id	sid	eid	entity	type
ejpam-3886	1	1	european journal	ORG
ejpam-3886	2	1	14	DATE
ejpam-3886	2	2	1	CARDINAL
ejpam-3886	2	3	2021	DATE
ejpam-3886	2	4	164	CARDINAL
ejpam-3886	2	5	new york	GPE
ejpam-3886	2	6	mohamed a. farahat1,2,∗	PERSON
ejpam-3886	2	7	t. al-bogamy1 1	PERSON
ejpam-3886	2	8	taif university	ORG
ejpam-3886	2	9	taif	GPE
ejpam-3886	2	10	el-haweiah	GPE
ejpam-3886	2	11	kingdom	GPE
ejpam-3886	2	12	saudi arabia	GPE
ejpam-3886	2	13	ksa	GPE
ejpam-3886	2	14	2	CARDINAL
ejpam-3886	2	15	al-azhar university	ORG
ejpam-3886	2	16	cairo	GPE
ejpam-3886	2	17	egypt	GPE
ejpam-3886	3	1	nil(r	ORG
ejpam-3886	3	2	α(e	PERSON
ejpam-3886	4	1	∈	ORG
ejpam-3886	4	2	2020	DATE
ejpam-3886	5	1	16d25	CARDINAL
ejpam-3886	5	2	16p60	DATE
ejpam-3886	5	3	16w60	CARDINAL
ejpam-3886	5	4	1	CARDINAL
ejpam-3886	6	1	nicholson	PERSON
ejpam-3886	6	2	watters	PERSON
ejpam-3886	6	3	6	CARDINAL
ejpam-3886	7	1	zero	CARDINAL
ejpam-3886	14	1	6	CARDINAL
ejpam-3886	15	1	6	CARDINAL
ejpam-3886	16	1	1	CARDINAL
ejpam-3886	16	2	6	CARDINAL
ejpam-3886	16	3	3.2	CARDINAL
ejpam-3886	19	1	salem	GPE
ejpam-3886	20	1	al.	PERSON
ejpam-3886	20	2	9	CARDINAL
ejpam-3886	20	3	2015	DATE
ejpam-3886	21	1	al-harthy	PERSON
ejpam-3886	21	2	3	CARDINAL
ejpam-3886	21	3	2017	DATE
ejpam-3886	22	1	m. a. farahat	PERSON
ejpam-3886	24	1	164	CARDINAL
ejpam-3886	24	2	2021	CARDINAL
ejpam-3886	25	1	m. a. farahat	PERSON
ejpam-3886	25	2	al-bogamy	ORG
ejpam-3886	26	1	j. pure	PERSON
ejpam-3886	26	2	14	CARDINAL
ejpam-3886	26	3	1	CARDINAL
ejpam-3886	26	4	2021	CARDINAL
ejpam-3886	26	5	164	CARDINAL
ejpam-3886	26	6	165	CARDINAL
ejpam-3886	27	1	8	CARDINAL
ejpam-3886	27	2	2017	DATE
ejpam-3886	28	1	al-bogamy	ORG
ejpam-3886	28	2	2	CARDINAL
ejpam-3886	28	3	2018	DATE
ejpam-3886	29	1	2	CARDINAL
ejpam-3886	33	1	2	CARDINAL
ejpam-3886	33	2	2	CARDINAL
ejpam-3886	34	1	1	CARDINAL
ejpam-3886	35	1	2	CARDINAL
ejpam-3886	36	1	3	CARDINAL
ejpam-3886	37	1	4	CARDINAL
ejpam-3886	39	1	2	CARDINAL
ejpam-3886	39	2	r.	NORP
ejpam-3886	39	3	2	CARDINAL
ejpam-3886	39	4	r.	NORP
ejpam-3886	39	5	3	CARDINAL
ejpam-3886	39	6	nr(x	PERSON
