DOCSLIB.ORG

Evaluation and Analyses of Some Finite Element and Finite Difference Procedures for Time-Dependent Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Evaluation and Analyses of Some Finite Element and Finite Difference Procedures for Time-Dependent Problems 'DO NOT RETURN TO LIB R A R / EVALUATION AND ANALYSES OF SOME FINITE ELEMENT AND FINITE DIFFERENCE PROCEDURES FOR TIME-DEPENDENT PROBLEMS by C. S. Desai, J. T. Oden, L. D. Johnson Soils and Pavements Laboratory U. S. Army Engineer Waterways Experiment Station P. O. Box 631, Vicksburg, Miss. 39180 April 1975 Final Report Approved For Public Release; Distribution Unlimited TA Prepared for Office, Chief of Engineers, U. S. Army 7 Washington, D. C. 2 0 3 14 .W34m S-75-7 1975 PROPERTY OF SUREAU OF RECLAMATION BUAEA.y..?,ÌlECLAMAT,0N DENVER LIBRARY -TA 92031039 Unclassified ^ 5 ^ 0 , \ SECURITY CLASSIFICATION OF THIS PAGE (When Data Enter'd) s -7^7 BEFORE COMPLETING FORM 2. GOVT ACCESSION NO. 3. RECIPIENT’S CATALOG NUMBER h-?v 1. R E P O R T N U M B ER Miscellaneous Paper S-75-7 5. TYPE OF REPORT & PERIOD COVERED 4. T IT L E (end Subtitle) Final report EVALUATION AND ANALYSES OF SOME F] [NITE ELEMENT AND FINITE DIFFERENCE PROCEDURES I PO R T I M E - 6. PERFORMING ORG. REPORT NUMBER DEPENDENT PROBLEMS 8. CONTRACT OR GRANT NUMBERfe) 7. AUTHOR^; C. S. Desai J. T. Oden’ L . D . Johnson 10. PROGRAM ELEMENT, PROJECT, TASK 9. PERFORMING ORGANIZATION NAME AND AUU k ESS . AREA & WORK UNIT NUMBERS U. S. Army Engineer Waterways Experiment Station Soils and Pavements Laboratory P. 0. Box 631, Vicksburg, Miss. 39180 12. R E P O R T D A T E 11. CONTROLLING OFFICE NAME AND ADDRESS Office, Chief of Engineers, U . S. Army April 1975 Washington, D, C. 2031^- 13. N U M BER O F PA G ES 107 15. S E C U R IT Y CLASS, (of this report) 14. M O N ITO R IN G AG EN C Y NA M E & ADDRESS(7f different from Controlling Office) Unclassified 15a DECLASSIFICATION/DOWNGRADING SCHEDULE 16. D IS T R IB U T IO N S T A T E M E N T (of thia Report) Approved for public release; distribution unlimited. 17. D IS T R IB U T IO N S T A TE M E N T' (at the abstract entered ,n Block 20, It dltterent Iron, Report) 18. SUPPLEMENTARY NOTES 19. K E Y W O RDS (Continue on reverse side if necessary and identify by block number) Finite element method Finite difference method Time dependence Soil mechanics -------- ----------- ” . , r t ____ _ _ nrrsst } riser, Hfv hv h lo c k num ber) A large number of scnemes using une i u u w --- - o (FD) methods have been proposed. Numerical and physical properties of these schemes, vital for their use as general procedures and significant from the viewpoint of the user, are often not examined adequately. One of the aims f this study was to examine numerical properties of a selected^number of schemes for problems relevant to geotechnical engineering. In this initial phase, the aim has been partially fulfilled by studying the problem of one-dimensional EDITION OF t NOV 65 IS OBSOLETE DD 1473 Unclassified ________ ___ SECURITY CLASSIFICATION OF THIS PAGE < ™ « > uera Entered) Unclassified. SECURITY CLASSIFICATION OF THIS P»GE(TW.«. D.lm Bnt.fdi 20. ABSTRACT (Continued). consolidation and wave propagation. Another aim of the research was to examine the concept of space-time finite elements for some time-dependent problems. A portion of Section I involves quantitative analyses of such factors as con- sta^ i t y , and computational efforts for a number of FD and FE scheme* for the consolidation problem governed by a linear parabolic equation Also examined are the effects of the order of the approximation models and heavier i ^ s o l u t i o n at different locations in space and in time. Mathematical stability criteria are also derived for two time integration schemes with two approximation models for the linear parabolic equation. In Section II, Prof. T. Oden has presented mathematical derivations for error estimates of FE ap­ proximations to the diffusion equation. The numerical characteristics have been examined and error estimates have been obtained by using semigroup theo­ retic, energy, and L? methods. The work presented in this section is father fundamental in nature~and can provide basis for needed investigations towards numerical analysis of FE procedures. Part V of Section I contains formulations of the concept of space-time finite elements for dynamical systems, and for owo equations of fluid flow through porous media. A simple example of the srngle-degree-of-freedom system