id	sid	eid	entity	type
ma-218	1	1	2024	CARDINAL
ma-218	2	1	j. math	PERSON
ma-218	4	1	4 (	CARDINAL
ma-218	4	2	2024	CARDINAL
ma-218	5	1	10.28924	CARDINAL
ma-218	5	2	lawal umar1,∗	PERSON
ma-218	5	3	yusuf ibrahim2	PERSON
ma-218	5	4	m.s. lawan3 1department	ORG
ma-218	5	5	zaria kaduna	PERSON
ma-218	5	6	nigeria	GPE
ma-218	6	1	saadatu rimi university of education	ORG
ma-218	6	2	kumbotso kano	ORG
ma-218	6	3	nigeria	GPE
ma-218	7	1	3department	CARDINAL
ma-218	7	2	kaduna polytechnic kaduna	GPE
ma-218	7	3	nigeria	GPE
ma-218	9	1	two	CARDINAL
ma-218	11	1	1	CARDINAL
ma-218	11	2	b∗	ORG
ma-218	11	3	〉	CARDINAL
ma-218	11	4	j ∈ b∗	PERSON
ma-218	11	5	∈	ORG
ma-218	12	1	6=	CARDINAL
ma-218	12	2	2b ∗	CARDINAL
ma-218	12	3	〉	CARDINAL
ma-218	13	1	d(v1	DATE
ma-218	14	1	v2 −	GPE
ma-218	14	2	∈	ORG
ma-218	14	3	1.1	CARDINAL
ma-218	14	4	bifunctions	ORG
ma-218	15	1	1.1	CARDINAL
ma-218	16	1	16 jan 2024	CARDINAL
ma-218	19	1	1	CARDINAL
ma-218	20	1	j. math	PERSON
ma-218	22	1	10.28924	CARDINAL
ma-218	22	2	1.1	CARDINAL
ma-218	22	3	d(v1, v2	PERSON
ma-218	22	4	0,∀v2	CARDINAL
ma-218	23	1	1.2	CARDINAL
ma-218	23	2	1.2	CARDINAL
ma-218	23	3	gep	ORG
ma-218	23	4	1.2).if	CARDINAL
ma-218	23	5	1.1	CARDINAL
ma-218	24	1	3]:find	CARDINAL
ma-218	24	2	d(v1, v2	PERSON
ma-218	24	3	0,∀v2	CARDINAL
ma-218	25	1	1.3	CARDINAL
ma-218	25	2	1.3	CARDINAL
ma-218	25	3	1.1	CARDINAL
ma-218	26	1	〉	CARDINAL
ma-218	26	2	1.4	CARDINAL
ma-218	26	3	1.4	CARDINAL
ma-218	26	4	1.4	CARDINAL
ma-218	27	1	1.1	CARDINAL
ma-218	28	1	6	CARDINAL
ma-218	28	2	tv1 = v1}	PERSON
ma-218	28	3	∈	ORG
ma-218	29	1	n→∞	CARDINAL
ma-218	29	2	− tvn ‖=	ORG
ma-218	29	3	0	CARDINAL
ma-218	30	1	f̂	CARDINAL
ma-218	30	2	6=	CARDINAL
ma-218	30	3	∈	ORG
ma-218	30	4	quasi−φ−asymptotically	ORG
ma-218	30	5	6=	CARDINAL
ma-218	30	6	{kn} ⊂	PERSON
ma-218	31	1	1,∞	CARDINAL
ma-218	31	2	kn −→ 1	PERSON
ma-218	31	3	t nv	GPE
ma-218	32	1	∈	ORG
ma-218	32	2	n ≥ 1	DATE
ma-218	32	3	1.2	CARDINAL
ma-218	33	1	6	CARDINAL
ma-218	35	1	〉	CARDINAL
ma-218	35	2	0	CARDINAL
ma-218	35	3	∀τ1	GPE
ma-218	35	4	∈	ORG
ma-218	35	5	0	CARDINAL
ma-218	38	1	∀τ1	ORG
ma-218	38	2	∈	ORG
ma-218	38	3	∃l	CARDINAL
ma-218	38	4	0	CARDINAL
ma-218	38	5	∈	ORG
ma-218	40	1	1	CARDINAL
ma-218	41	1	1.3	CARDINAL
ma-218	43	1	6	CARDINAL
ma-218	43	2	∈	ORG
ma-218	45	1	j. math	PERSON
ma-218	47	1	10.28924	CARDINAL
ma-218	47	2	16	CARDINAL
ma-218	48	1	4	CARDINAL
ma-218	48	2	5	DATE
ma-218	48	3	12	CARDINAL
ma-218	49	1	13	CARDINAL
ma-218	49	2	un	ORG
ma-218	49	3	1−	TIME
ma-218	49	4	17	CARDINAL
ma-218	50	1	17	CARDINAL
ma-218	50	2	7	CARDINAL
ma-218	50	3	8	DATE
ma-218	50	4	11	DATE
ma-218	50	5	18	DATE
ma-218	50	6	20	CARDINAL
ma-218	51	1	kazmiand ali	PERSON
ma-218	52	1	10	CARDINAL
ma-218	54	1	alansari et al	PERSON
ma-218	55	1	1	CARDINAL
ma-218	55	2	zn}generated	PERSON
ma-218	57	1	− wngµn	PERSON
ma-218	57	2	un = j−1(δnjzn	ORG
ma-218	58	1	1−	ORDINAL
ma-218	59	1	1−	LANGUAGE
ma-218	60	1	− u	PERSON
ma-218	60	2	jxn − jx0	PERSON
ma-218	60	3	0	CARDINAL
ma-218	60	4	1	CARDINAL
ma-218	60	5	0,∞	DATE
ma-218	61	1	0	CARDINAL
ma-218	61	2	1	CARDINAL
ma-218	62	1	0	CARDINAL
ma-218	63	1	xn}converges	PERSON
ma-218	64	1	6	CARDINAL
ma-218	65	1	j. math	PERSON
ma-218	67	1	10.28924	CARDINAL
ma-218	67	2	ma.4.8 4	PERSON
ma-218	67	3	x0	ORG
ma-218	69	1	ωn	CARDINAL
ma-218	70	1	−1(jωn −	PERSON
ma-218	70	2	j−1(δn,0jωn	PERSON
ma-218	70	3	zn =	ORG
