DOCSLIB.ORG

Integral Equations for Machine Learning Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Integral Equations for Machine Learning Problems Integral Equations For Machine Learning Problems Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Qichao Que, M.S. Graduate Program in Computer Science and Engineering The Ohio State University 2016 Dissertation Committee: Dr. Mikhail Belkin, Advisor Dr. Yusu Wang, co-Advisor Dr. Yoonkyung Lee Dr. DeLiang Wang Copyright by Qichao Que 2016 ABSTRACT Supervised learning algorithms have achieved significant success in the last decade. To further improve learning performance, we still need to have a better understand- ing of semi-supervised learning algorithms for leveraging a large amount of unlabeled data. In this dissertation, a new approach for semi-supervised learning will be dis- cussed, which takes advantage of unlabeled data information through an integral operator associated with a kernel function. More specifically, several problems in machine learning are formulated as a regularized Fredholm integral equation, which has been well studied in the literature of inverse problems. Under this framework, we propose several simple and easily implementable algorithms with sound theoretical guarantees. First, a new framework for supervised learning is proposed, referred as the Fred- holm learning. It allows a natural way to incorporate unlabeled data and is flexible on the choice of regularizations. In particular, we connect this new learning framework to the classical algorithm of radial basis function networks, and more specifically, an- alyze two common forms of regularization procedures for RBF networks, one based on the square norm of coefficients in a network and another one using centers ob- tained by the k-means clustering. We provide a theoretical analysis of these methods as well as a number of experimental results, pointing out very competitive empiri- cal performance as well as certain advantages over the standard kernel methods in terms of both flexibility (incorporating unlabeled data) and computational complex- ii ity. Moreover, the Fredholm learning algorithm could be interpreted as a special form of kernel methods using a data-dependent kernel. Our analysis shows that Fredholm kernels achieve noise suppressing effects under a new assumption for semi-supervised learning, termed the \noise assumption". q We also address the problem of estimating the probability density ratio function p , which could be used for solving the covariate shift problem in transfer learning, given the marginal distribution p for training data and q for testing data. Our approach is q based on reformulating the problem of estimating p as an inverse problem in terms of a Fredholm integral equation. This formulation, combined with the techniques of regularization and kernel methods, leads to a principled kernel-based framework for constructing algorithms and for analyzing them theoretically. The resulting family of algorithms, termed the FIRE algorithm for the Fredholm Inverse Regularized Estima- tor, is flexible, simple and easy to implement. More importantly, several encouraging experimental results are presented, especially applications to classification and semi- supervised learning within the covariate shift framework. We also show how the hyper-parameters in the FIRE algorithm can be chosen in a completely unsupervised manner. iii This work is dedicated to my family. iv ACKNOWLEDGMENTS I would like to express my gratitude to my advisor, Dr. Mikhail Belkin, without whom this dissertation would not have been possible. I consider it a great honor to work with him, motivated by his dedication and critical thinking towards research. I have learned so much from his in-depth understanding on machine learning and I feel deeply indebted to him for helping me edit my papers and his warm encouragements when I struggled to make progress on my PhD studies. I am also grateful to have the opportunity to work with Professor Yusu Wang. Her knowledgeable advice and attention to details really benefited my research sig- nificantly, and had led to several important work for this dissertation. I would like to thank Dr. Tao Shi and Dr. Brian Kulis for serving on my candidate committee, who gave me insightful suggestions on my dissertation proposal. Dr. Brian Kulis has stimulated the idea of connecting kernel LSH with kernel PCA. I would also like to thank Professor DeLiang Wang and Professor Yoonkyung Lee for serving on my dissertation committee and providing their insightful feedback on this dissertation. I also appreciate the hospitality of Herbert Edelsbrunner and IST Austria for hosting me briefly during the time I worked on one of my thesis work on density ratio estimation. Many thanks to Jitong Chen, Siyuan Ma, Yuanlong Shao, Xin Tong, Yuwen Zhuang, Dr. Yuxuan Wang, Dr. Xiaojia Zhao, Dr. Kun Han, Dr. Yanzhang He, Dr. Qingpeng Niu and other fellow students for being great friends and making the life v in graduate school