REVIEW ARTICLE ABSTRACT. Interpolation of generalized Biaxisymmetric potentials. D. Kumar* G.L. `Reddy**
|
|
- Cori Black
- 5 years ago
- Views:
Transcription
1 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 eo fo a eal valued geealized biaxisyetic Potetial (GBASP which is egula i the ope hype sphee about the oigi The lowe (p q ode ad lowe geealized (p q type have bee chaacteized i tes of these appoxiatio eos Key wods: Tasfiite diaete poxiate ode Lagage itepolatio polyoials ad exteal polyoials *00 Matheatics Subject classificatio: Piay 30 E0 Secoday 4A0 o EOL fo Uivesity of Hydeabad Idia e-ail: ggleddy@yahooco **Depatet of Matheatics Addis Ababa Uivesity Addis Ababa Ethiopia
2 Ethiop J Educ & Sc Vol No Septebe INTRODUCTION Let F αβ be a eal-valued egula solutio to the geealized biaxially syetic potetial equatio x + y + ( + (β + α β α x F x + x F y y F y = 0 α > β > - subject to the Cauchy data (0 y = ( x y = 0 which is satisfied a the sigula lies i the ope hype sphee ; x + y < Such fuctios with eve haoic extesios ae efeed to as geealized biaxisyetic potetials (GBASP havig local expasios of the fo F ( x y = =0 ites of the coplete set a R ( x y R ( x y P x y x + y = (x + y [ ] ( /( / ( = 0 3 P of biaxisyetic haoic potetials whee Let the opeato f(z = =0 K uiquely associated eve aalytic fuctio z P ae Jacobi Polyoials ([] [5] = 0 a z = x + iy C/ oto GBASP F α β ( x y = a R ( x y Followig McCoy [] fo Kaoowide s itegal fo Jacobi polyoials F ( x y π = K α β (f = 0 0 f ( ς µ α β (t s dsdt
3 Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 3 Whee µ α β (t s = γ ζ = x y t ixyt + cos s ( t α - β - t β+ (sis α γ = (α + (α - β + β The ivese opeato K applies othogoality of Jacobi polyoials ([] p8 ad Poisso Keel ([] p to uiquely defie the tasfo f(z = + K (F αβ β = α ( ξ ( ξ whee υ αβ ( I ξ = S αβ (Iξ ( ξ α ( + ξ β I α ( + I S αβ (I ξ = η α β F + + η αβ = (α + β + / α+β+ (α+ (β + z F υ α β ( ξ dξ β α + β + α + β + 3 I( + ξ ; ; β + ; ( + I Hee the oalizatio K αβ ( = K ( = is tae place The Keel S αβ (I ξ is aalytic o I < fo - ξ Let E be a copact set is the coplex plae ad ξ ( = {ξ 0 ξ ξ } be a syste of ( + poits of the set E such that V(ξ ( = o j ξ j ξ ad (j (ξ ( = = 0 j ξ j ξ j = 0 Agai let η ( = {η 0 η η η } be the syste of ( + poits i E such that V V(η ( = sup ξ ( E V(ξ ( ad (0 (η ( (j (η ( fo j = Such a syste always exists ad is called the th exteal syste of E The polyoials L j (z η ( = = 0 j z η ηj η j = 0
4 Ethiop J Educ & Sc Vol No Septebe ae called Lagage exteal polyoials ad the liit d d(e = called the tasfiite diaete of E li ( + V is Let C(E (f C(E is holoophic i the iteio of E ad cotiuous o E deote the algeba of aalytic fuctios o the set E Let the Chebyshev o be defied fo f C(E ad F α β C as follows: e (f: E e (f = whee f g = ad E α β ; if f g g h sup f(x g(x x E F E (F α β = if{ F αβ - G αβ G αβ F αβ - G αβ = sup x + y = F αβ (x y G αβ (x y H } = 0 The set h cotais all eal polyoials of degee at ost ad the set H cotais all eal biaxisyetic haoic polyoials of degee at ost The opeatos K αβ ad K establish oe-oe equivalece of sets h ad H McCoy [] coected classical ode ad type of eal-valued etie fuctios GBASP F αβ ad the associate f espectively with eve polyoial appoxiatio eo defied i [- ] He obtaied the esults fo GBASP of Sato idex [3] usig the esults obtaied by Reddy [] It has bee oticed that these esults fail to copae the gowth of those etie GBASPs which have sae positive fiite ode but thei types ae ifiity Fo the view poit of icludig this class of etie GBASPS we shall utilize the cocept of poxiate ode Recetly Kua ad Kasaa [9] studied the (p q-ode ad geealized (p q-type of GBASP fo eve polyoial appoxiatio eo defied o E to iclude the eal valued etie GBASPs of slow gowth ad fast gowth These esults obviously leave a big class of eal-valued etie GBASP such as study of lowe (p q-ode ad geealized
