Spectrum of The Direct Sum of Operators. 1. Introduction
|
|
|
- Claire Sherman
- 5 years ago
- Views:
Transcription
1 Specu of The Diec Su of Opeaos by E.OTKUN ÇEVİK ad Z.I.ISMILOV Kaadeiz Techical Uivesiy, Faculy of Scieces, Depae of Maheaics 6080 Tabzo, TURKEY e-ail adess : zaeddi@yahoo.co bsac: I his wok, a coecio bewee soe specal popeies of diec su of opeaos i he diec su of Hilbe spaces ad is coodiae opeaos has bee ivesigaed. Keywods: Diec su of Hilbe spaces ad opeaos; specu ad esolve ses; coiuous ad copac opeaos; disceeess of specu; asypoics of eigevalues; 000 MS Subjec Classificaio: Ioducio I is kow ha ifiie diec su of Hilbe spaces H, ad ifiie diec su of opeaos i H, ae defied as ad H H u u : u H,, u u H H, D u u H : u D,, u u H, : D H H (see []). The geeal heoy of liea closed opeaos i Hilbe spaces ad is applicaios o physical pobles has bee ivesigaed by ay aheaicias (fo exaple, see []). Howeve, ay physical pobles of oday aisig i he odellig of pocesses of ulipaicle uau echaics, uau field heoy ad i he physics of igid bodies suppo o sudy a heoy of liea diec su of opeaos i he diec su of Hilbe spaces (see []-[6] ad efeeces i i). I his pape, a coecio bewee specu, esolve ses, disceeess of he specu (sec. ) ad asyoical behaviou of he eigevalues (sec. 3) of diec su of opeaos defied i he diec su of Hilbe spaces ad suiable popeies of coodiae opeaos has bee esablished. The obaied esuls has bee suppoed wih applicaios.
2 These ad elaed pobles i he case coiuous diec su of he Hilbe space opeaos have bee ivesigaed i woks Bu i hese woks has o bee cosideed a coecio bewee pas of he specu of diec su opeao ad suiable pas of he specu hei coodiae opeaos.i his pape give shap foulaes i he his sese.. O he specu of diec su of opeaos I his secio, he elaioship bewee he specu ad esolve ses of he diec su of opeaos ad is coodiae opeaos will be ivesigaed. Fis of all i will be ivesigaed he coiuiy ad copacess popeies of he opeao i H H i case whe LH fo each. I is easy o see ha he followig poposiios ae ue i geeal. Theoe.. Le, H H ad fo ay he ecessay ad sufficie codiio is sup. LH. I ode fo LH I addiio, i his case whe LH i is ue sup (see Theoe.. Le C H li 0. fo each Fuheoe, he followig ai esul ca be poved. C H if ad oly if. I his case Theoe.3. Fo he pas of specu ad esolve ses of he opeao H H he followig clais ae ue i Hilbe space
3 Poof. The validiy of fis clai of give elaios is clea. Moeove, i is easy o pove he fouh eualiy usig he heoe.. Now we pove he secod elaio o he coiuous specu. Le c. I his case by he defiiio of coiuous specu E is a oe-ooe opeao, R E H ad R E is dese i H. Coseuely, fo ay a opeao E is a oe-o-oe opeao i H, hee exiss such ha R E H ad fo ay bu liea aifold R E This eas ha is dese i H o O he coay, ow suppose ha fo he poi he above elaio is saisfied. Coseuely, eihe fo ay c, o ad hee exis such ha c. Tha is, fo ay d fo his i iplies ha he opeao. Hece R E H is a oe-o-oe opeao, R E H ad. c R E H. is a oe-o-oe opeao, R E H ad 3
