Integration - A Functional Approach

Integration - A Functional Approach

Author: Klaus Bichteler

Publisher: Springer Science & Business Media

Published: 2010-11-23

Total Pages: 203

ISBN-13: 303480055X

DOWNLOAD EBOOK

This book covers Lebesgue integration and its generalizations from Daniell's point of view, modified by the use of seminorms. Integrating functions rather than measuring sets is posited as the main purpose of measure theory. From this point of view Lebesgue's integral can be had as a rather straightforward, even simplistic, extension of Riemann's integral; and its aims, definitions, and procedures can be motivated at an elementary level. The notion of measurability, for example, is suggested by Littlewood's observations rather than being conveyed authoritatively through definitions of (sigma)-algebras and good-cut-conditions, the latter of which are hard to justify and thus appear mysterious, even nettlesome, to the beginner. The approach taken provides the additional benefit of cutting the labor in half. The use of seminorms, ubiquitous in modern analysis, speeds things up even further. The book is intended for the reader who has some experience with proofs, a beginning graduate student for example. It might even be useful to the advanced mathematician who is confronted with situations - such as stochastic integration - where the set-measuring approach to integration does not work.


A Modern Approach to Functional Integration

A Modern Approach to Functional Integration

Author: John R. Klauder

Publisher: Springer Science & Business Media

Published: 2010-11-08

Total Pages: 292

ISBN-13: 0817647910

DOWNLOAD EBOOK

This text takes advantage of recent developments in the theory of path integration and attempts to make a major paradigm shift in how the art of functional integration is practiced. The techniques developed in the work will prove valuable to graduate students and researchers in physics, chemistry, mathematical physics, and applied mathematics who find it necessary to deal with solutions to wave equations, both quantum and beyond. A Modern Approach to Functional Integration offers insight into a number of contemporary research topics, which may lead to improved methods and results that cannot be found elsewhere in the textbook literature. Exercises are included in most chapters, making the book suitable for a one-semester graduate course on functional integration.


Functional Integration

Functional Integration

Author: Pierre Cartier

Publisher: Cambridge University Press

Published: 2006-11-30

Total Pages: 7

ISBN-13: 1139462881

DOWNLOAD EBOOK

In this text, Cartier and DeWitt-Morette, using their complementary interests and expertise, successfully condense and apply the essentials of Functional Integration to a great variety of systems, showing this mathematically elusive technique to be a robust, user friendly and multipurpose tool.


Functional Integration

Functional Integration

Author: Cécile Dewitt-Morette

Publisher: Springer Science & Business Media

Published: 2013-11-11

Total Pages: 436

ISBN-13: 1489903194

DOWNLOAD EBOOK

The program of the Institute covered several aspects of functional integration -from a robust mathematical foundation to many applications, heuristic and rigorous, in mathematics, physics, and chemistry. It included analytic and numerical computational techniques. One of the goals was to encourage cross-fertilization between these various aspects and disciplines. The first week was focused on quantum and classical systems with a finite number of degrees of freedom; the second week on field theories. During the first week the basic course, given by P. Cartier, was a presentation of a recent rigorous approach to functional integration which does not resort to discretization, nor to analytic continuation. It provides a definition of functional integrals simpler and more powerful than the original ones. Could this approach accommodate the works presented by the other lecturers? Although much remains to be done before answering "Yes," there seems to be no major obstacle along the road. The other courses taught during the first week presented: a) a solid introduction to functional numerical techniques (A. Sokal) and their applications to functional integrals encountered in chemistry (N. Makri). b) integrals based on Poisson processes and their applications to wave propagation (S. K. Foong), in particular a wave-restorer or wave-designer algorithm yielding the initial wave profile when one can only observe its distortion through a dissipative medium. c) the formulation of a quantum equivalence principle (H. Kleinert) which. given the flat space theory, yields a well-defined quantum theory in spaces with curvature and torsion.


Functional Integration and Quantum Physics

Functional Integration and Quantum Physics

Author: Barry Simon

Publisher: American Mathematical Soc.

Published: 2005

Total Pages: 322

ISBN-13: 0821835823

DOWNLOAD EBOOK

Focuses on probabilistic foundations of the Feynman-Kac formula. Starting with main examples of Gaussian processes (the Brownian motion, the oscillatory process, and the Brownian bridge), this book presents four different proofs of the Feynman-Kac formula.


A Radical Approach to Lebesgue's Theory of Integration

A Radical Approach to Lebesgue's Theory of Integration

Author: David M. Bressoud

Publisher: Cambridge University Press

Published: 2008-01-21

Total Pages: 15

ISBN-13: 0521884748

DOWNLOAD EBOOK

Meant for advanced undergraduate and graduate students in mathematics, this introduction to measure theory and Lebesgue integration is motivated by the historical questions that led to its development. The author tells the story of the mathematicians who wrestled with the difficulties inherent in the Riemann integral, leading to the work of Jordan, Borel, and Lebesgue.


