A New Result On A,p n,δ k -Summabilty

Size: px
Start display at page:

Download "A New Result On A,p n,δ k -Summabilty"

Transcription

1 OSR Joual of Matheatics (OSR-JM) e-ssn: , p-ssn: x. Volue 0, ssue Ve. V. (Feb. 204), PP A New Result O A,p,δ -Suabilty Ripeda Kua &Aditya Kua Raghuashi Depatet of Matheatics FTM Uiesity Moadabad, U.P., dia, Abstact: this pape we hae established a ew theoe o Ap,, ad iteestig esults ad peious ow esults as a coollay. Keywods:, N p -suability, A -suability, A, -suability, Ap,, ifiite seies. Matheatical classificatio :40D25, 40E05, 40F05 & 40C0.. NTRODUCTON: -suability which gies soe ew -suability ad Let a be a gie ifiite seies with the sequece of patial su ( s ) ad let A ( a ) be a oal atix of o zeo diagoal eties. The A defies the sequece to sequece tasfoatio appig the sequeces s ( s ) to As ( A( s)), A (s) A s whee The seies whee a is said to suable A, if (RHOADES ad SAVAS [3]) (.) A ( s) (.2) A ( s) A ( s) A ( s) ad it is said to be suable A,, 0 ad 0 if Let ( ) p be a sequece of positie ubes such that A (.3) P p as (.4) 0 whee Pi pi 0, i ad a is said to be suable A, p, if (ÖZARSLAN, [2]) P p Ad is said to be suable A, p,,. A ( s) if P A ( s) p (.6) p f P, 0, Ap,, -suability is the sae as A -suability also if we tae a, the Ap, -suability is the sae as N, p -suability (BOR []). A sequece ( b ) of positie ubes is said to be -quasi ootoe, if b 0 ultiately ad b whee ( ) is a sequece of positie ubes (SAVAS [4]). ad a sequece ( d ) is said to be alost iceasig if thee exist a positie iceasig sequece ( c ) ad two positie costats A ad B such that Ac d Bc fo each. (.5) p 56 Page

2 A New Result O A,p,δ -Suabilty. KNOWN RESULT: Coceig with absolute atix suability facto SAVAS [5] has poed the followig theoe. Theoe 2. Let A be a lowe tiagula o Noal atix with o-egatie eties satisfyig a,0 (2.) a, fo a (2.2) a O() (2.3) whee A associates with two lowe tiagula atices A & Â defied. ad a O( a ) (2.4) a, O( ) (2.5) a a,, 0,,2 ad a a a,, 2,3, f ( X ) is a alost iceasig sequece such that, X X O ad (2.6) 0 as (2.7) Suppose that thee exist a sequece of ubes ( A ) such that it is -quasi X, AX is coeget ad t O X ( ) whee t a, the the seies a is suable A,, ad 0.. MAN RESULT: The goal of this pape is to geealize the theoe (2.) fo Ap,, -suability. Theoe 3. follows ootoe with f A ( a ) is ay oal atix associated with two lowe sub-atices A ( a ) ad A ( a ) a a,, 0,,2 ad, â a a. whee 0,0 0,0 0,0 f the coditios (2.8) (2.9) as i (3.) i a a a (3.2) a,0 (3.3) 57 Page

3 a, a, fo ad let ( p ) be the sequece of positie ubes such that, P O( p ) as A New Result O A,p,δ -Suabilty (3.4) p a O P (3.5) P P a O a p p P P a, O p p P f { X } is a alost iceasig sequece such that X O( X ) p Suppose that thee exist a sequece of ubes ( A ) such that it is -quasi AX is coeget ad A fo all, if whee t a ae satisfied the the seies p P (3.6) (3.7) ad 0 as. X, ootoe with (3.8) P t O( X ) p (3.9) a issuable A, p,,, 0. V. LEMMA: We eed the followig leas fo the poof of theoe (3.). Lea 4.. Ude the coditio of theoe, we hae ((SAVAS [4]) X O() Lea 4.2 (SAVAS [5]) X is a alost iceasig sequece such that Let { } X X O f ( A ) is -quasi ootoe with X, A X is coeget, the X A X A ad O() (4.) Let { y } be the th te of the A -tasfo of V. PROOF OF THEOREM: iai the, i Page

