1	11	CARDINAL
cana-1164	396	1	0 1	MONEY
cana-1164	399	1	−𝜆9	ORG
cana-1164	401	1	−𝜆2(𝜆	EVENT
cana-1164	402	1	1)5(𝜆 + 1)2	DATE
cana-1164	402	2	0	CARDINAL
cana-1164	402	3	0	DATE
cana-1164	402	4	−1,−1	PERSON
cana-1164	402	5	1	CARDINAL
cana-1164	402	6	1	CARDINAL
cana-1164	402	7	1	CARDINAL
cana-1164	402	8	1	CARDINAL
cana-1164	402	9	1	CARDINAL
cana-1164	403	1	11	CARDINAL
cana-1164	404	1	11	CARDINAL
cana-1164	405	1	0 1	MONEY
cana-1164	406	1	4.1	CARDINAL
cana-1164	407	1	γ(𝑛	LOC
cana-1164	409	1	γ(𝑛	LOC
cana-1164	410	1	γ(𝑛	LOC
cana-1164	413	1	γ(𝑛	LOC
cana-1164	414	1	4.2	CARDINAL
cana-1164	415	1	γ(𝑛	LOC
cana-1164	415	2	𝑃γ1(𝜆	PERSON
cana-1164	415	3	⋅ 𝑃γ2(𝜆	PERCENT
cana-1164	415	4	⋅ 𝑃γ𝑠(𝜆	GPE
cana-1164	415	5	𝑃γ1(𝜆	PERSON
cana-1164	415	6	𝑃γ2(𝜆	PERSON
cana-1164	415	7	𝑃γ3(𝜆),⋯	GPE
cana-1164	417	1	γ(𝑛	LOC
cana-1164	417	2	γ(𝑛	LOC
cana-1164	420	1	𝑘))|	NORP
cana-1164	420	2	1 ≤	CARDINAL
cana-1164	421	1	1074	CARDINAL
cana-1164	422	1	6s	CARDINAL
cana-1164	422	2	2024	CARDINAL
cana-1164	422	3	111	CARDINAL
cana-1164	424	1	0	CARDINAL
cana-1164	424	2	− 𝜆𝐼𝑛	PERSON
cana-1164	424	3	− 𝜆𝐼𝑛2	PERSON
cana-1164	427	1	𝜆𝐼𝑛𝑠	ORG
cana-1164	428	1	𝑃γ1(𝜆	PERSON
cana-1164	428	2	⋅ 𝑃γ2(𝜆	PERCENT
cana-1164	428	3	⋅ 𝑃γ𝑠(𝜆	GPE
cana-1164	429	1	4.3	CARDINAL
cana-1164	430	1	γ(𝑛	LOC
cana-1164	431	1	γ(𝑛	LOC
cana-1164	432	1	at least one	CARDINAL
cana-1164	432	2	1 ≤	MONEY
cana-1164	434	1	0	CARDINAL
cana-1164	434	2	⋅ 𝑣	ORG
cana-1164	434	3	0	CARDINAL
cana-1164	434	4	+ ⋅	ORG
cana-1164	434	5	⋅ 𝑣1	ORG
cana-1164	434	6	1	CARDINAL
cana-1164	434	7	2,⋯	CARDINAL
cana-1164	434	8	at least one	CARDINAL
cana-1164	434	9	γ	GPE
cana-1164	435	1	𝐴γ𝑖 + ⋅	ORG
cana-1164	435	2	1 ≤	MONEY
cana-1164	435	3	0	CARDINAL
cana-1164	435	4	0	CARDINAL
cana-1164	436	1	0	CARDINAL
cana-1164	437	1	⋅ 𝑣/	ORG
cana-1164	438	1	0	CARDINAL
cana-1164	438	2	0	CARDINAL
cana-1164	438	3	1074	CARDINAL
cana-1164	439	1	6s	CARDINAL
cana-1164	439	2	2024	CARDINAL
cana-1164	439	3	112	CARDINAL
cana-1164	440	1	at least one	CARDINAL
cana-1164	441	1	γ𝑖.	DATE
cana-1164	441	2	4.4	CARDINAL
cana-1164	442	1	γ(𝑛	LOC
cana-1164	442	2	γ𝑠.	DATE
cana-1164	442	3	1	CARDINAL
cana-1164	443	1	1 ≤	CARDINAL
cana-1164	447	1	γ(𝑛	LOC
cana-1164	447	2	1	CARDINAL
cana-1164	447	3	1	CARDINAL
cana-1164	447	4	1	CARDINAL
cana-1164	447	5	0	CARDINAL
cana-1164	448	1	𝜆𝐼𝑛𝑖	ORG
cana-1164	450	1	𝜆𝐼𝑛𝑖	ORG
cana-1164	450	2	𝐶1	CARDINAL
cana-1164	450	3	𝜆𝐼𝑛𝑖	ORG
cana-1164	451	1	1	CARDINAL
cana-1164	451	2	1 ≤	CARDINAL
cana-1164	452	1	1	CARDINAL
cana-1164	453	1	1	CARDINAL
cana-1164	454	1	at least two	CARDINAL
cana-1164	454	2	1	CARDINAL
cana-1164	454	3	𝐴γ𝑖 + ⋅	FAC
cana-1164	454	4	𝐴γ𝑖 + ⋅	ORG
cana-1164	454	5	1 ⋅	MONEY
