Algebraic Probability Theory

Algebraic Probability Theory

Author: Imre Z. Ruzsa

Publisher: John Wiley & Sons Incorporated

ISBN: UOM:39015014348950

Category: Mathematics

Page: 272

View: 980

A large part of probability theory is the study of operations on, and convergence of, probability distributions. The most frequently used operations turn the set of distributions into a semigroup. A considerable part of probability theory can be expressed, proved, sometimes even understood in terms of the abstract theory of topological semigroups. The authors 'algebraic probability theory' is a field where problems stem mainly from probability theory, have an arithmetical flair and are often dressed in terms of algebra, while the tools employed frequently belong to the theory of (complex) functions and abstract harmonic analysis. It lies at the cross-roads of numerous mathematical theories, and should serve as a catalyst to further research.

Algebraic Probability Theory

Algebraic Probability Theory

Author: Imre Z. Ruzsa

Publisher: John Wiley & Sons Incorporated

ISBN: MINN:31951000422281T

Category: Mathematics

Page: 272

View: 660

A large part of probability theory is the study of operations on, and convergence of, probability distributions. The most frequently used operations turn the set of distributions into a semigroup. A considerable part of probability theory can be expressed, proved, sometimes even understood in terms of the abstract theory of topological semigroups. The authors 'algebraic probability theory' is a field where problems stem mainly from probability theory, have an arithmetical flair and are often dressed in terms of algebra, while the tools employed frequently belong to the theory of (complex) functions and abstract harmonic analysis. It lies at the cross-roads of numerous mathematical theories, and should serve as a catalyst to further research.

Probability Theory II

Probability Theory II

Author: M. Loeve

Publisher: Springer Science & Business Media

ISBN: 9780387902623

Category: Mathematics

Page: 416

View: 494

This book is intended as a text for graduate students and as a reference for workers in probability and statistics. The prerequisite is honest calculus. The material covered in Parts Two to Five inclusive requires about three to four semesters of graduate study. The introductory part may serve as a text for an undergraduate course in elementary probability theory. Numerous historical marks about results, methods, and the evolution of various fields are an intrinsic part of the text. About a third of the second volume is devoted to conditioning and properties of sequences of various types of dependence. The other two thirds are devoted to random functions; the last Part on Elements of random analysis is more sophisticated.

Probability Theory I

Probability Theory I

Author: M. Loeve

Publisher: Springer Science & Business Media

ISBN: 0387902104

Category: Mathematics

Page: 428

View: 775

This fourth edition contains several additions. The main ones con cern three closely related topics: Brownian motion, functional limit distributions, and random walks. Besides the power and ingenuity of their methods and the depth and beauty of their results, their importance is fast growing in Analysis as well as in theoretical and applied Proba bility. These additions increased the book to an unwieldy size and it had to be split into two volumes. About half of the first volume is devoted to an elementary introduc tion, then to mathematical foundations and basic probability concepts and tools. The second half is devoted to a detailed study of Independ ence which played and continues to playa central role both by itself and as a catalyst. The main additions consist of a section on convergence of probabilities on metric spaces and a chapter whose first section on domains of attrac tion completes the study of the Central limit problem, while the second one is devoted to random walks. About a third of the second volume is devoted to conditioning and properties of sequences of various types of dependence. The other two thirds are devoted to random functions; the last Part on Elements of random analysis is more sophisticated. The main addition consists of a chapter on Brownian motion and limit distributions.

Mathematics of the 19th Century

Mathematics of the 19th Century

Author: A.N. Kolmogorov

Publisher: Springer Science & Business Media

ISBN: 3764364416

Category: Mathematics

Page: 308

View: 685

This multi-authored effort, Mathematics of the nineteenth century (to be fol lowed by Mathematics of the twentieth century), is a sequel to the History of mathematics from antiquity to the early nineteenth century, published in three volumes from 1970 to 1972. 1 For reasons explained below, our discussion of twentieth-century mathematics ends with the 1930s. Our general objectives are identical with those stated in the preface to the three-volume edition, i. e. , we consider the development of mathematics not simply as the process of perfecting concepts and techniques for studying real-world spatial forms and quantitative relationships but as a social process as well. Mathematical structures, once established, are capable of a certain degree of autonomous development. In the final analysis, however, such immanent mathematical evolution is conditioned by practical activity and is either self-directed or, as is most often the case, is determined by the needs of society. Proceeding from this premise, we intend, first, to unravel the forces that shape mathe matical progress. We examine the interaction of mathematics with the social structure, technology, the natural sciences, and philosophy. Through an anal ysis of mathematical history proper, we hope to delineate the relationships among the various mathematical disciplines and to evaluate mathematical achievements in the light of the current state and future prospects of the science. The difficulties confronting us considerably exceeded those encountered in preparing the three-volume edition.

Cylindric-like Algebras and Algebraic Logic

Cylindric-like Algebras and Algebraic Logic

Author: Hajnal Andréka

Publisher: Springer Science & Business Media

ISBN: 9783642350252

Category: Mathematics

Page: 474

View: 376

Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.​

Selected Papers on Analysis and Related Topics

Selected Papers on Analysis and Related Topics

Author:

Publisher: American Mathematical Soc.