ejpam-3886	40	1	∈	ORG
ejpam-3886	41	1	ab ∈	ORG
ejpam-3886	41	2	r)⇔ ba	MONEY
ejpam-3886	41	3	∈	ORG
ejpam-3886	41	4	4	CARDINAL
ejpam-3886	41	5	two	CARDINAL
ejpam-3886	41	6	r. 3	LOC
ejpam-3886	41	7	1899–1968	CARDINAL
ejpam-3886	45	1	7	CARDINAL
ejpam-3886	46	1	m. a. farahat	PERSON
ejpam-3886	46	2	al-bogamy	ORG
ejpam-3886	47	1	j. pure	PERSON
ejpam-3886	47	2	14	CARDINAL
ejpam-3886	47	3	1	CARDINAL
ejpam-3886	47	4	2021	CARDINAL
ejpam-3886	47	5	164	CARDINAL
ejpam-3886	47	6	first	ORDINAL
ejpam-3886	47	7	david hilbert	PERSON
ejpam-3886	47	8	1862–1943	CARDINAL
ejpam-3886	48	1	1	CARDINAL
ejpam-3886	48	2	r.	NORP
ejpam-3886	49	1	r.	NORP
ejpam-3886	51	1	∈	ORG
ejpam-3886	51	2	r.	NORP
ejpam-3886	51	3	1	CARDINAL
ejpam-3886	52	1	α(1	GPE
ejpam-3886	52	2	1	CARDINAL
ejpam-3886	52	3	1	CARDINAL
ejpam-3886	53	1	1)x+	CARDINAL
ejpam-3886	53	2	α(1	GPE
ejpam-3886	53	3	1	CARDINAL
ejpam-3886	53	4	1	CARDINAL
ejpam-3886	54	1	5	CARDINAL
ejpam-3886	55	1	0	CARDINAL
ejpam-3886	55	2	∈ end(r,+	ORG
ejpam-3886	60	1	1	CARDINAL
ejpam-3886	60	2	5	CARDINAL
ejpam-3886	61	1	∈	ORG
ejpam-3886	61	2	xnr = n∑ i=0	PERSON
ejpam-3886	63	1	9	CARDINAL
ejpam-3886	63	2	8	CARDINAL
ejpam-3886	63	3	5	CARDINAL
ejpam-3886	63	4	1	CARDINAL
ejpam-3886	63	5	4	CARDINAL
ejpam-3886	68	1	1	CARDINAL
ejpam-3886	68	2	1	CARDINAL
ejpam-3886	68	3	1	CARDINAL
ejpam-3886	69	1	∈	ORG
ejpam-3886	69	2	0⇔	CARDINAL
ejpam-3886	70	1	0	CARDINAL
ejpam-3886	71	1	∈	ORG
ejpam-3886	74	1	m. a. farahat	PERSON
ejpam-3886	74	2	al-bogamy	ORG
ejpam-3886	75	1	j. pure	PERSON
ejpam-3886	75	2	14	CARDINAL
ejpam-3886	75	3	1	CARDINAL
ejpam-3886	75	4	2021	CARDINAL
ejpam-3886	75	5	164	CARDINAL
ejpam-3886	77	1	1	CARDINAL
ejpam-3886	78	1	1	CARDINAL
ejpam-3886	78	2	1	CARDINAL
ejpam-3886	78	3	2	CARDINAL
ejpam-3886	78	4	1	CARDINAL
ejpam-3886	78	5	3	CARDINAL
ejpam-3886	78	6	∈	ORG
ejpam-3886	78	7	∈	ORG
ejpam-3886	81	1	first	ORDINAL
ejpam-3886	82	1	2	CARDINAL
ejpam-3886	84	1	∈	ORG
ejpam-3886	86	1	⊆	CARDINAL
ejpam-3886	86	2	integer k.	PERSON
ejpam-3886	87	1	⊆	CARDINAL
ejpam-3886	88	1	∈	ORG
ejpam-3886	88	2	0	CARDINAL
ejpam-3886	89	1	3	CARDINAL