more enjoyable. Last but not least, I have to thank my parents for their unconditional support and encouragement, helping me go through difficulties during the PhD years. Special thanks to my girlfriend Ms. Yue Hu for sharing my frustrations and bringing me the joyful moments. Without them, this thesis would not have been possible. vi VITA Oct. 1987 . Born in Jiande, Zhejiang, China. Jul. 2010 . B.S. in Mathematics and Ap- plied Mathematics, Zhejiang Uni- versity, Hangzhou, China. May. 2014 . M.S. in Computer Science and Engineering, The Ohio State Uni- versity, Columbus, OH. PUBLICATIONS • Q. Que and M. Belkin, \Back to the future: Radial Basis Function networks re- visited." International Conference on Artificial Intelligence and Statistics (AIS- TATS), to appear, 2016. • K. Jiang, Q. Que and B. Kulis, \Revisiting Kernelized Locality-Sensitive Hash- ing for Improved Large-Scale Image Retrieval." Computer Vision and Pattern Recognition (CVPR), 2015. • Q. Que, M. Belkin, Y. Wang, \Learning with Fredholm Kernels." Advances in Neural Information Processing Systems 27 (NIPS), 2014. • Q. Que and M. Belkin, \Inverse Density as an Inverse Problem: The Fredholm vii Equation Approach." Advances in Neural Information Processing Systems 26 (NIPS), 2013. • T. K. Dey, X. Ge, Q. Que, I. Safa, L. Wang and Y. Wang, \Feature-Preserving Reconstruction of Singular Surfaces." Europraphics Symposium on Geometry Processing(Euro SGP), 2012. • M. Belkin, Q. Que, Y. Wang and X. Zhou, \Toward Understanding Complex Spaces: Graph Laplacians on Manifolds with Singularities and Boundaries." Conference on Learning Theory, (COLT) , 2012. FIELDS OF STUDY Major Field: Computer Science and Engineering viii TABLE OF CONTENTS Abstract . ii Dedication . iv Acknowledgments . v Vita . vii List of Tables . xi List of Figures . xiii 1 Introduction . 1 1.1 Learning With An Integral Equation . .3 1.2 Contribution of This Dissertation . .6 1.3 Collaboration and Prior Publications . .8 2 Supervised Learning with Fredholm Equations . 9 2.1 The Fredholm Learning Framework . 10 2.2 Choice of Regularization . 12 2.3 Related Work . 16 3 Revisiting Radial Basis Function Network . 19 3.1 RBF Network and Fredholm Learning . 20 3.2 Generalization Bounds for RBF Network . 22 3.3 RBF Networks for Semi-supervised Learning . 25 3.3.1 An Empirical Example . 26 3.3.2 Generalization Analysis for Semi-supervised RBF Network . 27 3.4 k-means RBF Networks . 29 3.4.1 The Algorithm . 29 3.4.2 Analysis for k-means RBF . 32 3.4.3 Denoising Effect of k-means . 34 ix 3.5 Experiments . 35 3.5.1 Kernel Machines and RBF Networks . 35 3.5.2 RBF Network With k-means Centers . 36 3.5.3 Regularization Effect of k-means . 38 4 Fredholm Kernel and The Noise Assumption . 40 4.1 The Noise Assumption . 40 4.2 Fredholm Learning and Kernel Approximation . 42 4.3 Theoretical Results for Fredholm Kernel . 45 4.3.1 Linear Kernel . 46 4.3.2 Gaussian Kernel . 48 4.4 Experiments . 49 4.4.1 Noise and Cluster Assumptions . 50 4.4.2 Real-world Data Sets for Testing Noise Assumption . 51 5 Fredholm Equations for Covariate Shift . 57 5.1 Covariate Shift and Fredholm Integral Equation . 57 5.1.1 Related Work . 62 5.2 The FIRE Algorithms . 64 5.2.1 Algorithms for The Type I Setting. 65 5.2.2 Algorithms for The Type II Setting . 67 5.2.3 Comparison Between Type I and Type II Settings . 68 5.3 Theoretical Analysis: Bounds and Convergence Rates . 69 5.4 Proofs of Main Results . 72 5.5 Experiments . 77 5.5.1 Experimental Setting and Model Selection . 77 5.5.2 Data Sets and Resampling . 80 5.5.3 Testing The FIRE Algorithm . 81 5.5.4 Supervised Learning: Regression and Classification . 85 5.5.5 Simulated Examples . 91 6 Conclusion and Future Work . 95 6.1 Future Work . 97 Appendix A Fredholm Learning Framework . 99 Appendix B Integral Equations for Covariate Shift . 125 Bibliography . 131 x LIST OF TABLES 3.1 Experiment results for comparison between standard kernel methods and l2 regularized RBF networks for supervised learning problems. 36 3.2 Experiment results for comparison between standard kernel methods and l2 regularized RBF networks for semi-supervised learning problems. 37 3.3 Experiment results for comparison between standard kernel methods and k-means RBF networks for supervised learning problems. 39 4.1 Classification errors of different classifiers on the linear toy example. 52 4.2 Classification errors of different classifiers on the circular toy example. 52 4.3 Classification errors of various methods on different text data sets. 54 4.4 Classification errors on Webkb with different numbers of labeled points. 54 4.5 Classification errors of nonlinear classifiers on the handwriting digits recognition data sets, trained with small number of labeled points. 55 4.6 Classification errors of nonlinear classifiers on the MNIST data set cor- rupted with Gaussian noise, trained with different numbers of labeled data. 55 5.1 Performance comparison for different density ratio estimation algo- rithms on the USPS data set with resampling using PCA.