4 A New Result O A,p,δ -Suabilty ad 0 Y a s i i i0 a a i i0 i0 0 i a a 0 a, i i a a (5.) y y y ( a a ) a, 0 a a (5.2) we ay wite a y a a a a a a a a 2 at a t a, ( ) t a, t ( ) T T T T (5.3) (say),,2,3,4 To coplete the poof, it is sufficiet, by Miowsi's iequality, to show that Usig Hölde's iequality ad (5.3), we get P T, p P T,, fo,2,3,4 p (5.4) P a t p P, O() a t p P a a p O() t P P a a p p O() t 59 Page

5 P P a a t p p O() P P t a a, p p O() P O() a t p P P O() a t a t p p P P t t p p O() ( O() A X O() X A New Result O A,p,δ -Suabilty O(). (5.5) Agai, usig the hypothesis of the theoe (3.) ad Lea (4.), usig Hölde's iequality P 2 T,2 2 p P a ( ) t p 2, P a, t 2 p P, 2 p O() a t P,, 2 p O() a t a fo (Rhoades ad Saas[3]). Hece a Ma, P P 2, 2 p p O() a a t P P t a a, p p O() P t a, p O() P O() t p 60 Page

6 A New Result O A,p,δ -Suabilty P P p t p p P O() P P P t t p p p O() O(). P O() X O() X p P O() X O() A X O() A X p O() Next usig the hypothesis of the theoe (3.) ad Hölde's iequality P 3 T,3 2 p Fially P p 2 a t, P a, t 2 p P t a, a, 2 p O() P P a t a, 2 p p O() P P a t a, p p O() P P t a a, p p O() P O() t a, p O() P t p P p O() ( t O() ( ) X O() X O() 6 Page

7 A New Result O A,p,δ -Suabilty P 4 T,4 p P ( ) a t p P O() a t p P P O() a a t p p P O() a t p O(), as i the poof of. This copletes the poof of theoe. V. COROLLARY: This theoe hae the followig esults as a coollay. Coollay 6. P Taig the theoe (3.) educes to theoe (2.). p Coollay 6.2 P Taig, ad 0 the theoe (3.) is A -suable. p Coollay 6.3 p Taig a, ad 0 the theoe (3.) is N, p -suable. P REFERENCES: [] BOR, H; O local popety of N, p, -suability of factoed Fouie seies, J. Math. Aal. Appl. 79 (993). [2] OZARSLAN, H.S.; O absolute atix suability ethods, Matheatical Cou. 2, (2007). [3] RHOADES, B,E. ad SAVAS, E.; O A -suability factos, Alco, Math. Hug, (2006). [4] SAVAS, E.; A suability facto theoe fo absolute suability iolig -quasi -ootoe ad alost iceasig sequece, Math. Coput. Model, 48, (2008). [5] SAVAS, E.; A ote o absolute suability facto theoe ad alost iceasig sequeces, acadeic, Joual, Vol. 8 (203) Page

ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES. 1. Introduction

ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES. 1. Introduction Joural of Classical Aalysis Volue 3, Nuber 2 208), 33 39 doi:0.753/jca-208-3-09 ON ABSOLUTE MATRIX SUMMABILITY FACTORS OF INFINITE SERIES AHMET KARAKAŞ Abstract. I the preset paper, a geeral theore dealig

More information

SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE

SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE Faulty of Siees ad Matheatis, Uivesity of Niš, Sebia Available at: http://www.pf.i.a.yu/filoat Filoat 22:2 (28), 59 64 SOME NEW SEQUENCE SPACES AND ALMOST CONVERGENCE Saee Ahad Gupai Abstat. The sequee

More information

Lacunary Almost Summability in Certain Linear Topological Spaces

Lacunary Almost Summability in Certain Linear Topological Spaces BULLETIN of te MLYSİN MTHEMTİCL SCİENCES SOCİETY Bull. Malays. Mat. Sci. Soc. (2) 27 (2004), 27 223 Lacuay lost Suability i Cetai Liea Topological Spaces BÜNYMIN YDIN Cuuiyet Uivesity, Facutly of Educatio,

More information

International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017

International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017 Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Coe Metic Saces, Coe Rectagula Metic Saces ad Coo Fixed Poit Theoes M. Sivastava; S.C. Ghosh Deatet of Matheatics, D.A.V. College

