Contemporary Mathematics, Volume: 632
2015; 371 pp; softcover
ISBN-13: 978-0-8218-9860-4
Expected publication date is March 30, 2015.
This volume contains the proceedings of the 11th International Conference on Finite Fields and their Applications (Fq11), held July 22-26, 2013, in Magdeburg, Germany.
Finite Fields are fundamental structures in mathematics. They lead to interesting deep problems in number theory, play a major role in combinatorics and finite geometry, and have a vast amount of applications in computer science.
Papers in this volume cover these aspects of finite fields as well as applications in coding theory and cryptography.
Graduate students and research mathematicians interested in number theory, coding theory, cryptography, and finite geometry.
Pure and Applied Undergraduate Texts,Volume: 23
2015; 144 pp; hardcover
ISBN-13: 978-1-4704-2018-5
Expected publication date is May 19, 2015.
This book offers a rigorous and coherent introduction to the five basic number systems of mathematics, namely natural numbers, integers, rational numbers, real numbers, and complex numbers. It is a subject that many mathematicians believe should be learned by any student of mathematics including future teachers.
The book starts with the development of Peano arithmetic in the first chapter which includes mathematical induction and elements of recursion theory. It proceeds to an examination of integers that also covers rings and ordered integral domains. The presentation of rational numbers includes material on ordered fields and convergence of sequences in these fields. Cauchy and Dedekind completeness properties of the field of real numbers are established, together with some properties of real continuous functions. An elementary proof of the Fundamental Theorem of Algebra is the highest point of the chapter on complex numbers. The great merit of the book lies in its extensive list of exercises following each chapter. These exercises are designed to assist the instructor and to enhance the learning experience of the students.
Undergraduate students interested in foundations of algebra and analysis.
Natural numbers
Integers
Rational numbers
Real numbers
Complex numbers
Sets, relations, functions
Bibliography
Index
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Graduate Studies in Mathematics, Volume: 161
2015; approx. 364 pp; hardcover
ISBN-13: 978-0-8218-5198-2
Expected publication date is May 4, 2015.
Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts.
Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics.
This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of the six chapters concludes with problems that will help the readers to practice their tropical skills, and to gain access to the research literature.
Undergraduate and graduate students and research mathematicians interested in algebraic geometry and combinatorics.
Tropical islands
Building blocks
Tropical varieties
Tropical rain forest
Tropical garden
Toric connections
Bibliography
Index
Graduate Studies in Mathematics, Volume: 163
2015; 641 pp; hardcover
ISBN-13: 978-0-8218-9854-3
Expected publication date is May 4, 2015.
This book provides a self contained, thorough introduction to the analytic and probabilistic methods of number theory. The prerequisites being reduced to classical contents of undergraduate courses, it offers to students and young researchers a systematic and consistent account on the subject. It is also a convenient tool for professional mathematicians, who may use it for basic references concerning many fundamental topics.
Deliberately placing the methods before the results, the book will be of use beyond the particular material addressed directly. Each chapter is complemented with bibliographic notes, useful for descriptions of alternative viewpoints, and detailed exercises, often leading to research problems.
This third edition of a text that has become classical offers a renewed and considerably enhanced content, being expanded by more than 50 percent. Important new developments are included, along with original points of view on many essential branches of arithmetic and an accurate perspective on up-to-date bibliography.
Graduate students and research mathematicians interested in number theory, analysis, and probability.
Elementary methods
Some tools from real analysis
Prime numbers
Arithmetic functions
Average orders
Sieve methods
Extremal orders
The method of van der Corput
Diophantine approximation
Complex analysis methods
The Euler gamma function
Generating functions: Dirichlet series
Summation formulae
The Riemann zeta function
The prime number theorem and the Riemann hypothesis
The Selberg-Delange method
Two arithmetic applications
Tauberian theorems
Primes in arithmetic progressions
Probabilistic methods
Densities
Limiting distributions of arithmetic functions
Normal order
Distribution of additive functions and mean values of multiplicative functions
Friable integers. The saddle-point method
Integers free of small factors
Bibliography
Index