Best Approximation By Linear Superpositions Approximate Nomography

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Best Approximation by Linear Superpositions (approximate Nomography)

S. I͡A. Khavinson
Best Approximation by Linear Superpositions approximate Nomography Book PDF

Author : S. I͡A. Khavinson
Publisher : American Mathematical Soc.
Page : 188 pages
File Size : 51,5 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 082189773X

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Best Approximation by Linear Superpositions (approximate Nomography) by S. I͡A. Khavinson Pdf

This book deals with problems of approximation of continuous or bounded functions of several variables by linear superposition of functions that are from the same class and have fewer variables. The main topic is the space of linear superpositions D considered as a sub-space of the space of continous functions C(X) on a compact space X. Such properties as density of D in C(X), its closedness, proximality, etc. are studied in great detail. The approach to these and other problems based on duality and the Hahn-Banach theorem is emphasized. Also, considerable attention is given to the discussion of the Diliberto-Straus algorithm for finding the best approximation of a given function by linear superpositions.

Best Approximation by Linear Superpositions (approximate Nomography)

S. I͡A. Khavinson
Best Approximation by Linear Superpositions approximate Nomography Book PDF

Author : S. I͡A. Khavinson
Publisher : American Mathematical Soc.
Page : 175 pages
File Size : 44,7 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 0821804227

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Best Approximation by Linear Superpositions (approximate Nomography) by S. I͡A. Khavinson Pdf

This book deals with problems of approximation of continuous or bounded functions of several variables by linear superposition of functions that are from the same class and have fewer variables. The main topic is the space of linear superpositions $D$ considered as a subspace of the space of continuous functions $C(X)$ on a compact space $X$. Such properties as density of $D$ in $C(X)$, its closedness, proximality, etc. are studied in great detail. The approach to these and other problems based on duality and the Hahn-Banach theorem is emphasized. Also, considerable attention is given to the discussion of the Diliberto-Straus algorithm for finding the best approximation of a given function by linear superpositions.

Ridge Functions and Applications in Neural Networks

Vugar E. Ismailov
Ridge Functions and Applications in Neural Networks Book PDF

Author : Vugar E. Ismailov
Publisher : American Mathematical Society
Page : 186 pages
File Size : 49,6 Mb
Release : 2021-12-17
Category : Mathematics
ISBN : 9781470467654

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Ridge Functions and Applications in Neural Networks by Vugar E. Ismailov Pdf

Recent years have witnessed a growth of interest in the special functions called ridge functions. These functions appear in various fields and under various guises. They appear in partial differential equations (where they are called plane waves), in computerized tomography, and in statistics. Ridge functions are also the underpinnings of many central models in neural network theory. In this book various approximation theoretic properties of ridge functions are described. This book also describes properties of generalized ridge functions, and their relation to linear superpositions and Kolmogorov's famous superposition theorem. In the final part of the book, a single and two hidden layer neural networks are discussed. The results obtained in this part are based on properties of ordinary and generalized ridge functions. Novel aspects of the universal approximation property of feedforward neural networks are revealed. This book will be of interest to advanced graduate students and researchers working in functional analysis, approximation theory, and the theory of real functions, and will be of particular interest to those wishing to learn more about neural network theory and applications and other areas where ridge functions are used.

Selected Topics in Complex Analysis

Vladimir Ya. Eiderman,Mikhail V. Samokhin
Selected Topics in Complex Analysis Book PDF

Author : Vladimir Ya. Eiderman,Mikhail V. Samokhin
Publisher : Springer Science & Business Media
Page : 225 pages
File Size : 40,5 Mb
Release : 2006-03-30
Category : Mathematics
ISBN : 9783764373405

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Selected Topics in Complex Analysis by Vladimir Ya. Eiderman,Mikhail V. Samokhin Pdf

This volume opens with a paper by V.P. Havin that presents a comprehensive survey of the work of mathematician S.Ya. Khavinson. It includes a complete bibliography, previously unpublished, of 163 items, and twelve peer-reviewed research and expository papers by leading mathematicians, including the joint paper by Khavinson and T.S. Kuzina. The emphasis is on the usage of tools from functional analysis, potential theory, algebra, and topology.

Ridge Functions

Allan Pinkus
Ridge Functions Book PDF

Author : Allan Pinkus
Publisher : Cambridge University Press
Page : 218 pages
File Size : 44,5 Mb
Release : 2015-08-07
Category : Computers
ISBN : 9781107124394

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Ridge Functions by Allan Pinkus Pdf

Presents the state of the art in the theory of ridge functions, providing a solid theoretical foundation.