Load more

Recommended publications
  • On the Stochastic Richards Equation
    Dissertation zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften am Fachbereich Mathematik und Informatik der Freien Universit¨at Berlin On the Stochastic Richards Equation vorgelegt von Ralf Forster Berlin 2011 Betreuer: Prof. Dr. Ralf Kornhuber, Berlin Gutachter: Prof. Dr. Ralf Kornhuber, Berlin Prof. Dr. Omar Knio, Baltimore Tag der Disputation: 8. M¨arz 2011 Contents Introduction 1 1 Deterministic Richards equation in homogeneous soil 7 1.1 Basic tools in hydrology and Kirchhoff transformation . ...... 7 1.2 Boundary conditions and weak formulation . .. 12 2 The stochastic Richards equation 16 2.1 Randomfields............................... 16 2.2 Tensorspaces............................... 18 2.3 Weakformulation............................. 22 3 Discretization 29 3.1 TheKarhunen–Lo`eveexpansion. 29 3.1.1 Definition ............................. 30 3.1.2 Independent KL expansions and reformulation . 36 3.1.3 Computationalaspects. 42 3.2 Time discretization and convex minimization . .... 44 3.2.1 Timediscretization. 44 3.2.2 Formulation as a convex minimization problem . 46 3.2.3 Variationalinclusions . 51 3.3 Polynomial chaos and finite elements . 57 3.3.1 (Generalized)Polynomialchaos . 58 3.3.2 Finiteelementsinspace . 63 3.3.3 Approximation of the nonlinear functional . .. 65 3.3.4 Aconvergenceresultforthelimitcases . 71 3.4 Numericalexperiments. 78 3.4.1 Resultsforonespacedimension. 79 3.4.2 Resultsfortwospacedimensions . 81 4 Numerical solution 85 4.1 CommonmethodsforSPDEs . 85 4.1.1 StochasticGalerkinmethod . 86 4.1.2 Stochasticcollocation . 90 4.1.3 MonteCarlomethod. 92 4.2 Nonlinear Block Gauß–Seidel and minimization . .... 95 4.2.1 NonlinearBlockGauß–Seidel . 95 4.2.2 TransformationofthetensorproductPC . 99 4.2.3 Blockminimization. 106 4.3 Monotonemultigridmethod. 107 4.3.1 Multilevelcorrections .