is solved quantitatively. Since the proiect had to be postponed, no quantitative analyses and definitive conclusions were derived for the concept of space-time elements. The topic of evaluation of different numerical schemes is of vital importance. This is particularly so because a large number of schemes have often been proposed without adequate guidelines for the user to assist hit, in selectinglhe o p t i ™ scheme his specific needs. It is felt that the work initiated in this study should be analysed^and^ev^!uatedf0r <* P ™ « - ! ¡«Prance he - ________Unclassified___________ _ SECURITY CLASSIFICATION OF THIS PAGE(Tf7ien Data Entered) PREFACE The study of the concept of finite elements in the time domain and evaluation of some finite element (FE) and finite difference (FD) schemes was initiated by the Soils and Pavements Laboratory, U. S. Army Engineer Waterways Experiment Station (WES), Vicksburg, Mississippi. This work was sponsored by the Research in Military Engineering and Con­ struction (RMEC) group of the Office, Chief of Engineers, U. S. Army. The investigations described in Section I were performed by Dr. C. S. Desai, Research Group, Soil Mechanics Division, now with the Department of Civil Engineering, Virginia Polytechnic Institute and State University, Blacksburg, Va. Prof. J. T. Oden, consultant to the project, University of Texas, Austin, Tex., contributed Section II, and provided valuable advice and suggestions. Dr. L. D. Johnson, Research Group, Soil Mechanics Division, assisted in computational work related to the analyses of one-dimensional consolidation problems. Prof. R. L. Lytton, Texas A&M University, collaborated on the derivation pre­ sented in Part IV. Section I of this report was prepared by Dr. C. S. Desai and Section II was prepared by Prof. J. T. Oden. The project was postponed after completion of the initial phase described herein. During the investigation, Mr. J. P. Sale was the Chief, Soils and Pavements Laboratory, and Mr. C. L. McAnear was Chief, Soil Mechan­ ics Divison. Directors of the WES during the investigation were BG E. D. Peixotto, CE, and COL G. H. Hilt, CE. Mr. F. R. Brown was Technical Director. iii CONTENTS Page iii PREFACE ............................................................ 1 SECTION I .............................................................................................................................................. • • PART I: INTRODUCTION............................... .............. 3 PART II: EFFECT OF ORDER OF APPROXIMATION MODELS ................ 5 Finite Element Formulation................................... 5 Numerical Characteristics and Comparisons .................. 13 Applications................................................. 22 Comments...................................................... 31 PART III: EVALUATION OF FINITE ELEMENT AND FINITE DIFFERENCE SCHEMES................................................. 32 Finite Difference Schemes ................................... 32 Finite Element Method ....................................... 33 Comparisons ................................................. 33 Illustrative Example......................................... 1+0 Comments. ................................................. 1+1 PART IV: DERIVATION OF STABILITY CRITERIA FOR TWO TIME INTEGRATION SCHEMES FOR LINEAR PARABOLIC EQUATION .... 1+3 Linear Model................................................. 1+3 1+ 1+ Cubic M o d e l ........ ......................................... Discussion.................................................... 1+7 PART V: FINITE ELEMENTS IN THE TIME DOMAIN ....................... 1+8 Dynamical System................ ............................ 1+9 Application ................................................. 52 Fluid Flow Problems ......................................... 51+ 59 SECTION I I .................................................................................................................................................... PART VI: SOME ASPECTS OF THE THEORY OF FINITE ELEMENT APPROXIMATIONS OF THE DIFFUSION EQUATION................ 61 6l Introduction.......... ............ .. • • ................... Finite Element Approximation and Interpolation. .^........ 6 3 Finite Element Approximation of the Diffusion Equation. 68 Error Estimates for the Diffusion Equation Using Semigroup Theoretic Results ............................... 70 v Error Estimates for the Diffusion Equation Using the Energy Method . ............ ................ .. 83 Error Estimates for the Diffusion Equation Using L2 M e t h o d s ................ .................... .............. 87 PART VII: CONCLUSIONS AND RECOMMENDATIONS . .................. 97 REFERENCES ........................................... ...... 98 TABLES.1-5 vi SECTION I by C. S. Desai, L. D. Johnson EVALUATION AND ANALYSES OF SOME FINITE ELEMENT AND FINITE DIFFERENCE PROCEDURES FOR TIME-DEPENDENT PROBLEMS PART I: INTRODUCTION 1. A large number of finite element (FE) and finite difference (FD) formulations has been proposed for solution of engineering problems The question facing the engineer who intends to use these methods is which procedure will provide sufficient accuracy most economically. A number of factors, such as numerical characteristics of the procedures, discretization schemes, physical requirements,
Load more

Recommended publications
  • On the Use of Boundary Integral Equations and Linear Operators in Room Acoustics
    Guimarães - Portugal paper ID: 169 /p.1 On the Use of Boundary Integral Equations and Linear Operators in Room Acoustics D. Alarcão and J. L. Bento Coelho CAPS – Instituto Superior Técnico, Av. Rovisco Pais, P-1049-001 Lisbon, Portugal, [email protected] RESUMO: Este artigo introduz alguns conceitos de base sobre equações integrais e mostra a sua aplicação na determinação da propagação de energia acústica em recintos fechados. Mostra-se que as equações integrais resultantes podem ser formuladas na linguagem dos operadores lineares, daí resultando uma notação simplificada em que as propriedades algébricas das equações que determinam a propagação da energia são mais facilmente caracterizadas. São apresentados alguns métodos gerais de resolução de equações integrais tal como métodos aproximados e métodos de bases vectoriais finitas. Apresentam-se, ainda, as definições necessárias envolvidas na descrição de campos de energia acústica, que servem de ponto de partida à aplicação das técnicas apresentadas. ABSTRACT: This paper introduces some basic concepts of boundary integral equations and their application for the determination of the propagation of sound energy inside enclosures. Linear operators are shown to provide a simplified notation and to emphasize the algebraic properties of the resulting integral equations. Some general methods of solving linear operator and integral equations are reviewed and discussed, such as approximation methods and finite basis methods. In addition, some of the necessary definitions involved in describing acoustic energy fields for applying these techniques in the field of room acoustics prediction are presented. 1. INTRODUCTION Energy based methods offer an interesting and valid alternative mid and high frequency technique to classical predictive methods such as FEM, BEM and others.
    [Show full text]
  • AMATH 731: Applied Functional Analysis Lecture Notes
    AMATH 731: Applied Functional Analysis Lecture Notes Sumeet Khatri November 24, 2014 Table of Contents List of Tables ................................................... v List of Theorems ................................................ ix List of Definitions ................................................ xii Preface ....................................................... xiii 1 Review of Real Analysis .......................................... 1 1.1 Convergence and Cauchy Sequences...............................1 1.2 Convergence of Sequences and Cauchy Sequences.......................1 2 Measure Theory ............................................... 2 2.1 The Concept of Measurability...................................3 2.1.1 Simple Functions...................................... 10 2.2 Elementary Properties of Measures................................ 11 2.2.1 Arithmetic in [0, ] .................................... 12 1 2.3 Integration of Positive Functions.................................. 13 2.4 Integration of Complex Functions................................. 14 2.5 Sets of Measure Zero......................................... 14 2.6 Positive Borel Measures....................................... 14 2.6.1 Vector Spaces and Topological Preliminaries...................... 14 2.6.2 The Riesz Representation Theorem........................... 14 2.6.3 Regularity Properties of Borel Measures........................ 14 2.6.4 Lesbesgue Measure..................................... 14 2.6.5 Continuity Properties of Measurable Functions...................