Linear Holomorphic Partial Differential Equations and Classical Potential Theory

Dmitry Khavinson,Erik Lundberg
Linear Holomorphic Partial Differential Equations and Classical Potential Theory Book PDF

Author : Dmitry Khavinson,Erik Lundberg
Publisher : American Mathematical Soc.
Page : 214 pages
File Size : 44,5 Mb
Release : 2018-07-09
Category : Differential equations, Linear
ISBN : 9781470437800

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Linear Holomorphic Partial Differential Equations and Classical Potential Theory by Dmitry Khavinson,Erik Lundberg Pdf

Why do solutions of linear analytic PDE suddenly break down? What is the source of these mysterious singularities, and how do they propagate? Is there a mean value property for harmonic functions in ellipsoids similar to that for balls? Is there a reflection principle for harmonic functions in higher dimensions similar to the Schwarz reflection principle in the plane? How far outside of their natural domains can solutions of the Dirichlet problem be extended? Where do the continued solutions become singular and why? This book invites graduate students and young analysts to explore these and many other intriguing questions that lead to beautiful results illustrating a nice interplay between parts of modern analysis and themes in “physical” mathematics of the nineteenth century. To make the book accessible to a wide audience including students, the authors do not assume expertise in the theory of holomorphic PDE, and most of the book is accessible to anyone familiar with multivariable calculus and some basics in complex analysis and differential equations.

Sign-based Methods in Linear Statistical Models

M. V. Boldin,Galina I. Simonova,I_Uri_ Nikolaevich Ti_urin
Sign based Methods in Linear Statistical Models Book PDF

Author : M. V. Boldin,Galina I. Simonova,I_Uri_ Nikolaevich Ti_urin
Publisher : American Mathematical Soc.
Page : 252 pages
File Size : 53,6 Mb
Release : 1997-04-22
Category : Mathematics
ISBN : 0821897764

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Sign-based Methods in Linear Statistical Models by M. V. Boldin,Galina I. Simonova,I_Uri_ Nikolaevich Ti_urin Pdf

For nonparametric statistics, the last half of this century was the time when rank-based methods originated, were vigorously developed, reached maturity, and received wide recognition. The rank-based approach in statistics consists in ranking the observed values and using only the ranks rather than the original numerical data. In fitting relationships to observed data, the ranks of residuals from the fitted dependence are used. The signed-based approach is based on the assumption that random errors take positive or negative values with equal probabilities. Under this assumption, the sign procedures are distribution-free. These procedures are robust to violations of model assumptions, for instance, to even a considerable number of gross errors in observations. In addition, sign procedures have fairly high relative asymptotic efficiency, in spite of the obvious loss of information incurred by the use of signs instead of the corresponding numerical values. In this work, sign-based methods in the framework of linear models are developed. In the first part of the book, there are linear and factor models involving independent observations. In the second part, linear models of time series, primarily autoregressive models, are considered.

Linear and Nonlinear Perturbations of the Operator Div

Viktor Grigorʹevich Osmolovskiĭ
Linear and Nonlinear Perturbations of the Operator Div Book PDF

Author : Viktor Grigorʹevich Osmolovskiĭ
Publisher : American Mathematical Soc.
Page : 126 pages
File Size : 47,8 Mb
Release : 1997-01-01
Category : Mathematics
ISBN : 0821897748

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Linear and Nonlinear Perturbations of the Operator Div by Viktor Grigorʹevich Osmolovskiĭ Pdf

This book presents results onboundary-value problems for L and the theory of nonlinear perturbations of L. Specifically, necessary and sufficient solvability conditions in explicit form are found for various boundary-value problems for the operator L. an analog of the Weyl decomposition is proved.

Ordinary Differential Equations with Constant Coefficient

Serge_ Konstantinovich Godunov
Ordinary Differential Equations with Constant Coefficient Book PDF

Author : Serge_ Konstantinovich Godunov
Publisher : American Mathematical Soc.
Page : 298 pages
File Size : 53,8 Mb
Release : 1997-08-19
Category : Mathematics
ISBN : 0821897799

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Ordinary Differential Equations with Constant Coefficient by Serge_ Konstantinovich Godunov Pdf

This book presents the theory of ordinary differential equations with constant coefficients. The exposition is based on ideas developing the Gelfand-Shilov theorem on the polynomial representation of a matrix exponential. Boundary value problems for ordinary equations, Green matrices, Green functions, the Lopatinskii condition, and Lyapunov stability are considered. This volume can be used for practical study of ordinary differential equations using computers. In particular, algorithms and computational procedures, including the orthogonal sweep method, are described. The book also deals with stationary optimal control systems described by systems of ordinary differential equations with constant coefficients. The notions of controllability, observability, and stabilizability are analyzed, and some questions on the matrix Lure-Riccati equations are studied.

Probability Theory

Anatoli_ I_A_kovlevich Dorogovt_s_ev,A. Ya. Dorogovtsev,D. S. Silvestrov,A. V. Skorokhod,M. I. Yadrenko
Probability Theory Book PDF

Author : Anatoli_ I_A_kovlevich Dorogovt_s_ev,A. Ya. Dorogovtsev,D. S. Silvestrov,A. V. Skorokhod,M. I. Yadrenko
Publisher : American Mathematical Soc.
Page : 362 pages
File Size : 52,6 Mb
Release : 2011-06-21
Category : Mathematics
ISBN : 9780821868669

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Probability Theory by Anatoli_ I_A_kovlevich Dorogovt_s_ev,A. Ya. Dorogovtsev,D. S. Silvestrov,A. V. Skorokhod,M. I. Yadrenko Pdf

This book of problems is intended for students in pure and applied mathematics. There are problems in traditional areas of probability theory and problems in the theory of stochastic processes, which has wide applications in the theory of automatic control, queuing and reliability theories, and in many other modern science and engineering fields. Answers to most of the problems are given, and the book provides hints and solutions for more complicated problems.