ma-218	71	1	un	ORG
ma-218	71	2	trnzn	ORG
ma-218	71	3	∈	ORG
ma-218	71	4	un	ORG
ma-218	72	1	− u	PERSON
ma-218	72	2	jxn − jx0	PERSON
ma-218	72	3	1	CARDINAL
ma-218	73	1	0	CARDINAL
ma-218	73	2	1	CARDINAL
ma-218	73	3	0,∞	DATE
ma-218	73	4	0	CARDINAL
ma-218	73	5	1	CARDINAL
ma-218	74	1	0	CARDINAL
ma-218	75	1	kazmi	PERSON
ma-218	75	2	ali	PERSON
ma-218	76	1	10	CARDINAL
ma-218	77	1	1	CARDINAL
ma-218	78	1	6	CARDINAL
ma-218	78	2	1.4	CARDINAL
ma-218	78	3	two	CARDINAL
ma-218	78	4	quasi−φ−asymptotically	ORG
ma-218	79	1	kazmi	PERSON
ma-218	79	2	ali	PERSON
ma-218	80	1	10	CARDINAL
ma-218	81	1	1	CARDINAL
ma-218	82	1	6	CARDINAL
ma-218	83	1	2	CARDINAL
ma-218	83	2	∈	ORG
ma-218	83	3	0	CARDINAL
ma-218	83	4	2	CARDINAL
ma-218	83	5	0	CARDINAL
ma-218	85	1	2 ≤ 1−	QUANTITY
ma-218	85	2	∀τ1	GPE
ma-218	85	3	∈ w	ORG
ma-218	86	1	2	CARDINAL
ma-218	86	2	∀τ1	ORG
ma-218	86	3	∈ w	ORG
ma-218	86	4	6=	CARDINAL
ma-218	87	1	∀τ1	PERSON
ma-218	87	2	∈ w	ORG
ma-218	87	3	∀τ1	GPE
ma-218	87	4	∈	ORG
ma-218	88	1	jτ2〉+ ‖τ2 ‖2	PERSON
ma-218	88	2	∀τ1	ORG
ma-218	89	1	φ	ORG
ma-218	89	2	6	CARDINAL
ma-218	90	1	∀τ1	GPE
ma-218	90	2	∈	ORG
ma-218	92	1	1− λ)φ(τ1	DATE
ma-218	92	2	∀τ1	GPE
ma-218	92	3	∈	ORG
ma-218	95	1	∈	ORG
ma-218	97	1	j. math	PERSON
ma-218	99	1	10.28924	CARDINAL
ma-218	99	2	5	CARDINAL
ma-218	99	3	2.1	CARDINAL
ma-218	101	1	∀τ1	GPE
ma-218	101	2	∈	ORG
ma-218	101	3	2.2	CARDINAL
ma-218	102	1	9	CARDINAL
ma-218	102	2	6=	CARDINAL
ma-218	103	1	∃	ORG
ma-218	103	2	φ(τ0	ORG
ma-218	103	3	2.3	CARDINAL
ma-218	104	1	15	CARDINAL
ma-218	106	1	quasi−φ−asymptotically	ORG
ma-218	108	1	lemma 2.4	ORG
ma-218	109	1	14	CARDINAL
ma-218	109	2	6=	CARDINAL
ma-218	110	1	1.4	CARDINAL
ma-218	111	1	2.5	CARDINAL
ma-218	112	1	19	CARDINAL
ma-218	112	2	2−uniformly	CARDINAL
ma-218	113	1	∈	ORG
ma-218	114	1	two	CARDINAL
ma-218	115	1	2.6	CARDINAL
ma-218	116	1	19	CARDINAL
ma-218	116	2	two	CARDINAL
ma-218	116	3	2	CARDINAL
ma-218	116	4	∈	ORG
ma-218	117	1	2.7	CARDINAL
ma-218	118	1	9	CARDINAL
ma-218	118	2	un	ORG
ma-218	118	3	un	ORG
ma-218	119	1	lim	PERSON
ma-218	119	2	n→∞	CARDINAL
ma-218	119	3	φ(un	GPE
ma-218	119	4	0	CARDINAL
ma-218	119	5	lim n→∞ ‖ un − vn ‖= 0	ORG
ma-218	119	6	2.8	CARDINAL
ma-218	120	1	lemma 2.7	ORG
ma-218	120	2	un	ORG
ma-218	120	3	2.9	CARDINAL
ma-218	121	1	2	CARDINAL
ma-218	121	2	6=	CARDINAL
ma-218	122	1	φ(v	GPE
ma-218	124	1	∈	ORG
ma-218	124	2	∈	ORG
ma-218	124	3	jτ1 − jv	ORG
ma-218	124	4	〉	CARDINAL
ma-218	124	5	∀u ∈	ORG
ma-218	124	6	1	CARDINAL
ma-218	124	7	3	CARDINAL
ma-218	124	8	d(v	ORG
ma-218	125	1	1.e	DATE
ma-218	125	2	d(v	ORG
ma-218	126	1	+d(u	PERSON
ma-218	126	2	7→ d(v	QUANTITY
ma-218	126	3	∈	ORG
ma-218	127	1	2	CARDINAL
ma-218	128	1	j. math	PERSON
ma-218	130	1	10.28924	CARDINAL
ma-218	130	2	ϑ(v	PERSON
ma-218	130	3	ϑ(v	PERSON
ma-218	130	4	u)− ϑ(u	PERSON
ma-218	131	1	+ ϑ(u	PERSON
ma-218	131	2	second	ORDINAL
ma-218	132	1	2.10	CARDINAL
ma-218	133	1	1	CARDINAL
ma-218	133	2	6	DATE
ma-218	135	1	1	CARDINAL
ma-218	135	2	2	CARDINAL
ma-218	135	3	∈	ORG
ma-218	136	1	d(u	PERSON
ma-218	137	1	1	CARDINAL
ma-218	137	2	ju − jτ1〉+ ψ(u	PERSON
ma-218	140	1	∈	ORG
ma-218	140	2	trτ2	ORDINAL
ma-218	143	1	trτ2	ORDINAL
ma-218	143	2	jτ1 − jτ2〉	PERSON
ma-218	143	3	1.1	CARDINAL
ma-218	143	4	trτ1	PERSON
ma-218	144	1	∈	ORG
ma-218	146	1	j −1τ∗1	PERSON
ma-218	146	2	2.11	CARDINAL
ma-218	147	1	2	CARDINAL
ma-218	148	1	1 +	CARDINAL