4 O he ohe had he siple calculaios show ha c c c c p c. By he siilaly idea ca be poved he validiy of he hid eualiy of he heoe. Exaple.4. Coside he followig ulipoi diffeeial opeao fo fis ode u u, H L, a, b, a b a... : D H H, D u W : u a u b,, ad H H. Fo ay opeao ki ad ae oal, p : k b a eigevecos accodig o he eigevalue k,, k ae i he fo ad u c exp a,, c \ 0 []. k k k k I his case uk ck k a d ck b a exp. H The coefficies c k ay be choose such ha he las seies o be covege. This eas ha. Fo his ad Theoe. i is obaied ha k p p p. Defiiio.5[]. Le T be a liea closed ad desely defied opeao i ay Hilbe space. If T ad fo T he esolve opeao R T C T : DT is called a opeao wih discee specu. Noe ha he followig esuls ae ue., he opeao I is clea ha if he opeao fo evey he opeao is a opeao wih discee specu i is also i H. The followig poposiio is poved by usig he heoe.. H H, he 4
5 Theoe.6. If, R is a opeao wih discee specu i H,, ad li 0, he is a opeao wih discee specu i H. Poof: I his case fo each we have R C H, :. Now we defie he opeao K R i H. I his case fo evey u u D have K Eu R E u R E u R E u u ad E K u E R u E R u E R u E R u u These elaios show ha R R Fuheoe, we defie he followig opeaos K : H H, i he fo K u : R u, R u,..., R u,0,0,..., u u H. we Now he covegece i opeao o of he opeaos ivesigaed. Fo he u u H we have K o he opeao K will be K u Ku R u R u R u sup H H H H sup R u Fo his i is obaied ha H K K sup R,. 5
6 This eas ha seuece of he opeaos K coveges i opeao o o he opeao K. The by he ipoa heoe of he heoy of copac opeaos K C H ay K C H. Exaple.7. Coside ha he followig faily of he opeaos i he fo d : S, S S 0, S C, d : D L L, a, b, sup b a, D u W, : u b W u a, W W, whee [], because fo L L,,, is ay Hilbe space ad W is a uiay opeao i, (fo his see []). Fo ay a opeao Fo he is oal wih discee specu ad ad sufficiely lage a siple calculaio shows ha S E a S E b a S E b s R f e E W e W e f s ds SEs e f sds, f L, a O he ohe had he followig esiaes ae ue. S E s S E s a L a e f s ds e f s ds d S E s S E s e ds d f s ds e ds d f L a a s S s s e e ds d f e ds d f L L a a b a a b e f L,. 4 6
7 E W e W W e e S E b a S E b a S E b a S E b s 0 0 b a S E b a e e,. b a L e f s ds e f,.3 Hece fo. ad.3 we have S E a S E b a S E b s e E W e W e f s ds L a S a S E b a S E b s e e d E W e W e f s ds b a b a L ba e e e f,.4 4 L whee, is he eal pa of ad is he fis eigevalue of he opeao S,. Theefoe, fo esiaes. ad.4 he followig esul is obaied., sup b a ad S as, he Poposiio.8. If 0 as. R Coseuely, he opeao is a opeao wih discee specu i L L. 3. sypoical behaviou of he eigevalues I his secio asypoical behaviou fo he eigevalues of he opeao will be ivesigaed i a special case. i H H 7
8 Theoe 3.. ssue ha eh opeao specu ad fo i, j, i j i H ad i H, ae opeaos wih discee, i j c, 0 c,,, c, is saisfied. If ad hee exiss such ha if 0, he, 0, as. Poof: Fis of all oe ha by he Theoe.3 Hee i is deoed by NT. p p, ha is, a ube of eigevalues of he soe T ; :, 0 liea closed opeao T i ay Hilbe space wih odules of hese eigevalues less ha o eual o, 0. This fucio akes values i he se of o-egaive iege ubes ad i case of ubouded opeao T i is odeceasig ad eds o Sice fo evey i, j, i j I his case i is clea ha, i j c c c as., he N ; N ; N ;, 0. The las seies is uifoly covege i, o. The li c c Theefoe ;, 0, as N c c c. The he followig asypoic behaviou of eigevalues of he opeao, 0, as is ue. i H Reak 3.. If i he above heoe he coefficies, saisfy he followig codiio if 0, he fo evey 0 if, N ; as Reak 3.3. If he evey fiiely ay ses of he faily, 8 i coplex plae iesec i he fiiely ay pois, he i ca be poved ha clai of he Theoe 3. is valid i his case oo. Exaple 3.4: Le H H, H,, : D H H, u : c u,,,,,,,,, 0, u u D c c c k k c k k k