    [Show full text]
  • A Abstract Factorization Approach, 887–889 Abstract Interpolation
    Index A Bergman-Hodge type decomposition, Abstract factorization approach, 887–889 1386–1387 Abstract interpolation problem, 693 Bergman kernel, 73–85 Abstract operator adjoints, 905 Bergman projection, 74 Accretive L-system, 799 Bergman space, 39–40, 1116 Accumulative system, 801 Besov-Sobolev space, 1137–1138 Adele ring, 1288–1291 Bessel functions, 568–569, 1660 Adele von Neumann algebra, 1308–1310 Beurling type theorems, 1160–1161 Adele W-probability space, 1328–1333 Beurling–Lax–Halmos theorem, 998 Admissible kernel, 135 Beurling–Malliavin majorant, 583 Algebraic analysis, 1555, 1603 Beurling–Malliavin multiplier theorem, Algebraic isomorphism, 1168–1169 582–591 Algebraic Riccati equations, 441–445, Beurling’s representations, 1059–1061 463–467 Bezout equation, 460–461, 891 Ambiguity transformation, 847, 859–861 Biholomorphic mappings, 80–83 Ando dilation, 1106–1108 Birth-and-death processes, 519 Andô’s theorem, 225, 982 Blaschke condition, 494–495 Antisymmetric Fock space, 1142 Blecher-Effros-Ruan axioms, 226 Arithmetic algebra, 1251, 1297–1298 Block-column operator matrix, 662 Arithmetic-function(s), 1246–1282 Block-tridiagonal system, 921–923 tensor(ing), 1264–1269 BL.V /, 1814 tensor product algebra, 1265 Bochner-Martinelli formula, 1826 Arithmetic p-prime probability space, Bognár-Krámli factorization, 230–232 1262 Borichev’s construction, 585–588 Arveson space, 1778, 1781 Boundary triplets, 183–215 Associated pairs, 733–734 Boundary value(s), 1222 Asymptotic behavior, 345–368 Boundary value problems, 1370–1390 Atomization procedure, 601–603 Bounded real lemma, 455–456 Azukawa pseudometric function, 78 Brangesian complementary space, 647 Brehmer conditions, 1105 BR.V /, 1814 B B.V /, 1814 Backward multishift, 1098 Backward shift, 1095 Banach algebra, 887 C Banach space, 875–879, 1183 Canonical coherent states (CCS), 116–117 Bandlimited functions, 835, 838–842 Canonical (co-prime) external forms, 910 Beals’ approach, 401–402 Canonical factorizations, 928 Berezin transform, 121, 1017 Canonical forms, 433–435, 907–908 © Springer Basel 2015 1853 D.
    [Show full text]
  • Book of Abstracts
    PANTONE 179C Proceedings of the VOLUME V Mathematicians International Congress of Proceedings of the International Congress of Mathematicians Rio de Janeiro 2018 Book of Abstracts Alexander Arbieto Editor C C M M Y Y CM CM MY MY CY CY CMY CMY K K ISBN 978-981-3272-93-4 9 789813 272934 a SBM On conciseness of words in residually finite groups, Pavel Shumyatsky Univer- sity of Brasilia [email protected] A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group G X, it always follows that w(G) is finite. P. Hall asked whether every word is concise.2 Due to Ivanov the answer to this problem is known to be negative. On the other hand, for residually finite groups the problem remains wide open. We recently discovered that the Lie-theoretic techniques created by Zelmanov in his solution of the restricted Burnside problem can be used to prove conciseness in residually finite groups of many words. Our talk will be about recent developments with respect to that problem. Using MIR-max algorithm to monitor performance of donor-funded projects in developing countries based on DHS data, Elias Mwakilama and Martin Pais- ley Staffordshire University, UK and University of Malawi, Malawi emwakilama@ cc.ac.mw Most developing countries in Africa are faced with problems of ill-health, poverty, human rights violation, and non-access to human growth services such as clean water.
    [Show full text]
  • Signature Redacted 7
    ON INTEGRAL EQUATIONS Their Solution by Iteration and Analytic Continuation by ALAN J. PERLIS B. S., Carnegie Institute of Technology (1942) S. M., Massachusetts Institute of Technology (1949) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREIENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY (1950) Signature Redacted Signature of Author................W........................... Department of Mathematics, May 12, 1950 Signature Redacted Certified Thesis Supervisor Signature Redacted * *9*00 @ 00*0* - 0i0*0 **00 * e*0 Oe e 0. 0 0C 0e* O OC C e OC C 0C 0e C O C * *CCS Chairman, Departmental Committee on Graduate Students 7/ Table of Contents; Page Acknowledgment............. ...... ................. Introduction. ........................................ 1 I. Iterative proceeaures....................... 12 II. Remarks on the Resolvent as an Analytic function..36 52 III. Solution by Analytic Ontinuation................. -___ Acknowledgment The author wishes to express his sincere gratitude and appreciation to Professor Philip Franklin for his encouragement, assistance, and guidance in the preparation and development of this thesis. ii Abstract This thesis is concerned with the linear integral equation with fixed finite limits, known as the Fredholm equation of the second kind. The kernel is not assumed to be symmetric; but, as with all other functions involved, is assumed continuous in the unit square 0h X3 I A 1. Other than the Fredholm procedure involving infinite determinants, solutions for large ) have not been exhibited, at least in forms not demanding prior explicit knowledge of the characteristic values and characteristic functions. From the standpoint of computing, it is desirable to obtain solutions requiring only evaluation of the kernels and iterated kernels of the equation.
