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Distributionen und Hilbertraumoperatoren: Mathematische Methoden der Physik

Das Buch bietet eine Einführung in die zum Studium der Theoretischen Physik notwendigen mathematischen Grundlagen. Der erste Teil des Buches beschäftigt sich mit der Theorie der Distributionen und vermittelt daneben einige Grundbegriffe der linearen Funktionalanalysis. Der zweite Teil baut darauf auf und gibt eine auf das Wesentliche beschränkte Einführung in die Theorie der linearen Operatoren in Hilbert-Räumen. Beide Teile werden von je einer Übersicht begleitet, die die zentralen Ideen und Begriffe knapp erläutert und den Inhalt kurz beschreibt. In den Anhängen werden einige grundlegende Konstruktionen und Konzepte der Funktionalanalysis dargestellt und wichtige Konsequenzen entwickelt.

Distributionen und ihre Anwendung in der Physik

Das vorliegende Buch stellt eine Einführung in die Theorie der Distributionen (verallge­ meinerte Funktionen) und ihrer Anwendungen in der Physik dar. Der zum Verständnis der Theorie notwendige topologische Apparat wurde auf ein Minimum reduziert. Lediglich das erste Kapitel gibt eine Einführung in die Theorie der abzählbar normierten Räume. Es wird angenommen, daß der Leser vertraut mit den elementaren Begriffen der Funktionalanalysis (Hilbert- und Banachraum) ist. Das Buch enthält die bereits klassisch gewordenen Kapitel der Theorie der Distributionen, wie: Lokale Eigenschaften von Distributionen, Distributionen mit kompaktem Träger, temperierte Distributionen, Regularisierung divergenter Integrale, Fourier- und Fourier­ Laplace-Transformation, den Satz von Paley-Wiener-Schwartz, Distributionen als Rand­ werte analytischer Funktionen usw. In Kapitel 11 werden Distributionen untersucht, die auf Flächen konzentriert sind; insbesondere auf dem Lichtkegel konzentrierte Distri­ butionen. In den Kapiteln 8, 9, 10 werden verschiedene Anwendungen der Theorie der Distributionen in der relativistischen Physik (Feldtheorie) entwickelt. Kapitel 12 schließlich enthält Probleme der Theorie der Distributionen im Hilbertraum und ihre Anwendungen in der Quantenphysik (Vertauschungsrelationen, Fock-Raum, Quanten­ feldtheorie usw.). Das Buch wendet sich sowohl an Mathematiker, die auch die Anwendungen der Theorie der Distributionen in der Physik kennenlernen wollen; als auch an Physiker, die sich für die Theorie der Distributionen als Teilgebiet der mathematischen und theoretischen Physik interessieren. Das vorliegende Buch entstand aus Vorlesungen, die ich im Jahre 1970 als Humboldt­ Stipendiat an der Universität München gehalten habe. Mein besonderer Dank gilt daher an dieser Stelle Herrn Prof. Dr. W. Güttinger für die Unterstützung in meinen ersten Arbeitsjahren in Deutschland.

Distributionen und Operatoren: Ihre Anwendung in Naturwissenschaft und Technik

Distributions: Theory and Applications (Cornerstones)

This textbook is an application-oriented introduction to the theory of distributions, a powerful tool used in mathematical analysis. The treatment emphasizes applications that relate distributions to linear partial differential equations and Fourier analysis problems found in mechanics, optics, quantum mechanics, quantum field theory, and signal analysis. The book is motivated by many exercises, hints, and solutions that guide the reader along a path requiring only a minimal mathematical background.

Distributions

This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same &“weak&” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting.This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.

Distributions

This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same &“weak&” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting.This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent sub-sequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.

Distributions and Nonlinear Partial Differential Equations (Lecture Notes in Mathematics #684)

Distributions and Operators: (pdf) (Graduate Texts in Mathematics #252)

Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R (Chapman & Hall/CRC The R Series)

This is a book about statistical distributions, their properties, and their application to modelling the dependence of the location, scale, and shape of the distribution of a response variable on explanatory variables. It will be especially useful to applied statisticians and data scientists in a wide range of application areas, and also to those interested in the theoretical properties of distributions. This book follows the earlier book ‘Flexible Regression and Smoothing: Using GAMLSS in R’, [Stasinopoulos et al., 2017], which focused on the GAMLSS model and software. GAMLSS (the Generalized Additive Model for Location, Scale, and Shape, [Rigby and Stasinopoulos, 2005]), is a regression framework in which the response variable can have any parametric distribution and all the distribution parameters can be modelled as linear or smooth functions of explanatory variables. The current book focuses on distributions and their application. Key features: Describes over 100 distributions, (implemented in the GAMLSS packages in R), including continuous, discrete and mixed distributions. Comprehensive summary tables of the properties of the distributions. Discusses properties of distributions, including skewness, kurtosis, robustness and an important classification of tail heaviness. Includes mixed distributions which are continuous distributions with additional specific values with point probabilities. Includes many real data examples, with R code integrated in the text for ease of understanding and replication. Supplemented by the gamlss website. This book will be useful for applied statisticians and data scientists in selecting a distribution for a univariate response variable and modelling its dependence on explanatory variables, and to those interested in the properties of distributions.