Functional Integrals in Quantum Field Theory and Statistical Physics

Functional Integrals in Quantum Field Theory and Statistical Physics

Author: V.N. Popov

Publisher: Springer Science & Business Media

Published: 2001-11-30

Total Pages: 316

ISBN-13: 9781402003073

DOWNLOAD EBOOK

Functional integration is one of the most powerful methods of contempo rary theoretical physics, enabling us to simplify, accelerate, and make clearer the process of the theoretician's analytical work. Interest in this method and the endeavour to master it creatively grows incessantly. This book presents a study of the application of functional integration methods to a wide range of contemporary theoretical physics problems. The concept of a functional integral is introduced as a method of quantizing finite-dimensional mechanical systems, as an alternative to ordinary quantum mechanics. The problems of systems quantization with constraints and the manifolds quantization are presented here for the first time in a monograph. The application of the functional integration methods to systems with an infinite number of degrees of freedom allows one to uniquely introduce and formulate the diagram perturbation theory in quantum field theory and statistical physics. This approach is significantly simpler than the widely accepted method using an operator approach.


Vector Integration and Stochastic Integration in Banach Spaces

Vector Integration and Stochastic Integration in Banach Spaces

Author: Nicolae Dinculeanu

Publisher: John Wiley & Sons

Published: 2000-02-04

Total Pages: 482

ISBN-13: 9780471377382

DOWNLOAD EBOOK

A breakthrough approach to the theory and applications of stochastic integration The theory of stochastic integration has become an intensely studied topic in recent years, owing to its extraordinarily successful application to financial mathematics, stochastic differential equations, and more. This book features a new measure theoretic approach to stochastic integration, opening up the field for researchers in measure and integration theory, functional analysis, probability theory, and stochastic processes. World-famous expert on vector and stochastic integration in Banach spaces Nicolae Dinculeanu compiles and consolidates information from disparate journal articles-including his own results-presenting a comprehensive, up-to-date treatment of the theory in two major parts. He first develops a general integration theory, discussing vector integration with respect to measures with finite semivariation, then applies the theory to stochastic integration in Banach spaces. Vector Integration and Stochastic Integration in Banach Spaces goes far beyond the typical treatment of the scalar case given in other books on the subject. Along with such applications of the vector integration as the Reisz representation theorem and the Stieltjes integral for functions of one or two variables with finite semivariation, it explores the emergence of new classes of summable processes that make applications possible, including square integrable martingales in Hilbert spaces and processes with integrable variation or integrable semivariation in Banach spaces. Numerous references to existing results supplement this exciting, breakthrough work.


Lean Integration

Lean Integration

Author: John G. Schmidt

Publisher: Pearson Education

Published: 2010-05-18

Total Pages: 685

ISBN-13: 0321712390

DOWNLOAD EBOOK

Use Lean Techniques to Integrate Enterprise Systems Faster, with Far Less Cost and Risk By some estimates, 40 percent of IT budgets are devoted to integration. However, most organizations still attack integration on a project-by-project basis, causing unnecessary expense, waste, risk, and delay. They struggle with integration “hairballs”: complex point-to-point information exchanges that are expensive to maintain, difficult to change, and unpredictable in operation. The solution is Lean Integration. This book demonstrates how to use proven “lean” techniques to take control over the entire integration process. John Schmidt and David Lyle show how to establish “integration factories” that leverage the powerful benefits of repeatability and continuous improvement across every integration project you undertake. Drawing on their immense experience, Schmidt and Lyle bring together best practices; solid management principles; and specific, measurable actions for streamlining integration development and maintenance. Whether you’re an IT manager, project leader, architect, analyst, or developer, this book will help you systematically improve the way you integrate—adding value that is both substantial and sustainable. Coverage includes Treating integration as a business strategy and implementing management disciplines that systematically address its people, process, policy, and technology dimensions Providing maximum business flexibility and supporting rapid change without compromising stability, quality, control, or efficiency Applying improvements incrementally without “Boiling the Ocean” Automating processes so you can deliver IT solutions faster–while avoiding the pitfalls of automation Building in both data and integration quality up front, rather than inspecting quality in later More than a dozen in-depth case studies that show how real organizations are applying Lean Integration practices and the lessons they’ve learned Visit integrationfactory.com for additional resources, including more case studies, best practices, templates, software demos, and reference links, plus a direct connection to lean integration practitioners worldwide.


Measure, Integration & Real Analysis

Measure, Integration & Real Analysis

Author: Sheldon Axler

Publisher: Springer Nature

Published: 2019-11-29

Total Pages: 430

ISBN-13: 3030331431

DOWNLOAD EBOOK

This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online. For errata and updates, visit https://measure.axler.net/