cana-1164	455	1	1	CARDINAL
cana-1164	455	2	one	CARDINAL
cana-1164	456	1	4.5	CARDINAL
cana-1164	457	1	1	CARDINAL
cana-1164	457	2	γ(𝑛	LOC
cana-1164	460	1	γ(𝑛	LOC
cana-1164	460	2	γ(𝑛	LOC
cana-1164	461	1	𝑘))|	NORP
cana-1164	461	2	1 ≤	CARDINAL
cana-1164	462	1	0	CARDINAL
cana-1164	462	2	4.4	CARDINAL
cana-1164	462	3	1	CARDINAL
cana-1164	462	4	1	CARDINAL
cana-1164	463	1	− 𝜆𝐼𝑛	PERSON
cana-1164	465	1	− 𝜆𝐼𝑛2	PERSON
cana-1164	466	1	𝜆𝐼𝑛𝑠	ORG
cana-1164	466	2	1	CARDINAL
cana-1164	467	1	1	CARDINAL
cana-1164	468	1	1 + 1 + 1 +⋯+ 1⏟ 𝑠−𝑡𝑒𝑟𝑚𝑠	DATE
cana-1164	468	2	γ(𝑛	LOC
cana-1164	469	1	1	CARDINAL
cana-1164	469	2	γ(𝑛	LOC
cana-1164	470	1	1074	CARDINAL
cana-1164	471	1	6s	CARDINAL
cana-1164	471	2	2024	CARDINAL
cana-1164	471	3	113	CARDINAL
cana-1164	471	4	5	CARDINAL
cana-1164	473	1	γ(𝑛	LOC
cana-1164	474	1	γ(𝑛	LOC
cana-1164	475	1	1	CARDINAL
cana-1164	477	1	1	CARDINAL
cana-1164	477	2	combinatorics	ORG
cana-1164	477	3	1	CARDINAL
cana-1164	477	4	2012	DATE
cana-1164	477	5	9	CARDINAL
cana-1164	478	1	2	CARDINAL
cana-1164	478	2	r. b. bapat	PERSON
cana-1164	478	3	2010	DATE
cana-1164	479	1	3	CARDINAL
cana-1164	479	2	e. blanton jr.	PERSON
cana-1164	479	3	s. hurd	PERSON
cana-1164	479	4	j. mccranie	PERSON
cana-1164	479	5	square modulo n	PERSON
cana-1164	479	6	quarterly	DATE
cana-1164	479	7	34	CARDINAL
cana-1164	479	8	1992	DATE
cana-1164	479	9	322–334	CARDINAL
cana-1164	480	1	4	CARDINAL
cana-1164	480	2	s. bryant	PERSON
cana-1164	480	3	fermat	ORG
cana-1164	480	4	amer	PERSON
cana-1164	481	1	monthly	DATE
cana-1164	482	1	74	CARDINAL
cana-1164	482	2	1967	DATE
cana-1164	482	3	152–156	CARDINAL
cana-1164	483	1	5	CARDINAL
cana-1164	483	2	t. ju	GPE
cana-1164	483	3	m. wu	PERSON
cana-1164	483	4	zero	CARDINAL
cana-1164	483	5	czechoslovak mathematical journal	ORG
cana-1164	483	6	64	CARDINAL
cana-1164	483	7	611–628	CARDINAL
cana-1164	484	1	6	CARDINAL
cana-1164	484	2	w. carlip	PERSON
cana-1164	484	3	m. mincheva	PERSON
cana-1164	484	4	czechoslovak mathematical journal	ORG
cana-1164	484	5	58	DATE
cana-1164	484	6	2008	DATE
cana-1164	484	7	131	CARDINAL
cana-1164	485	1	7	CARDINAL
cana-1164	485	2	m. cvetkovic	PERSON
cana-1164	485	3	m. doob	PERSON
cana-1164	485	4	h. sachs	PERSON
cana-1164	485	5	1980	DATE
cana-1164	486	1	8	CARDINAL
cana-1164	486	2	p. goswami	PERSON
cana-1164	486	3	s. k. thakur	PERSON
cana-1164	486	4	g. c. ray	PERSON
cana-1164	486	5	42	CARDINAL
cana-1164	486	6	2	CARDINAL
cana-1164	486	7	2023	CARDINAL
cana-1164	486	8	457	CARDINAL
cana-1164	487	1	9	CARDINAL
cana-1164	487	2	w. y. jiang	PERSON
cana-1164	487	3	t. g. hua	PERSON
cana-1164	489	1	prc	GPE
cana-1164	489	2	36	CARDINAL
cana-1164	489	3	4	CARDINAL
