Monographs in Number Theory: Volume 11
Pages: 200
There is no surprise that arithmetic properties of integral ("whole") numbers are controlled by analytic functions of complex variable. At the same time, the values of analytic functions themselves happen to be interesting numbers, for which we often seek explicit expressions in terms of other "better known" numbers or try to prove that no such exist. This natural symbiosis of number theory and analysis is centuries old but keeps enjoying new results, ideas and methods.
The present book takes a semi-systematic review of analytic achievements in number theory ranging from classical themes about primes, continued fractions, transcendence of ƒÎ and resolution of Hilbert's seventh problem to some recent developments on the irrationality of the values of Riemann's zeta function, sizes of non-cyclotomic algebraic integers and applications of hypergeometric functions to integer congruences.
Our principal goal is to present a variety of different analytic techniques that are used in number theory, at a reasonably accessible ? almost popular ? level, so that the materials from this book can suit for teaching a graduate course on the topic or for a self-study. Exercises included are of varying difficulty and of varying distribution within the book (some chapters get more than other); they not only help the reader to consolidate their understanding of the material but also suggest directions for further study and investigation. Furthermore, the end of each chapter features brief notes about relevant developments of the themes discussed.
Numbers and q-Numbers
Prime Number Theorem
Riemann's Zeta Function and Its Multiple Generalisation
Continued Fractions
Dirichlet's Theorem on Primes in Arithmetic Progressions
Algebraic and Transcendental Numbers. The Transcendence of e and ƒÎ
Irrationality of Zeta Values
Hilbert's Seventh Problem
Schinzel?Zassenhaus Conjecture
Creative Microscoping
Bibliography
Index
Graduates, researchers and enthusiasts in number theory, complex analysis, and special functions; suitable for teaching graduate courses in number theory and self-study.
Pages: 280
ISBN: 978-981-127-766-5 (hardcover)
ISBN: 978-981-127-870-9 (softcover)
This book is translated from the Chinese version published by Science Press, Beijing, China, in 2017. It was written for the Chern class in mathematics of Nankai University and has been used as the textbook for the course Abstract Algebra for this class for more than five years. It has also been adapted in abstract algebra courses in several other distinguished universities across China.
The aim of this book is to introduce the fundamental theories of groups, rings, modules, and fields, and help readers set up a solid foundation for algebra theory. The topics of this book are carefully selected and clearly presented. This is an excellent mathematical exposition, well-suited as an advanced undergraduate textbook or for independent study. The book includes many new and concise proofs of classical theorems, along with plenty of basic as well as challenging exercises.
Groups
Rings
Modules
Fields
Textbook for an advanced undergraduate course on abstract algebra. Reference book for graduate students in physics, engineering, and computer science. Any students interested in abstract algebra.
Pages: 292
ISBN: 978-981-127-809-9 (hardcover)
Stirling numbers are one of the most known classes of special numbers in Mathematics, especially in Combinatorics and Algebra. They were introduced by Scottish mathematician James Stirling (1692?1770) in his most important work, Differential Method with a Tract on Summation and Interpolation of Infinite Series (1730). Stirling numbers have rich history; many arithmetic, number-theoretical, analytical and combinatorial connections; numerous classical properties; as well as many modern applications.
This book collects together much of the scattered material on the two subclasses of Stirling numbers to provide a holistic overview of the topic. From the combinatorial point of view, Stirling numbers of the second kind S(n,k) count the number of ways to partition a set of n different objects (i.e., a given n-set) into k non-empty subsets. Stirling numbers of the first kind s(n, k) give the number of permutations of n elements with k disjoint cycles. Both subclasses of Stirling numbers play an important role in Algebra: they form the coefficients, connecting well-known sets of polynomials.
This book is suitable for students and professionals, providing a broad perspective of the theory of this class of special numbers, and many generalizations and relatives of Stirling numbers, including Bell numbers and Lah numbers. Throughout the book, readers are presented with exercises to test and cement their understanding.
Notations
Preface
Preliminaries
Combinatorics of Partitions
Stirling Numbers of the Second Kind
Stirling Numbers of the First Kind
Generalizations and Relatives of Stirling Numbers
Zoo of Numbers
Mini Dictionary
Exercises
Teachers and students (esp. at university) interested in Combinatorics, Number Theory, General Algebra, Cryptography and related fields, as well as general audience of amateurs of Mathematics.
Pages: 440
ISBN: 978-981-128-008-5 (hardcover)
Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar?Moh?Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was introduced by Iitaka and Kawamata.
Readers will find the book covers vast basic material on an extremely rigorous level:
It begins with an introduction to algebraic geometry which comprises almost all results in commutative algebra and algebraic geometry.
Arguments frequently used in affine algebraic geometry are elucidated by treating affine lines embedded in the affine plane and automorphism theorem of the affine plane. There is also a detailed explanation on affine algebraic surfaces which resemble the affine plane in the ring-theoretic nature and for actions of algebraic groups.
The Jacobian conjecture for these surfaces is also considered by making use of the results and tools already presented in this book. The conjecture has been thought as one of the most unattackable problems even in dimension two.
Advanced results are collected in appendices of chapters so that readers can understand the main streams of arguments.
There are abundant problems in the first three chapters which come with hints and ideas for proof.
Preface
Introduction to Algebraic Geometry
Geometry on Affine Surfaces
Geometry and Topology of Polynomial Rings
Postscript
Mathematics students, both undergraduate and graduate, where knowledge of group, ring and linear algebra is required, and researchers. If the book is used as a textbook, it is for students in the beginning class of algebraic geometry and commutative algebra.