9 as, k is covege ad hee exiss such ha if. I his case, fo ay Now we obai he esolve opeao of is a liea oal opeao ad c. Le p. The fo he elaio E u v,, v H, i.e c u u v,. v I is esablished ha u,, c i.e R v,. v c O he ohe had sice c k, as, he fo ay v H H c c c R v v v Coseuely, fo ay v v H we have R (3.) c Moeove, i is kow ha a esolve opeao R, 9 is copac if ad oly if.sice c,, ad codiios o c, he he las codiio is saisfied. Hece fo ay R C H O he ohe had sice he seies. k is covege, he fo he ieualiy (3.) fo he, i is easy o see ha li R 0, R Hece by he Theoe.6 fo he i is esablished ha R C H he Theoe.3 i is ue ha p p. Fuheoe, he validiy of he elaio i j Theefoe by he Theoe 3., 0, as, i, j, i j is clea.. The by
10 Refeeces. Dufod, N., Schwaz, J. T., Liea Opeaos, I, II,Iesciece, New Yok, 958,963.. Tiosheko, S., Theoy of Elasic Sabiliy, McGaw-Hill, New Yok, Gaakhe, F. R., Kei, M. G., Oscillaig Maices ad Keels ad Sall Oscillaios of Mechaical Syses, Gosekheoizda, Moscow, 950, (i Russia). 4. Zel,., Su-Liouville Theoy, e. Mah. Soc., Mah. Suvey ad Moogaphs vol., US, Kochubei,. N., Syeic Opeaos ad Noclassical Specal Pobles, Ma. Zaeki, 5, 3 (979), Isailov, Z.I., Mulipoi Noal Diffeeial Opeaos fo Fis Ode, Opusc. Mah., 9, 4, (009), Naiak,M..,Foi S.V.,Coiuous diec sus of Hilbe spaces ad soe of hei applicaios,uspehi Ma.Nauk.,0,(64)955,-4.(i Russia). 8. Chow,T.R., specal heoy fo he diec su iegal of opeaos,mah..88(970), zoff,e..,specu ad diec iegal,tas.e.mah.soc.,97(974), Fialkow,L.., oe o diec sus of uasiilpoe opeaos,poc.e.mah.soc.,46(975),5-3.. Isailov, Z.I., Copac iveses of fis-ode oal diffeeial opeaos, J. Mah. al. ppl., 30,, (006), Gobachuk, V.I., Gobachuk M.L., Bouday Value Pobles fo Opeao Diffeeial Euaios, Kluwe cadeic Publishes, Dodech, Hue, J.K., Nachegaele, B., pplied alysis, Uivesiy of Califoia, Davis,
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
Relations on the Apostol Type (p, q)-frobenius-euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems
Tish Joal of Aalysis ad Nmbe Theoy 27 Vol 5 No 4 26-3 Available olie a hp://pbssciepbcom/ja/5/4/2 Sciece ad Edcaio Pblishig DOI:269/ja-5-4-2 Relaios o he Aposol Type (p -Fobeis-Ele Polyomials ad Geealizaios
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
The Central Limit Theorems for Sums of Powers of Function of Independent Random Variables
ScieceAsia 8 () : 55-6 The Ceal Limi Theoems fo Sums of Poes of Fucio of Idepede Radom Vaiables K Laipapo a ad K Neammaee b a Depame of Mahemaics Walailak Uivesiy Nakho Si Thammaa 86 Thailad b Depame of
Research Article On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series
Hidawi Publishig Copoaio Joual of Fucio Spaces Volue 5, Aicle ID 475, 9 pages hp://dx.doi.og/.55/5/475 Reseach Aicle O Poiwise Appoxiaio of Cojugae Fucios by Soe Hup Maix Meas of Cojugae Fouie Seies W.
