Bøger af Soon-Mo Jung

¿This book discusses the process by which Ulam's conjecture is proved, aptly detailing how mathematical problems may be solved by systematically combining interdisciplinary theories. It presents the state-of-the-art of various research topics and methodologies in mathematics, and mathematical analysis by presenting the latest research in emerging research areas, providing motivation for further studies. The book also explores the theory of extending the domain of local isometries by introducing a generalized span.For the reader, working knowledge of topology, linear algebra, and Hilbert space theory, is essential. The basic theories of these fields are gently and logically introduced. The content of each chapter provides the necessary building blocks to understanding the proof of Ulam¿s conjecture and are summarized as follows: Chapter 1 presents the basic concepts and theorems of general topology. In Chapter 2, essential concepts and theorems in vector space, normed space, Banach space, inner product space, and Hilbert space, are introduced. Chapter 3 gives a presentation on the basics of measure theory. In Chapter 4, the properties of first- and second-order generalized spans are defined, examined, and applied to the study of the extension of isometries. Chapter 5 includes a summary of published literature on Ulam¿s conjecture; the conjecture is fully proved in Chapter 6.

We intended to write this book simple and easy enough for those mathematics majors in their sophomore year or above while boiling down some of the contents to make it appropriate as a semester long number theory textbook without including quadratic congruences, sums of squares, and continued fractions. This book is not only suitable for those graduate students in mathematics as a number theory seminar reference who are not majoring in number theory, but also for those gifted juniors and seniors in high school as a guidance who are preparing for the International Mathematics Olympiad.

In an integrated and self-contained fashion, this book covers almost all classical results on the Hyers-Ulam-Rassias stability. It is arranged so that highly advanced undergraduate as well as graduate level students will be able to follow the materials.