5 Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 33 lowe (p q type which have ot bee cosideed ealie The ai of this pape is to exted the esults of [9] to lowe (p q-ode ad geealized lowe (p q-type The axiu odulii of GBASP ad associate ae defied as i coplex fuctio theoy M( f = ax f(z ad M( F α β = z = ax y x + = F αβ (x y A eal etie GBASP is said to be of (p q-ode ρ(p q ad lowe (p q ode λ(p q if it is of idex-pai (p q such that li sup if [ p] M ( F [ q ] = ρ( p q λ( p q ad the fuctio f(z havig (p q-ode ρ(p q (b < ρ(p q < is said to be of (p q type T(p q ad lowe (p q type t(p q if li sup if [ p ] M ( [ q ] ( whee b = if p = q b = 0 if p > q F ρ ( p q = T ( p q t( p q 0 t (p q T(p q A positive fuctio ρ p q ( defied o [ 0 0 > exp [q-] is said to be poxiate ode of a etie fuctio GBASP with idex-pai (p q if (i ρ p q ( ρ(p q as b < ρ < : (ii [q] ( ρ p q ( 0 as whee ρ p q ( deotes the deivative of ρ pq ( ad fo coveiece [q] ( = =0 i q [i] The existece of such copaiso fuctios o (p q-scale has bee established i [8] We ow defie geealized (p q-type T*(p q ad geealized lowe (p q-type t*(p q of F αβ with espect to a give poxiate ode ρ p q ( as li sup if [ p ] M ( [ q ] ( F ρ p a ( = T * ( p q t * ( p q 0 t* (p q T*(p q If the quatity t*(p q is diffeet fo zeo ad ifiite the ρ pq ( is said to be the lowe poxiate ode of a give etie fuctio GBASP ad t*(p q as its geealized lowe
6 Ethiop J Educ & Sc Vol No Septebe (p q-type Clealy lowe poxiate ode ad coespodig geealized lowe (p q- type of F αβ ae ot uiquely deteied Fo exaple if we add c/ [q] 0< c < to the poxiate ode ρ p q ( the ρ p q ( + c/ [q] is also a poxiate ode satisfyig (i ad (ii ad cosequetly the geealized lowe (p q-type tus out to be e c t*(p q [ q ] ρ ( A Sice ( is a ootoically iceasig fuctio [] of fo > 0 we ca defie φ(x to be the uique solutio of the equatio [ q ] ρ ( A x = ( if ad oly if φ(x = [q-] whee A = if (p q = ( ad A = 0 othewise Cosequetly it ca be show that [] φ( ηx ( ρ p q A li = η uifoly fo evey η 0 < η < x φ( x Let E be the lagest equipotetial cuve of E defied by E = {z C/ / ϕ(z d = } (if = d the E = E whee w = ϕ(z is holoophic ad aps the ubouded copoet of the copleet of E o w > such that ϕ( = ad ϕ ( > 0 Also we set M ( F = sup F αβ (z 0 fo > ad M ( f z E sup f(z z E Soe Basic Results Lea Let F αβ be eal valued etie fuctios GBASP with ap K αβ associate f The the (p q odes ad lowe (p q-odes of F αβ ad f ae idetical Futhe the espective geealized (p q-type ad geealized lowe (p q-type of F αβ ad f ae also equal Poof: Let us coside the elatio N αβ (ϒ = ax{ η s ( γ β / ξ } Fo ay ε > 0 we have [ p58] ( M( F αβ M( f M(ε - F αβ N αβ (ε Usig ( ad defiitio of (p q-ode lowe (p q-ode geealized (p q-type ad geealized lowe (p q-type of F αβ ad f we coclude the esult
7 Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 35 Lea : Let F αβ be a eal valued etie fuctio GBASP of (p q-ode ρ(p q ad lowe (p q-ode λ(p q The li sup if [ p ] M ( F [ q] = ρ( p q λ( p q ad fo ρ(p q (b < ρ(p q < T*(p q ad t*ip q ae give by li sup if [ p ] M ( [ q ] ( F ρ p q ( = T * ( p q t * ( p q 0 t* (p q T*(p q Poof: Taig the defiitio of (p q-ode lowe (p q-ode geealized (p q-type ad geealized lowe (p q type of etie GBASP ito accout the poof follows o the lies of Lea i [6] Theoe A If f C(E ca be exteded to a etie fuctio with idex-pai (p q lowe (p q-ode λ(p q (b < λ(p q < ad geealized lowe (p q-type t*(p q the fo e (f thee exists a etie fuctio g(z = = 0 + e ( f z such that λ(p q f = λ(p q g ad t* (p q f = β*t*(p q f whee β* = d -ρ(p q fo q = β* = fo q > Poof: It has bee show i Lea 3 ad 4 of [6] that the fuctio g(z = = 0 + e ( f z is a etie fuctio Wiiasi [6 p 66] has poved that fo ay ε > 0 ( e (f K M ( f ε de whee K is a costat ad d > 0 is the tasfiite diaete of E Usig ( i the powe seies expasio of g(z it is ifeed that g de ε = =0 e ( f de ε + KM ( f ε de =0 e ε KM ( f de ε e ε (