ma-218	148	2	∀τ1 ∈	PERSON
ma-218	148	3	∈	ORG
ma-218	148	4	3	CARDINAL
ma-218	148	5	3.1	CARDINAL
ma-218	149	1	2−uniformly	CARDINAL
ma-218	149	2	b∗	ORG
ma-218	149	3	γ ∈	ORG
ma-218	149	4	0	CARDINAL
ma-218	149	5	1	CARDINAL
ma-218	150	1	1	CARDINAL
ma-218	150	2	2	CARDINAL
ma-218	151	1	1	CARDINAL
ma-218	151	2	2	DATE
ma-218	151	3	two	CARDINAL
ma-218	151	4	quasi−φ−asymptotically	ORG
ma-218	152	1	∩	PERSON
ma-218	152	2	1.4	CARDINAL
ma-218	152	3	1.1	CARDINAL
ma-218	152	4	6=	CARDINAL
ma-218	153	1	j. math	PERSON
ma-218	155	1	10.28924	CARDINAL
ma-218	155	2	x0	ORG
ma-218	155	3	ωn	CARDINAL
ma-218	158	1	j−1(µn,0jωn	PERSON
ma-218	158	2	zn	DATE
ma-218	162	1	un	ORG
ma-218	162	2	trnzn	ORG
ma-218	162	3	un	ORG
ma-218	162	4	x0	ORG
ma-218	162	5	∀n ≥ 1	DATE
ma-218	162	6	3.1	CARDINAL
ma-218	162	7	0	CARDINAL
ma-218	162	8	1	CARDINAL
ma-218	163	1	0	CARDINAL
ma-218	163	2	1	CARDINAL
ma-218	163	3	0	CARDINAL
ma-218	163	4	1	CARDINAL
ma-218	163	5	n∑ i=0	PERSON
ma-218	163	6	1	CARDINAL
ma-218	164	1	1	CARDINAL
ma-218	164	2	n→∞	CARDINAL
ma-218	165	1	ηn,0	ORG
ma-218	165	2	1	CARDINAL
ma-218	165	3	0	CARDINAL
ma-218	166	1	0,∞	DATE
ma-218	166	2	n→∞	CARDINAL
ma-218	167	1	2	CARDINAL
ma-218	168	1	$ onto	MONEY
ma-218	168	2	ω	ORDINAL
ma-218	170	1	1	CARDINAL
ma-218	172	1	3.1	CARDINAL
ma-218	173	1	un	ORG
ma-218	173	2	1− k2 n	PRODUCT
ma-218	173	3	jun〉+ 2k2 n 〈u	PERSON
ma-218	175	1	x0	ORG
ma-218	175	2	∀n ≥ 1	DATE
ma-218	176	1	ω 6=	DATE
ma-218	176	2	2.3	CARDINAL
ma-218	176	3	2.4	CARDINAL
ma-218	176	4	2.10	CARDINAL
ma-218	176	5	ω isclosed	ORG
ma-218	177	1	2	CARDINAL
ma-218	177	2	ω ⊂	ORG
ma-218	177	3	∀n ≥ 1	DATE
ma-218	178	1	ω ⊂	ORG
ma-218	178	2	ω ⊂	ORG
ma-218	179	1	x̂ ∈ ω	ORG
ma-218	179	2	φ	ORG
ma-218	179	3	quasi−φ−asymptotically	ORG
ma-218	180	1	j. math	PERSON
ma-218	182	1	10.28924	CARDINAL
ma-218	183	1	φ(x̂	ORG
ma-218	183	2	un	ORG
ma-218	184	1	zn	ORG
ma-218	184	2	3.2	CARDINAL
ma-218	184	3	x̂	ORG
ma-218	198	1	〉	CARDINAL
ma-218	199	1	2	CARDINAL
ma-218	200	1	jsni yn〉+	PERSON
ma-218	201	1	ηn,0	ORG
ma-218	204	1	jsni yn〉+ ‖sni yn‖2	PERSON
ma-218	206	1	sni yn	PERSON
ma-218	208	1	+ kn n∑ i=1	PERSON
ma-218	208	2	3.3	CARDINAL
ma-218	208	3	quasi−φ−asymptotically	ORG
ma-218	208	4	φ	ORG
ma-218	210	1	x̂	ORG
ma-218	210	2	j−1(µn,0jωn	GPE
ma-218	213	1	x̂ ‖2 −2µn,0〈x̂	ORG
ma-218	213	2	〉	CARDINAL
ma-218	214	1	2	CARDINAL
ma-218	214	2	jt ni ωn〉+	PERSON
ma-218	214	3	µn,0	CARDINAL
ma-218	215	1	jt ni	PERSON
ma-218	219	1	+ kn n∑ i=1	PERSON
ma-218	221	1	+ kn n∑ i=1	PERSON
ma-218	221	2	ωn	GPE
ma-218	221	3	3.4	CARDINAL
ma-218	222	1	j. math	PERSON
ma-218	224	1	10.28924	CARDINAL
ma-218	224	2	3.3	CARDINAL
ma-218	224	3	3.4	CARDINAL
ma-218	224	4	φ(x̂	ORG
ma-218	224	5	un	ORG
ma-218	226	1	+ kn n∑	PERSON
ma-218	229	1	3.5	CARDINAL
ma-218	229	2	2.6	CARDINAL
ma-218	229	3	2.11	CARDINAL
ma-218	230	1	x̂	ORG
ma-218	230	2	j−1(jωn −	PERSON
ma-218	230	3	βnqωn	ORDINAL
ma-218	231	1	x̂	ORG
ma-218	231	2	x̂	ORG
ma-218	234	1	2〈j−1(jωn	CARDINAL
ma-218	234	2	〉	CARDINAL
ma-218	234	3	jωn)− 2βn〈j−1(jωn	PERSON
ma-218	234	4	− βnqωn)− x̂	PERSON
ma-218	234	5	〉	CARDINAL
ma-218	234	6	2〈ωn	CARDINAL
ma-218	234	7	2βn〈j−1(jωn	CARDINAL
ma-218	234	8	〉	CARDINAL
ma-218	234	9	2〈ωn	CARDINAL
ma-218	235	1	2βn〈j−1(jωn	CARDINAL
ma-218	235	2	2βnγ ‖ qωn‖2	DATE
ma-218	236	1	j−1(jωn −qωn)−	GPE
ma-218	236	2	ωn)− 2βnγ	CARDINAL