GENERALIZED FRACTIONAL INTEGRAL OPERATORS AND THEIR MODIFIED VERSIONS
GENERALIZED FRACTIONAL INTEGRAL OPERATORS AND THEIR MODIFIED VERSIONS HENDRA GUNAWAN Absac. Associaed o a fucio ρ :(, ) (, ), le T ρ be he opeao defied o a suiable fucio space by T ρ f(x) := f(y) dy, R
Available online at J. Math. Comput. Sci. 2 (2012), No. 4, ISSN:
Available olie a h://scik.og J. Mah. Comu. Sci. 2 (22), No. 4, 83-835 ISSN: 927-537 UNBIASED ESTIMATION IN BURR DISTRIBUTION YASHBIR SINGH * Deame of Saisics, School of Mahemaics, Saisics ad Comuaioal
EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar
Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded
New Results on Oscillation of even Order Neutral Differential Equations with Deviating Arguments
Advace i Pue Maheaic 9-53 doi: 36/ap3 Pubihed Oie May (hp://wwwscirpog/oua/ap) New Reu o Ociaio of eve Ode Neua Diffeeia Equaio wih Deviaig Ague Abac Liahog Li Fawei Meg Schoo of Maheaica Sye Sciece aiha
Supplementary Information
Supplemeay Ifomaio No-ivasive, asie deemiaio of he coe empeaue of a hea-geeaig solid body Dea Ahoy, Daipaya Saka, Aku Jai * Mechaical ad Aeospace Egieeig Depame Uivesiy of Texas a Aligo, Aligo, TX, USA.
S, we call the base curve and the director curve. The straight lines
Developable Ruled Sufaces wih Daboux Fame i iowsi -Space Sezai KIZILTUĞ, Ali ÇAKAK ahemaics Depame, Faculy of As ad Sciece, Ezica Uivesiy, Ezica, Tuey ahemaics Depame, Faculy of Sciece, Aau Uivesiy, Ezuum,
ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS. ( ρ( x y ) T ρ f(x) := f(y) R x y n dy, R x y n ρ( y )(1 χ )
Scieiae Mahemaicae Japoicae Olie, Vol., 24), 37 38 37 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS ERIDANI, HENDRA GUNAWAN 2 AND EIICHI NAKAI 3 Received Augus 29, 23; evised Apil 7, 24 Absac. We pove he
Outline. Review Homework Problem. Review Homework Problem II. Review Dimensionless Problem. Review Convection Problem
adial diffsio eqaio Febay 4 9 Diffsio Eqaios i ylidical oodiaes ay aeo Mechaical Egieeig 5B Seia i Egieeig Aalysis Febay 4, 9 Olie eview las class Gadie ad covecio boday codiio Diffsio eqaio i adial coodiaes
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
The Non-Truncated Bulk Arrival Queue M x /M/1 with Reneging, Balking, State-Dependent and an Additional Server for Longer Queues
Alied Maheaical Sciece Vol. 8 o. 5 747-75 The No-Tucaed Bul Aival Queue M x /M/ wih Reei Bali Sae-Deede ad a Addiioal Seve fo Loe Queue A. A. EL Shebiy aculy of Sciece Meofia Uiveiy Ey elhebiy@yahoo.co
On a Z-Transformation Approach to a Continuous-Time Markov Process with Nonfixed Transition Rates
Ge. Mah. Noes, Vol. 24, No. 2, Ocobe 24, pp. 85-96 ISSN 229-784; Copyigh ICSRS Publicaio, 24 www.i-css.og Available fee olie a hp://www.gema.i O a Z-Tasfomaio Appoach o a Coiuous-Time Maov Pocess wih Nofixed
SPECTRUM OF THE DIRECT SUM OF OPERATORS
Electroic Joural of Differetial Equatios, Vol. 202 (202), No. 20, pp. 8. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.ut.edu ftp ejde.math.txstate.edu SPECTRUM OF THE DIRECT SUM
Comparison between Fourier and Corrected Fourier Series Methods
Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1
A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY
U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 206 ISSN 223-7027 A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY İbrahim Çaak I his paper we obai a Tauberia codiio i erms of he weighed classical
Some Embedding Theorems and Properties of Riesz Potentials
meica Joual o ahemaics ad Saisics 3 3(6) 445-453 DOI 593/jajms336 Some Embeddig Theoems ad Poeies o iesz Poeials ahim zaev * Fuad N liyev 3 Isiue o ahemaics ad echaics o Naioal cademy o Scieces o zebaija;
FIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE
Mohia & Samaa, Vol. 1, No. II, December, 016, pp 34-49. ORIGINAL RESEARCH ARTICLE OPEN ACCESS FIED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE 1 Mohia S. *, Samaa T. K. 1 Deparme of Mahemaics, Sudhir Memorial
Energy Density / Energy Flux / Total Energy in 1D. Key Mathematics: density, flux, and the continuity equation.