    [Show full text]
  • Arxiv:1709.09646V1 [Math.FA] 27 Sep 2017 Mensional Aahspace Banach Us-Opeetdin Quasi-Complemented Iesoa Aahspace Banach Dimensional Basis? Schauder a with Quotient 2
    ON THE SEPARABLE QUOTIENT PROBLEM FOR BANACH SPACES J. C. FERRANDO, J. KA¸KOL, M. LOPEZ-PELLICER´ AND W. SLIWA´ To the memory of our Friend Professor Pawe l Doma´nski Abstract. While the classic separable quotient problem remains open, we survey gen- eral results related to this problem and examine the existence of a particular infinite- dimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c0 and ℓp, for 1 ≤ p ≤ ∞. Also recent results of Argyros, Dodos, Kanellopoulos [1] and Sliwa´ [66] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica. 1. Introduction One of unsolved problems of Functional Analysis (posed by S. Mazur in 1932) asks: Problem 1. Does any infinite-dimensional Banach space have a separable (infinite di- mensional) quotient? An easy application of the open mapping theorem shows that an infinite dimensional Banach space X has a separable quotient if and only if X is mapped on a separable Banach space under a continuous linear map. Seems that the first comments about Problem 1 are mentioned in [46] and [55]. It is already well known that all reflexive, or even all infinite-dimensional weakly compactly generated Banach spaces (WCG for short), have separable quotients. In [38, Theorem IV.1(i)] Johnson and Rosenthal proved that every infinite dimensional separable Banach arXiv:1709.09646v1 [math.FA] 27 Sep 2017 space admits a quotient with a Schauder basis.
    [Show full text]
  • Continuity of Convolution of Test Functions on Lie Groups
    Canad. J. Math. Vol. 66 (1), 2014 pp. 102–140 http://dx.doi.org/10.4153/CJM-2012-035-6 c Canadian Mathematical Society 2012 Continuity of Convolution of Test Functions on Lie Groups Lidia Birth and Helge Glockner¨ 1 1 1 Abstract. For a Lie group G, we show that the map Cc (G) × Cc (G) ! Cc (G), (γ; η) 7! γ ∗ η, taking a pair of test functions to their convolution, is continuous if and only if G is σ-compact. More generally, consider r; s; t 2 N0 [ f1g with t ≤ r + s, locally convex spaces E1, E2 and a continuous r s bilinear map b: E1 × E2 ! F to a complete locally convex space F. Let β : Cc (G; E1) × Cc(G; E2) ! t Cc(G; F), (γ; η) 7! γ ∗b η be the associated convolution map. The main result is a characterization of those (G; r; s; t; b) for which β is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed as well as convolution of compactly supported L1-functions and convolution of compactly supported Radon measures. 1 Introduction and Statement of Results It has been known since the beginnings of distribution theory that the bilinear convo- 1 n 1 n 1 n lution map β : Cc (R )×Cc (R ) ! Cc (R ), (γ; η) 7! γ ∗η (and even convolution 1 n 0 1 n 1 n C (R ) × Cc (R ) ! Cc (R )) is hypocontinuous [39, p. 167]. However, a proof for continuity of β was published only recently [29, Proposition 2.3].
