Focusing exclusively on the theory of finite fields, the book delves into the mathematical structures and properties that define these fields. It explores their applications in various areas, including coding theory and cryptography. With a systematic approach, the text provides in-depth explanations, examples, and exercises to enhance understanding, making it a valuable resource for students and researchers interested in advanced algebra and number theory.
Monte Carlo methods are numerical methods based on random sampling and quasi-Monte Carlo methods are their deterministic versions. This volume contains the refereed proceedings of the Second International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing which was held at the University of Salzburg (Austria) from July 9--12, 1996. The conference was a forum for recent progress in the theory and the applications of these methods. The topics covered in this volume range from theoretical issues in Monte Carlo and simulation methods, low-discrepancy point sets and sequences, lattice rules, and pseudorandom number generation to applications such as numerical integration, numerical linear algebra, integral equations, binary search, global optimization, computational physics, mathematical finance, and computer graphics. These proceedings will be of interest to graduate students and researchers in Monte Carlo and quasi-Monte Carlo methods, to numerical analysts, and to practitioners of simulation methods.
This textbook bridges basic number theory and recent advances in applied number theory, providing a unified account of four major application areas: cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation. It highlights the connections between these fields. Number theory, often referred to as the queen of mathematics by Carl-Friedrich Gauss, is celebrated for its beauty, elegant results, and proofs. While historically limited in real-life applications, it is now integral to everyday technologies such as supermarket bar code scanners, GPS systems, and online banking. The book begins with an introductory course on number theory in Chapter 1, making it accessible for undergraduates. Chapters 2-5 delve into the four main application areas and present advanced results without proofs, requiring more sophisticated mathematical skills. The final chapter reviews additional applications, including check-digit systems, quantum computation, and raster-graphics memory organization. This resource is valuable for upper-level undergraduates, graduates, and researchers in number theory.
Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples. This book collects the results of the workshops „Applications of algebraic curves“ and „Applications of finite fields“ of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.
This book represents the refereed proceedings of the Fifth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing which was held at the National University of Singapore in the year 2002. An important feature are invited surveys of the state of the art in key areas such as multidimensional numerical integration, low-discrepancy point sets, computational complexity, finance, and other applications of Monte Carlo and quasi-Monte Carlo methods. These proceedings also include carefully selected contributed papers on all aspects of Monte Carlo and quasi-Monte Carlo methods. The reader will be informed about current research in this very active area
This book represents the refereed proceedings of the Third International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing which was held at Claremont Graduate University in 1998.