ma-218	236	3	ωn)− 2βn	DATE
ma-218	236	4	2βn	ORDINAL
ma-218	236	5	3.6	CARDINAL
ma-218	236	6	2	CARDINAL
ma-218	236	7	φ(x̂	ORG
ma-218	236	8	3.7	CARDINAL
ma-218	236	9	3.7	CARDINAL
ma-218	236	10	3.5	CARDINAL
ma-218	236	11	φ(x̂	ORG
ma-218	236	12	un	ORG
ma-218	237	1	ηn,0	ORG
ma-218	238	1	2 nφ(x̂	QUANTITY
ma-218	238	2	un	ORG
ma-218	238	3	3.8	CARDINAL
ma-218	238	4	x̂ ∈	ORG
ma-218	238	5	ω ⊂	ORG
ma-218	238	6	cn+1	GPE
ma-218	239	1	ω ⊂	PERCENT
ma-218	239	2	∀n ≥ 1	DATE
ma-218	241	1	j. math	PERSON
ma-218	243	1	10.28924	CARDINAL
ma-218	243	2	10	CARDINAL
ma-218	243	3	3	CARDINAL
ma-218	243	4	un	ORG
ma-218	244	1	πcnx0	PERSON
ma-218	244	2	⊂	GPE
ma-218	244	3	∀n ≥ 1	DATE
ma-218	245	1	2.9	CARDINAL
ma-218	245	2	φ(xn	ORG
ma-218	245	3	x0	ORG
ma-218	245	4	x0	ORG
ma-218	245	5	φ(xn	ORG
ma-218	245	6	x0	ORG
ma-218	246	1	φ(xn	ORG
ma-218	246	2	x0	ORG
ma-218	246	3	x0	ORG
ma-218	246	4	x0	ORG
ma-218	246	5	φ(xn	ORG
ma-218	246	6	x0	ORG
ma-218	247	1	φ(xn	ORG
ma-218	247	2	x0	ORG
ma-218	248	1	φ(xn	ORG
ma-218	248	2	x0	ORG
ma-218	249	1	un	ORG
ma-218	250	1	2.9	CARDINAL
ma-218	250	2	φ(xm	PERSON
ma-218	251	1	πcnx0	PERSON
ma-218	251	2	x0)−	GPE
ma-218	251	3	φ(xn	ORG
ma-218	251	4	x0	ORG
ma-218	252	1	3.9	CARDINAL
ma-218	252	2	lemma 2.7	ORG
ma-218	252	3	lim n→∞ ‖ xm	ORG
ma-218	254	1	0	CARDINAL
ma-218	256	1	4	CARDINAL
ma-218	256	2	ωn −→ $	MONEY
ma-218	256	3	un −→ $	MONEY
ma-218	257	1	∈	ORG
ma-218	257	2	lim n→∞	ORG
ma-218	258	1	3.10	CARDINAL
ma-218	258	2	3.9	CARDINAL
ma-218	258	3	lim n→∞	FAC
ma-218	259	1	0	CARDINAL
ma-218	260	1	3.11	CARDINAL
ma-218	260	2	2.7	CARDINAL
ma-218	260	3	lim	PERSON
ma-218	260	4	n→∞	PRODUCT
ma-218	260	5	0	CARDINAL
ma-218	261	1	3.12	CARDINAL
ma-218	261	2	3.1	CARDINAL
ma-218	264	1	‖≤‖	GPE
ma-218	266	1	3.12	CARDINAL
ma-218	266	2	lim n→∞	ORG
ma-218	267	1	‖ωn	ORG
ma-218	268	1	0	CARDINAL
ma-218	269	1	3.13	CARDINAL
ma-218	269	2	3.10	CARDINAL
ma-218	269	3	3.13	CARDINAL
ma-218	269	4	lim	ORG
ma-218	270	1	3.14	CARDINAL
ma-218	270	2	2.8	CARDINAL
ma-218	270	3	3.13	CARDINAL
ma-218	270	4	ωn	CARDINAL
ma-218	270	5	lim	PERSON
ma-218	270	6	n→∞	PRODUCT
ma-218	271	1	0	CARDINAL
ma-218	272	1	3.15	CARDINAL
ma-218	273	1	j. math	PERSON
ma-218	275	1	10.28924	CARDINAL
ma-218	275	2	3.12	CARDINAL
ma-218	275	3	3.13	CARDINAL
ma-218	275	4	lim	PERSON
ma-218	276	1	ωn ‖= 0	DATE
ma-218	277	1	3.16	CARDINAL
ma-218	277	2	2.8	CARDINAL
ma-218	277	3	3.16	CARDINAL
ma-218	277	4	lim n→∞	PERSON
ma-218	278	1	0	CARDINAL
ma-218	279	1	3.17	CARDINAL
ma-218	279	2	x0 ∈	ORG
ma-218	279	3	⊂	GPE
ma-218	279	4	un	ORG
ma-218	279	5	3.17	CARDINAL
ma-218	279	6	lim n→∞	FAC
ma-218	279	7	un	ORG
ma-218	280	1	0	CARDINAL
ma-218	280	2	2.7	CARDINAL
ma-218	280	3	lim n→∞	LAW
ma-218	281	1	un	ORG
ma-218	281	2	0	CARDINAL
ma-218	282	1	3.18	CARDINAL
ma-218	283	1	3.12	CARDINAL
ma-218	283	2	3.18	CARDINAL
ma-218	283	3	lim n→∞	LAW
ma-218	284	1	un	ORG
ma-218	284	2	0	CARDINAL
ma-218	285	1	3.19	CARDINAL
ma-218	285	2	3.10	CARDINAL
ma-218	285	3	3.19	CARDINAL
ma-218	285	4	lim	PERSON
ma-218	285	5	n→∞	CARDINAL
ma-218	285	6	un	ORG
ma-218	286	1	3.20	CARDINAL
ma-218	286	2	x0 ∈	ORG
ma-218	286	3	⊂	GPE
ma-218	286	4	3.17	CARDINAL
ma-218	286	5	lim n→∞ φ(xn+1	FAC
ma-218	287	1	0	CARDINAL
ma-218	287	2	2.7	CARDINAL
ma-218	287	3	n→∞	CARDINAL
ma-218	288	1	zn	CARDINAL
ma-218	288	2	0	CARDINAL
ma-218	289	1	3.21	CARDINAL
ma-218	291	1	3.12	CARDINAL