More information

On the Circulant Matrices with. Arithmetic Sequence

On the Circulant Matrices with. Arithmetic Sequence It J Cotep Math Scieces Vol 5 o 5 3 - O the Ciculat Matices with Aithetic Sequece Mustafa Bahsi ad Süleya Solak * Depatet of Matheatics Educatio Selçuk Uivesity Mea Yeiyol 499 Koya-Tukey Ftly we have defied

More information

Structure and Some Geometric Properties of Nakano Difference Sequence Space

Structure and Some Geometric Properties of Nakano Difference Sequence Space Stuctue ad Soe Geoetic Poeties of Naao Diffeece Sequece Sace N Faied ad AA Baey Deatet of Matheatics, Faculty of Sciece, Ai Shas Uivesity, Caio, Egyt awad_baey@yahooco Abstact: I this ae, we exted the

More information

Some Integral Mean Estimates for Polynomials

Some Integral Mean Estimates for Polynomials Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi

More information

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity

More information

Some Topics on Weighted Generalized Inverse and Kronecker Product of Matrices

Some Topics on Weighted Generalized Inverse and Kronecker Product of Matrices Malaysia Soe Joual Topics of Matheatical o Weighted Scieces Geealized (): Ivese 9-22 ad Koece (27) Poduct of Matices Soe Topics o Weighted Geealized Ivese ad Koece Poduct of Matices Zeyad Abdel Aziz Al

More information

On Summability Factors for N, p n k

On Summability Factors for N, p n k Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet

More information

A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES

A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES Bulleti of Mathematical Aalysis ad Applicatios ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 8 Issue 42016), Pages 91-97. A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES ŞEBNEM YILDIZ Abstract.

More information

ABSOLUTE INDEXED SUMMABILITY FACTOR OF AN INFINITE SERIES USING QUASI-F-POWER INCREASING SEQUENCES

ABSOLUTE INDEXED SUMMABILITY FACTOR OF AN INFINITE SERIES USING QUASI-F-POWER INCREASING SEQUENCES Available olie a h://sciog Egieeig Maheaics Lees 2 (23) No 56-66 ISSN 249-9337 ABSLUE INDEED SUMMABILIY FACR F AN INFINIE SERIES USING QUASI-F-WER INCREASING SEQUENCES SKAIKRAY * RKJAI 2 UKMISRA 3 NCSAH

More information

Strong Result for Level Crossings of Random Polynomials

Strong Result for Level Crossings of Random Polynomials IOSR Joual of haacy ad Biological Scieces (IOSR-JBS) e-issn:78-8, p-issn:19-7676 Volue 11, Issue Ve III (ay - Ju16), 1-18 wwwiosjoualsog Stog Result fo Level Cossigs of Rado olyoials 1 DKisha, AK asigh

More information

Generalization of Horadam s Sequence

Generalization of Horadam s Sequence Tuish Joual of Aalysis ad Nube Theoy 6 Vol No 3-7 Available olie at http://pubssciepubco/tjat///5 Sciece ad Educatio Publishig DOI:69/tjat---5 Geealizatio of Hoada s Sequece CN Phadte * YS Valaulia Depatet

More information

On a Problem of Littlewood

On a Problem of Littlewood Ž. JOURAL OF MATHEMATICAL AALYSIS AD APPLICATIOS 199, 403 408 1996 ARTICLE O. 0149 O a Poblem of Littlewood Host Alze Mosbache Stasse 10, 51545 Waldbol, Gemay Submitted by J. L. Bee Received May 19, 1995

More information

Asymptotic Expansions of Legendre Wavelet

Asymptotic Expansions of Legendre Wavelet Asptotic Expasios of Legede Wavelet C.P. Pade M.M. Dixit * Depatet of Matheatics NERIST Nijuli Itaaga Idia. Depatet of Matheatics NERIST Nijuli Itaaga Idia. Astact A e costuctio of avelet o the ouded iteval

More information

Strong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA

Strong Result for Level Crossings of Random Polynomials. Dipty Rani Dhal, Dr. P. K. Mishra. Department of Mathematics, CET, BPUT, BBSR, ODISHA, INDIA Iteatioal Joual of Reseach i Egieeig ad aageet Techology (IJRET) olue Issue July 5 Available at http://wwwijetco/ Stog Result fo Level Cossigs of Rado olyoials Dipty Rai Dhal D K isha Depatet of atheatics

More information

Taylor Transformations into G 2

Taylor Transformations into G 2 Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that