8 Ethiop J Educ & Sc Vol No Septebe o g z de 0( + M ( f + Thus i view of above iequality ad Lea fo p ad q = λ(p g λ(p f ad t*(p g e ε ρ(p d ρ(p t*(p f ad fo p ad q > λ(p q g λ (p q f ad t*(p q g t*(p q f Sice ε is abitay both iequalities iply that fo all (p q (3 λ(p q g λ(p q f ad β*t*(p q g t*(p q f Futhe usig the iequality M ( f a 0 g(/d we obseve that fo q = λ(p f λ(p g ad t*(p f d -ρ(p t*(p g ad fo q > λ(p q f λ(p q g ad t*(p f t*(p q g Hece fo all idex-pais (p q (4 λ(p q f λ(p q g ad t*(p q f β*t*(p q g Cobiig (3 ad (4 we have λ(p q f = λ(p q g ad t*(p q f = β*t*(p q g Theoe B: Let f(z C(E The f(z ca be exteded to a etie fuctio of lowe (p q-ode λ(p q (b < λ(p q < if ad oly if fo (p q ( (5 λ(p q = (6 λ(p q = whee l(p q = ax [P χ (l(p q] ad { } ax [P χ (l*(p q] { } [ p ] li [ q ] e ( f ad l*(p q = li [ q ] [ p ] e e ( f ( f
9 Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 37 such that P χ (L(p q = ad χ χ { } = L( p q χ + L( p q ax( L( p q li if q < p < if p = q = if 3 p = q if p = q = Futhe (5 ad (6 hold fo (p q = ( also povided { } be the sequece of picipal idices such that - as Poof: By Lea 3 & 4 of [6] we coclude that f C(E ca be exteded to a etie fuctio if ad oly if g(z is a etie fuctio Moeove by Theoe A f(z ad g(z have the sae lowe (p q-ode Applyig Theoe by Jueja et al [4 p 6] to the fuctio g(z = = 0 + e ( f z the esult follows Reas (a Fo E = [- ] ad (p q = ( the esult (5 icludes a Theoe by Sigh [4] ad a esult ( by Massa [0] ad i additio fo (p q = ( (6 gives Theoe 5 by Reddy [] (b Also fo E = [- ] the esults (5 ad (6 give Theoe ad by Jueja [3] fo etie fuctios of Sato gowth [3] ie (p q = (p Theoe C Let f(z C(E The f ca be exteded to a etie fuctio of (p q-ode ρ(p q (b < ρ(p q < ad geealized lowe (p q-type t*(p q (0 < t*(p q < if ad oly if (7 t*(p q = β* ax { } li φ [ p ] ( [ q ] [ q ] e ( f ρ ( p q p
10 Ethiop J Educ & Sc Vol No Septebe ad futhe if the sequece of picipal idices { } satisfies - as the fo p = t * ( q ( q = β* ax { } li if φ( [ q ] [ A] e ( f ρ ( q A whee axiu is tae ove all iceasig sequece of positive iteges ad Poof: (p q = ( ρ( eρ( ρ ( /( ρ( ρ ( if if ( p q = ( ( p q = ( othewise Applyig Theoe of Kasaa et al [7] to the fuctio g(z = ad the esultig chaacteizatio of t*(p q g i tes of e (f ad the elatio t*(pqf = β*t*(p q g taig togethe pove the theoe = 0 e ( f z + Taig ρ pq ( = ρ(p q fo all > 0 ad φ(x = x ( ρ ( p q A we have the followig coollay which gives a foula fo lowe (p q-type t(p q i tes of appoxiatio eos of a etie fuctio f(z Coollay Let f(z C(E The f(z is the estictio of a etie fuctio havig (p q-ode ρ(p q (b < ρ(p q < ad lowe (p q-type t(p q (0 < t(p q < if ad oly if t( p q ( p q β* ax { } li if [ q ] e [ p ] ρ ( q A ( f O the doai E = [- ] ad fo appoxiatio eo e (f this coollay also icludes soe esults of Reddy[] espectively fo (p q = ( ad (p q = (
11 Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 39 3 Mai Results: Polyoial Appoxiatio of GBASP I this sectio we exaie the global existece of GBASP F αβ ad gowth of its c- o E (F αβ McCoy[] poved the followig theoe Theoe E Fo each GBASP F α β egula i the hype sphee thee is a uique ap K αβ associated with a eve fuctio f aalytic i the disc D R ad covesely Now we pove the followig Theoe Let F α β be eal valued GBASP egula i ad cotiuous o The F α β ca be exteded to a etie fuctio G ASP of lowe (p q-ode λ(p q (b < λ(p q < if ad oly if fo (p q ( 3 λ(p q = 3 λ(p q = whee l (p q = ax [p χ* (l (p q] ad { } ax [p χ * (l**(p q] { } if li [ q ] [ p ] E ( F ad l**(p q = li if [ p ] [ ] ( E F q E ( F such tht χ * χ *{ } = li if