ma-218	291	2	3.21	CARDINAL
ma-218	291	3	lim n→∞	LAW
ma-218	292	1	zn	DATE
ma-218	292	2	0	CARDINAL
ma-218	293	1	3.22	CARDINAL
ma-218	294	1	j. math	PERSON
ma-218	296	1	10.28924	CARDINAL
ma-218	296	2	3.22	CARDINAL
ma-218	296	3	lim	PERSON
ma-218	297	1	3.23	CARDINAL
ma-218	297	2	x0 ∈	ORG
ma-218	298	1	⊂	GPE
ma-218	298	2	3.17	CARDINAL
ma-218	298	3	lim	PERSON
ma-218	298	4	n→∞	PRODUCT
ma-218	300	1	2.7	CARDINAL
ma-218	300	2	lim n→∞	LAW
ma-218	301	1	0	CARDINAL
ma-218	302	1	3.24	CARDINAL
ma-218	304	1	3.12	CARDINAL
ma-218	304	2	3.24	CARDINAL
ma-218	304	3	lim n→∞	LAW
ma-218	305	1	0	CARDINAL
ma-218	306	1	3.25	CARDINAL
ma-218	306	2	3.10	CARDINAL
ma-218	306	3	3.25	CARDINAL
ma-218	306	4	lim	ORG
ma-218	308	1	3.26	CARDINAL
ma-218	308	2	x0 ∈	ORG
ma-218	308	3	⊂	GPE
ma-218	308	4	3.17	CARDINAL
ma-218	308	5	lim n→∞	FAC
ma-218	309	1	0	CARDINAL
ma-218	309	2	lemma 2.7	ORG
ma-218	309	3	lim n→∞	LAW
ma-218	310	1	0	CARDINAL
ma-218	311	1	3.27	CARDINAL
ma-218	312	1	3.12	CARDINAL
ma-218	312	2	3.27	CARDINAL
ma-218	312	3	lim n→∞	LAW
ma-218	314	1	0	CARDINAL
ma-218	315	1	3.28	CARDINAL
ma-218	315	2	3.10	CARDINAL
ma-218	315	3	3.28	CARDINAL
ma-218	315	4	lim n→∞	FAC
ma-218	316	1	3.29	CARDINAL
ma-218	317	1	j. math	PERSON
ma-218	319	1	10.28924	CARDINAL
ma-218	319	2	13	CARDINAL
ma-218	319	3	4	CARDINAL
ma-218	319	4	− sni yn ‖=	PERSON
ma-218	319	5	0	CARDINAL
ma-218	320	1	3.16	CARDINAL
ma-218	320	2	3.24	CARDINAL
ma-218	320	3	0	CARDINAL
ma-218	321	1	3.30	CARDINAL
ma-218	321	2	3.1	CARDINAL
ma-218	323	1	− jt ni ωn	PERSON
ma-218	324	1	− jt ni ωn ‖ −µn,0 ‖ jωn − jxn+1 ‖	PERSON
ma-218	324	2	− jt ni ωn ‖≤	PERSON
ma-218	326	1	3.30	CARDINAL
ma-218	326	2	lim n→∞ ‖ jxn+1	ORG
ma-218	327	1	− jt ni ωn ‖=	PERSON
ma-218	328	1	lim n→∞	LAW
ma-218	329	1	0	CARDINAL
ma-218	330	1	3.31	CARDINAL
ma-218	330	2	ωn	CARDINAL
ma-218	332	1	3.16	CARDINAL
ma-218	332	2	3.31	CARDINAL
ma-218	332	3	lim n→∞	ORG
ma-218	332	4	0	CARDINAL
ma-218	333	1	3.32	CARDINAL
ma-218	333	2	3.21	CARDINAL
ma-218	333	3	3.27	CARDINAL
ma-218	333	4	0	CARDINAL
ma-218	334	1	3.33	CARDINAL
ma-218	335	1	j. math	PERSON
ma-218	337	1	10.28924	CARDINAL
ma-218	337	2	3.1	CARDINAL
ma-218	343	1	− jsni yn	ORG
ma-218	348	1	− jsni yn‖ ≤	PERSON
ma-218	351	1	3.33	CARDINAL
ma-218	351	2	lim n→∞	LAW
ma-218	351	3	− jsni yn ‖=	ORG
ma-218	351	4	0	CARDINAL
ma-218	351	5	n→∞	CARDINAL
ma-218	352	1	− sni yn ‖=	PERSON
ma-218	352	2	0	CARDINAL
ma-218	353	1	3.34	CARDINAL
ma-218	354	1	− sni yn ‖≤‖	PERSON
ma-218	354	2	3.24	CARDINAL
ma-218	354	3	3.34	CARDINAL
ma-218	354	4	lim n→∞	LAW
ma-218	354	5	− sni yn ‖=	PERSON
ma-218	354	6	0	CARDINAL
ma-218	355	1	3.35	MONEY
ma-218	355	2	3.32	CARDINAL
ma-218	355	3	3.35	MONEY
ma-218	355	4	lim n→∞	LAW
ma-218	355	5	n→∞	CARDINAL
ma-218	356	1	− sni yn ‖=	PERSON
ma-218	356	2	0	CARDINAL
ma-218	357	1	5	CARDINAL
ma-218	357	2	∈	MONEY
ma-218	357	3	∩	PERSON
ma-218	358	1	1	CARDINAL
ma-218	358	2	ωn	CARDINAL
ma-218	359	1	3.14	CARDINAL
ma-218	359	2	3.32	CARDINAL
ma-218	359	3	lim n→∞ ‖ t	ORG
ma-218	359	4	0	CARDINAL
ma-218	360	1	3.36	CARDINAL
ma-218	361	1	j. math	PERSON
ma-218	363	1	10.28924	CARDINAL
ma-218	373	1	3.12	CARDINAL
ma-218	373	2	3.32	CARDINAL
ma-218	373	3	lim n→∞ ‖t	ORG
ma-218	374	1	0	CARDINAL
ma-218	374	2	3.36	CARDINAL
ma-218	374	3	lim n→∞ ‖t	ORG
ma-218	374	4	1	CARDINAL
ma-218	375	1	ωn −→ $	MONEY
ma-218	376	1	1	CARDINAL
ma-218	377	1	∈	MONEY