More information

Some Ideal Convergent Sequence Spaces Defined by a Sequence of Modulus Functions Over n-normed Spaces

Some Ideal Convergent Sequence Spaces Defined by a Sequence of Modulus Functions Over n-normed Spaces MATEMATIKA, 203, Volue 29, Nube 2, 87 96 c Depatent of Matheatical Sciences, UTM. Soe Ideal Convegent Sequence Spaces Defined by a Sequence of Modulus Functions Ove n-noed Spaces Kuldip Raj and 2 Sunil

More information

174. A Tauberian Theorem for (J,,fin) Summability*)

174. A Tauberian Theorem for (J,,fin) Summability*) No, 10] 807 174. A Tauberia Theore for (J,,fi) Suability*) By Kazuo IsHIGURo Departet of Matheatics, Hokkaido Uiversity, Sapporo (Co. by Kijiro KUNUGI, M.J,A., Dec. 12, 1964) 1. We suppose throughout ad

More information

Sums of Involving the Harmonic Numbers and the Binomial Coefficients

Sums of Involving the Harmonic Numbers and the Binomial Coefficients Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa

More information

Available online at J. Math. Comput. Sci. 4 (2014), No. 3, ISSN:

Available online at   J. Math. Comput. Sci. 4 (2014), No. 3, ISSN: Available olie at http://scik.org J. Math. Coput. Sci. (1, No. 3, 9-5 ISSN: 197-537 ON SYMMETRICAL FUNCTIONS WITH BOUNDED BOUNDARY ROTATION FUAD. S. M. AL SARARI 1,, S. LATHA 1 Departet of Studies i Matheatics,

More information

On Almost Increasing Sequences For Generalized Absolute Summability

On Almost Increasing Sequences For Generalized Absolute Summability Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute

More information

EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS

EVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal

More information

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES #A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet

More information

Existence and multiplicity of solutions to boundary value problems for nonlinear high-order differential equations

Existence and multiplicity of solutions to boundary value problems for nonlinear high-order differential equations Jounal of pplied Matheatics & Bioinfoatics, vol.5, no., 5, 5-5 ISSN: 79-66 (pint), 79-6939 (online) Scienpess Ltd, 5 Existence and ultiplicity of solutions to bounday value pobles fo nonlinea high-ode

More information

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.

More information

Modular Spaces Topology

Modular Spaces Topology Applied Matheatics 23 4 296-3 http://ddoiog/4236/a234975 Published Olie Septebe 23 (http://wwwscipog/joual/a) Modula Spaces Topology Ahed Hajji Laboatoy of Matheatics Coputig ad Applicatio Depatet of Matheatics

More information

REVIEW ARTICLE ABSTRACT. Interpolation of generalized Biaxisymmetric potentials. D. Kumar* G.L. `Reddy**

REVIEW ARTICLE ABSTRACT. Interpolation of generalized Biaxisymmetric potentials. D. Kumar* G.L. `Reddy** Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 9 REVIEW ARTICLE Itepolatio of geealized Biaxisyetic potetials D Kua* GL `Reddy** ABSTRACT I this pape we study the chebyshev ad itepolatio

More information

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as

Math 7409 Homework 2 Fall from which we can calculate the cycle index of the action of S 5 on pairs of vertices as Math 7409 Hoewok 2 Fall 2010 1. Eueate the equivalece classes of siple gaphs o 5 vetices by usig the patte ivetoy as a guide. The cycle idex of S 5 actig o 5 vetices is 1 x 5 120 1 10 x 3 1 x 2 15 x 1

More information

Some Remarks on the Boundary Behaviors of the Hardy Spaces

Some Remarks on the Boundary Behaviors of the Hardy Spaces Soe Reaks on the Bounday Behavios of the Hady Spaces Tao Qian and Jinxun Wang In eoy of Jaie Kelle Abstact. Soe estiates and bounday popeties fo functions in the Hady spaces ae given. Matheatics Subject

More information

International Journal of Mathematical Archive-3(5), 2012, Available online through ISSN

International Journal of Mathematical Archive-3(5), 2012, Available online through   ISSN Iteatioal Joual of Matheatical Achive-3(5,, 8-8 Available olie though www.ija.ifo ISSN 9 546 CERTAIN NEW CONTINUED FRACTIONS FOR THE RATIO OF TWO 3 ψ 3 SERIES Maheshwa Pathak* & Pakaj Sivastava** *Depatet

More information

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics

More information

Research Article Approximation of Signals (Functions) by Trigonometric Polynomials in L p -Norm

Research Article Approximation of Signals (Functions) by Trigonometric Polynomials in L p -Norm Intenational Matheatics and Matheatical Sciences, Aticle ID 267383, 6 pages http://dx.doi.og/.55/24/267383 Reseach Aticle Appoxiation of Signals (Functions) by Tigonoetic Polynoials in L p -No M. L. Mittal

More information

ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS

ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, 39-43 ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh-202002, Idia sdqaar@rediffail.co,

More information

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.