Introduction to Complex Analysis

Junjiro Noguchi
Introduction to Complex Analysis Book PDF

Author : Junjiro Noguchi
Publisher : American Mathematical Soc.
Page : 268 pages
File Size : 45,7 Mb
Release : 2008-04-09
Category : Mathematics
ISBN : 0821889605

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Introduction to Complex Analysis by Junjiro Noguchi Pdf

This book describes a classical introductory part of complex analysis for university students in the sciences and engineering and could serve as a text or reference book. It places emphasis on rigorous proofs, presenting the subject as a fundamental mathematical theory. The volume begins with a problem dealing with curves related to Cauchy's integral theorem. To deal with it rigorously, the author gives detailed descriptions of the homotopy of plane curves. Since the residue theorem is important in both pure and applied mathematics, the author gives a fairly detailed explanation of how to apply it to numerical calculations; this should be sufficient for those who are studying complex analysis as a tool.

An Introduction to Algebraic Geometry

Kenji Ueno
An Introduction to Algebraic Geometry Book PDF

Author : Kenji Ueno
Publisher : American Mathematical Soc.
Page : 266 pages
File Size : 48,6 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821811443

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An Introduction to Algebraic Geometry by Kenji Ueno Pdf

This introduction to algebraic geometry allows readers to grasp the fundamentals of the subject with only linear algebra and calculus as prerequisites. After a brief history of the subject, the book introduces projective spaces and projective varieties, and explains plane curves and resolution of their singularities. The volume further develops the geometry of algebraic curves and treats congruence zeta functions of algebraic curves over a finite field. It concludes with a complex analytical discussion of algebraic curves. The author emphasizes computation of concrete examples rather than proofs, and these examples are discussed from various viewpoints. This approach allows readers to develop a deeper understanding of the theorems.

Real Analysis

Satoru Igari
Real Analysis Book PDF

Author : Satoru Igari
Publisher : American Mathematical Soc.
Page : 276 pages
File Size : 52,6 Mb
Release : 1998
Category : Mathematics
ISBN : 0821821040

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Real Analysis by Satoru Igari Pdf

This introduction to real analysis is based on a series of lectures by the author at Tohoku University. The text covers real numbers, the notion of general topology, and a brief treatment of the Riemann integral, followed by chapters on the classical theory of the Lebesgue integral on Euclidean spaces; the differentiation theorem and functions of bounded variation; Lebesgue spaces; distribution theory; the classical theory of the Fourier transform and Fourier series; and wavelet theory.

Mathematics of Fractals

Masaya Yamaguchi,Masayoshi Hata,Jun Kigami
Mathematics of Fractals Book PDF

Author : Masaya Yamaguchi,Masayoshi Hata,Jun Kigami
Publisher : American Mathematical Soc.
Page : 104 pages
File Size : 49,6 Mb
Release : 1997
Category : Mathematics
ISBN : 0821805371

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Mathematics of Fractals by Masaya Yamaguchi,Masayoshi Hata,Jun Kigami Pdf

This book aims at providing a handy explanation of the notions behind the self-similar sets called "fractals" and "chaotic dynamical systems". The authors emphasize the beautiful relationship between fractal functions (such as Weierstrass's) and chaotic dynamical systems; these nowhere-differentiable functions are generating functions of chaotic dynamical systems. These functions are shown to be in a sense unique solutions of certain boundary problems. The last chapter of the book treats harmonic functions on fractal sets.

Mathematics of Information and Coding

Te Sun Han,Kingo Kobayashi
Mathematics of Information and Coding Book PDF

Author : Te Sun Han,Kingo Kobayashi
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 44,7 Mb
Release : 2002
Category : Computers
ISBN : 0821842560

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Mathematics of Information and Coding by Te Sun Han,Kingo Kobayashi Pdf

This book is intended to provide engineering and/or statistics students, communications engineers, and mathematicians with the firm theoretic basis of source coding (or data compression) in information theory. Although information theory consists of two main areas, source coding and channel coding, the authors choose here to focus only on source coding. The reason is that, in a sense, it is more basic than channel coding, and also because of recent achievements in source coding and compression. An important feature of the book is that whenever possible, the authors describe universal coding methods, i.e., the methods that can be used without prior knowledge of the statistical properties of the data. The authors approach the subject of source coding from the very basics to the top frontiers in an intuitively transparent, but mathematically sound, manner. The book serves as a theoretical reference for communication professionals and statisticians specializing in information theory. It will also serve as an excellent introductory text for advanced-level and graduate students taking elementary or advanced courses in telecommunications, electrical engineering, statistics, mathematics, and computer science.