ma-218	379	1	∈	MONEY
ma-218	379	2	∩	PERSON
ma-218	380	1	1.4	CARDINAL
ma-218	381	1	ωn	CARDINAL
ma-218	383	1	3.13	CARDINAL
ma-218	383	2	3.22	CARDINAL
ma-218	383	3	lim n→∞	LAW
ma-218	384	1	zn	DATE
ma-218	384	2	0	CARDINAL
ma-218	385	1	3.37	CARDINAL
ma-218	385	2	lim n→∞	LAW
ma-218	385	3	0	CARDINAL
ma-218	386	1	3.38	CARDINAL
ma-218	386	2	x̂ ∈ ω	ORG
ma-218	386	3	3.2	CARDINAL
ma-218	386	4	3.3	CARDINAL
ma-218	386	5	3.4	CARDINAL
ma-218	386	6	3.6	CARDINAL
ma-218	386	7	φ(x̂	ORG
ma-218	387	1	φ(x̂	ORG
ma-218	387	2	ωn)− 2βn	DATE
ma-218	387	3	2βn	ORDINAL
ma-218	392	1	ωn)− 2βnηn,0	CARDINAL
ma-218	392	2	2βn	ORDINAL
ma-218	393	1	2βn	ORDINAL
ma-218	393	2	qωn ‖2	GPE
ma-218	393	3	2βnηn,0	CARDINAL
ma-218	393	4	2βn	ORDINAL
ma-218	393	5	k2 nφ(x̂	PERSON
ma-218	393	6	zn	ORG
ma-218	393	7	3.39	CARDINAL
ma-218	394	1	j. math	PERSON
ma-218	396	1	10.28924	CARDINAL
ma-218	397	1	jzn〉+	GPE
ma-218	397	2	zn	DATE
ma-218	398	1	1)‖x̂‖2	CARDINAL
ma-218	398	2	− 2k2 n 〈x̂	PERSON
ma-218	399	1	x̂ ‖2	ORG
ma-218	400	1	n − 1)〈x̂	ORG
ma-218	400	2	− 2k2 n 〈x̂	PERSON
ma-218	403	1	〉	CARDINAL
ma-218	405	1	〉	CARDINAL
ma-218	410	1	x̂ ‖2	ORG
ma-218	410	2	1	CARDINAL
ma-218	411	1	n ‖ x̂	ORG
ma-218	412	1	zn	DATE
ma-218	413	1	kn −→ 1	PERSON
ma-218	413	2	3.37	CARDINAL
ma-218	413	3	3.38	CARDINAL
ma-218	413	4	lim n→∞	ORG
ma-218	414	1	0	CARDINAL
ma-218	415	1	3.40	CARDINAL
ma-218	415	2	2βn	ORDINAL
ma-218	415	3	0	CARDINAL
ma-218	415	4	3.39	CARDINAL
ma-218	415	5	3.40	CARDINAL
ma-218	415	6	n→∞	PRODUCT
ma-218	415	7	0	CARDINAL
ma-218	416	1	3.41	CARDINAL
ma-218	419	1	3.40	CARDINAL
ma-218	419	2	∈	MONEY
ma-218	420	1	∈	MONEY
ma-218	420	2	1.4	CARDINAL
ma-218	421	1	∈	MONEY
ma-218	421	2	1.1	CARDINAL
ma-218	422	1	un	ORG
ma-218	422	2	‖≤‖	GPE
ma-218	425	1	3.19	CARDINAL
ma-218	425	2	3.22	CARDINAL
ma-218	425	3	lim n→∞ ‖	LAW
ma-218	425	4	un	ORG
ma-218	425	5	zn	CARDINAL
ma-218	425	6	0	CARDINAL
ma-218	426	1	lim n→∞ ‖	ORG
ma-218	426	2	jun − jzn ‖=	PERSON
ma-218	426	3	0	CARDINAL
ma-218	427	1	3.42	CARDINAL
ma-218	427	2	3.42	CARDINAL
ma-218	427	3	lim n→∞	LAW
ma-218	427	4	jun − jzn	PERSON
ma-218	427	5	0	CARDINAL
ma-218	428	1	3.43	CARDINAL
ma-218	428	2	un	ORG
ma-218	428	3	trnzn	ORG
ma-218	428	4	h(un	PERSON
ma-218	428	5	+ 1 rn	QUANTITY
ma-218	428	6	un	ORG
ma-218	428	7	jun − jzn〉+	PERSON
ma-218	428	8	ϑ(v	PERSON
ma-218	428	9	ϑ(un	ORG
ma-218	428	10	un	ORG
ma-218	428	11	0	CARDINAL
ma-218	429	1	j. math	PERSON
ma-218	431	1	10.28924	CARDINAL
ma-218	432	1	d(un	GPE
ma-218	432	2	〉	CARDINAL
ma-218	433	1	1 rn	QUANTITY
ma-218	434	1	un	ORG
ma-218	435	1	jun − jzn	PERSON
ma-218	435	2	〉	CARDINAL
ma-218	435	3	ϑ(v	PERSON
ma-218	435	4	un	ORG
ma-218	435	5	un	ORG
ma-218	435	6	≥ h(v	PERSON
ma-218	435	7	ϑ(v	PERSON
ma-218	435	8	un	ORG
ma-218	435	9	un	ORG
ma-218	436	1	n −→∞	PERSON
ma-218	436	2	d4	ORG
ma-218	436	3	3.43	CARDINAL
ma-218	438	1	0	CARDINAL
ma-218	438	2	1	CARDINAL
ma-218	439	1	1− s)$.	PRODUCT
ma-218	439	2	∈	ORG
ma-218	440	1	d4	ORG
ma-218	440	2	0	CARDINAL
ma-218	440	3	1− s)h(vs	DATE
ma-218	441	1	1− s	PRODUCT
ma-218	442	1	1− s	PRODUCT
ma-218	442	2	ϑ(v	PERSON
ma-218	442	3	0	CARDINAL
ma-218	443	1	+ ϑ(v	PERSON
ma-218	443	2	0	CARDINAL
ma-218	443	3	∈	ORG
ma-218	443	4	1.1	CARDINAL
ma-218	444	1	6	CARDINAL
ma-218	445	1	∈ ω ⊂	ORG
ma-218	445	2	φ(xn	ORG
ma-218	445	3	x0	ORG
ma-218	445	4	x0	ORG
ma-218	445	5	∀n ≥ 0	DATE
ma-218	446	1	x0	ORG
