الصفحة 1
الصفحة 1
Eigenvalues, Inequalities, and Ergodic Theory
A problem of broad interest – the estimation of the spectral gap for matrices or differential operators (Markov chains or diffusions) – is covered in this book. The area has a wide range of applications, and provides a tool to describe the phase transitions and the effectiveness of random algorithms. In particular, the book studies a subset of the general problem, taking some approaches that have, up till now, only appeared largely in the Chinese literature.Eigenvalues, Inequalities and Ergodic Theory serves as an introduction to this developing field, and provides an overview of the methods used, in an accessible and concise manner.
Abstraction, refinement and proof for probabilistic systems
Probabilistic techniques are increasingly being employed in computer programs and systems because they can increase efficiency in sequential algorithms, enable otherwise nonfunctional distribution applications, and allow quantification of risk and safety in general. This makes operational models of how they work, and logics for reasoning about them, extremely important. Abstraction, Refinement and Proof for Probabilistic Systems presents a rigorous approach to modeling and reasoning about computer systems that incorporate probability. Its foundations lie in traditional Boolean sequential-program logic—but its extension to numeric rather than merely true-or-false judgments takes it much further, into areas such as randomized algorithms, fault tolerance, and, in distributed systems, almost-certain symmetry breaking. The presentation begins with the familiar "assertional" style of program development and continues with increasing specialization: Part I treats probabilistic program logic, including many examples and case studies; Part II sets out the detailed semantics; and Part III applies the approach to advanced material on temporal calculi and two-player games.