ejpam-3886	90	1	f(x	ORG
ejpam-3886	91	1	∈	ORG
ejpam-3886	93	1	f(x	ORG
ejpam-3886	94	1	∈	ORG
ejpam-3886	94	2	0 ≤	MONEY
ejpam-3886	96	1	f(x	ORG
ejpam-3886	97	1	integer k	PERSON
ejpam-3886	97	2	0	CARDINAL
ejpam-3886	99	1	2 +	DATE
ejpam-3886	102	1	0	CARDINAL
ejpam-3886	102	2	α(k−1)n(an)xkn	PERSON
ejpam-3886	103	1	anα n(an)α2n(an)	ORG
ejpam-3886	103	2	α(k−1)n(an	ORG
ejpam-3886	104	1	∈	ORG
ejpam-3886	106	1	2 +	DATE
ejpam-3886	107	1	n−1	ORG
ejpam-3886	108	1	0	CARDINAL
ejpam-3886	108	2	q+	PERSON
ejpam-3886	108	3	q+ anx n	PERSON
ejpam-3886	110	1	q+ anx n	PERSON
ejpam-3886	110	2	aj ∈	PERSON
ejpam-3886	113	1	x;α	GPE
ejpam-3886	114	1	qk	ORG
ejpam-3886	114	2	2 +	DATE
ejpam-3886	115	1	k m. a. farahat	PERSON
ejpam-3886	115	2	al-bogamy	ORG
ejpam-3886	116	1	j. pure	PERSON
ejpam-3886	116	2	14	CARDINAL
ejpam-3886	116	3	1	CARDINAL
ejpam-3886	116	4	2021	CARDINAL
ejpam-3886	116	5	164	CARDINAL
ejpam-3886	117	1	x;α	GPE
ejpam-3886	118	1	k−1)(n−1)(an−1	GPE
ejpam-3886	119	1	∈	ORG
ejpam-3886	119	2	⊆	CARDINAL
ejpam-3886	120	1	0	CARDINAL
ejpam-3886	120	2	0 ≤	MONEY
ejpam-3886	121	1	0	CARDINAL
ejpam-3886	121	2	0 ≤	MONEY
ejpam-3886	122	1	2 +	DATE
ejpam-3886	123	1	0(a0	CARDINAL
ejpam-3886	124	1	0(a1	CARDINAL
ejpam-3886	129	1	n∑ i=0	ORG
ejpam-3886	136	1	aj ∈	PERSON
ejpam-3886	137	1	0	CARDINAL
ejpam-3886	137	2	0 ≤	MONEY
ejpam-3886	138	1	vi3 ui3	ORG
ejpam-3886	139	1	ai1	GPE
ejpam-3886	139	2	ai2	PERSON
ejpam-3886	139	3	⊆	CARDINAL
ejpam-3886	139	4	s.	PERSON
ejpam-3886	139	5	0	CARDINAL
ejpam-3886	140	1	f(x	ORG
ejpam-3886	142	1	1	CARDINAL
ejpam-3886	143	1	f(x	ORG
ejpam-3886	144	1	2 +	DATE
ejpam-3886	146	1	one	CARDINAL
ejpam-3886	146	2	1	CARDINAL
ejpam-3886	146	3	2	CARDINAL
ejpam-3886	146	4	acc	ORG
ejpam-3886	146	5	dcc	ORG
ejpam-3886	146	6	f(x	ORG
ejpam-3886	147	1	∈	ORG
ejpam-3886	148	1	one	CARDINAL
ejpam-3886	148	2	1	CARDINAL
ejpam-3886	148	3	2	CARDINAL
ejpam-3886	149	1	3	CARDINAL
ejpam-3886	150	1	4	CARDINAL
ejpam-3886	151	1	r.	NORP
ejpam-3886	151	2	ab ∈	ORG
ejpam-3886	153	1	ab ∈	ORG
ejpam-3886	153	2	∈	ORG
ejpam-3886	154	1	f(x	ORG
ejpam-3886	156	1	∈	ORG