Distributions for Modeling Location, Scale, and Shape: Using GAMLSS in R (Chapman & Hall/CRC The R Series)

This is a book about statistical distributions, their properties, and their application to modelling the dependence of the location, scale, and shape of the distribution of a response variable on explanatory variables. It will be especially useful to applied statisticians and data scientists in a wide range of application areas, and also to those interested in the theoretical properties of distributions. This book follows the earlier book ‘Flexible Regression and Smoothing: Using GAMLSS in R’, [Stasinopoulos et al., 2017], which focused on the GAMLSS model and software. GAMLSS (the Generalized Additive Model for Location, Scale, and Shape, [Rigby and Stasinopoulos, 2005]), is a regression framework in which the response variable can have any parametric distribution and all the distribution parameters can be modelled as linear or smooth functions of explanatory variables. The current book focuses on distributions and their application. Key features: Describes over 100 distributions, (implemented in the GAMLSS packages in R), including continuous, discrete and mixed distributions. Comprehensive summary tables of the properties of the distributions. Discusses properties of distributions, including skewness, kurtosis, robustness and an important classification of tail heaviness. Includes mixed distributions which are continuous distributions with additional specific values with point probabilities. Includes many real data examples, with R code integrated in the text for ease of understanding and replication. Supplemented by the gamlss website. This book will be useful for applied statisticians and data scientists in selecting a distribution for a univariate response variable and modelling its dependence on explanatory variables, and to those interested in the properties of distributions.

Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets (Applied and Numerical Harmonic Analysis)

A comprehensive exposition on analytic methods for solving science and engineering problems, written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important to practioners and researchers. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise.

Distributions in the Physical and Engineering Sciences, Volume 1: Distributional and Fractal Calculus, Integral Transforms and Wavelets (Applied and Numerical Harmonic Analysis)

Distributions in the Physical and Engineering Sciences is a comprehensive exposition on analytic methods for solving science and engineering problems which is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important to practitioners and researchers. The goal of the book is to give the reader, specialist and non-specialist usable and modern mathematical tools in their research and analysis. This new text is intended for graduate students and researchers in applied mathematics, physical sciences and engineering. The careful explanations, accessible writing style, and many illustrations/examples also make it suitable for use as a self-study reference by anyone seeking greater understanding and proficiency in the problem solving methods presented. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise. The present, softcover reprint is designed to make this classic textbook available to a wider audience.

Distributions in the Physical and Engineering Sciences, Volume 2: Linear and Nonlinear Dynamics in Continuous Media (Applied and Numerical Harmonic Analysis)

Distributions in the Physical and Engineering Sciences is a comprehensive exposition on analytic methods for solving science and engineering problems. It is written from the unifying viewpoint of distribution theory and enriched with many modern topics which are important for practitioners and researchers. The goal of the books is to give the reader, specialist and non-specialist, useable and modern mathematical tools in their research and analysis. Volume 2: Linear and Nonlinear Dynamics of Continuous Media continues the multivolume project which endeavors to show how the theory of distributions, also called the theory of generalized functions, can be used by graduate students and researchers in applied mathematics, physical sciences, and engineering. It contains an analysis of the three basic types of linear partial differential equations--elliptic, parabolic, and hyperbolic--as well as chapters on first-order nonlinear partial differential equations and conservation laws, and generalized solutions of first-order nonlinear PDEs. Nonlinear wave, growing interface, and Burger’s equations, KdV equations, and the equations of gas dynamics and porous media are also covered. The careful explanations, accessible writing style, many illustrations/examples and solutions also make it suitable for use as a self-study reference by anyone seeking greater understanding and proficiency in the problem solving methods presented. The book is ideal for a general scientific and engineering audience, yet it is mathematically precise. Features· Application oriented exposition of distributional (Dirac delta) methods in the theory of partial differential equations. Abstract formalism is keep to a minimum.· Careful and rich selection of examples and problems arising in real-life situations. Complete solutions to all exercises appear at the end of the book.· Clear explanations, motivations, and illustration of all necessary mathematical concepts.