12 Ethiop J Educ & Sc Vol No Septebe Futhe (3 ad (3 hold fo (p q = ( also povide { } be the sequece of picipal idices such that - as Poof: By Theoe E F αβ is etie if ad oly if the associate f is etie Moeove lowe (p q-ode of the associate agees by Theoe B Usig Coollay ([4] p 6 ad Lea Theoe follows Theoe Let F αβ be eal valued GBASP egula i ad cotiuous o The F αβ ca be exteded to a etie fuctio GBASP of (p q-ode ρ(p q (b< ρ(p q < ad geealized lowe (p q-type t*(p q (0 < t*(p q < if ad oly if t*(p q = β* ax { } φ( li if [ q ] E [ p ] ad futhe if the sequece of picipal idices { } satisfies - as the fo p = t( q ( q = β* ax { } ( F α β ρ ( q p 3 ρ ( q A φ( li if [ A] E ( F whee axiu is tae ove all iceasig sequece of positive iteges Poof: Iequalities (4 ad (8 of [9] show that the associate f eets the sae liitig equieets as GBASP Usig Theoe C fo eve f ad Lea the poof of the theoe follows REFERENCES Asey R Othogoal Polyoials ad Special Fuctios Regioal cofeece seies i Applied Matheatics SIAM Philadelphia 975
13 Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 4 Eistei-Mathews SM ad Kasaa HS Poxiate ode ad type of etie fuctios of seveal coplex vaiables Isael J Math (Jeusale 9(995 o Jueja OP Appoxiatio of a etie fuctio J Appox Theoy No 4 ( Jueja OP Kapoo GP ad Bajapi SK O the (p q-ode ad lowe (p q- ode of a etie fuctio JReie agew Math 8( Kasaa HS The geealized type of etie fuctios with idex-pai (p q Coet Math No 9( Kasaa HS ad Kua D O appoxiatio ad itepolatio of etie fuctios with idex-pai (p q Publicatios Mathetiques 38( Kasaa HS Kua D ad Sivastava GS O the geealized lowe type of etie gap powe seies with idex-pai (p q UUDM Repot (990 8 Kasaa HS ad Sahai A The poxiate ode of etie Diichlet seies Coplex Vaiables: Theoy ad Applicatio (New Yo USA 9(987 o Kua D ad Kasaa HS Appoxiatio ad itepolatio of geealized Biaxisyetic Potetials Paae Matheatical J (999 o Massa S Reas o the gowth of a etie fuctio ad the degee of appoxiatio Riv Mat Uiv Paa(4 7(98-6 McCoy PA Polyoial appoxiatio of geealized biaxisyetic potetials J Appox Theoy 5( Reddy AR Appoxiatio of a etie fuctio J Appox Theoy 3( Sato D O the ate of gowth of etie fuctios of fast gowth Bull Ae Math Soe 69( Sigh JP Appoxaitio of etie fuctios YoohaaMathJ 9( Szego G Othogoal Polyoial Colloquiu Publicatios 3 AeMath Soc Povidece RI Wiiasi T N Appoxiatio ad itepolatio of etie fuctios A Polo Math 9(
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 informationStrong 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 informationStrong 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 informationRange 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 informationA New Result On A,p n,δ k -Summabilty
OSR Joual of Matheatics (OSR-JM) e-ssn: 2278-5728, p-ssn:239-765x. Volue 0, ssue Ve. V. (Feb. 204), PP 56-62 www.iosjouals.og A New Result O A,p,δ -Suabilty Ripeda Kua &Aditya Kua Raghuashi Depatet of
More informationSOME 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 informationAsymptotic 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 informationInternational 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 informationGeneralized 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 informationGeneralization 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 informationConditional 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 informationMath 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 informationMultivector Functions