ejpam-3886	158	1	= aδ(b	PERSON
ejpam-3886	159	1	x;α	GPE
ejpam-3886	161	1	k(x	PERSON
ejpam-3886	162	1	m. a. farahat	PERSON
ejpam-3886	162	2	al-bogamy	ORG
ejpam-3886	163	1	j. pure	PERSON
ejpam-3886	163	2	14	CARDINAL
ejpam-3886	163	3	1	CARDINAL
ejpam-3886	163	4	2021	CARDINAL
ejpam-3886	163	5	164	CARDINAL
ejpam-3886	163	6	∈	ORG
ejpam-3886	167	1	x;α	GPE
ejpam-3886	168	1	aα2(b	ORG
ejpam-3886	169	1	p(x	PERSON
ejpam-3886	169	2	q(x	PERSON
ejpam-3886	170	1	∈	ORG
ejpam-3886	172	1	x;α	GPE
ejpam-3886	173	1	aδ2(b	GPE
ejpam-3886	180	1	ab ∈	ORG
ejpam-3886	182	1	f(x	ORG
ejpam-3886	183	1	n∑ i=0	PERSON
ejpam-3886	184	1	j=0	GPE
ejpam-3886	184	2	j ∈	PERSON
ejpam-3886	187	1	∈	ORG
ejpam-3886	187	2	0 ≤	MONEY
ejpam-3886	187	3	0 ≤ j ≤ m. proof	QUANTITY
ejpam-3886	188	1	f(x	ORG
ejpam-3886	189	1	n∑ i=0	PERSON
ejpam-3886	190	1	j=0	GPE
ejpam-3886	190	2	j ∈	PERSON
ejpam-3886	191	1	3	CARDINAL
ejpam-3886	191	2	∆n+m	GPE
ejpam-3886	191	3	1	CARDINAL
ejpam-3886	191	4	n(bm−1	GPE
ejpam-3886	192	1	2	CARDINAL
ejpam-3886	193	1	n(bm−2	ORG
ejpam-3886	196	1	3	CARDINAL
ejpam-3886	196	2	4	CARDINAL
ejpam-3886	196	3	∈	ORG
ejpam-3886	197	1	bman ∈	PERSON
ejpam-3886	198	1	bm	ORG
ejpam-3886	199	1	lemma 4	ORG
ejpam-3886	199	2	an−1bm ∈	QUANTITY
ejpam-3886	200	1	lemma 4	ORG
ejpam-3886	201	1	∈	ORG
ejpam-3886	201	2	0 ≤	MONEY
ejpam-3886	201	3	0 ≤ j ≤ m. conversely	QUANTITY
ejpam-3886	201	4	f(x	ORG
ejpam-3886	202	1	n∑ i=0	PERSON
ejpam-3886	203	1	j=0	GPE
ejpam-3886	203	2	j ∈	PERSON
ejpam-3886	204	1	∈	ORG
ejpam-3886	204	2	0 ≤	MONEY
ejpam-3886	206	1	0 ≤ j ≤ m.	QUANTITY
ejpam-3886	207	1	3	CARDINAL
ejpam-3886	207	2	lemma 4	ORG
ejpam-3886	211	1	1	CARDINAL
ejpam-3886	215	1	m. a. farahat	PERSON
ejpam-3886	215	2	al-bogamy	ORG
ejpam-3886	216	1	j. pure	PERSON
ejpam-3886	216	2	14	CARDINAL
ejpam-3886	216	3	1	CARDINAL
ejpam-3886	216	4	2021	CARDINAL
ejpam-3886	216	5	164	CARDINAL
ejpam-3886	219	1	∈	ORG
ejpam-3886	221	1	j = 〈	PERSON
ejpam-3886	222	1	j = r	PERSON
ejpam-3886	222	2	1	CARDINAL
ejpam-3886	223	1	ϕ(x	LOC
ejpam-3886	225	1	j ∈ l	PERSON
ejpam-3886	226	1	k∑ i=0 bix	ORG
ejpam-3886	227	1		GPE