It is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial. Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable

More information

Some remarks on the paper Some elementary inequalities of G. Bennett

Some remarks on the paper Some elementary inequalities of G. Bennett Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries

More information

Conditional Convergence of Infinite Products

Conditional Convergence of Infinite Products Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this

More information

ON CERTAIN CLASS OF ANALYTIC FUNCTIONS

ON CERTAIN CLASS OF ANALYTIC FUNCTIONS ON CERTAIN CLASS OF ANALYTIC FUNCTIONS Nailah Abdul Rahma Al Diha Mathematics Depatmet Gils College of Educatio PO Box 60 Riyadh 567 Saudi Aabia Received Febuay 005 accepted Septembe 005 Commuicated by

More information

Using Difference Equations to Generalize Results for Periodic Nested Radicals

Using Difference Equations to Generalize Results for Periodic Nested Radicals Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =

More information

Range Symmetric Matrices in Minkowski Space

Range Symmetric Matrices in Minkowski Space BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608

More information

Available online through ISSN

Available online through  ISSN Intenational eseach Jounal of Pue Algeba -() 01 98-0 Available online though wwwjpainfo ISSN 8 907 SOE ESULTS ON THE GOUP INVESE OF BLOCK ATIX OVE IGHT OE DOAINS Hanyu Zhang* Goup of athematical Jidong

More information

On ARMA(1,q) models with bounded and periodically correlated solutions

On ARMA(1,q) models with bounded and periodically correlated solutions Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,

More information

Generalized Fixed Point Theorem. in Three Metric Spaces

Generalized Fixed Point Theorem. in Three Metric Spaces It. Joural of Math. Aalysis, Vol. 4, 00, o. 40, 995-004 Geeralized Fixed Poit Thee i Three Metric Spaces Kristaq Kikia ad Luljeta Kikia Departet of Matheatics ad Coputer Scieces Faculty of Natural Scieces,

More information

Quadratic Harmonic Number Sums

Quadratic Harmonic Number Sums Applied Matheatics E-Notes, (), -7 c ISSN 67-5 Available fee at io sites of http//www.ath.nthu.edu.tw/aen/ Quadatic Haonic Nube Sus Anthony Sofo y and Mehdi Hassani z Received July Abstact In this pape,

More information

Relation (12.1) states that if two points belong to the convex subset Ω then all the points on the connecting line also belong to Ω.

Relation (12.1) states that if two points belong to the convex subset Ω then all the points on the connecting line also belong to Ω. Lectue 6. Poectio Opeato Deiitio A.: Subset Ω R is cove i [ y Ω R ] λ + λ [ y = z Ω], λ,. Relatio. states that i two poits belog to the cove subset Ω the all the poits o the coectig lie also belog to Ω.

More information

A GENERALIZATION OF A CONJECTURE OF MELHAM. 1. Introduction The Fibonomial coefficient is, for n m 1, defined by

A GENERALIZATION OF A CONJECTURE OF MELHAM. 1. Introduction The Fibonomial coefficient is, for n m 1, defined by A GENERALIZATION OF A CONJECTURE OF MELHAM EMRAH KILIC 1, ILKER AKKUS, AND HELMUT PRODINGER 3 Abstact A genealization of one of Melha s conectues is pesented Afte witing it in tes of Gaussian binoial coefficients,

More information

Complementary Dual Subfield Linear Codes Over Finite Fields

Complementary Dual Subfield Linear Codes Over Finite Fields 1 Complemetay Dual Subfield Liea Codes Ove Fiite Fields Kiagai Booiyoma ad Somphog Jitma,1 Depatmet of Mathematics, Faculty of Sciece, Silpao Uivesity, Naho Pathom 73000, hailad e-mail : ai_b_555@hotmail.com

