Research Article On the Upper Bounds of Eigenvalues for a Class of Systems of Ordinary Differential Equations with Higher Order
|
|
- Bernadette Quinn
- 5 years ago
- Views:
Transcription
1 Hndw Publshng Corporton Interntonl Journl of Dfferentl Equtons Volume 0, Artcle ID 7703, pges do:055/0/7703 Reserch Artcle On the Upper Bounds of Egenvlues for Clss of Systems of Ordnry Dfferentl Equtons wth Hgher Order Go J, L-N Hung, nd We Lu College of Scence, Unversty of Shngh for Scence nd Technology, Shngh 00093, Chn Correspondence should be ddressed to Go J, Receved 4 My 0; Revsed 6 July 0; Accepted 9 July 0 Acdemc Edtor: Bshr Ahmd Copyrght q 0 Go J et l Ths s n open ccess rtcle dstrbuted under the Cretve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, nd reproducton n ny medum, provded the orgnl work s properly cted The estmte of the upper bounds of egenvlues for clss of systems of ordnry dfferentl equtons wth hgher order s consdered by usng the clculus theory Severl results bout the upper bound nequltes of the n th egenvlue re obtned by the frst n egenvlues The estmte coeffcents do not hve ny relton to the geometrc mesure of the domn Ths knd of problem s nterestng nd sgnfcnt both n theory of systems of dfferentl equtons nd n pplctons to mechncs nd physcs Introducton In mny physcl settngs, such s the vbrtons of the generl homogeneous or nonhomogeneous strng, rod nd plte cn yeld the Sturm-Louvlle egenvlue problems or other egenvlue problems However, t s not esy to get the ccurte vlues by the nlytc method Sometmes, t s necessry to consder the estmtons of the egenvlues And snce 960s, the problems of the egenvlue estmtes hd become one of the hotspots of the dfferentl equtons There re lots of chevements bout the upper bounds of rbtrry egenvlues for the dfferentl equtons nd unformly ellptc opertors wth hgher orders 9 However, there re few chevements ssocted wth the estmtes of the egenvlues for systems of dfferentl equtons wth hgher order In the followng, we wll obtn some nequltes concernng the egenvlue λ n n terms of λ,λ,,λ n n the systems of ordnry dfferentl equtons wth hgher order In fct, the egenvlue problems hve gret strong prctcl bckgrounds nd mportnt theoretcl vlues 0,
2 Interntonl Journl of Dfferentl Equtons Let, b R be bounded domn nd t be n nteger The followng egenvlue problems re studed: t D t D t y D t y n D t y n λs x y, t D t D t y D t y n D t y n λs x y, t D t n D t y n D t y nn D t y n λs x yn, D k y D k y b 0,,,n, k 0,,,,t, where D d/, D k condtons: d k / k, j x, j,,,n nd s x stsfes the followng j x C t, b, j x j x,, j,,,n; for the rbtrry ξ ξ,ξ,,ξ n R n, we hve μ ξ j x ξ ξ j μ ξ, x, b,,j where μ μ > 0, μ,μ re both constnts; 3 s x C, b, nd there re constnts ν ν, such tht 0 <ν s x ν Accordng to the theores of the dfferentl equtons,, the egenvlues of re ll postve rel numbers, nd they re dscrete We chnge to the form of mtrx Let y T y y y n D t y x x n x D t y x x n x, D t y T, A x 3 D t y n n x n x nn x By vrtue of j x j x, therefore A T x A x, cn be chnged nto the followng form: t D t A x D t y T λs x y T, 4 y k y k b 0, k 0,,,,t 5 Obvously, 4-5 s equvlent to