ejpam-3886	227	2	j  ∈	FAC
ejpam-3886	228	1	1	CARDINAL
ejpam-3886	228	2	∈	ORG
ejpam-3886	228	3	0 ≤	MONEY
ejpam-3886	230	1	0 ≤ j ≤ n. consequently	QUANTITY
ejpam-3886	230	2	bia ∈ nil(r	PERSON
ejpam-3886	230	3	0 ≤	MONEY
ejpam-3886	231	1	0 ≤	MONEY
ejpam-3886	233	1	r.	NORP
ejpam-3886	233	2	∈	ORG
ejpam-3886	234	1	∈	ORG
ejpam-3886	235	1	∈	ORG
ejpam-3886	236	1	/∈ l.	PERSON
ejpam-3886	236	2	l+	EVENT
ejpam-3886	237	1	f(x	ORG
ejpam-3886	239	1	j=0	GPE
ejpam-3886	239	2	j ∈	PERSON
ejpam-3886	239	3	1	CARDINAL
ejpam-3886	239	4	+ rb0	PERSON
ejpam-3886	240	1	0	CARDINAL
ejpam-3886	240	2	1	CARDINAL
ejpam-3886	240	3	∈	PRODUCT
ejpam-3886	242	1	0	CARDINAL
ejpam-3886	244	1	j	PERSON
ejpam-3886	244	2	r.	NORP
ejpam-3886	244	3	⊆	CARDINAL
ejpam-3886	244	4	∈	ORG
ejpam-3886	245	1	1	CARDINAL
ejpam-3886	245	2	⊆	CARDINAL
ejpam-3886	247	1	i=0	ORG
ejpam-3886	248	1	j ∈ l	PERSON
ejpam-3886	248	2	k∑ i=0 mix i	ORG
ejpam-3886	248	3		GPE
ejpam-3886	248	4	j  ∈	FAC
ejpam-3886	249	1	1	CARDINAL
ejpam-3886	249	2	∈	ORG
ejpam-3886	249	3	0 ≤	MONEY
ejpam-3886	249	4	0 ≤	QUANTITY
ejpam-3886	249	5	j. consequently	PERSON
ejpam-3886	249	6	bia ∈ nil(r	PERSON
ejpam-3886	249	7	m. a. farahat	PERSON
ejpam-3886	249	8	al-bogamy	ORG
ejpam-3886	250	1	j. pure	PERSON
ejpam-3886	250	2	14	CARDINAL
ejpam-3886	250	3	1	CARDINAL
ejpam-3886	250	4	2021	CARDINAL
ejpam-3886	250	5	164	CARDINAL
ejpam-3886	250	6	0 ≤	MONEY
ejpam-3886	250	7	i. hence	PERSON
ejpam-3886	251	1	∈	ORG
ejpam-3886	251	2	0 ≤	MONEY
ejpam-3886	251	3	ϕ(x	LOC
ejpam-3886	252	1	2	CARDINAL
ejpam-3886	252	2	∈	ORG
ejpam-3886	253	1	∈	ORG
ejpam-3886	256	1	ϕ(x	LOC
ejpam-3886	258	1	j ∈ l	PERSON
ejpam-3886	259	1	k∑ i=0 bix	ORG
ejpam-3886	260	1		GPE
ejpam-3886	260	2	j  ∈	FAC
ejpam-3886	261	1	1	CARDINAL
ejpam-3886	261	2	∈	ORG
ejpam-3886	261	3	0 ≤	MONEY
ejpam-3886	263	1	0 ≤ j ≤ n. consequently	QUANTITY
ejpam-3886	263	2	bia ∈ nil(r	PERSON
ejpam-3886	263	3	0 ≤	MONEY
ejpam-3886	263	4	∈	ORG
ejpam-3886	264	1	bia2	ORG
ejpam-3886	265	1	bian	NORP
ejpam-3886	265	2	0 ≤	MONEY
ejpam-3886	266	1	∈	ORG
ejpam-3886	266	2	0 ≤	MONEY
ejpam-3886	266	3	k.	PERSON
