This book introduces the notion of an E-semigroup, a generalization of the known concept of E_O-semigroup. These objects are families of endomorphisms of a von Neumann algebra satisfying certain natural algebraic and continuity conditions. Its thorough approach is ideal for graduate students and research mathematicians.

In these notes the author presents a complete theory of classification of E0-semigroups by product systems of correspondences. As an application of his theory, he answers the fundamental question if a Markov semigroup admits a dilation by a cocycle perturbations of noise: It does if and only if it is spatial.

This volume contains the proceedings of the conference on Advances in Quantum Dynamics. The purpose of the conference was to assess the current state of knowledge and to outline future research directions of quantum dynamical semigroups on von Neumann algebras. Since the appearance of the landmark papers by F. Murray and J. von Neumann, On the Rings of Operators, von Neumann algebras have been used as a mathematical model in the study of time evolution of quantum mechanical systems.Following the work of M. H. Stone, von Neumann, and others on the structure of one-parameter groups of unitary transformations, many researchers have made fundamental contributions to the understanding of time-reversible dynamical systems. This book deals with the mathematics of time-irreversiblesystems, also called dissipative systems. The time parameter is the half-line, and the transformations are now endomorphisms as opposed to automorphisms. For over a decade, W. B. Arveson and R. T. Powers have pioneered the effort to understand the structure of irreversible quantum dynamical systems on von Neumann algebras. Their papers in this volume serve as an excellent introduction to the theory. Also included are contributions in other areas which have had an impact on the theory, such asBrownian motion, dilation theory, quantum probability, and free probability. The volume is suitable for graduate students and research mathematicians interested in the dynamics of quantum systems and corresponding topics in the theory of operator algebras.

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

This book offers a presentation of some new trends in operator theory and operator algebras, with a view to their applications. It consists of separate papers written by some of the leading practitioners in the field. The content is put together by the three editors in a way that should help students and working mathematicians in other parts of the mathematical sciences gain insight into an important part of modern mathematics and its applications. While different specialist authors are outlining new results in this book, the presentations have been made user friendly with the aid of tutorial material. In fact, each paper contains three things: a friendly introduction with motivation, tutorial material, and new research. The authors have strived to make their results relevant to the rest of mathematics. A list of topics discussed in the book includes wavelets, frames and their applications, quantum dynamics, multivariable operator theory, $C*$-algebras, and von Neumann algebras. Some longer papers present recent advances on particular, long-standing problems such as extensions and dilations, the Kadison-Singer conjecture, and diagonals of self-adjoint operators.

A longstanding problem in Gabor theory is to identify time-frequency shifting lattices aZ×bZ and ideal window functions χI on intervals I of length c such that {e−2πinbtχI(t−ma): (m,n)∈Z×Z} are Gabor frames for the space of all square-integrable functions on the real line. In this paper, the authors create a time-domain approach for Gabor frames, introduce novel techniques involving invariant sets of non-contractive and non-measure-preserving transformations on the line, and provide a complete answer to the above abc-problem for Gabor systems.

This Festschrift volume, published in honor of Yuri Gurevich on the occasion of his 70th birthday, contains contributions, written by his colleagues. The collection of articles herein begins with an academic biography, an annotated list of Yuri's publications and reports, and a personal tribute by Jan Van den Bussche. These are followed by 28 technical contributions. These articles - though they cover a broad range of topics - represent only a fraction of Yuri Gurevich’s multiple areas of interest.

During the early part of the last century, Ferdinand Georg Frobenius (1849-1917) raised he following problem, known as the Frobenius Problem (FP): given relatively prime positive integers a1,...,an, find the largest natural number (called the Frobenius number and denoted by g(a1,...,an) that is not representable as a nonnegative integer combination of a1,...,an, . At first glance FP may look deceptively specialized. Nevertheless it crops up again and again in the most unexpected places and has been extremely useful in investigating many different problems. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight such methods, ideas, viewpoints and applications to a broader audience.

This is an introduction to those aspects of parallel programming and parallel algorithms that relate to a single topic: the prefix problem. This approach provides intense development of a single computational tool used in many parallel computations, showing and discussing its techniques. The text may be used for graduate courses.