3 Interntonl Journl of Dfferentl Equtons 3 Suppose tht 0 < λ λ λ n re egenvlues of 4-5, y, y,,y n, re the correspondng egenfunctons nd stsfy the followng weghted orthogonl condtons:, j, s x y y T j δ j 0, / j,, j,, 6 Multplyng y n sdes of 4, byusng 5 nd ntegrton by prts, we hve λ D t y A x D t y T,,, 7 From, we hve b D t y D t y D t y T λ,,, 8 μ For fxed n,let Φ xy b j y j,,,,n, 9 j where b j xs x y y T j Obvously, b j b j, nd Φ re weghted orthogonl to y, y,,y n Furthermore, Φ Φ b 0,, j,,,n We cn use the well-known Rylegh theorem, to obtn t Φ D t A x D t Φ T λ n 0 s x Φ It s esy to see tht t D t A x D t Φ T t td t A x D t y T t td t A x D t y T t xd t A x D t y T t n b j D t A x D t y T j t td t A x D t y T j t td t A x D t y T λ xs x y T s x λ j b j y T j j
4 4 Interntonl Journl of Dfferentl Equtons We hve Φ t D t A x D t Φ T λ xs x Φ y T t t t t Φ D t A x D t y T s x Φ λ j b j y T j j Φ D t A x D t y T In ddton, usng the fct tht Φ re weghted orthogonl to y, y,,y n nd s x Φ xs x Φ y T, 3 we know tht the lst term of s equl to zero Thus, we hve Φ t D t A x D t Φ T λ s x Φ t t t t Φ D t A x D t y T Φ D t A x D t y T 4 Set I t t J t t Φ D t A x D t y T Φ D t A x D t y T,, I J I, J 5 From 4, we hve Φ t D t A x D t Φ T λ s x Φ I J 6 By usng 0 nd 6, one cn gve λ n s x Φ λ s x Φ I J 7
5 Interntonl Journl of Dfferentl Equtons 5 Substtutng λ n for λ n 7, weget λ n λ n s x Φ I J 8 In order to get the estmtons of the egenvlues, we only need to show the estmtes bout I,J, nd n b s x Φ Lemms Lemm Suppose tht the egenfunctons y of 4-5 correspond to the egenvlues λ Then one hs Dp y ν / p Dp y p/ p, p,,,t ; Dy ν /t λ /μ /t Proof By nducton If p, usng ntegrton by prts nd the Schwrz nequlty, we hve b b Dy Dy Dy Dy T y D y T / / y D y T b / ν / D y T Therefore, when p, s true If for p k, s true, tht s, D k b k/ k / k y ν D k y For p k, usng ntegrton by prts, the Schwrz nequlty nd the result when p k, one cn gve b D k b y D k y D k y D k y T / D k b y D k y T / ν / k D k b y / k/ k D k y 3
6 6 Interntonl Journl of Dfferentl Equtons By further clcultng, one cn gve D k b k / k / k y ν D k y 4 Therefore, when p k, s true Usng nd the nductve method, we hve D p y ν / p p/ p D p y ν / p p/ p D p y 5 ν p/t p/t D t y From 8 nd 5, weget D p y ν p/t D t y p/t ν p/t λ μ p/t, p,,,t 6 Tkng p, we hve /t Dy ν /t λ 7 μ So Lemm s true Lemm Let λ,λ,,λ n be the egenvlues of 4-5 Then one hs I J t t μ /t ν /t 8 μ λ /t
7 Interntonl Journl of Dfferentl Equtons 7 Proof Snce I t t t t Φ D t A x D t y T xy b j y j D t A x D t y T j t t t t xy D t A x D t y T j b j y j D t A x D t y T 9 t D t y A x D t y T t xd t y A x D t y T t J t t j b j D t y j A x D t y T Φ D t A x D t y T, t t D t y A x D t y T t xd t y A x D t y T t j b j D t y j A x D t y T, 0 we hve I J I J t t td t y A x D t y T t D t y A x D t y T,j b j D t y j A x D t y T D t y j A x D t y T By j x j x, the lst term of s zero Then we cn get I J t td t y A x D t y T t D t y A x D t y T
8 8 Interntonl Journl of Dfferentl Equtons Usng, Lemm, nd 6, we hve D t y A x D t y T μ D t y μ ν /t λ μ /t 3 Usng, the Schwrz nequlty, Lemm, nd 6, one cn gve b D t y A x D t y T μ D t y / /t μ ν /t λ μ D t y T / 4 Therefore, we obtn I J t t μ /t ν /t μ λ /t 5 Lemm 3 If Φ nd λ,,,n s bove, then one hs Proof By the defnton of Φ, one hs s x Φ μ/t ν /t n λ /t 4ν 6 Φ Dy T xy Dy T,j b j y j Dy T 7 Usng b j b j nd y jdy T y Dy T j, t s esy to see tht the lst term of 7 s zero Then we hve Φ Dy T xy Dy T 8 Usng ntegrton by prts, one cn gve xy Dy T y xy Dy T xy Dy T, 9 y 0