More information

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to

( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special

More information

SHIFTED HARMONIC SUMS OF ORDER TWO

SHIFTED HARMONIC SUMS OF ORDER TWO Commu Koea Math Soc 9 0, No, pp 39 55 http://dxdoiog/03/ckms0939 SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of

More information

International Journal of Mathematical Archive-5(3), 2014, Available online through ISSN

International Journal of Mathematical Archive-5(3), 2014, Available online through   ISSN Iteatioal Joual of Mathematical Achive-5(3, 04, 7-75 Available olie though www.ijma.ifo ISSN 9 5046 ON THE OSCILLATOY BEHAVIO FO A CETAIN CLASS OF SECOND ODE DELAY DIFFEENCE EQUATIONS P. Mohakuma ad A.

More information

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges

More information

Generalized Fibonacci-Lucas Sequence

Generalized Fibonacci-Lucas Sequence Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash

More information

ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES

ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES M aheaical I equaliies & A pplicaios Volue 19, Nube 1 (216), 287 296 doi:1.7153/ia-19-21 ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES W. ŁENSKI AND B. SZAL (Couicaed by

More information

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION

THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia

More information

PROGRESSION AND SERIES

PROGRESSION AND SERIES INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of

More information

The value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng

The value of Banach limits on a certain sequence of all rational numbers in the interval (0,1) Bao Qi Feng The value of Baach limits o a certai sequece of all ratioal umbers i the iterval 0, Bao Qi Feg Departmet of Mathematical Scieces, Ket State Uiversity, Tuscarawas, 330 Uiversity Dr. NE, New Philadelphia,

More information

Al Lehnen Madison Area Technical College 10/5/2014

Al Lehnen Madison Area Technical College 10/5/2014 The Correlatio of Two Rado Variables Page Preliiary: The Cauchy-Schwarz-Buyakovsky Iequality For ay two sequeces of real ubers { a } ad = { b } =, the followig iequality is always true. Furtherore, equality

More information

A 2nTH ORDER LINEAR DIFFERENCE EQUATION

A 2nTH ORDER LINEAR DIFFERENCE EQUATION A 2TH ORDER LINEAR DIFFERENCE EQUATION Doug Aderso Departmet of Mathematics ad Computer Sciece, Cocordia College Moorhead, MN 56562, USA ABSTRACT: We give a formulatio of geeralized zeros ad (, )-discojugacy

More information

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio

More information

Lecture 6: October 16, 2017

Lecture 6: October 16, 2017 Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of

More information

A question of Gol dberg concerning entire functions with prescribed zeros

A question of Gol dberg concerning entire functions with prescribed zeros A question of Gol dbeg concening entie functions with pescibed zeos Walte Begweile Abstact Let (z j ) be a sequence of coplex nubes satisfying z j as j and denote by n() the nube of z j satisfying z j.

More information

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The

More information

Generalized k-normal Matrices

Generalized k-normal Matrices Iteatioal Joual of Computatioal Sciece ad Mathematics ISSN 0974-389 Volume 3, Numbe 4 (0), pp 4-40 Iteatioal Reseach Publicatio House http://wwwiphousecom Geealized k-omal Matices S Kishamoothy ad R Subash

More information

JORDAN CANONICAL FORM AND ITS APPLICATIONS

JORDAN CANONICAL FORM AND ITS APPLICATIONS JORDAN CANONICAL FORM AND ITS APPLICATIONS Shivani Gupta 1, Kaajot Kau 2 1,2 Matheatics Depatent, Khalsa College Fo Woen, Ludhiana (India) ABSTRACT This pape gives a basic notion to the Jodan canonical

More information

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3

More information

SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR

SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR Joural of the Alied Matheatics Statistics ad Iforatics (JAMSI) 5 (9) No SOME PROPERTIES OF CERTAIN MULTIVALENT ANALYTIC FUNCTIONS USING A DIFFERENTIAL OPERATOR SP GOYAL AND RAKESH KUMAR Abstract Here we

More information

PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.

PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1. Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6

More information

Green Functions. January 12, and the Dirac delta function. 1 x x

Green Functions. January 12, and the Dirac delta function. 1 x x Gee Fuctios Jauay, 6 The aplacia of a the Diac elta fuctio Cosie the potetial of a isolate poit chage q at x Φ = q 4πɛ x x Fo coveiece, choose cooiates so that x is at the oigi. The i spheical cooiates,

More information

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations 5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview

More information

Binomial transform of products

Binomial transform of products Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {

More information

KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE WITH TWO NEGATIVE PARAMETERS. Received April 13, 2010; revised August 18, 2010

KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE WITH TWO NEGATIVE PARAMETERS. Received April 13, 2010; revised August 18, 2010 Scientiae Matheaticae Japonicae Online, e-200, 427 439 427 KANTOROVICH TYPE INEQUALITIES FOR THE DIFFERENCE WITH TWO NEGATIVE PARAMETERS Young Ok Ki, Jun Ichi Fujii, Masatoshi Fujii + and Yuki Seo ++ Received

More information

CERTAIN CONGRUENCES FOR HARMONIC NUMBERS Kotor, Montenegro

CERTAIN CONGRUENCES FOR HARMONIC NUMBERS Kotor, Montenegro MATHEMATICA MONTISNIGRI Vol XXXVIII (017) MATHEMATICS CERTAIN CONGRUENCES FOR HARMONIC NUMBERS ROMEO METROVIĆ 1 AND MIOMIR ANDJIĆ 1 Maritie Faculty Kotor, Uiversity of Moteegro 85330 Kotor, Moteegro e-ail:

More information

FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES

FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES Joual of Appled Matheatcs ad Coputatoal Mechacs 7, 6(), 59-7 www.ac.pcz.pl p-issn 99-9965 DOI:.75/jac.7..3 e-issn 353-588 FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL

More information

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P. Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a

More information

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009

Discrete Mathematics: Lectures 8 and 9 Principle of Inclusion and Exclusion Instructor: Arijit Bishnu Date: August 11 and 13, 2009 Discrete Matheatics: Lectures 8 ad 9 Priciple of Iclusio ad Exclusio Istructor: Arijit Bishu Date: August ad 3, 009 As you ca observe by ow, we ca cout i various ways. Oe such ethod is the age-old priciple

More information

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a

Jacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi

More information

Iterative Techniques for Solving Ax b -(3.8). Assume that the system has a unique solution. Let x be the solution. Then x A 1 b.

Iterative Techniques for Solving Ax b -(3.8). Assume that the system has a unique solution. Let x be the solution. Then x A 1 b. Iterative Techiques for Solvig Ax b -(8) Cosider solvig liear systems of them form: Ax b where A a ij, x x i, b b i Assume that the system has a uique solutio Let x be the solutio The x A b Jacobi ad Gauss-Seidel

More information

SOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS

SOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz

More information

Common Fixed Points for Multifunctions Satisfying a Polynomial Inequality

Common Fixed Points for Multifunctions Satisfying a Polynomial Inequality BULETINUL Uiversităţii Petrol Gaze di Ploieşti Vol LXII No /00 60-65 Seria Mateatică - Iforatică - Fizică Coo Fixed Poits for Multifuctios Satisfyig a Polyoial Iequality Alexadru Petcu Uiversitatea Petrol-Gaze

More information

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions

ECE 901 Lecture 4: Estimation of Lipschitz smooth functions ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig

More information

A note on random minimum length spanning trees

A note on random minimum length spanning trees A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu

More information

S. K. VAISH AND R. CHANKANYAL. = ρ(f), b λ(f) ρ(f) (1.1)

S. K. VAISH AND R. CHANKANYAL. = ρ(f), b λ(f) ρ(f) (1.1) TAMKANG JOURNAL OF MATHEMATICS Volume 35, Number, Witer 00 ON THE MAXIMUM MODULUS AND MAXIMUM TERM OF COMPOSITION OF ENTIRE FUNCTIONS S. K. VAISH AND R. CHANKANYAL Abstract. We study some growth properties

More information

Central limit theorem for functions of weakly dependent variables

Central limit theorem for functions of weakly dependent variables Int. Statistical Inst.: Poc. 58th Wold Statistical Congess, 2011, Dublin (Session CPS058 p.5362 Cental liit theoe fo functions of weakly dependent vaiables Jensen, Jens Ledet Aahus Univesity, Depatent

More information

A Common Fixed Point Theorem Using Compatible Mappings of Type (A-1)

A Common Fixed Point Theorem Using Compatible Mappings of Type (A-1) Aals of Pure ad Applied Mathematics Vol. 4, No., 07, 55-6 ISSN: 79-087X (P), 79-0888(olie) Published o 7 September 07 www.researchmathsci.org DOI: http://dx.doi.org/0.457/apam.v4a8 Aals of A Commo Fixed