Distributions in the Physical and Engineering Sciences, Volume 3: Random and Anomalous Fractional Dynamics in Continuous Media (Applied and Numerical Harmonic Analysis)

Continuing the authors’ multivolume project, this text considers the theory of distributions from an applied perspective, demonstrating how effective a combination of analytic and probabilistic methods can be for solving problems in the physical and engineering sciences. Volume 1 covered foundational topics such as distributional and fractional calculus, the integral transform, and wavelets, and Volume 2 explored linear and nonlinear dynamics in continuous media. With this volume, the scope is extended to the use of distributional tools in the theory of generalized stochastic processes and fields, and in anomalous fractional random dynamics. Chapters cover topics such as probability distributions; generalized stochastic processes, Brownian motion, and the white noise; stochastic differential equations and generalized random fields; Burgers turbulence and passive tracer transport in Burgers flows; and linear, nonlinear, and multiscale anomalous fractional dynamics in continuous media. The needs of the applied-sciences audience are addressed by a careful and rich selection of examples arising in real-life industrial and scientific labs and a thorough discussion of their physical significance. Numerous illustrations generate a better understanding of the core concepts discussed in the text, and a large number of exercises at the end of each chapter expand on these concepts.Distributions in the Physical and Engineering Sciences is intended to fill a gap in the typical undergraduate engineering/physical sciences curricula, and as such it will be a valuable resource for researchers and graduate students working in these areas. The only prerequisites are a three-four semester calculus sequence (including ordinary differential equations, Fourier series, complex variables, and linear algebra), and some probability theory, but basic definitions and facts are covered as needed. An appendix also provides background material concerning the Dirac-delta and other distributions.

Distributions (Large Print)

This diagram has two line graphs, one at the top and one at the bottom of the page which show skewed distributions. There is a locator dot shown, which will be at the top left of the page when the image is the right way up. Both y-axes show frequency, and the position of the mode, median and mean, indicated by three different sorts of line, labelled in the key at the bottom of the page. The graph at the top shows a negatively skewed distribution, where most of the scores occur at the higher values. The line on this graph rises gradually to a peak and then falls steeply thereafter, so that it is skewed towards the right. The graph at the bottom of the page shows a positively skewed distribution where most of the scores occur at the lower values. The line on this graph rises sharply to a peak and then falls gradually thereafter, so that it is skewed to the left.

Distributions of Correlation Coefficients

An important problem in personnel psychology, namely, the psychometric problem known as "validity generalization" is addressed in this volume. From a statistical point of view, the problem is how to make statements about a population correlation coefficient based on inferences from a collection of sample correlation coefficients. The first part of the book examines the largely ad hoc procedures which have been used to determine validity generalization. The second part develops a new model formulated from the perspective of finite mixture theory and, in addition, illustrates its use in several applications.

Distributions, Partial Differential Equations, and Harmonic Analysis (Universitext)

The aim of this book is to offer, in a concise, rigorous, and largely self-contained manner, a rapid introduction to the theory of distributions and its applications to partial differential equations and harmonic analysis. The book is written in a format suitable for a graduate course spanning either over one-semester, when the focus is primarily on the foundational aspects, or over a two-semester period that allows for the proper amount of time to cover all intended applications as well. It presents a balanced treatment of the topics involved, and contains a large number of exercises (upwards of two hundred, more than half of which are accompanied by solutions), which have been carefully chosen to amplify the effect, and substantiate the power and scope, of the theory of distributions. Graduate students, professional mathematicians, and scientifically trained people with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. Throughout, a special effort has been made to develop the theory of distributions not as an abstract edifice but rather give the reader a chance to see the rationale behind various seemingly technical definitions, as well as the opportunity to apply the newly developed tools (in the natural build-up of the theory) to concrete problems in partial differential equations and harmonic analysis, at the earliest opportunity.The main additions to the current, second edition, pertain to fundamental solutions (through the inclusion of the Helmholtz operator, the perturbed Dirac operator, and their iterations) and the theory of Sobolev spaces (built systematically from the ground up, exploiting natural connections with the Fourier Analysis developed earlier in the monograph).