9 Interntonl Journl of Dfferentl Equtons 9 By /ν y /ν, we hve xy Dy T y ν From 8 nd, we cn get Φ Dy T n ν Usng the Schwrz nequlty, Lemm, nd 3, we hve n 4ν s x Φ s x Φ ν /t μ /t Dy s x λ /t 3 By further clcultng, we cn esly get Lemm 3 3 Mn Results Theorem 3 If λ,,,n re the egenvlues of 4-5,then λ n λ n 4t t μ ν μ ν n λ n 4t t μ ν λ μ ν n λ /t λ /t ; 3 3 Proof From 8, we cn get λ n λ n I J s x Φ 33 Usng Lemms nd 3, we cn esly get 3 In 3, Replcng λ wth λ n, by further clcultng, we cn get 3 Theorem 3 For n, one hs λ /t μ ν n λ n λ 4t t μ ν λ /t 34
10 0 Interntonl Journl of Dfferentl Equtons Proof Choosng the prmeter σ>λ n,usng 7, one cn gve λ n s x Φ σ s x Φ λ σ s x Φ I J 35 By nd the Young nequlty, we obtn n ν δ σ λ s x Φ δ σ λ Dy s x, 36 where δ>0 s constnt to be determned Set V s x Φ, T σ λ s x Φ 37 Usng Lemm, 35,nd 36, we cn get the followng results, respectvely, λ n σ V T I J, n δt ν δ μ /t ν /t 38 σ λ λ /t 39 In order to get the mnmum of the rght of 39, we cn tke / δ T / μ /t ν /t σ λ λ /t 30 By 39,nd 30, we cn esly get T μ/t ν /t n 4ν λ /t σ λ 3 Usng Lemm, 38,nd 3, we hve λ n σ V μ/t ν /t n 4ν λ /t σ λ t t μ /t ν /t μ λ /t, 3 tht s, λ n σ V t t μ /t ν /t μ λ /t μ/t ν /t n 4ν λ /t σ λ 33
11 Interntonl Journl of Dfferentl Equtons Let the rght term of 33 be f σ It s esy to see tht lm f σ, σ lm f σ t t μ /t σ λ ν /t n μ λ /t > 0 34 Hence, there s σ 0 λ n,, such tht λ /t μ ν n σ 0 λ 4t t μ ν λ /t 35 On the other hnd, lettng g σ λ /t, σ λ 36 we hve g λ /t σ σ λ 0 37 It mples tht g σ s the monotone decresng nd contnuous functon, nd ts vlue rnge s 0, Therefore, there exts exctly one σ 0 to stsfy 35 From 33, we know tht σ 0 >λ n Replcng σ 0 wth λ n n 35, we cn get the result References L E Pyne, G Poly, nd H F Wenberger, Sur le quotent de deux frquences propres conscutves, Comptes Rendus de l AcdémedesScencesdePrs, vol 4, pp 97 99, 955 L E Pyne, G Poly, nd H F Wenberger, On the rto of consecutve egenvlues, Journl of Mthemtcs nd Physcs, vol 35, pp 89 98, G N Hle nd M H Protter, Inequltes for egenvlues of the Lplcn, Indn Unversty Mthemtcs Journl, vol 9, no 4, pp , G N Hle nd R Z Yeb, Inequltes for egenvlues of the bhrmonc opertor, Pcfc Journl of Mthemtcs, vol, no, pp 5 33, S M Hook, Domn ndependent upper bounds for egenvlues of ellptc opertors, Trnsctons of the Amercn Mthemtcl Socety, vol 38, no, pp 65 64, Z C Chen nd C L Qn, On the dfference of consecutve egenvlues of unformly ellptc opertors of hgher orders, Chnese Annls of Mthemtcs B, vol 4, no 4, pp , Z C Chen nd C L Qn, On the upper bound of egenvlues for ellptc equtons wth hgher orders, Journl of Mthemtcl Anlyss nd Applctons, vol 86, no 3, pp 8 834, G J, X P Yng, nd C L Qn, On the upper bound of second egenvlues for unformly ellptc opertors of ny orders, Act Mthemtce Applcte Snc, vol 9, no, pp 07 6, G J nd X P Yng, The upper bounds of rbtrry egenvlues for unformly ellptc opertors wth hgher orders, Act Mthemtce Applcte Snc, vol, no 4, pp , M H Protter, Cn one her the shpe of drum? SIAM Revew, vol 9, no, pp 85 97, 987
12 Interntonl Journl of Dfferentl Equtons R Cournt nd D Hlbert, Methods of Mthemtcl Physcs, Interscence Publshers, New York, NY, USA, 989 H Weyl, Rmfctons, old nd new, of the egenvlue problem, Bulletn of the Amercn Mthemtcl Socety, vol 56, pp 5 39, 950