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : , MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +

More information

Assignment 5: Solutions

Assignment 5: Solutions McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics

More information

Int. Journal of Math. Analysis, Vol. 6, 2012, no. 31, S. Panayappan

Int. Journal of Math. Analysis, Vol. 6, 2012, no. 31, S. Panayappan It Joural of Math Aalysis, Vol 6, 0, o 3, 53 58 O Power Class ( Operators S Paayappa Departet of Matheatics Goveret Arts College, Coibatore 6408 ailadu, Idia paayappa@gailco N Sivaai Departet of Matheatics

More information

Global asymptotic stability in a rational dynamic equation on discrete time scales

Global asymptotic stability in a rational dynamic equation on discrete time scales Iteatioal Joual of Egieeig Reseach & Sciece (IJOER) ISSN: [395-699] [Vol-, Issue-, Decebe- 6] Global asyptotic stability i a atioal dyaic euatio o discete tie scales a( t) b( ( t)) ( ( t)), t T c ( ( (

More information

Fixed Point Theorems for Expansive Mappings in G-metric Spaces

Fixed Point Theorems for Expansive Mappings in G-metric Spaces Turkish Joural of Aalysis ad Number Theory, 7, Vol. 5, No., 57-6 Available olie at http://pubs.sciepub.com/tjat/5//3 Sciece ad Educatio Publishig DOI:.69/tjat-5--3 Fixed Poit Theorems for Expasive Mappigs

More information

For this purpose, we need the following result:

For this purpose, we need the following result: 9 Lectue Sigulities of omplex Fuctio A poit is clled sigulity of fuctio f ( z ) if f ( z ) is ot lytic t the poit. A sigulity is clled isolted sigulity of f ( z ), if f ( z ) is lytic i some puctued disk

More information

Combinatorial Interpretation of Raney Numbers and Tree Enumerations

Combinatorial Interpretation of Raney Numbers and Tree Enumerations Ope Joual of Discete Matheatics, 2015, 5, 1-9 Published Olie Jauay 2015 i SciRes. http://www.scip.og/joual/ojd http://dx.doi.og/10.4236/ojd.2015.51001 Cobiatoial Itepetatio of Raey Nubes ad Tee Eueatios

More information

Recursion. Algorithm : Design & Analysis [3]

Recursion. Algorithm : Design & Analysis [3] Recusio Algoithm : Desig & Aalysis [] I the last class Asymptotic gowth ate he Sets Ο, Ω ad Θ Complexity Class A Example: Maximum Susequece Sum Impovemet of Algoithm Compaiso of Asymptotic Behavio Aothe

More information

DANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD

DANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito

More information

MATH 312 Midterm I(Spring 2015)

MATH 312 Midterm I(Spring 2015) MATH 3 Midterm I(Sprig 05) Istructor: Xiaowei Wag Feb 3rd, :30pm-3:50pm, 05 Problem (0 poits). Test for covergece:.. 3.. p, p 0. (coverges for p < ad diverges for p by ratio test.). ( coverges, sice (log

More information

Metric Space Properties

Metric Space Properties Metric Space Properties Math 40 Fial Project Preseted by: Michael Brow, Alex Cordova, ad Alyssa Sachez We have already poited out ad will recogize throughout this book the importace of compact sets. All

More information

Math 104: Homework 2 solutions

Math 104: Homework 2 solutions Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does

More information

The degree sequences and spectra of scale-free random graphs

The degree sequences and spectra of scale-free random graphs The degee sequeces ad specta of scale-fee ado gaphs Joatha Joda Uivesity of Sheffield 30th July 004 Abstact We ivestigate the degee sequeces of scale-fee ado gaphs. We obtai a foula fo the liitig popotio

More information

A New Criterion for Stability of Delayed Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks

A New Criterion for Stability of Delayed Takagi-Sugeno Fuzzy Cohen-Grossberg Neural Networks Iteatioal Joual of Coputig Acadeic Reseach (IJCAR) ISSN 305-9184, Volue 7, Nube 3 (Jue 018), pp.43-50 MEACSE Publicatios http://www.eacse.og/ijca A New Citeio fo Stability of Delayed Takagi-Sugeo Fuzzy

More information