    [Show full text]
  • Approximation Properties and Eigenvalue Problems for Banach Spaces
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Pakistan Research Repository Approximation properties and eigenvalue problems for Banach spaces Qaisar Latif February 12, 2015 Approximation properties and eigenvalue problems for Banach spaces Name: Qaisar Latif Year of Admission: 2009 Registration No.: 105{GCU{PHD{SMS{09 Abdus Salam School of Mathematical Sciences GC University, Lahore, Pakistan. Approximation properties and eigenvalue problems for Banach spaces Submitted to Abdus Salam School of Mathematical Sciences GC University Lahore, Pakistan in the partial fulfillment of the requirements for the award of degree of Doctor of Philosophy in Mathematics By Name: Qaisar Latif Year of Admission: 2009 Registration No.: 105{GCU{PHD{SMS{09 Abdus Salam School of Mathematical Sciences GC University, Lahore, Pakistan. DECLARATION I, Mr. Qaisar Latif Registration No. 105{GCU{PHD{SMS{09 student at Abdus Salam School of Mathematical Sciences GC University in the subject of Mathe- matics, hereby declare that the matter printed in this thesis, titled \Approximation properties and eigenvalue problems for Banach spaces" is my own work and that 1. No direct major work had already been done by me or anybody else on this topic; I worked on, for the PhD degree. 2. The work, I am submitting for the Ph. D. degree has not already been submitted elsewhere and shall not in future be submitted by me for obtaining similar degree from any other institution. Dated: |||||||||-, Signature of Deponent: |||||||||||| RESEARCH COMPLETION CERTIFICATE It is certified that the research work contained in this thesis, titled \Approximation properties and eigenvalue problems for Banach spaces" has been carried out and completed by Mr.
    [Show full text]
  • Stochastic Analysis of Gaussian Processes Via Fredholm
    STOCHASTIC ANALYSIS OF GAUSSIAN PROCESSES VIA FREDHOLM REPRESENTATION TOMMI SOTTINEN AND LAURI VIITASAARI Abstract. We show that every separable Gaussian process with inte- grable variance function admits a Fredholm representation with respect to a Brownian motion. We extend the Fredholm representation to a transfer principle and develop stochastic analysis by using it. We show the convenience of the Fredholm representation by giving applications to equivalence in law, bridges, series expansions, stochastic differential equations and maximum likelihood estimations. 1. Introduction The stochastic analysis of Gaussian processes that are not semimartin- gales is challenging. One way to overcome the challenge is to represent the Gaussian process under consideration, X say, in terms of a Brownian motion and then develop a transfer principle so that that the stochastic analysis can be done in the “Brownian level” and then transfered back into the level of X. One of the most studied representation in terms of a Brownian motion is the so-called Volterra representation. A Gaussian Volterra process is a process that can be represented as t (1.1) Xt = K(t,s)dWs, t ∈ [0, T ], Z0 where W is a Brownian motion and K ∈ L2([0, T ]2). Here the integration goes only upto t, hence the name “Volterra”. This Volterra nature is very convenient: it means that the filtration of X is included in the filtration of the underlying Brownian motion W . Gaussian Volterra processes and their stochastic analysis has been studied, e.g., in [2] and [11], just to mention arXiv:1410.2230v5 [math.PR] 22 Mar 2016 few. Apparently, the most famous Gaussian process admitting Volterra representation is the fractional Brownian motion and its stochastic analysis indeed has been developed mostly by using its Volterra representation, see e.g.