13 Advnces n Opertons Reserch Hndw Publshng Corporton Volume 04 Advnces n Decson Scences Hndw Publshng Corporton Volume 04 Journl of Appled Mthemtcs Algebr Hndw Publshng Corporton Hndw Publshng Corporton Volume 04 Journl of Probblty nd Sttstcs Volume 04 The Scentfc World Journl Hndw Publshng Corporton Hndw Publshng Corporton Volume 04 Interntonl Journl of Dfferentl Equtons Hndw Publshng Corporton Volume 04 Volume 04 Submt your mnuscrpts t Interntonl Journl of Advnces n Combntorcs Hndw Publshng Corporton Mthemtcl Physcs Hndw Publshng Corporton Volume 04 Journl of Complex Anlyss Hndw Publshng Corporton Volume 04 Interntonl Journl of Mthemtcs nd Mthemtcl Scences Mthemtcl Problems n Engneerng Journl of Mthemtcs Hndw Publshng Corporton Volume 04 Hndw Publshng Corporton Volume 04 Volume 04 Hndw Publshng Corporton Volume 04 Dscrete Mthemtcs Journl of Volume 04 Hndw Publshng Corporton Dscrete Dynmcs n Nture nd Socety Journl of Functon Spces Hndw Publshng Corporton Abstrct nd Appled Anlyss Volume 04 Hndw Publshng Corporton Volume 04 Hndw Publshng Corporton Volume 04 Interntonl Journl of Journl of Stochstc Anlyss Optmzton Hndw Publshng Corporton Hndw Publshng Corporton Volume 04 Volume 04
CHOVER-TYPE LAWS OF THE ITERATED LOGARITHM FOR WEIGHTED SUMS OF ρ -MIXING SEQUENCES
CHOVER-TYPE LAWS OF THE ITERATED LOGARITHM FOR WEIGHTED SUMS OF ρ -MIXING SEQUENCES GUANG-HUI CAI Receved 24 September 2004; Revsed 3 My 2005; Accepted 3 My 2005 To derve Bum-Ktz-type result, we estblsh
More informationLOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN FRACTAL HEAT TRANSFER
Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 S3 LOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN
More informationCase Study of Markov Chains Ray-Knight Compactification
Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and
More informationA Family of Multivariate Abel Series Distributions. of Order k
Appled Mthemtcl Scences, Vol. 2, 2008, no. 45, 2239-2246 A Fmly of Multvrte Abel Seres Dstrbutons of Order k Rupk Gupt & Kshore K. Ds 2 Fculty of Scence & Technology, The Icf Unversty, Agrtl, Trpur, Ind
More informationThe Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
More informationStatistics and Probability Letters
Sttstcs nd Probblty Letters 79 (2009) 105 111 Contents lsts vlble t ScenceDrect Sttstcs nd Probblty Letters journl homepge: www.elsever.com/locte/stpro Lmtng behvour of movng verge processes under ϕ-mxng
More informationTwo Coefficients of the Dyson Product
Two Coeffcents of the Dyson Product rxv:07.460v mth.co 7 Nov 007 Lun Lv, Guoce Xn, nd Yue Zhou 3,,3 Center for Combntorcs, LPMC TJKLC Nnk Unversty, Tnjn 30007, P.R. Chn lvlun@cfc.nnk.edu.cn gn@nnk.edu.cn
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationRank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
More information90 S.S. Drgomr nd (t b)du(t) =u()(b ) u(t)dt: If we dd the bove two equltes, we get (.) u()(b ) u(t)dt = p(; t)du(t) where p(; t) := for ll ; t [; b]:
RGMIA Reserch Report Collecton, Vol., No. 1, 1999 http://sc.vu.edu.u/οrgm ON THE OSTROWSKI INTEGRAL INEQUALITY FOR LIPSCHITZIAN MAPPINGS AND APPLICATIONS S.S. Drgomr Abstrct. A generlzton of Ostrowsk's
More informationLecture notes. Fundamental inequalities: techniques and applications
Lecture notes Fundmentl nequltes: technques nd pplctons Mnh Hong Duong Mthemtcs Insttute, Unversty of Wrwck Eml: m.h.duong@wrwck.c.uk Jnury 4, 07 Abstrct Inequltes re ubqutous n Mthemtcs (nd n rel lfe.
More informationGAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
More informationDCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
More informationSharp integral inequalities involving high-order partial derivatives. Journal Of Inequalities And Applications, 2008, v. 2008, article no.