ecure Phys 375 Eergy Desiy / Eergy Flu / oal Eergy i D Overview ad Moivaio: Fro your sudy of waves i iroducory physics you should be aware ha waves ca raspor eergy fro oe place o aoher cosider he geeraio
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,
The Nehari Manifold for a Class of Elliptic Equations of P-laplacian Type. S. Khademloo and H. Mohammadnia. afrouzi
Wold Alied cieces Joal (8): 898-95 IN 88-495 IDOI Pblicaios = h x g x x = x N i W whee is a eal aamee is a boded domai wih smooh boday i R N 3 ad< < INTRODUCTION Whee s ha is s = I his ae we ove he exisece
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
1 Notes on Little s Law (l = λw)
Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i
Numerical Solution of Sine-Gordon Equation by Reduced Differential Transform Method
Poceedigs of he Wold Cogess o Egieeig Vol I WCE, July 6-8,, Lodo, U.K. Nueical Soluio of Sie-Godo Equaio by Reduced Diffeeial Tasfo Mehod Yıldıay Kesi, İbahi Çağla ad Ayşe Beül Koç Absac Reduced diffeeial
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
Existence and Smoothness of Solution of Navier-Stokes Equation on R 3
Ieaioal Joual of Mode Noliea Theoy ad Applicaio, 5, 4, 7-6 Published Olie Jue 5 i SciRes. hp://www.scip.og/joual/ijma hp://dx.doi.og/.436/ijma.5.48 Exisece ad Smoohess of Soluio of Navie-Sokes Equaio o
Degree of Approximation of Fourier Series
Ieaioal Mahemaical Foum Vol. 9 4 o. 9 49-47 HIARI Ld www.m-hiai.com h://d.doi.og/.988/im.4.49 Degee o Aoimaio o Fouie Seies by N E Meas B. P. Padhy U.. Misa Maheda Misa 3 ad Saosh uma Naya 4 Deame o Mahemaics
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
In this section we will study periodic signals in terms of their frequency f t is said to be periodic if (4.1)
Fourier Series Iroducio I his secio we will sudy periodic sigals i ers o heir requecy is said o be periodic i coe Reid ha a sigal ( ) ( ) ( ) () or every, where is a uber Fro his deiiio i ollows ha ( )
Some Properties of Semi-E-Convex Function and Semi-E-Convex Programming*
The Eighh Ieraioal Symposium o Operaios esearch ad Is Applicaios (ISOA 9) Zhagjiajie Chia Sepember 2 22 29 Copyrigh 29 OSC & APOC pp 33 39 Some Properies of Semi-E-Covex Fucio ad Semi-E-Covex Programmig*
Comparing Different Estimators for Parameters of Kumaraswamy Distribution
Compaig Diffee Esimaos fo Paamees of Kumaaswamy Disibuio ا.م.د نذير عباس ابراهيم الشمري جامعة النهرين/بغداد-العراق أ.م.د نشات جاسم محمد الجامعة التقنية الوسطى/بغداد- العراق Absac: This pape deals wih compaig
CHARACTERIZATIONS OF THE NON-UNIFORM IN TIME ISS PROPERTY AND APPLICATIONS
CHARACTERIZATIONS OF THE NON-UNIFORM IN TIME ISS PROPERTY AND APPLICATIONS I. Karafyllis ad J. Tsiias Depare of Maheaics, Naioal Techical Uiversiy of Ahes, Zografou Capus 578, Ahes, Greece Eail: jsi@ceral.ua.gr.