cana-1164	489	4	2016) 676682	DATE
cana-1164	490	1	10	CARDINAL
cana-1164	490	2	e. miller	PERSON
cana-1164	490	3	c. reiter	PERSON
cana-1164	490	4	modulo p	ORG
cana-1164	491	1	34	CARDINAL
cana-1164	491	2	226	CARDINAL
cana-1164	492	1	11	CARDINAL
cana-1164	492	2	m. haris mateen	PERSON
cana-1164	492	3	m. khalid mahmood	PERSON
cana-1164	492	4	punjab	NORP
cana-1164	492	5	univ	NORP
cana-1164	493	1	j. math	PERSON
cana-1164	493	2	51	DATE
cana-1164	493	3	2019	CARDINAL
cana-1164	493	4	93	CARDINAL
cana-1164	494	1	12	CARDINAL
cana-1164	494	2	e. a. osba	PERSON
cana-1164	494	3	s. a. addasi	PERSON
cana-1164	494	4	n.a.	GPE
cana-1164	494	5	jaradeh	PRODUCT
cana-1164	494	6	zero	CARDINAL
cana-1164	494	7	modulo n	ORG
cana-1164	494	8	taylor & francis	ORG
cana-1164	494	9	36	CARDINAL
cana-1164	494	10	3865-3877	CARDINAL
cana-1164	495	1	13	CARDINAL
cana-1164	495	2	m. rahmati	PERSON
cana-1164	495	3	11	CARDINAL
cana-1164	495	4	1 (2017	CARDINAL
cana-1164	495	5	47	CARDINAL
cana-1164	496	1	14	CARDINAL
cana-1164	496	2	148	CARDINAL
cana-1164	496	3	1996	DATE
cana-1164	496	4	317	CARDINAL
cana-1164	497	1	15	CARDINAL
cana-1164	497	2	j. skowronek-kaziow	PERSON
cana-1164	497	3	zero	CARDINAL
cana-1164	497	4	108	CARDINAL
cana-1164	497	5	2008	DATE
cana-1164	497	6	165	CARDINAL
cana-1164	498	1	16	CARDINAL
cana-1164	498	2	j. skowronek-kaziow	PERSON
cana-1164	498	3	z. gora	PERSON
cana-1164	498	4	czechoslovak mathematical journal	ORG
cana-1164	498	5	59	CARDINAL
cana-1164	499	1	134	CARDINAL
cana-1164	499	2	2009	DATE
cana-1164	499	3	39	CARDINAL
cana-1164	500	1	17	CARDINAL
cana-1164	500	2	m. krizek	PERSON
cana-1164	500	3	czechoslovak mathematical journal	ORG
cana-1164	500	4	54	CARDINAL
cana-1164	500	5	2004	DATE
cana-1164	500	6	465	CARDINAL
cana-1164	501	1	18	CARDINAL
cana-1164	501	2	m. krizek	PERSON
cana-1164	501	3	moduli	CARDINAL
cana-1164	501	4	306	CARDINAL
cana-1164	501	5	2006	DATE
cana-1164	501	6	2174-2185	CARDINAL
cana-1164	502	1	19	CARDINAL
cana-1164	502	2	bdtf tud	PERSON
cana-1164	502	3	közl	PERSON
cana-1164	502	4	8 (1992	DATE
cana-1164	502	5	71	CARDINAL
cana-1164	502	6	hungarian	NORP
cana-1164	503	1	20	CARDINAL
cana-1164	503	2	s. k. thakur	PERSON
cana-1164	503	3	p. goswami	PERSON
cana-1164	503	4	g. c. ray	PERSON
cana-1164	503	5	≡b	PERSON
cana-1164	503	6	29	CARDINAL
cana-1164	503	7	3	CARDINAL
cana-1164	503	8	2023	CARDINAL
cana-1164	503	9	525	CARDINAL
cana-1164	504	1	21	CARDINAL
cana-1164	504	2	s. k. thakur	PERSON
cana-1164	504	3	p. goswami	PERSON
cana-1164	504	4	g. c. ray	PERSON
cana-1164	504	5	2)n	CARDINAL
cana-1164	505	1	22	CARDINAL
cana-1164	505	2	c. vasudeva	PERSON
cana-1164	505	3	81-224	CARDINAL