    [Show full text]
  • Operator Theory a Comprehensive Course in Analysis, Part 4
    Operator Theory A Comprehensive Course in Analysis, Part 4 Barry Simon Operator Theory A Comprehensive Course in Analysis, Part 4 http://dx.doi.org/10.1090/simon/004 Operator Theory A Comprehensive Course in Analysis, Part 4 Barry Simon Providence, Rhode Island 2010 Mathematics Subject Classification. Primary 47-01, 34-01, 46-01; Secondary 81Q10, 34L05, 35P05, 42C05, 46H05, 22B05. For additional information and updates on this book, visit www.ams.org/bookpages/simon Library of Congress Cataloging-in-Publication Data Simon, Barry, 1946– Operator theory / Barry Simon. pages cm. — (A comprehensive course in analysis ; part 4) Includes bibliographical references and indexes. ISBN 978-1-4704-1103-9 (alk. paper) 1. Mathematical analysis—Textbooks. 2. Operator theory. I. Title. QA300.S5285 2015 515.724—dc23 2015033104 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s).
    [Show full text]
  • 12.2% 116,000 120M Top 1% 154 4,000
    We are IntechOpen, the world’s leading publisher of Open Access books Built by scientists, for scientists 4,000 116,000 120M Open access books available International authors and editors Downloads Our authors are among the 154 TOP 1% 12.2% Countries delivered to most cited scientists Contributors from top 500 universities Selection of our books indexed in the Book Citation Index in Web of Science™ Core Collection (BKCI) Interested in publishing with us? Contact [email protected] Numbers displayed above are based on latest data collected. For more information visit www.intechopen.com Chapter Introductory Chapter: Frontier Research on Integral Equations and Recent Results Francisco Bulnes “Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country” - David Hilbert 1. General discussion The themes of recent research are focused on nonlinear integral equations [1], the new numerical and adaptive methods of resolution of integral equations [2], the generalization of Fredholm integral equations [3] of second kind, integral equations in time scales and the spectral densities [3, 4], operator theories for nonsymmetric and symmetric kernels [1, 5], extension problems to Banach algebras to kernels of integral equations [5–7], singular integral equations [10], special treatments to solve Fredholm integral equations of first and second kinds, nondegenerate kernels [3, 6] and symbols of integral equations [7], topological methods for the resolution of integral equations and representation problems of operators of integral equations. Now, well, the field of the integral equations is not finished yet, not much less with the integral equations for which the Fredholm theorem is worth [fredholm], nor with the completely continuous operators, since there exist other integral equations devel- oped of the Hilbert theory respect to the Fredholm discussion, and studies on singular integral equations, also by Hilbert, Wiener and others [8].
    [Show full text]
  • Linear Integral Equations Theory and Technique
    Linear Integral Equations Theory and Technique RAM P. KANWAL Pennsylvania State University University Park, Pennsylvania ACADEMIC PRESS 1971 New York and London COPYRIGHT © 1971, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London W1X 6BA LIBRARY OF CONGRESS CATALOG CARD NUMBER: 77-156268 AMS (MOS) 1970 Subject Classification: 45A05, 45B05, 45C05, 45D05, 45E05, 45E10, 45E99, 45F05, 35C15, 44A25, 44A35, 47G05 PRINTED IN THE UNITED STATES OF AMERICA IN LOVING MEMORY OF MY GRANDMOTHER PREFACE Many physical problems which are usually solved by differential equation methods can be solved more effectively by integral equation methods. Indeed, the latter have been appearing in current literature with increasing frequency and have provided solutions to problems heretofore not solvable by standard methods of differential equations. Such problems abound in many applied fields, and the types of solutions explored here will be useful particularly in applied mathematics, theoretical mechanics, and mathematical physics. Each section of the book contains a selection of examples based on the technique of that section, thus making it possible to use the book as the text for a beginning graduate course. The latter part of the book will be equally useful to research workers who encounter boundary value problems. The level of mathematical knowledge required of the reader is no more than that taught in undergraduate applied mathematics courses.