Ttle Sharp ntegral nequaltes nvolvng hgh-order partal dervatves Authors Zhao, CJ; Cheung, WS Ctaton Journal Of Inequaltes And Applcatons, 008, v. 008, artcle no. 5747 Issued Date 008 URL http://hdl.handle.net/07/569
More informationThe Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
More informationKatholieke Universiteit Leuven Department of Computer Science
Updte Rules for Weghted Non-negtve FH*G Fctorzton Peter Peers Phlp Dutré Report CW 440, Aprl 006 Ktholeke Unverstet Leuven Deprtment of Computer Scence Celestjnenln 00A B-3001 Heverlee (Belgum) Updte Rules
More informationSequences of Intuitionistic Fuzzy Soft G-Modules
Interntonl Mthemtcl Forum, Vol 13, 2018, no 12, 537-546 HIKARI Ltd, wwwm-hkrcom https://doorg/1012988/mf201881058 Sequences of Intutonstc Fuzzy Soft G-Modules Velyev Kemle nd Huseynov Afq Bku Stte Unversty,
More informationTHE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR
REVUE D ANALYSE NUMÉRIQUE ET DE THÉORIE DE L APPROXIMATION Tome 32, N o 1, 2003, pp 11 20 THE COMBINED SHEPARD ABEL GONCHAROV UNIVARIATE OPERATOR TEODORA CĂTINAŞ Abstrct We extend the Sheprd opertor by
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the
More informationInternational Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl
More informationINTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
More informationLAPLACE TRANSFORM SOLUTION OF THE PROBLEM OF TIME-FRACTIONAL HEAT CONDUCTION IN A TWO-LAYERED SLAB
Journl of Appled Mthemtcs nd Computtonl Mechncs 5, 4(4), 5-3 www.mcm.pcz.pl p-issn 99-9965 DOI:.75/jmcm.5.4. e-issn 353-588 LAPLACE TRANSFORM SOLUTION OF THE PROBLEM OF TIME-FRACTIONAL HEAT CONDUCTION
More informationNotes on convergence of an algebraic multigrid method
Appled Mthemtcs Letters 20 (2007) 335 340 www.elsever.com/locte/ml Notes on convergence of n lgebrc multgrd method Zhohu Hung,, Peln Sh b Stte Key Lbortory of Spce Wether, Center for Spce Scence nd Appled
More informationResearch Article Relative Smooth Topological Spaces
Advances n Fuzzy Systems Volume 2009, Artcle ID 172917, 5 pages do:10.1155/2009/172917 Research Artcle Relatve Smooth Topologcal Spaces B. Ghazanfar Department of Mathematcs, Faculty of Scence, Lorestan
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationFormulated Algorithm for Computing Dominant Eigenvalue. and the Corresponding Eigenvector
Int. J. Contemp. Mth. Scences Vol. 8 23 no. 9 899-9 HIKARI Ltd www.m-hkr.com http://dx.do.org/.2988/jcms.23.3674 Formulted Algorthm for Computng Domnnt Egenlue nd the Correspondng Egenector Igob Dod Knu
More informationComparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method
Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method
More informationUNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
More informationResearch Article Green s Theorem for Sign Data
Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID 539359, 10 pages do:10.5402/2012/539359 Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of
More information4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
More informationExistence of Two Conjugate Classes of A 5 within S 6. by Use of Character Table of S 6
Internatonal Mathematcal Forum, Vol. 8, 2013, no. 32, 1591-159 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/mf.2013.3359 Exstence of Two Conjugate Classes of A 5 wthn S by Use of Character Table
More informationJean Fernand Nguema LAMETA UFR Sciences Economiques Montpellier. Abstract
Stochstc domnnce on optml portfolo wth one rsk less nd two rsky ssets Jen Fernnd Nguem LAMETA UFR Scences Economques Montpeller Abstrct The pper provdes restrctons on the nvestor's utlty functon whch re
More informationMany-Body Calculations of the Isotope Shift
Mny-Body Clcultons of the Isotope Shft W. R. Johnson Mrch 11, 1 1 Introducton Atomc energy levels re commonly evluted ssumng tht the nucler mss s nfnte. In ths report, we consder correctons to tomc levels
More informationThe Jacobsthal and Jacobsthal-Lucas Numbers via Square Roots of Matrices
Internatonal Mathematcal Forum, Vol 11, 2016, no 11, 513-520 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/mf20166442 The Jacobsthal and Jacobsthal-Lucas Numbers va Square Roots of Matrces Saadet Arslan
More informationApplied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
More informationReproducing Kernel Hilbert Space for. Penalized Regression Multi-Predictors: Case in Longitudinal Data
Interntonl Journl of Mthemtcl Anlyss Vol. 8, 04, no. 40, 95-96 HIKARI Ltd, www.m-hkr.com http://dx.do.org/0.988/jm.04.47 Reproducng Kernel Hlbert Spce for Penlzed Regresson Mult-Predctors: Cse n Longudnl
More informationLecture 36. Finite Element Methods
CE 60: Numercl Methods Lecture 36 Fnte Element Methods Course Coordntor: Dr. Suresh A. Krth, Assocte Professor, Deprtment of Cvl Engneerng, IIT Guwht. In the lst clss, we dscussed on the ppromte methods
More informationRemember: Project Proposals are due April 11.