ma-218	447	1	n→∞	CARDINAL
ma-218	447	2	φ(xn	ORG
ma-218	447	3	x0	ORG
ma-218	447	4	x0	ORG
ma-218	449	1	3.2	CARDINAL
ma-218	450	1	2−uniformly	CARDINAL
ma-218	450	2	b∗	ORG
ma-218	450	3	1	CARDINAL
ma-218	450	4	2	CARDINAL
ma-218	451	1	1	CARDINAL
ma-218	451	2	2	DATE
ma-218	451	3	two	CARDINAL
ma-218	452	1	j. math	PERSON
ma-218	454	1	10.28924	CARDINAL
ma-218	454	2	∩	PERSON
ma-218	454	3	1.1	CARDINAL
ma-218	455	1	6=	CARDINAL
ma-218	455	2	x0	ORG
ma-218	457	1	ωn	CARDINAL
ma-218	459	1	j−1(µn,0jωn	PERSON
ma-218	463	1	un	ORG
ma-218	463	2	trnzn	ORG
ma-218	463	3	un	ORG
ma-218	463	4	x0	ORG
ma-218	463	5	∀n ≥ 1	DATE
ma-218	463	6	0	CARDINAL
ma-218	463	7	1	CARDINAL
ma-218	464	1	0	CARDINAL
ma-218	464	2	1	CARDINAL
ma-218	464	3	0	CARDINAL
ma-218	464	4	1	CARDINAL
ma-218	464	5	n∑ i=0	PERSON
ma-218	464	6	1	CARDINAL
ma-218	465	1	1	CARDINAL
ma-218	465	2	n→∞	CARDINAL
ma-218	466	1	ηn,0	ORG
ma-218	466	2	1	CARDINAL
ma-218	466	3	0	CARDINAL
ma-218	468	1	$ onto	MONEY
ma-218	468	2	ω	ORDINAL
ma-218	469	1	3.3	CARDINAL
ma-218	470	1	2−uniformly	CARDINAL
ma-218	470	2	b∗	ORG
ma-218	471	1	1	CARDINAL
ma-218	471	2	2	DATE
ma-218	471	3	two	CARDINAL
ma-218	471	4	quasi−φ−asymptotically	ORG
ma-218	472	1	∩	PERSON
ma-218	472	2	1.2	CARDINAL
ma-218	472	3	6=	CARDINAL
ma-218	472	4	x0	ORG
ma-218	472	5	ωn	CARDINAL
ma-218	474	1	j−1(µn,0jωn	PERSON
ma-218	478	1	un	ORG
ma-218	478	2	trnzn	ORG
ma-218	478	3	un	ORG
ma-218	478	4	x0	ORG
ma-218	478	5	∀n ≥ 1	DATE
ma-218	479	1	j. math	PERSON
ma-218	481	1	10.28924	CARDINAL
ma-218	482	1	0	CARDINAL
ma-218	482	2	1	CARDINAL
ma-218	483	1	0	CARDINAL
ma-218	483	2	1	CARDINAL
ma-218	483	3	0	CARDINAL
ma-218	483	4	1	CARDINAL
ma-218	483	5	n∑ i=0	PERSON
ma-218	483	6	1	CARDINAL
ma-218	484	1	1	CARDINAL
ma-218	484	2	n→∞	CARDINAL
ma-218	485	1	ηn,0	ORG
ma-218	485	2	1	CARDINAL
ma-218	485	3	0	CARDINAL
ma-218	487	1	$ onto	MONEY
ma-218	487	2	ω	ORDINAL
ma-218	488	1	4	CARDINAL
ma-218	489	1	0	CARDINAL
ma-218	489	2	1	CARDINAL
ma-218	490	1	2u∀u	CARDINAL
ma-218	490	2	∈	ORG
ma-218	491	1	ϑ(u	PERSON
ma-218	491	2	d(u	PERSON
ma-218	491	3	q(u	GPE
ma-218	491	4	2u	CARDINAL
ma-218	491	5	ti(u	ORG
ma-218	491	6	1 i+1u	CARDINAL
ma-218	491	7	0.9 2n	CARDINAL
ma-218	491	8	1 2	CARDINAL
ma-218	491	9	0.9	CARDINAL
ma-218	491	10	1 2	DATE
ma-218	491	11	1 2	CARDINAL
ma-218	491	12	1	CARDINAL
ma-218	491	13	1 3	DATE
ma-218	491	14	2 3 so that ∑ni=0	DATE
ma-218	491	15	3.1	CARDINAL
ma-218	491	16	∈ ω	ORG
ma-218	493	1	1	CARDINAL
ma-218	493	2	2	CARDINAL
ma-218	494	1	0	CARDINAL
ma-218	494	2	6=	CARDINAL
ma-218	495	1	0	CARDINAL
ma-218	495	2	6=	CARDINAL
ma-218	495	3	1 2	CARDINAL
ma-218	495	4	two	CARDINAL
ma-218	495	5	1	CARDINAL
ma-218	497	1	0	CARDINAL
ma-218	498	1	ω	CARDINAL
ma-218	498	2	0	CARDINAL
ma-218	498	3	6=	CARDINAL
ma-218	498	4	3.1	CARDINAL
ma-218	498	5	4.1	CARDINAL
ma-218	498	6	x0	ORG
ma-218	499	1	ωn	CARDINAL
ma-218	501	1	1 2ωn	CARDINAL
ma-218	501	2	+ 1 2(n+1)ωn	DATE
ma-218	501	3	1 3yn + 2 3(n+1)vn	DATE
ma-218	501	4	un	ORG
ma-218	501	5	0	CARDINAL
ma-218	501	6	2	CARDINAL
ma-218	501	7	∀n ≥ 1	DATE
ma-218	502	1	0 1	CARDINAL
ma-218	502	2	1 ωn	CARDINAL
ma-218	503	1	4.1	CARDINAL
ma-218	504	1	j. math	PERSON
ma-218	506	1	10.28924	CARDINAL
ma-218	506	2	xn}converges	PERSON
ma-218	506	3	0} as	MONEY
ma-218	506	4	1	CARDINAL
ma-218	507	1	1.0	CARDINAL
ma-218	507	2	0.5	CARDINAL
ma-218	507	3	1	CARDINAL