    [Show full text]
  • Approximation Properties of Locally Convex Spaces and the Problem Of
    Approximation Properties of Locally Convex Spaces and the Problem of Uniqueness of the Trace of a Linear Operator ∗ G. L. Litvinov Selecta Mathematica Sovietica 0272-9903/92/010025-16 $1.50 + 0.20/0 Vol. 11, No. 1 (1992), p. 25-40. ©1992 Birkh¨auser Verlag, Basel 1. Introduction Linear positive operators in Hilbert spaces that have a trace were described for the first time in studies by von Neumann devoted to the mathematical apparatus of quantum mechanics. Subsequently, Schatten and von Neumann [13] described all the linear continuous operators that have a trace in Hilbert spaces. An operator that possesses a trace was shown to correspond in one- to-one fashion with a bivalent tensor, and it was proved that the trace of an operator could be defined as the result of a total contraction of the corresponding tensor. This construction was generalized by Ruston to the case of Banach spaces and, independently of Ruston, Grothendieck [4] genera- lized the construction to the case of arbitrary locally convex spaces. It was ′ arXiv:1108.1721v1 [math.FA] 8 Aug 2011 shown that given an inductive topological tensor product E ⊗¯E of some space E and the space E′ dual to it, a linear operator in E could be associated with every element of this product. However, the “operators with trace” obtained in this way are not, generally speaking, continuous; a narrower class is formed by the Fredholm operators in the sense of [4], which are always weakly continuous. Finally, an even narrower class is formed by the nuclear operators (which are continuous).
    [Show full text]
  • Lecture Notes in Physics
    p. VOPD> ph .r3 o· l~ 2, MA!1~ ~IS·l.f L £tC 3'6 o_~.~----~ 0- 0. __• _ _____ Lecture Notes in Physics Edited by J. Ehlers, Munchen, K. Hepp, Zurich R. Kippenhahn, Munchen, H. A. Weidenmuller, Heidelberg and J. Zittartz, Koln Managing Editor: W. Beiglbock, Heidelberg 123- i. ,$ dB ;gIS! !!\lP' d! -tp '* +9 A?++& ei Dieter H."Mayer" ,The Ruelle-Araki Transfer Operator in Classical Statistical Mechani( 311 + F -6-9 q..- , ...... -%"...cl· d }1M 'ws:* •• •• • APPENDIX A. GROTHENDI.ECK'S THEORY OF NUCLEAR OPERATORS IN BANACH SPACES In this appendix we recall the fundamentals of Grothendieck's the- ory about nuclear operators in Banach spaces as far as they are neces- sary for the considerations in this work. A much more extensive pre­ sentation of this theory can be found in ~ 1] and fi 28] • A.l. The projective topological tensor product of Banach spaces Let E, F be two complex Banach spaces with norms II liE and II I~. Let E ® F be the tensor product of these. two spaces with the norm II II'll defined by = inf z:::=.1Ie)..11 E II f .11 F' (A1) IIx "71" {i} ). ;!. where the infimum has to be taken over all possible finite representa- ,- !~ tions of x € E ® F in the form I, ~ ,i :1 x =Le. ®f. , (A2) {i}). ). ,- ij with e. E and f. e F• ). e ). The completion of the space E.® F under this norm is denoted by ~ E~ ., F and is called the projective topological tensor product of the \ -71" ~ two spaces E and F.
    [Show full text]
  • Introductory Chapter: Frontier Research on Integral Equations and Recent Results Francisco Bulnes
    Chapter Introductory Chapter: Frontier Research on Integral Equations and Recent Results Francisco Bulnes “Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country” - David Hilbert 1. General discussion The themes of recent research are focused on nonlinear integral equations [1], the new numerical and adaptive methods of resolution of integral equations [2], the generalization of Fredholm integral equations [3] of second kind, integral equations in time scales and the spectral densities [3, 4], operator theories for nonsymmetric and symmetric kernels [1, 5], extension problems to Banach algebras to kernels of integral equations [5–7], singular integral equations [10], special treatments to solve Fredholm integral equations of first and second kinds, nondegenerate kernels [3, 6] and symbols of integral equations [7], topological methods for the resolution of integral equations and representation problems of operators of integral equations. Now, well, the field of the integral equations is not finished yet, not much less with the integral equations for which the Fredholm theorem is worth [fredholm], nor with the completely continuous operators, since there exist other integral equations devel- oped of the Hilbert theory respect to the Fredholm discussion, and studies on singular integral equations, also by Hilbert, Wiener and others [8]. Arise numerical and approximate methods on the big vastness that give the Banach algebras, even using probabilistic measures to solve some integral equations
    [Show full text]