How can one be assured that computer codes that solve differential equations are correct? Standard practice using benchmark testing no longer provides full coverage because today's production codes solve more complex equations using more powerful algorithms. By verifying the order-of-accuracy of the numerical algorithm implemented in the code, one can detect most any coding mistake that would prevent correct solutions from being computed.
Verification of Computer Codes in Computational Science and Engineering sets forth a powerful alternative called OVMSP: Order-Verification via the...

In this second edition of a Carus Monograph Classic, Steven G. Krantz, a leading worker in complex analysis and a winner of the Chauvenet Prize for outstanding mathematical exposition, develops material on classical non-Euclidean geometry. He shows how it can be developed in a natural way from the invariant geometry of the complex disk. He also introduces the Bergmann kernel and metric and provides profound applications, some of which have never appeared in print before. In general, the new edition represents a considerable polishing and re-thinking of the original successful volume. A...

Tracing a path from the earliest beginnings of Fourier series through to the latest research A Panorama of Harmonic Analysis discusses Fourier series of one and several variables, the Fourier transform, spherical harmonics, fractional integrals, and singular integrals on Euclidean space. The climax is a consideration of ideas from the point of view of spaces of homogeneous type, which culminates in a discussion of wavelets. This book is intended for graduate students and advanced undergraduates, and mathematicians of whatever background who want a clear and concise overview of the subject of...

A quick and easy-to-use introduction to the key topics in complex variables, for mathematicians and non-mathematicians alike.

Mathematicians are expected to publish their work: in journals, conference proceedings, and books. It is vital to advancing their careers. This is a guidebook to publishing mathematics. It describes both the general setting of mathematical publishing and t

Addresses the major issues of graduate school, including choosing a program, passing the qualifying exams, finding an advisor, writing a thesis, and getting that first job. This book also contains a description of the basic elements of a mathematical education, as well as a glossary and appendices on the structure of a typical department.

A successful mathematical career involves doing good mathematics, but also requires a range of skills that aren't normally taught in graduate school. The purpose of this book is to provide guidance to the professional mathematician in how to develop and survive in the profession.

The purpose of this book is to teach the basic principles of problem solving, including both mathematical and nonmathematical problems. The book aims to help students to translate verbal discussions into analytical data, learn problem solving methods for attacking collections of analytical questions or data, and build a personal arsenal of solutions and internalized problem-solving techniques. After many solved problems are given, a challenge problem is presented, and additional problems are included for readers to tackle at the end of each chapter.

This work departs from earlier treatments of the subject by emphasizing integral formulas, the geometric theory of pseudoconvexity, estimates, partial differential equations, approximation theory, the boundary behavior of holomorphic functions, inner functions, invariant metrics, and mapping theory. While due homage is paid to the more traditional algebraic theory (for example, sheaves, and Cousin problems), the student with a background in real and complex variable theory, harmonic analysis, and differential equations should be most comfortable with this treatment.