    [Show full text]
  • Random Linear Functionals
    RANDOM LINEAR FUNCTIONALS BY R. M. DUDLEYS) Let S be a real linear space (=vector space). Let Sa be the linear space of all linear functionals on S (not just continuous ones even if S is topological). Let 86{Sa, S) or @{S) denote the smallest a-algebra of subsets of Sa such that the evaluation x -> x(s) is measurable for each s e S. We define a random linear functional (r.l.f.) over S as a probability measure P on &(S), or more formally as the pair (Sa, P). In the cases we consider, S is generally infinite-dimensional, and in some ways Sa is too large for convenient handling. Various alternate characterizations of r.l.f.'s are useful and will be discussed in §1 below. If 7 is a topological linear space let 7" be the topological dual space of all con- tinuous linear functions on T. If 7" = S, then a random linear functional over S defines a " weak distribution " on 7 in the sense of I. E. Segal [23]. See the discussion of "semimeasures" in §1 below for the relevant construction. A central question about an r.l.f. (Sa, P), supposing S is a topological linear space, is whether P gives outer measure 1 to S'. Then we call the r.l.f. canonical, and as is well known P can be restricted to S' suitably (cf. 2.3 below). For simplicity, suppose that for each s e S, j x(s)2 dP(x) <oo. Let C(s)(x) = x(s).
    [Show full text]
  • On Limits of Sequences of Resolvent Kernels for Subkernels
    ON LIMITS OF SEQUENCES OF RESOLVENT KERNELS FOR SUBKERNELS IGOR M. NOVITSKII Abstract. In this paper, we approximate to continuous bi-Carleman kernels vanishing at in- finity by sequences of their subkernels of Hilbert-Schmidt type and try to construct the resolvent kernels for these kernels as limits of sequences of the resolvent kernels for the approximating subkernels. 1. Introduction In the general theory of integral equations of the second kind in L2 = L2(R), i.e., equations of the form f(s) λ T (s,t)f(t) dt = g(s) for almost every s R, (1.1) − R ∈ Z it is customary to call an integral kernel T |λ a resolvent kernel for T at λ if the integral operator 2 −1 it induces on L is the Fredholm resolvent T|λ := T (I λT ) of the integral operator T , which 2 − is induced on L by the kernel T . Once the resolvent kernel T |λ has been constructed, one can express the L2-solution f to equation (1.1) in a direct and simple fashion as f(s)= g(s)+ λ T |λ(s,t)g(t) dt for almost every s R, R ∈ Z regardless of the particular choice of the function g of L2. Here it should be noted that, in general, the property of being an integral operator is not shared by Fredholm resolvents of integral operators, and there is even an example, given in [17] (see also [18, Section 5, Theorem 8]), of an integral operator having the property that at each non-zero regular value of the parameter λ, its Fredholm resolvent is not an integral operator.
    [Show full text]
  • Essential Self-Adjointness of the Symplectic Dirac Operators
    Essential Self-Adjointness of the Symplectic Dirac Operators by A. Nita B.A., University of California, Irvine, 2007 M.A., University of Colorado, Boulder, 2010 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics 2016 This thesis entitled: Essential Self-Adjointness of the Symplectic Dirac Operators written by A. Nita has been approved for the Department of Mathematics Prof. Alexander Gorokhovsky Prof. Carla Farsi Date The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii Nita, A. (Ph.D., Mathematics) Essential Self-Adjointness of the Symplectic Dirac Operators Thesis directed by Prof. Alexander Gorokhovsky The main problem we consider in this thesis is the essential self-adjointness of the symplectic Dirac operators D and D~ constructed by Katharina Habermann in the mid 1990s. Her construc- tions run parallel to those of the well-known Riemannian Dirac operators, and show that in the symplectic setting many of the same properties hold. For example, the symplectic Dirac operators are also unbounded and symmetric, as in the Riemannian case, with one important difference: the bundle of symplectic spinors is now infinite-dimensional, and in fact a Hilbert bundle. This infinite dimensionality makes the classical proofs of essential self-adjointness fail at a crucial step, 2 n namely in local coordinates the coefficients are now seen to be unbounded operators on L (R ).