UNSTEADY HELICAL FLOWS OF A MAXWELL FLUID
PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Seies A, OF THE ROMANIAN ACADEMY Volue 5, Nube /4,. - UNSTEADY HELICAL FLOWS OF A MAXWELL FLUID Cosai FETECAU, Coia FETECAU Techical Uivesiy of Iasi,
The shortest path between two truths in the real domain passes through the complex domain. J. Hadamard
Complex Analysis R.G. Halbud R.Halbud@ucl.ac.uk Depamen of Mahemaics Univesiy College London 202 The shoes pah beween wo uhs in he eal domain passes hough he complex domain. J. Hadamad Chape The fis fundamenal
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
Generalized Fibonacci-Type Sequence and its Properties
Geelized Fibocci-Type Sequece d is Popeies Ompsh Sihwl shw Vys Devshi Tuoil Keshv Kuj Mdsu (MP Idi Resech Schol Fculy of Sciece Pcific Acdemy of Highe Educio d Resech Uivesiy Udipu (Rj Absc: The Fibocci
THE 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
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
Mapping 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
). So the estimators mainly considered here are linear
6 Ioic Ecooică (4/7 Moe Geel Cedibiliy Models Vigii ATANASIU Dee o Mheics Acdey o Ecooic Sudies e-il: vigii_siu@yhooco This couicio gives soe exesios o he oigil Bühl odel The e is devoed o sei-lie cedibiliy
BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics
Biod Prasad Dhaal / BIBCHANA 9 (3 5-58 : BMHSS,.5 (Olie Publicaio: Nov., BIBCHANA A Mulidisciliary Joural of Sciece, Techology ad Mahemaics ISSN 9-76 (olie Joural homeage: h://ejol.ifo/idex.h/bibchana
Calculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.
Limi of a fucio.. Oe-Sided..... Ifiie limis Verical Asympoes... Calculaig Usig he Limi Laws.5 The Squeeze Theorem.6 The Precise Defiiio of a Limi......7 Coiuiy.8 Iermediae Value Theorem..9 Refereces..
N! AND THE GAMMA FUNCTION
N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio
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,
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
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
ECSE Partial fraction expansion (m<n) 3 types of poles Simple Real poles Real Equal poles
ECSE- Lecue. Paial facio expasio (m
( ) 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
Extremal graph theory II: K t and K t,t
Exremal graph heory II: K ad K, Lecure Graph Theory 06 EPFL Frak de Zeeuw I his lecure, we geeralize he wo mai heorems from he las lecure, from riagles K 3 o complee graphs K, ad from squares K, o complee
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
MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES. Boundary Value Problem for the Higher Order Equation with Fractional Derivative
Malaysia Joural of Maheaical Scieces 7(): 3-7 (3) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural hoepage: hp://eispe.up.edu.y/joural Boudary Value Proble for he Higher Order Equaio wih Fracioal Derivaive
S n. = n. Sum of first n terms of an A. P is
PROGREION I his secio we discuss hree impora series amely ) Arihmeic Progressio (A.P), ) Geomeric Progressio (G.P), ad 3) Harmoic Progressio (H.P) Which are very widely used i biological scieces ad humaiies.