I: J. Math. Aal. ad Appl., ol. 24, No. 3, c Academic Pess (968) 467 473. Multivecto Fuctios David Hestees I a pevious pape [], the fudametals of diffeetial ad itegal calculus o Euclidea -space wee expessed
More informationOn 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 informationMapping Radius of Regular Function and Center of Convex Region. Duan Wenxi
d Iteatioal Cofeece o Electical Compute Egieeig ad Electoics (ICECEE 5 Mappig adius of egula Fuctio ad Cete of Covex egio Dua Wexi School of Applied Mathematics Beijig Nomal Uivesity Zhuhai Chia 363463@qqcom
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
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
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 informationSome 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 informationFIXED 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 informationOn 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 informationON 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 informationModular 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 informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationUsing 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 informationCHAPTER 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 informationSOME 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 informationOn composite conformal mapping of an annulus to a plane with two holes
O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy
More informationEVALUATION 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 informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC POTENTIALS IN L p -NORM, p 1
TJMM 3 (211), No. 2, 13-11 ON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC POTENTIALS IN L p -NORM, p 1 HUZOOR H. KHAN AND RIFAQAT ALI Abstract. The real valued regular solutio of geeralized
More informationON 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 informationAdvanced Physical Geodesy
Supplemetal Notes Review of g Tems i Moitz s Aalytic Cotiuatio Method. Advaced hysical Geodesy GS887 Chistophe Jekeli Geodetic Sciece The Ohio State Uivesity 5 South Oval Mall Columbus, OH 4 7 The followig
More informationALMOST 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 informationSHIFTED 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 informationGreen 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 informationSequences 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 informationComplementary 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 informationStructure 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 information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationOn 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 informationTaylor 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 informationInternational 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 informationSolutions for Math 411 Assignment #8 1
Solutios for Math Assigmet #8 A8. Fid the Lauret series of f() 3 2 + i (a) { < }; (b) { < < }; (c) { < 2 < 3}; (d) {0 < + < 2}. Solutio. We write f() as a sum of partial fractios: f() 3 2 + ( ) 2 ( + )
More informationS. 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 informationFRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION
Joual of Rajastha Academy of Physical Scieces ISSN : 972-636; URL : htt://aos.og.i Vol.5, No.&2, Mach-Jue, 26, 89-96 FRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION Jiteda Daiya ad Jeta Ram