Distributions, Partial Differential Equations, and Harmonic Analysis (Universitext)

​The theory of distributions constitutes an essential tool in the study of partial differential equations. This textbook would offer, in a concise, largely self-contained form, a rapid introduction to the theory of distributions and its applications to partial differential equations, including computing fundamental solutions for the most basic differential operators: the Laplace, heat, wave, Lam\'e and Schrodinger operators.​

Distributions (UEB Contracted)

This diagram has two line graphs, one at the top and one at the bottom of the page which show skewed distributions. There is a locator dot shown, which will be at the top left of the page when the image is the right way up. Both y-axes show frequency, and the position of the mode, median and mean, indicated by three different sorts of line, labelled in the key at the bottom of the page. The graph at the top shows a negatively skewed distribution, where most of the scores occur at the higher values. The line on this graph rises gradually to a peak and then falls steeply thereafter, so that it is skewed towards the right. The graph at the bottom of the page shows a positively skewed distribution where most of the scores occur at the lower values. The line on this graph rises sharply to a peak and then falls gradually thereafter, so that it is skewed to the left.

Distributions (UEB Uncontracted)

This diagram has two line graphs, one at the top and one at the bottom of the page which show skewed distributions. There is a locator dot shown, which will be at the top left of the page when the image is the right way up. Both y-axes show frequency, and the position of the mode, median and mean, indicated by three different sorts of line, labelled in the key at the bottom of the page. The graph at the top shows a negatively skewed distribution, where most of the scores occur at the higher values. The line on this graph rises gradually to a peak and then falls steeply thereafter, so that it is skewed towards the right. The graph at the bottom of the page shows a positively skewed distribution where most of the scores occur at the lower values. The line on this graph rises sharply to a peak and then falls gradually thereafter, so that it is skewed to the left.

Distributions with given Marginals and Moment Problems

The last decade has seen a remarkable development of the "Marginal and Moment Problems" as a research area in Probability and Statistics. Its attractiveness stemmed from its lasting ability to provide a researcher with difficult theoretical problems that have direct consequences for appli­ cations outside of mathematics. The relevant research aims centered mainly along the following lines that very frequently met each other to provide sur­ prizing and useful results : -To construct a probability distribution (to prove its existence, at least) with a given support and with some additional inner stochastic property defined typically either by moments or by marginal distributions. -To study the geometrical and topological structure of the set of prob­ ability distributions generated by such a property mostly with the aim to propose a procedure that would result in a stochastic model with some optimal properties within the set of probability distributions. These research aims characterize also, though only very generally, the scientific program of the 1996 conference "Distributions with given marginals and moment problems" held at the beginning of September in Prague, Czech Republic, to perpetuate the tradition and achievements of the closely related 1990 Roma symposium "On Frechet Classes" 1 and 1993 Seattle" AMS Summer Conference on Marginal Problem".

Distributions With Given Marginals and Statistical Modelling

Distributivity-like Results in the Medieval Traditions of Euclid's Elements: Between Geometry and Arithmetic (SpringerBriefs in History of Science and Technology)

This book provides a fresh view on an important and largely overlooked aspect of the Euclidean traditions in the medieval mathematical texts, particularly concerning the interrelations between geometry and arithmetic, and the rise of algebraic modes of thought. It appeals to anyone interested in the history of mathematics in general and in history of medieval and early modern science.

Divergence Operator and Related Inequalities (SpringerBriefs in Mathematics)

This Brief is mainly devoted to two classical and related results: the existence of a right inverse of the divergence operator and the so-called Korn Inequalities. It is well known that both results are fundamental tools in the analysis of some classic differential equations, particularly in those arising in fluid dynamics and elasticity. Several connections between these two topics and improved Poincaré inequalities are extensively treated. From simple key ideas the book is growing smoothly in complexity. Beginning with the study of these problems on star-shaped domains the arguments are extended first to John domains and then to Hölder α domains where the need of weighted spaces arises naturally. In this fashion, the authors succeed in presenting in an unified and concise way several classic and recent developments in the field. These features certainly makes this Brief useful for students, post-graduate students, and researchers as well.

Divergent Series, Summability and Resurgence I: Monodromy and Resurgence (Lecture Notes in Mathematics #2153)

Providing an elementary introduction to analytic continuation and monodromy, the first part of this volume applies these notions to the local and global study of complex linear differential equations, their formal solutions at singular points, their monodromy and their differential Galois groups. The Riemann-Hilbert problem is discussed from Bolibrukh’s point of view. The second part expounds 1-summability and Ecalle’s theory of resurgence under fairly general conditions. It contains numerous examples and presents an analysis of the singularities in the Borel plane via “alien calculus”, which provides a full description of the Stokes phenomenon for linear or non-linear differential or difference equations. The first of a series of three, entitled Divergent Series, Summability and Resurgence, this volume is aimed at graduate students, mathematicians and theoretical physicists interested in geometric, algebraic or local analytic properties of dynamical systems. It includes useful exercises with solutions. The prerequisites are a working knowledge of elementary complex analysis and differential algebra.