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
Transistor configurations: There are three main ways to place a FET/BJT in an architecture:
F3 Mo 0. Amplifie Achiecues Whe a asiso is used i a amplifie, oscillao, file, seso, ec. i will also be a eed fo passive elemes like esisos, capacios ad coils o povide biasig so ha he asiso has he coec
An interesting result about subset sums. Nitu Kitchloo. Lior Pachter. November 27, Abstract
A ieresig resul abou subse sums Niu Kichloo Lior Pacher November 27, 1993 Absrac We cosider he problem of deermiig he umber of subses B f1; 2; : : :; g such ha P b2b b k mod, where k is a residue class
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
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
ES 330 Electronics II Homework 03 (Fall 2017 Due Wednesday, September 20, 2017)
Pae1 Nae Soluios ES 330 Elecroics II Hoework 03 (Fall 017 ue Wedesday, Sepeber 0, 017 Proble 1 You are ive a NMOS aplifier wih drai load resisor R = 0 k. The volae (R appeari across resisor R = 1.5 vols
The Solutions of Initial Value Problems for Nonlinear Fourth-Order Impulsive Integro-Differential Equations in Banach Spaces
WSEAS TRANSACTIONS o MATHEMATICS Zhag Lglg Y Jgy Lu Juguo The Soluos of Ial Value Pobles fo Nolea Fouh-Ode Ipulsve Iego-Dffeeal Equaos Baach Spaces Zhag Lglg Y Jgy Lu Juguo Depae of aheacs of Ta Yua Uvesy
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
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
Cameras and World Geometry
Caeas ad Wold Geoe How all is his woa? How high is he caea? Wha is he caea oaio w. wold? Which ball is close? Jaes Has Thigs o eebe Has Pihole caea odel ad caea (pojecio) ai Hoogeeous coodiaes allow pojecio
Technical 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
E will be denoted by n
JASEM ISSN 9-8362 All rigs reserved Full-ex Available Olie a p:// wwwbiolieorgbr/ja J Appl Sci Eviro Mg 25 Vol 9 3) 3-36 Corollabiliy ad Null Corollabiliy of Liear Syses * DAVIES, I; 2 JACKREECE, P Depare
Multivector 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
On imploding cylindrical and spherical shock waves in a perfect gas
J. Fluid Mech. (2006), vol. 560, pp. 103 122. c 2006 Cambidge Uivesiy Pess doi:10.1017/s0022112006000590 Pied i he Uied Kigdom 103 O implodig cylidical ad spheical shock waves i a pefec gas By N. F. PONCHAUT,
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.
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
BINOMIAL THEOREM OBJECTIVE PROBLEMS in the expansion of ( 3 +kx ) are equal. Then k =
wwwskshieduciocom BINOMIAL HEOREM OBJEIVE PROBLEMS he coefficies of, i e esio of k e equl he k /7 If e coefficie of, d ems i e i AP, e e vlue of is he coefficies i e,, 7 ems i e esio of e i AP he 7 7 em
STK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
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
On Interval Valued Generalized Difference Classes Defined by Orlicz Function
Tuih oua of Aayi ad Numbe Theoy, 03, Vo., No., 48-53 Avaiabe oie a hp://pub.ciepub.com/ja///0 Sciece ad ducaio Pubihig DOI:0.69/ja---0 O Ieva Vaued Geeaized Diffeece Cae Defied by Oicz Fucio Ayha i, Bipa
Approximately Quasi Inner Generalized Dynamics on Modules. { } t t R
Joural of Scieces, Islamic epublic of Ira 23(3): 245-25 (22) Uiversiy of Tehra, ISSN 6-4 hp://jscieces.u.ac.ir Approximaely Quasi Ier Geeralized Dyamics o Modules M. Mosadeq, M. Hassai, ad A. Nikam Deparme
arxiv: v4 [math.pr] 20 Jul 2016
![arxiv: v4 [math.pr] 20 Jul 2016](/thumbs/94/121800553.jpg "arxiv: v4 [math.pr] 20 Jul 2016")
Submied o he Aals of Applied Pobabiliy ε-strong SIMULATION FOR MULTIDIMENSIONAL STOCHASTIC DIFFERENTIAL EQUATIONS VIA ROUGH PATH ANALYSIS axiv:1403.5722v4 [mah.pr] 20 Jul 2016 By Jose Blache, Xiyu Che
The Connection between the Basel Problem and a Special Integral
Applied Mahemaics 4 5 57-584 Published Olie Sepember 4 i SciRes hp://wwwscirporg/joural/am hp://ddoiorg/436/am45646 The Coecio bewee he Basel Problem ad a Special Iegral Haifeg Xu Jiuru Zhou School of
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
Consider the time-varying system, (14.1)
Leue 4 // Oulie Moivaio Equivale Defiiios fo Lyapuov Sabiliy Uifomly Sabiliy ad Uifomly Asympoial Sabiliy 4 Covese Lyapuov Theoem 5 Ivaiae- lie Theoem 6 Summay Moivaio Taig poblem i ool, Suppose ha x (
Notes 03 largely plagiarized by %khc
1 1 Discree-Time Covoluio Noes 03 largely plagiarized by %khc Le s begi our discussio of covoluio i discree-ime, sice life is somewha easier i ha domai. We sar wih a sigal x[] ha will be he ipu io our
BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
ECE-314 Fall 2012 Review Questions
ECE-34 Fall 0 Review Quesios. A liear ime-ivaria sysem has he ipu-oupu characerisics show i he firs row of he diagram below. Deermie he oupu for he ipu show o he secod row of he diagram. Jusify your aswer.