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationOn the Explicit Determinants and Singularities of r-circulant and Left r-circulant Matrices with Some Famous Numbers
O the Explicit Detemiats Sigulaities of -ciculat Left -ciculat Matices with Some Famous Numbes ZHAOLIN JIANG Depatmet of Mathematics Liyi Uivesity Shuaglig Road Liyi city CHINA jzh08@siacom JUAN LI Depatmet
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationlim za n n = z lim a n n.
Lecture 6 Sequeces ad Series Defiitio 1 By a sequece i a set A, we mea a mappig f : N A. It is customary to deote a sequece f by {s } where, s := f(). A sequece {z } of (complex) umbers is said to be coverget
More informationGeneralized 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 informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationGeneralized Near Rough Probability. in Topological Spaces
It J Cotemp Math Scieces, Vol 6, 20, o 23, 099-0 Geealized Nea Rough Pobability i Topological Spaces M E Abd El-Mosef a, A M ozae a ad R A Abu-Gdaii b a Depatmet of Mathematics, Faculty of Sciece Tata
More informationSteiner Hyper Wiener Index A. Babu 1, J. Baskar Babujee 2 Department of mathematics, Anna University MIT Campus, Chennai-44, India.
Steie Hype Wiee Idex A. Babu 1, J. Baska Babujee Depatmet of mathematics, Aa Uivesity MIT Campus, Cheai-44, Idia. Abstact Fo a coected gaph G Hype Wiee Idex is defied as WW G = 1 {u,v} V(G) d u, v + d
More informationOn Some Fractional Integral Operators Involving Generalized Gauss Hypergeometric Functions
Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 93-9466 Vol. 5, Issue (Decembe ), pp. 3 33 (Peviously, Vol. 5, Issue, pp. 48 47) Applicatios ad Applied Mathematics: A Iteatioal Joual (AAM) O
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationBinomial 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 informationTHE GRAVITATIONAL POTENTIAL OF A MULTIDIMENSIONAL SHELL
THE GRAVITATIONAL POTENTIAL OF A MULTIDIMENSIONAL SHELL BY MUGUR B. RĂUŢ Abstact. This pape is a attept to geealize the well-kow expessio of the gavitatioal potetial fo oe tha thee diesios. We used the
More informationJacobi 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 informationDANIEL 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 informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistat Professor Departmet of Mathematics Uiversity Of Kalyai West Begal, Idia E-mail : sahoopulak1@gmail.com 1 Module-2: Stereographic Projectio 1 Euler
More informationSome 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 informationf(1), and so, if f is continuous, f(x) = f(1)x.
2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.
More informationA Negative Result. We consider the resolvent problem for the scalar Oseen equation
O Osee Resolvet Estimates: A Negative Result Paul Deurig Werer Varhor 2 Uiversité Lille 2 Uiversität Kassel Laboratoire de Mathématiques BP 699, 62228 Calais cédex Frace paul.deurig@lmpa.uiv-littoral.fr
More informationAssignment 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 informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationCommon Coupled Fixed Point of Mappings Satisfying Rational Inequalities in Ordered Complex Valued Generalized Metric Spaces
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn:319-765x Volume 10, Issue 3 Ver II (May-Ju 014), PP 69-77 Commo Coupled Fixed Poit of Mappigs Satisfyig Ratioal Iequalities i Ordered Complex
More informationTHE 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 informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationRECIPROCAL POWER SUMS. Anthony Sofo Victoria University, Melbourne City, Australia.
