This book, which presupposes familiarity only with the most elementary concepts of arithmetic (divisibility properties, greatest common divisor, etc.), is an expanded version of a series of lectures for graduate students on elementary number theory. Topics include: Compositions and Partitions; Arithmetic Functions; Distribution of Primes; Irrational Numbers; Congruences; Diophantine Equations; Combinatorial Number Theory; and Geometry of Numbers. Three sections of problems (which include exercises as well as unsolved problems) complete the text.
Functional Inequalities Markov Semigroups and Spectral Theory
In this book, the functional inequalities are introduced to describe: (i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principle eigenvalue, and the spectral gap; (ii) the semigroup properties: the uniform intergrability, the compactness, the convergence rate, and the existence of density; (iii) the reference measure and the intrinsic metric: the concentration, the isoperimetic inequality, and the transportation cost inequality.
Satan, Cantor, And Infinity and Other Mind-Boggling Puzzles
The author of What Is the Name of This Book? presents a compilation of more than two hundred challenging new logic puzzles--ranging from simple brainteasers to complex mathematical paradoxes.
The application of graph theory to modelling systems began in several scientific areas, among them statistical physics (the study of large particle systems), genetics (studying inheritable properties of natural species), and interactions in contingency tables. The use of graphical models in statistics has increased considerably in these and other areas such as artificial intelligence, and the theory has been greatly developed and extended. This is the first comprehensive and authoritative account of the theory of graphical models.
This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.