Bonformtcs ecture Notes Announcements Remember: Project Proposls re due Aprl. Clss 22 Aprl 4, 2002 A. Hdden Mrov Models. Defntons Emple - Consder the emple we tled bout n clss lst tme wth the cons. However,
More informationQuiz: Experimental Physics Lab-I
Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled
More informationThe binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence
Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the
More informationReactor Control Division BARC Mumbai India
A Study of Frctonl Schrödnger Equton-composed v Jumre frctonl dervtve Joydp Bnerjee 1, Uttm Ghosh, Susmt Srkr b nd Shntnu Ds 3 Uttr Bunch Kjl Hr Prmry school, Ful, Nd, West Bengl, Ind eml- joydp1955bnerjee@gml.com
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationResearch Article The Solution of Two-Point Boundary Value Problem of a Class of Duffing-Type Systems with Non-C 1 Perturbation Term
Hndaw Publshng Corporaton Boundary Value Problems Volume 9, Artcle ID 87834, 1 pages do:1.1155/9/87834 Research Artcle The Soluton of Two-Pont Boundary Value Problem of a Class of Duffng-Type Systems wth
More information523 P a g e. is measured through p. should be slower for lesser values of p and faster for greater values of p. If we set p*
R. Smpth Kumr, R. Kruthk, R. Rdhkrshnn / Interntonl Journl of Engneerng Reserch nd Applctons (IJERA) ISSN: 48-96 www.jer.com Vol., Issue 4, July-August 0, pp.5-58 Constructon Of Mxed Smplng Plns Indexed
More informationCOMPLEX NUMBERS INDEX
COMPLEX NUMBERS INDEX. The hstory of the complex numers;. The mgnry unt I ;. The Algerc form;. The Guss plne; 5. The trgonometrc form;. The exponentl form; 7. The pplctons of the complex numers. School
More informationON SIMPSON S INEQUALITY AND APPLICATIONS. 1. Introduction The following inequality is well known in the literature as Simpson s inequality : 2 1 f (4)
ON SIMPSON S INEQUALITY AND APPLICATIONS SS DRAGOMIR, RP AGARWAL, AND P CERONE Abstrct New neultes of Smpson type nd ther pplcton to udrture formule n Numercl Anlyss re gven Introducton The followng neulty
More informationVariable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng
More informationDennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1
Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationψ ij has the eigenvalue
Moller Plesset Perturbton Theory In Moller-Plesset (MP) perturbton theory one tes the unperturbed Hmltonn for n tom or molecule s the sum of the one prtcle Foc opertors H F() where the egenfunctons of
More informationChapter 2 Introduction to Algebra. Dr. Chih-Peng Li ( 李 )
Chpter Introducton to Algebr Dr. Chh-Peng L 李 Outlne Groups Felds Bnry Feld Arthetc Constructon of Glos Feld Bsc Propertes of Glos Feld Coputtons Usng Glos Feld Arthetc Vector Spces Groups 3 Let G be set
More informationReview of linear algebra. Nuno Vasconcelos UCSD
Revew of lner lgebr Nuno Vsconcelos UCSD Vector spces Defnton: vector spce s set H where ddton nd sclr multplcton re defned nd stsf: ) +( + ) (+ )+ 5) λ H 2) + + H 6) 3) H, + 7) λ(λ ) (λλ ) 4) H, - + 8)
More informationFUNDAMENTALS ON ALGEBRA MATRICES AND DETERMINANTS
Dol Bgyoko (0 FUNDAMENTALS ON ALGEBRA MATRICES AND DETERMINANTS Introducton Expressons of the form P(x o + x + x + + n x n re clled polynomls The coeffcents o,, n re ndependent of x nd the exponents 0,,,
More informationLinear and Nonlinear Optimization
Lner nd Nonlner Optmzton Ynyu Ye Deprtment of Mngement Scence nd Engneerng Stnford Unversty Stnford, CA 9430, U.S.A. http://www.stnford.edu/~yyye http://www.stnford.edu/clss/msnde/ Ynyu Ye, Stnford, MS&E
More informationResearch Article An Extension of Stolarsky Means to the Multivariable Case
Internatonal Mathematcs and Mathematcal Scences Volume 009, Artcle ID 43857, 14 pages do:10.1155/009/43857 Research Artcle An Extenson of Stolarsky Means to the Multvarable Case Slavko Smc Mathematcal
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationIntroduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
More informationInvestigation phase in case of Bragg coupling
Journl of Th-Qr Unversty No.3 Vol.4 December/008 Investgton phse n cse of Brgg couplng Hder K. Mouhmd Deprtment of Physcs, College of Scence, Th-Qr, Unv. Mouhmd H. Abdullh Deprtment of Physcs, College