#A39 INTEGERS () RECIPROCAL POWER SUMS Athoy Sofo Victoia Uivesity, Melboue City, Austalia. athoy.sofo@vu.edu.au Received: /8/, Acceted: 6//, Published: 6/5/ Abstact I this ae we give a alteative oof ad
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More information2012 GCE A Level H2 Maths Solution Paper Let x,
GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs
More informationREAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS
REAL ANALYSIS II: PROBLEM SET 1 - SOLUTIONS 18th Feb, 016 Defiitio (Lipschitz fuctio). A fuctio f : R R is said to be Lipschitz if there exists a positive real umber c such that for ay x, y i the domai
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationFinite 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 information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationSequences and Limits
Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q
More informationProgression. 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 informationMathematical Methods for Physics and Engineering
Mathematical Methods for Physics ad Egieerig Lecture otes Sergei V. Shabaov Departmet of Mathematics, Uiversity of Florida, Gaiesville, FL 326 USA CHAPTER The theory of covergece. Numerical sequeces..
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationA New Type of q-szász-mirakjan Operators
Filoat 3:8 07, 567 568 https://doi.org/0.98/fil7867c Published by Faculty of Scieces ad Matheatics, Uiversity of Niš, Serbia Available at: http://www.pf.i.ac.rs/filoat A New Type of -Szász-Miraka Operators
More informationBy 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 informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationMeasure and Measurable Functions
3 Measure ad Measurable Fuctios 3.1 Measure o a Arbitrary σ-algebra Recall from Chapter 2 that the set M of all Lebesgue measurable sets has the followig properties: R M, E M implies E c M, E M for N implies
More informationChapter 8. Euler s Gamma function
Chapter 8 Euler s Gamma fuctio The Gamma fuctio plays a importat role i the fuctioal equatio for ζ(s) that we will derive i the ext chapter. I the preset chapter we have collected some properties of the
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationThe Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1
460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that
More informationSome Properties of the K-Jacobsthal Lucas Sequence
Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas
More informationBeyond simple iteration of a single function, or even a finite sequence of functions, results
A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad are
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationThe natural exponential function
The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality.....................................2
More informationWeek 5-6: The Binomial Coefficients
Wee 5-6: The Biomial Coefficiets March 6, 2018 1 Pascal Formula Theorem 11 (Pascal s Formula For itegers ad such that 1, ( ( ( 1 1 + 1 The umbers ( 2 ( 1 2 ( 2 are triagle umbers, that is, The petago umbers
More informationOptimally Sparse SVMs
A. Proof of Lemma 3. We here prove a lower boud o the umber of support vectors to achieve geeralizatio bouds of the form which we cosider. Importatly, this result holds ot oly for liear classifiers, but
More informationFIXED POINTS OF n-valued MULTIMAPS OF THE CIRCLE
FIXED POINTS OF -VALUED MULTIMAPS OF THE CIRCLE Robert F. Brow Departmet of Mathematics Uiversity of Califoria Los Ageles, CA 90095-1555 e-mail: rfb@math.ucla.edu November 15, 2005 Abstract A multifuctio
More informationMinimal order perfect functional observers for singular linear systems
Miimal ode efect fuctioal obseves fo sigula liea systems Tadeusz aczoek Istitute of Cotol Idustial lectoics Wasaw Uivesity of Techology, -66 Waszawa, oszykowa 75, POLAND Abstact. A ew method fo desigig
More informationGreatest term (numerically) in the expansion of (1 + x) Method 1 Let T
BINOMIAL THEOREM_SYNOPSIS Geatest tem (umeically) i the epasio of ( + ) Method Let T ( The th tem) be the geatest tem. Fid T, T, T fom the give epasio. Put T T T ad. Th will give a iequality fom whee value
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
More information