    [Show full text]
  • Abstracts of the 41St Annual Iranian Mathematics Conference
    Abstracts of the 41st Annual Iranian Mathematics Conference 12-15 September 2010, University of Urmia, Urmia, Iran Department of Mathematics Urmia University, Urmia-Iran ii Organizing Committee: M. A. Abolfathi R. Aghalari M. A. Asadi Gh. Azadi H. Azanchiler H. Behravesh P. Darania A. Ebadian K. H. Ghahremanlou M. Ghasemi Gh. Gholami A. R. Haghighi A. S. Janfada S. Ostadbashi B. Rezaei R. Sazeedeh Kh. Shahbazpour S. Shams J. Shokri S. Sohrabi S. Bohluli N. H. Ghahremanlou M. Heidarvand Sh. Mehrang Z. Mir Ali Ashrafi N. Mousavi A. Paknafs S. Roozegari N. Shafighi E. Yusefzadeh iii Scientific Committee: R. Aghalari M. A. Asadi Gh. Azadi H. Azanchiler H. Behravesh P. Darania A. Ebadian K. H. Ghahremanlou M. Ghasemi Gh. Gholami K. Ivaz A. S. Janfada M. Jelodari Mamaghani E. Mahmoodian M. S. Moslehian S. Ostadbashi B. Rezaei R. Sazeedeh S. Shams S. Sohrabi iv Supporting Organizations: H. Behravesh (Secretary of the Scientific Committee) B. Esmaielzadeh (Executive director of W. Azarbayjan P.T.T) H. Goudarzi (Dean Faculty of Sciences) Jalali (Sh. Rajaie Tarbiat Moallem Chancellor) V. Jalalzadeh (State Governor) M. Madadi (Rersearch Assistant of the State Governor) I. Mirzaie (Chancellor of Urmia Technology University) H. Molaie (Vise Chancellor of Elm o Fan College of Higher Education) S. Ostadbashi (Secretary of the Organizing Committee) H. Rezaie (Science and Technology Park Principal) N. Samadi (Financial Office Vise Chancellor) Samarghandi (Chief of the Educational Organization) H. Sedghi (Chansellor of Urmia University) A. Talebi (Research Vise Chancellor) M. Taleie (Student Vise Chancellor) v Invited Speakers: M. Asghari Iran D. Bakic Croatia V. D. Breaz Romania A.
    [Show full text]
  • On Finiteness Spaces and Extensional Presheaves Over the Lawvere Theory of Polynomials Thomas Ehrhard
    On finiteness spaces and extensional presheaves over the Lawvere theory of polynomials Thomas Ehrhard To cite this version: Thomas Ehrhard. On finiteness spaces and extensional presheaves over the Lawvere theory ofpoly- nomials. 2007. hal-00165230 HAL Id: hal-00165230 https://hal.archives-ouvertes.fr/hal-00165230 Preprint submitted on 25 Jul 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On finiteness spaces and extensional presheaves over the Lawvere theory of polynomials Thomas Ehrhard Preuves, Programmes, Syst`emes (UMR 7126) Universit´eParis Diderot – Paris 7 CNRS [email protected] July 25, 2007 Abstract We define a faithful functor from a cartesian closed category of linearly topologized vector spaces over a field and generalized polynomial functions to the category of “extensional” presheaves over the Lawvere theory of polynomial functions, and show that, under some conditions on the field, this functor is full and preserves the cartesian closed structure. Introduction We introduced in [Ehr05] the notion of finiteness space for defining a new denotational model1 of classical linear logic [Gir87]. Finiteness spaces are “discrete” objects presenting similarities with coherence [Gir87] or hypercoherence [Ehr93] spaces. There are many differences however.