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
More informationPrinciple Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
More informationChemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F 0 E 0 F E Q W
More informationCOMPLEX NUMBER & QUADRATIC EQUATION
MCQ COMPLEX NUMBER & QUADRATIC EQUATION Syllus : Comple numers s ordered prs of rels, Representton of comple numers n the form + nd ther representton n plne, Argnd dgrm, lger of comple numers, modulus
More informationCHI-SQUARE DIVERGENCE AND MINIMIZATION PROBLEM
CHI-SQUARE DIVERGENCE AND MINIMIZATION PROBLEM PRANESH KUMAR AND INDER JEET TANEJA Abstrct The mnmum dcrmnton nformton prncple for the Kullbck-Lebler cross-entropy well known n the lterture In th pper
More informationStudy of Trapezoidal Fuzzy Linear System of Equations S. M. Bargir 1, *, M. S. Bapat 2, J. D. Yadav 3 1
mercn Interntonl Journl of Reserch n cence Technology Engneerng & Mthemtcs vlble onlne t http://wwwsrnet IN (Prnt: 38-349 IN (Onlne: 38-3580 IN (CD-ROM: 38-369 IJRTEM s refereed ndexed peer-revewed multdscplnry
More information6.6 The Marquardt Algorithm
6.6 The Mqudt Algothm lmttons of the gdent nd Tylo expnson methods ecstng the Tylo expnson n tems of ch-sque devtves ecstng the gdent sech nto n tetve mtx fomlsm Mqudt's lgothm utomtclly combnes the gdent
More informationChemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F E F E + Q! 0
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationResearch Article A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations
Hndaw Publshng Corporaton Advances n Dfference Equatons Volume 008, Artcle ID 695495, pages do:0.55/008/695495 Research Artcle A Generalzed Sum-Dfference Inequalty and Applcatons to Partal Dfference Equatons
More informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationJens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for
More informationTHE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS
Dscussones Mathematcae Graph Theory 27 (2007) 401 407 THE CHVÁTAL-ERDŐS CONDITION AND 2-FACTORS WITH A SPECIFIED NUMBER OF COMPONENTS Guantao Chen Department of Mathematcs and Statstcs Georga State Unversty,
More informationFINITE NEUTROSOPHIC COMPLEX NUMBERS. W. B. Vasantha Kandasamy Florentin Smarandache
INITE NEUTROSOPHIC COMPLEX NUMBERS W. B. Vsnth Kndsmy lorentn Smrndche ZIP PUBLISHING Oho 11 Ths book cn be ordered from: Zp Publshng 1313 Chespeke Ave. Columbus, Oho 31, USA Toll ree: (61) 85-71 E-ml:
More informationResearch Article Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem
Mathematcal Problems n Engneerng Volume 2012, Artcle ID 871741, 16 pages do:10.1155/2012/871741 Research Artcle Global Suffcent Optmalty Condtons for a Specal Cubc Mnmzaton Problem Xaome Zhang, 1 Yanjun
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationValuated Binary Tree: A New Approach in Study of Integers
Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach
More informationANALOG CIRCUIT SIMULATION BY STATE VARIABLE METHOD
U.P.B. Sc. Bull., Seres C, Vol. 77, Iss., 25 ISSN 226-5 ANAOG CIRCUIT SIMUATION BY STATE VARIABE METHOD Rodc VOICUESCU, Mh IORDACHE 22 An nlog crcut smulton method, bsed on the stte euton pproch, s presented.
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationVoting Games with Positive Weights and. Dummy Players: Facts and Theory
Appled Mathematcal Scences, Vol 10, 2016, no 53, 2637-2646 HIKARI Ltd, wwwm-hkarcom http://dxdoorg/1012988/ams201667209 Votng Games wth Postve Weghts and Dummy Players: Facts and Theory Zdravko Dmtrov
More informationarxiv:math/ v1 [math.nt] 13 Dec 2005
LARGE SIEVE INEQUALITIES WITH QUADRATIC AMPLITUDES LIANGYI ZHAO rxv:mth/0570v [mth.nt] 3 Dec 005 Abstrct. In ths pper, we develop lrge seve type nequlty wth qudrtc mpltude. We use the double lrge seve
More informationQuantum Codes from Generalized Reed-Solomon Codes and Matrix-Product Codes
1 Quntum Codes from Generlzed Reed-Solomon Codes nd Mtrx-Product Codes To Zhng nd Gennn Ge Abstrct rxv:1508.00978v1 [cs.it] 5 Aug 015 One of the centrl tsks n quntum error-correcton s to construct quntum
More informationSYMMETRY CONCEPT APPLIED IN MOLECULAR SPECTROSCOPY