ejpam-3886	267	1	0 ≤	MONEY
ejpam-3886	268	1	ϕ(x	LOC
ejpam-3886	273	1	i=0	ORG
ejpam-3886	274	1	∈	ORG
ejpam-3886	275	1	2	CARDINAL
ejpam-3886	279	1	1	CARDINAL
ejpam-3886	281	1	2	CARDINAL
ejpam-3886	282	1	∈	ORG
ejpam-3886	282	2	0 ≤	MONEY
ejpam-3886	282	3	k.	PERSON
ejpam-3886	282	4	0 ≤	MONEY
ejpam-3886	286	1	i=0	ORG
ejpam-3886	288	1	∈	ORG
ejpam-3886	289	1	172	CARDINAL
ejpam-3886	289	2	zero	CARDINAL
ejpam-3886	291	1	2	CARDINAL
ejpam-3886	295	1	3	CARDINAL
ejpam-3886	299	1	4	CARDINAL
ejpam-3886	302	1	5	CARDINAL
ejpam-3886	305	1	zero	CARDINAL
ejpam-3886	306	1	6	CARDINAL
ejpam-3886	309	1	1	CARDINAL
ejpam-3886	309	2	s. annin	PERSON
ejpam-3886	311	1	j. alg	PERSON
ejpam-3886	312	1	3:193–205, 2004	DATE
ejpam-3886	313	1	2	CARDINAL
ejpam-3886	313	2	s. al-bogamy	PERSON
ejpam-3886	313	3	hurwitz series	PERSON
ejpam-3886	315	1	j. pure	PERSON
ejpam-3886	315	2	11(1):244–259	DATE
ejpam-3886	315	3	2018	DATE
ejpam-3886	316	1	3	CARDINAL
ejpam-3886	316	2	n. al-harthy	PERSON
ejpam-3886	319	1	j. math	PERSON
ejpam-3886	320	1	46(5):1–6	DATE
ejpam-3886	320	2	2017	DATE
ejpam-3886	321	1	4	CARDINAL
ejpam-3886	321	2	c. hong	PERSON
ejpam-3886	321	3	n. kim	PERSON
ejpam-3886	321	4	t. kwak	PERSON
ejpam-3886	322	1	baer	ORG
ejpam-3886	323	1	j. pure	PERSON
ejpam-3886	323	2	2000	DATE
ejpam-3886	324	1	5	CARDINAL
ejpam-3886	324	2	a. leory	PERSON
ejpam-3886	324	3	j. matczuk	PERSON
ejpam-3886	326	1	25(8):2459–2516, 1997	DATE
ejpam-3886	327	1	6	CARDINAL
ejpam-3886	327	2	w. nicholson	PERSON
ejpam-3886	327	3	j. watters	PERSON
ejpam-3886	330	1	amer	PERSON
ejpam-3886	331	1	102:443–450, 1988	DATE
ejpam-3886	333	1	noncommutative	NORP
ejpam-3886	334	1	ann	PERSON
ejpam-3886	334	2	34(3):480–508, 1933	DATE
ejpam-3886	335	1	8	CARDINAL
ejpam-3886	338	1	czec	GPE
ejpam-3886	340	1	j.	PERSON
ejpam-3886	340	2	67(142):389–398	DATE
ejpam-3886	340	3	2017	DATE
ejpam-3886	341	1	9	CARDINAL
ejpam-3886	341	2	r. salem	GPE
ejpam-3886	341	3	m. farahat	PERSON
ejpam-3886	341	4	h. abd-elmalk	PERSON
ejpam-3886	344	1	j. math	PERSON
ejpam-3886	345	1	sci.	ORG
ejpam-3886	345	2	2015	DATE