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,
MATH 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
BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
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
BE.430 Tutorial: Linear Operator Theory and Eigenfunction Expansion
BE.43 Tuorial: Liear Operaor Theory ad Eigefucio Expasio (adaped fro Douglas Lauffeburger) 9//4 Moivaig proble I class, we ecouered parial differeial equaios describig rasie syses wih cheical diffusio.
One of the common descriptions of curvilinear motion uses path variables, which are measurements made along the tangent t and normal n to the path of
Oe of he commo descipios of cuilie moio uses ph ibles, which e mesuemes mde log he ge d oml o he ph of he picles. d e wo ohogol xes cosideed sepely fo eey is of moio. These coodies poide ul descipio fo
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
CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE
Fameal Joal of Mahemaic a Mahemaical Sciece Vol. 7 Ie 07 Page 5- Thi pape i aailable olie a hp://.fi.com/ Pblihe olie Jaa 0 07 CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE Caolo
TAKA KUSANO. laculty of Science Hrosh tlnlersty 1982) (n-l) + + Pn(t)x 0, (n-l) + + Pn(t)Y f(t,y), XR R are continuous functions.
Iera. J. Mah. & Mah. Si. Vol. 6 No. 3 (1983) 559-566 559 ASYMPTOTIC RELATIOHIPS BETWEEN TWO HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS TAKA KUSANO laculy of Sciece Hrosh llersy 1982) ABSTRACT. Some asympoic
Ideal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory
Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable
A 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
INF 5460 Electronic noise Estimates and countermeasures. Lecture 13 (Mot 10) Amplifier Architectures
NF 5460 lecoic oise simaes ad couemeasues Lecue 3 (Mo 0) Amplifie Achiecues Whe a asiso is used i a amplifie, oscillao, file, seso, ec. i will also be a eed fo passive elemes like esisos, capacios ad coils
Highly connected coloured subgraphs via the Regularity Lemma
Highly coeced coloued subgaphs via he Regulaiy Lemma Hey Liu 1 Depame of Mahemaics, Uivesiy College Lodo, Gowe See, Lodo WC1E 6BT, Uied Kigdom Yuy Peso 2 Isiu fü Ifomaik, Humbold-Uivesiä zu Beli, Ue de
Lecture 15 First Properties of the Brownian Motion
Lecure 15: Firs Properies 1 of 8 Course: Theory of Probabiliy II Term: Sprig 2015 Isrucor: Gorda Zikovic Lecure 15 Firs Properies of he Browia Moio This lecure deals wih some of he more immediae properies
Recurrence Relations for the Product, Ratio and Single Moments of Order Statistics from Truncated Inverse Weibull (IW) Distribution
Recuece Relatios fo the Poduct, Ratio ad Sigle Moets of Ode Statistics fo Tucated Ivese Weiull (IW) Distiutio ISSN 684 8403 Joual of Statistics Vol: 3, No. (2006) Recuece Relatios fo the Poduct, Ratio
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
Solution. 1 Solutions of Homework 6. Sangchul Lee. April 28, Problem 1.1 [Dur10, Exercise ]
![Solution. 1 Solutions of Homework 6. Sangchul Lee. April 28, Problem 1.1 [Dur10, Exercise ]](/thumbs/90/101616821.jpg "Solution. 1 Solutions of Homework 6. Sangchul Lee. April 28, Problem 1.1 [Dur10, Exercise ]")
Soluio Sagchul Lee April 28, 28 Soluios of Homework 6 Problem. [Dur, Exercise 2.3.2] Le A be a sequece of idepede eves wih PA < for all. Show ha P A = implies PA i.o. =. Proof. Noice ha = P A c = P A c