SYMMETRY CONCEPT APPLED N MOLECULAR SPECTROSCOPY 1 D.O. DOROHO, C.E. HRETCANU, 3 M.C. CRASMAREANU 1 Fculty of Physcs, Al.. Cuz Unversty, ş, Romn Food Engneerng Fculty, Ştefn cel Mre Unversty, Sucev, Romn
More informationInternational Journal of Algebra, Vol. 8, 2014, no. 5, HIKARI Ltd,
Internatonal Journal of Algebra, Vol. 8, 2014, no. 5, 229-238 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ja.2014.4212 On P-Duo odules Inaam ohammed Al Had Department of athematcs College of Educaton
More informationSymmetries and Conservation Laws in Classical Mechanics
Symmetres nd Conservton Lws n Clsscl Mechncs Wllm Andrew Astll September 30, 0 Abstrct Ths pper wll provde detled explorton nd explnton of symmetres n clsscl mechncs nd how these symmetres relte to conservton
More informationSUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION
talan journal of pure appled mathematcs n. 33 2014 (63 70) 63 SUPER PRINCIPAL FIBER BUNDLE WITH SUPER ACTION M.R. Farhangdoost Department of Mathematcs College of Scences Shraz Unversty Shraz, 71457-44776
More informationNumerical Solution of Freholm-Volterra Integral Equations by Using Scaling Function Interpolation Method
Aled Mthetcs 3 4 4-9 htt://ddoorg/436/34a3 Pulshed Onlne Jnury 3 (htt://wwwscrorg/ournl/) uercl Soluton of Frehol-Volterr Integrl Equtons y Usng Sclng Functon Interolton Method Yousef Al-Jrrh En-Bng Ln
More informationOn the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers
Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 DOI 086/s3660-06-0997-0 R E S E A R C H Open Access On the spectral norm of r-crculant matrces wth the Pell and Pell-Lucas numbers Ramazan
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationFitting a Polynomial to Heat Capacity as a Function of Temperature for Ag. Mathematical Background Document
Fttng Polynol to Het Cpcty s Functon of Teperture for Ag. thetcl Bckground Docuent by Theres Jul Zelnsk Deprtent of Chestry, edcl Technology, nd Physcs onouth Unversty West ong Brnch, J 7764-898 tzelns@onouth.edu
More informationFractional Euler-Lagrange Equations of Order ( α, β ) for Lie Algebroids
Studes n Mthemtcl Scences Vol. 1 No. 1 2010 pp. 13-20 www.cscnd.org ISSN 1923-8444 [Prnt] ISSN 1923-8452 [Onlne] www.cscnd.net Frctonl Euler-Lgrnge Equtons of Order ( α β ) for Le Algerods El-Nuls Ahmd
More informationDefinition of Tracking
Trckng Defnton of Trckng Trckng: Generte some conclusons bout the moton of the scene, objects, or the cmer, gven sequence of mges. Knowng ths moton, predct where thngs re gong to project n the net mge,
More informationNumbers Related to Bernoulli-Goss Numbers
ursh Journl of Anlyss n Nuber heory, 4, Vol., No., -8 Avlble onlne t htt://ubs.sceub.co/tnt///4 Scence n Eucton Publshng OI:.69/tnt---4 Nubers Relte to Bernoull-Goss Nubers Mohe Oul ouh Benough * érteent
More informationSUM PROPERTIES FOR THE K-LUCAS NUMBERS WITH ARITHMETIC INDEXES
Avlble ole t http://sc.org J. Mth. Comput. Sc. 4 (04) No. 05-7 ISSN: 97-507 SUM PROPERTIES OR THE K-UCAS NUMBERS WITH ARITHMETIC INDEXES BIJENDRA SINGH POOJA BHADOURIA AND OMPRAKASH SIKHWA * School of
More informationChapter 3 Describing Data Using Numerical Measures
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
More informationApplied Stochastic Processes
STAT455/855 Fall 23 Appled Stochastc Processes Fnal Exam, Bref Solutons 1. (15 marks) (a) (7 marks) The dstrbuton of Y s gven by ( ) ( ) y 2 1 5 P (Y y) for y 2, 3,... The above follows because each of
More informationElectrochemical Thermodynamics. Interfaces and Energy Conversion
CHE465/865, 2006-3, Lecture 6, 18 th Sep., 2006 Electrochemcl Thermodynmcs Interfces nd Energy Converson Where does the energy contrbuton F zϕ dn come from? Frst lw of thermodynmcs (conservton of energy):
More informationDynamic Power Management in a Mobile Multimedia System with Guaranteed Quality-of-Service
Dynmc Power Mngement n Moble Multmed System wth Gurnteed Qulty-of-Servce Qnru Qu, Qng Wu, nd Mssoud Pedrm Dept. of Electrcl Engneerng-Systems Unversty of Southern Clforn Los Angeles CA 90089 Outlne! Introducton
More informationOn Graphs with Same Distance Distribution
Appled Mathematcs, 07, 8, 799-807 http://wwwscrporg/journal/am ISSN Onlne: 5-7393 ISSN Prnt: 5-7385 On Graphs wth Same Dstance Dstrbuton Xulang Qu, Xaofeng Guo,3 Chengy Unversty College, Jme Unversty,
More information