While optimality conditions for optimal control problems with state constraints have been extensively investigated in the literature the results pertaining to numerical methods are relatively scarce. This book fills the gap by providing a family of new methods. Among others, a novel convergence analysis of optimal control algorithms is introduced. The analysis refers to the topology of relaxed controls only to a limited degree and makes little use of Lagrange multipliers corresponding to state constraints. This approach enables the author to provide global convergence analysis of first order and superlinearly convergent second order methods. Further, the implementation aspects of the methods developed in the book are presented and discussed. The results concerning ordinary differential equations are then extended to control problems described by differential-algebraic equations in a comprehensive way for the first time in the literature.

This book deals with the study of convex functions and of their behavior from the point of view of stability with respect to perturbations. Convex functions are considered from the modern point of view that underlines the geometrical aspect: thus a function is defined as convex whenever its graph is a convex set. A primary goal of this book is to study the problems of stability and well-posedness, in the convex case. Stability means that the basic parameters of a minimum problem do not vary much if we slightly change the initial data. On the other hand, well-posedness means that points with values close to the value of the problem must be close to actual solutions. In studying this, one is naturally led to consider perturbations of functions and of sets. This approach fits perfectly with the idea of regarding functions as sets. Thus the second part of the book starts with a short, yet rather complete, overview of the so-called hypertopologies, i.e. topologies in the closed subsets of a metric space. While there exist numerous classic texts on the issue of stability, there only exists one book on hypertopologies [Beer 1993]. The current book differs from Beers in that it contains a much more condensed explication of hypertopologies and is intended to help those not familiar with hypertopologies learn how to use them in the context of optimization problems.

The only elementary text to provide a complete introduction to diffusion theory and the various analytical and numerical methods of solution. Presents an integrated set of real-life problems taken mainly from metallurgy and device processing, and offers an overview of the solution of diffusion problems in practical cases, with clear explanations of the interrelationships between mathematical solutions, and the underlying physics and chemistry. Covers oxidation theory, error functions, Laplace transforms, and similarity, a topic of current research interest. Also covers Green's functions and integral equations, rarely discussed in introductory texts.

This 2nd volume in the series History of the Theory of Numbers presents material related to Diophantine Analysis. This series is the work of a distinguished mathematician who taught at the University of Chicago for 4 decades and is celebrated for his many contributions to number theory and group theory. 1919 edition.

This 1st volume in the series History of the Theory of Numbers presents the material related to the subjects of divisibility and primality. This series is the work of a distinguished mathematician who taught at the University of Chicago for 4 decades and is celebrated for his many contributions to number theory and group theory. 1919 edition.

Over 300 sequences and many unsolved problems and conjectures related to them are presented herein. The book contains definitions, unsolved problems, questions, theorems corollaries, formulae, conjectures, examples, mathematical criteria, etc. ( on integer sequences, numbers, quotients, residues, exponents, sieves, pseudo-primes/squares/cubes/factorials, almost primes, mobile periodicals, functions, tables, prime/square/factorial bases, generalized factorials, generalized palindromes, etc. ).

Prime numbers beckon to the beginner, the basic notion of primality being accessible to a child. Yet, some of the simplest questions about primes have stumped humankind for millennia. In this book, the authors concentrate on the computational aspects of prime numbers, such as recognizing primes and discovering the fundamental prime factors of a given number. Over 100 explicit algorithms cast in detailed pseudocode are included in the book. Applications and theoretical digressions serve to illuminate, justify, and underscore the practical power of these algorithms. The 2nd edition adds new material on primality and algorithms and updates all the numerical records, such as the largest prime, etc. It has been revised throughout.

In elementary mathematics, there are many difficult and interesting problems not connected with the name of an individual, but rather possessing the character of "mathematical folklore". Such problems are scattered throughout the wide literature of popular (or, simply, entertaining!) mathematics, and often it is very difficult to establish the source of a particular problem.