ma-218	507	4	m. alansari	PERSON
ma-218	507	5	r. ali	PERSON
ma-218	507	6	m. farid	PERSON
ma-218	507	7	j. ineq	PERSON
ma-218	509	1	2020	DATE
ma-218	509	2	2020	DATE
ma-218	510	1	42.[2	CARDINAL
ma-218	512	1	15–50.[3	DATE
ma-218	513	1	e. blum	PERSON
ma-218	513	2	w. oettli	PERSON
ma-218	514	1	63(1994	CARDINAL
ma-218	514	2	123–145.[4	CARDINAL
ma-218	514	3	e.r.	GPE
ma-218	514	4	c. hendrich	PERSON
ma-218	514	5	douglas-racheord	PERSON
ma-218	516	1	256	CARDINAL
ma-218	516	2	2015	CARDINAL
ma-218	516	3	e.r. csetnek	GPE
ma-218	516	4	forwardbackward	ORG
ma-218	516	5	algor	PERSON
ma-218	517	1	71	DATE
ma-218	517	2	2016	CARDINAL
ma-218	517	3	519–540.[6	CARDINAL
ma-218	517	4	m. farid	PERSON
ma-218	517	5	r. ali	PERSON
ma-218	517	6	k.r. kazmi	GPE
ma-218	517	7	37	CARDINAL
ma-218	517	8	2023	CARDINAL
ma-218	517	9	6133-6150.[7	CARDINAL
ma-218	517	10	s. gupta	PERSON
ma-218	517	11	s. husain	PERSON
ma-218	517	12	mishra	PERSON
ma-218	517	13	− h	ORG
ma-218	517	14	filomat	GPE
ma-218	517	15	31	CARDINAL
ma-218	517	16	2017	CARDINAL
ma-218	517	17	6529–6542.[8	CARDINAL
ma-218	517	18	j. iqbal	PERSON
ma-218	517	19	mashra w.a. mir	PERSON
ma-218	517	20	a.h. dar	PERSON
ma-218	517	21	m. ishiyak	PERSON
ma-218	517	22	l. rathour	PERSON
ma-218	517	23	georgian	NORP
ma-218	518	1	j. 3 (	PERSON
ma-218	518	2	2022),533–542.[9	CARDINAL
ma-218	518	3	s. kamimura	PERSON
ma-218	518	4	w. takahashi	PERSON
ma-218	518	5	j. optim.13	PERSON
ma-218	518	6	2002	DATE
ma-218	518	7	938–945	CARDINAL
ma-218	520	1	j. math	PERSON
ma-218	522	1	10.28924	CARDINAL
ma-218	523	1	10	CARDINAL
ma-218	523	2	k.r.	GPE
ma-218	523	3	kazmi	PERSON
ma-218	523	4	r. ali	PERSON
ma-218	523	5	111	CARDINAL
ma-218	523	6	2017	CARDINAL
ma-218	523	7	l. umar	PERSON
ma-218	523	8	y. ibrahim	PERSON
ma-218	523	9	kabir	PERSON
ma-218	523	10	uzbek	NORP
ma-218	524	1	j. 66	PERSON
ma-218	524	2	2022	CARDINAL
ma-218	524	3	101–118.[12	CARDINAL
ma-218	524	4	l. umar	PERSON
ma-218	524	5	kabir	PERSON
ma-218	524	6	i.u. haruna	PERSON
ma-218	524	7	generalized f projection	ORG
ma-218	525	1	sci.	ORG
ma-218	525	2	1 (	CARDINAL
ma-218	525	3	2022	CARDINAL
ma-218	525	4	j. comp	PERSON
ma-218	527	1	219	CARDINAL
ma-218	527	2	2008	DATE
ma-218	528	1	223–236.[14	CARDINAL
ma-218	528	2	k. nakajo	PERSON
ma-218	530	1	271	CARDINAL
ma-218	530	2	2017	CARDINAL
ma-218	531	1	x.l. qin	PERSON
ma-218	531	2	y.j. cho	GPE
ma-218	531	3	s.m. kang	PERSON
ma-218	531	4	j. comp	PERSON
ma-218	533	1	225	CARDINAL
ma-218	533	2	2009	DATE
ma-218	533	3	20–30.[16	CARDINAL
ma-218	534	1	ussr	NORP
ma-218	537	1	4 (1964)1–7.[17	CARDINAL
ma-218	537	2	w. takahashi	PERSON
ma-218	537	3	k. zembayashi	PERSON
ma-218	538	1	70	DATE
ma-218	538	2	2009	DATE
ma-218	538	3	45–57.[18	CARDINAL
ma-218	538	4	vandana	ORG
ma-218	538	5	r. dubey	PERSON
ma-218	538	6	deepmala	GPE
ma-218	538	7	l.n. mishra	PERSON
ma-218	538	8	v.n. mishra	PERSON
ma-218	538	9	amer	PERSON
ma-218	539	1	j. oper.	PERSON
ma-218	540	1	8 (	PERCENT
ma-218	540	2	2018	DATE
ma-218	540	3	293–311.[19	CARDINAL
ma-218	541	1	xu	PERSON
ma-218	544	1	16 (1991)	DATE
ma-218	545	1	c. zalinesco	GPE
ma-218	547	1	95	CARDINAL
ma-218	547	2	1983	DATE
ma-218	547	3	344–374.[21	CARDINAL
ma-218	549	1	72	DATE
ma-218	549	2	2010	DATE
ma-218	549	3	2136–2146	CARDINAL
ma-218	550	1	1	CARDINAL
ma-218	550	2	2	CARDINAL
ma-218	550	3	3	CARDINAL
ma-218	550	4	4	CARDINAL