The familiar wave equation is the most fundamental hyperbolic partial differential equation. Other hyperbolic equations, both linear and nonlinear, exhibit many wave-like phenomena. The primary theme of this book is the mathematical investigation of such wave phenomena. The exposition begins with derivations of some wave equations, including waves in an elastic body, such as those observed in connection with earthquakes. Certain existence results are proved early on, allowing the later analysis to concentrate on properties of solutions. The existence of solutions is established using methods...

The asymptotic properties of the likelihood ratio play an important part in solving problems in statistics for various schemes of observations. In this work, the author describes the asymptotic methods for parameter estimation and hypothesis testing based on asymptotic properties of the likelihood ratios in the case where an observed stochastic process is a semimartingale. Chapter 1 gives the general basic notions and results of the theory under consideration. Chapters 2 and 3 are devoted to the problem of distinguishing between two simple statistical hypotheses. In Chapter 2, certain types...

A minimal length curve joining two points in a surface is called a geodesic. One may trace the origin of the problem of finding geodesics back to the birth of calculus. Many contemporary mathematical problems, as in the case of geodesics, may be formulated as variational problems in surfaces or in a more generalized form on manifolds. One may characterize geometric variational problems as a field of mathematics that studies global aspects of variational problems relevant in the geometry and topology of manifolds. For example, the problem of finding a surface of minimal area spanning a given...

Presents the theory of functions spaces, known as Sobolev spaces, which are widely used in the theory of partial differential equations, mathematical physics, and numerous applications. This book also covers the variational method of solution of boundary v

The study of asymptotic solutions to nonlinear systems of partial differential equations is a powerful tool in the analysis of such systems and their applications in physics, mechanics, and engineering. In this book, the authors propose a method of asymptotic analysis of solutions, which can be applied in the case of the so-called smoothed shock waves, nonlinear waves which vary fast in a neighborhood of the front and slowly outside of this neighborhood. The proposed method, based on the study of geometric objects associated to the front, can be viewed as a generalization of the geometric...

This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only one-year courses of algebra and analysis.

Focusing on one of the major branches of probability theory, this book treats the large class of processes with continuous sample paths that possess the Markov property. The exposition is based on the theory of stochastic analysis, which uses such notions as stochastic differentials and stochastic integrals.

This book reproduces the doctoral thesis written by a remarkable mathematician, Sergei V. Kerov. His untimely death at age 54 left the mathematical community with an extensive body of work and this one-of-a-kind monograph. Here, he gives a clear and lucid account of results and methods of asymptotic representation theory. The book is a unique source of information on an important topic of current research. Asymptotic representation theory of symmetric groups deals with problems of two types: asymptotic properties of representations of symmetric groups of large order and representations of the...

Information geometry provides the mathematical sciences with a fresh framework of analysis. This book presents a comprehensive introduction to the mathematical foundation of information geometry. It provides an overview of many areas of applications, such

Presents an exposition of the fundamentals of the theory of Abelian functions based on the methods of the classical theory of functions. This book covers such topics as theta functions, Jacobians, and Picard varieties.