ISBN: 0821839284

Category: Mathematics

Page: 178

View: 353

This volume contains translations of papers that originally appeared in the Japanese journal Sugaku. The papers range over a variety of topics, including operator algebras, analysis, and statistics. This volume is suitable for graduate students and research mathematicians interested in analysis and its applications.

Probability Measures on Groups X

Probability Measures on Groups X

Author: H. Heyer

Publisher: Springer Science & Business Media

ISBN: 9781489923646

Category: Mathematics

Page: 498

View: 833

The present volume contains the transactions of the lOth Oberwolfach Conference on "Probability Measures on Groups". The series of these meetings inaugurated in 1970 by L. Schmetterer and the editor is devoted to an intensive exchange of ideas on a subject which developed from the relations between various topics of mathematics: measure theory, probability theory, group theory, harmonic analysis, special functions, partial differential operators, quantum stochastics, just to name the most significant ones. Over the years the fruitful interplay broadened in various directions: new group-related structures such as convolution algebras, generalized translation spaces, hypercomplex systems, and hypergroups arose from generalizations as well as from applications, and a gradual refinement of the combinatorial, Banach-algebraic and Fourier analytic methods led to more precise insights into the theory. In a period of highest specialization in scientific thought the separated minds should be reunited by actively emphasizing similarities, analogies and coincidences between ideas in their fields of research. Although there is no real separation between one field and another - David Hilbert denied even the existence of any difference between pure and applied mathematics - bridges between probability theory on one side and algebra, topology and geometry on the other side remain absolutely necessary. They provide a favorable ground for the communication between apparently disjoint research groups and motivate the framework of what is nowadays called "Structural probability theory".

Mathematics of the 19th Century

Mathematics of the 19th Century

Author: KOLMOGOROV

Publisher: Birkhäuser

ISBN: 9783034851121

Category: Mathematics

Page: 308

View: 381

This multi-authored effort, Mathematics of the nineteenth century (to be fol lowed by Mathematics of the twentieth century), is a sequel to the History of mathematics fram antiquity to the early nineteenth century, published in three 1 volumes from 1970 to 1972. For reasons explained below, our discussion of twentieth-century mathematics ends with the 1930s. Our general objectives are identical with those stated in the preface to the three-volume edition, i. e. , we consider the development of mathematics not simply as the process of perfecting concepts and techniques for studying real-world spatial forms and quantitative relationships but as a social process as weIl. Mathematical structures, once established, are capable of a certain degree of autonomous development. In the final analysis, however, such immanent mathematical evolution is conditioned by practical activity and is either self-directed or, as is most often the case, is determined by the needs of society. Proceeding from this premise, we intend, first, to unravel the forces that shape mathe matical progress. We examine the interaction of mathematics with the social structure, technology, the natural sciences, and philosophy. Throughan anal ysis of mathematical history proper, we hope to delineate the relationships among the various mathematical disciplines and to evaluate mathematical achievements in the light of the current state and future prospects of the science. The difficulties confronting us considerably exceeded those encountered in preparing the three-volume edition.

The Logico-Algebraic Approach to Quantum Mechanics

The Logico-Algebraic Approach to Quantum Mechanics

Author: C.A. Hooker

Publisher: Springer Science & Business Media

ISBN: 9789401017954

Category: Science

Page: 622

View: 998

The twentieth century has witnessed a striking transformation in the un derstanding of the theories of mathematical physics. There has emerged clearly the idea that physical theories are significantly characterized by their abstract mathematical structure. This is in opposition to the tradi tional opinion that one should look to the specific applications of a theory in order to understand it. One might with reason now espouse the view that to understand the deeper character of a theory one must know its abstract structure and understand the significance of that struc ture, while to understand how a theory might be modified in light of its experimental inadequacies one must be intimately acquainted with how it is applied. Quantum theory itself has gone through a development this century which illustrates strikingly the shifting perspective. From a collection of intuitive physical maneuvers under Bohr, through a formative stage in which the mathematical framework was bifurcated (between Schrödinger and Heisenberg) to an elegant culmination in von Neumann's Hilbert space formulation the elementary theory moved, flanked even at the later stage by the ill-understood formalisms for the relativistic version and for the field-theoretic altemative; after that we have a gradual, but constant, elaboration of all these quantal theories as abstract mathematical struc tures (their point of departure being von Neumann's formalism) until at the present time theoretical work is heavily preoccupied with the manip ulation of purely abstract structures.

Probability Theory and Applications

Probability Theory and Applications

Author: J. Galambos

Publisher: Springer Science & Business Media

ISBN: 9789401128179

Category: Mathematics

Page: 350

View: 564

This volume contains twenty-two original contributions by leading scientists in many important areas of probability theory and its applications. The material also includes significant new results. Together this collection of papers provides a good state-of-the-art survey of current research in the following areas: inequalities; limit theorems; renewal theory and reliability theory; characterizations of distributions; infinite divisibility of polynomials of normal variables; limiting distributions for order statistics; stochastic processes; functional equations in engineering model building; and probabilistic number theory.