    [Show full text]
  • International Journal of Pure and Applied Mathematics ————————————————————————– Volume 54 No
    International Journal of Pure and Applied Mathematics ————————————————————————– Volume 54 No. 3 2009, 359-374 INTEGRAL REPRESENTATIONS OF UNBOUNDED OPERATORS BY INFINITELY SMOOTH BI-CARLEMAN KERNELS Igor M. Novitskii Institute for Applied Mathematics Far-Eastern Branch of the Russian Academy of Sciences Dzerzhinskiy Street 54, Khabarovsk, 680 000, RUSSIA e-mail: [email protected] Abstract: In this paper, we establish that if a closed linear operator in a separable Hilbert space H is unitarily equivalent to a bi-Carleman integral operator in an appropriate L2(Y,µ), then that operator is unitarily equivalent to a bi-Carleman integral operator in L2(R), whose kernel T : R2 → C and two Carleman functions t(s) = T (s, ·), t′(s) = T (·,s) : R → L2(R) are infinitely smooth and vanish at infinity together with all partial and all strong derivatives, respectively. The implementing unitary operator (from H onto L2(R)) is found by direct construction. AMS Subject Classification: 47G10, 45P05, 47B33, 47B38 Key Words: closed linear operator, integral linear operator, Carleman in- tegral operator, bi-Carleman integral operator, characterization theorems for integral operators, linear integral equation 1. Introduction and the Main Result The present paper may hopefully prove interesting for researchers who are inter- ested in the development of the theory of non-compact, non-self-adjoint linear integral operators in L2 spaces (see [6], [10]). In applications of this theory, such Received: June 6, 2009 c 2009 Academic Publications 360 I.M. Novitskii as occur, for instance, in the theory of singular integral equations of the second kind, it is often desirable to have the kernel function with special properties that make its associated integral operator easier to work with.
    [Show full text]
  • Random Matrices
    Random Matrices Estelle Basor Mathematics Department California Polytechnic State University San Luis Obispo, 93407 Winter 1995 and Spring 2007 1 Contents 2 1 Introduction These notes are based on a series of lectures given by Estelle Basor in an applied analysis graduate course at Cal Poly first done in Winter of 1996. They were based on notes taken by the students in the class, typed by Mike Robertson, and then finally edited by Estelle Basor and Jon Jacobsen. The notes were again used in Spring of 2007 and corrections to the notes were provided by the students in the class and some other sections were added. These notes are not meant to be complete or perfect, but rather an informal set of notes describing a new and lively field in mathematics. It is the intention of the author that they provide a heuristic background so that one can begin to understand random matrices and consider future questions in the subject. The notes provide an introduction to the basics of the Gaussian Unitary Ensemble (GUE), a derivation of the semi-circle law, a derivation of the Painlev´eequation connected to the basic probabilites of the GUE and at the end a derivation of the distribution function for linear statistics. Many of the tools used are provided in the text, while others are missing. In particular, the asymptotics of the relevant orthogonal polynomials are used but not derived. Several properties of Hilbert spaces, operators, trace class operators are used and not proved. 3 4 2 The probability distribution for the eigenvalues of a random hermitian matrix Given a random Hermitian matrix what can we say about its eigenvalues? Some questions include: What is the largest one? How are they spaced? What is the probability that one does (or does not) lie in a given interval? This section will describe a probability distribution on the space of matrices and show how this distribution induces one on the space of eigenvalues.
    [Show full text]
  • Extension of Vector-Valued Integral Polynomials
    Extension of vector-valued integral polynomials Daniel Carando Departamento de Matem´atica, Universidad de San Andr´es, Vito Dumas 284 (B1644BID) Victoria, Buenos Aires, Argentina. and Silvia Lassalle ∗ Departamento de Matem´atica - Pab I, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina Abstract We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued Pietsch-integral polynomial on E extends to an X- valued Pietsch-integral polynomial on any space F containing E, with the same integral norm. This is not the case for Grothendieck-integral polynomials: they do not always extend to X-valued Grothendieck-integral polynomials. However, they are extendible to X-valued polynomials. The Aron-Berner extension of an integral polynomial is also studied. A canonical integral representation is given for domains not containing `1. Key words: Integral polynomials, extendibility Introduction In this note we study extendibility properties of Pietsch and Grothendieck inte- gral polynomials. Generally, polynomials on Banach spaces do not extend to larger spaces, even in the scalar valued case [20]. In other words, there is no Hahn-Banach ∗ Corresponding author Email addresses: [email protected] (Daniel Carando), [email protected] (Silvia Lassalle). Preprint submitted to Elsevier Science 9 February 2005 extension theorem for polynomials. However, since the symmetric injective tensor product respects subspaces, scalar-valued integral polynomials are extendible. For vector-valued polynomials, the word “extendible” needs to be properly defined. We say that a polynomial P : E → X is extendible if for any Banach space F con- taining E, there exists Pe : F → X extending P ([20], see also [5]).
    [Show full text]