This self-contained treatment of algebraic topology assumes only some knowledge of real numbers and real analysis. The first three chapters focus on the basics of point-set topology, offering background to students approaching the subject with no previous knowledge. Readers already familiar with point-set topology can proceed directly to Chapter 4, which examines the fundamental group as well as homology groups and continuous mapping, barycentric subdivision and excision, the homology sequence, and simplicial complexes.
Exercises form an integral part of the text; they include theorems...

Numerous worked examples and exercises highlight this unified treatment of the Hermitian operator theory in its Hilbert space setting. Its simple explanations of difficult subjects make it accessible to undergraduates as well as an ideal self-study guide.
Featuring full discussions of first and second order linear differential equations, the text introduces the fundamentals of Hilbert space theory and Hermitian differential operators. It derives the eigenvalues and eigenfunctions of classical Hermitian differential operators, develops the general theory of orthogonal bases in Hilbert...

Translated into many languages, this book was in continuous use as the standard university-level text for a quarter-century, until it was revised and enlarged by the author in 1952. World-renowned writer and researcher Nathan Altshiller-Court (1881-1968) was a professor of mathematics at the University of Oklahoma for more than thirty years. His revised introduction to modern geometry offers today's students the benefits of his many years of teaching experience.
The first part of the text stresses construction problems, proceeding to surveys of similitude and homothecy, properties of the...

The teaching of mathematics has undergone extensive changes in approach, with a shift in emphasis from rote memorization to acquiring an understanding of the logical foundations and methodology of problem solving. This book offers guidance in that direction, exploring arithmetic's underlying concepts and their logical development.
This volume's great merit lies in its wealth of explanatory material, designed to promote an informal and intuitive understanding of the rigorous logical approach to the number system. The first part explains and comments on axioms and definitions, making their...

A self-contained treatment of finite Markov chains and processes, this text covers both theory and applications. Author Marius Iosifescu, vice president of the Romanian Academy and director of its Center for Mathematical Statistics, begins with a review of relevant aspects of probability theory and linear algebra. Experienced readers may start with the second chapter, a treatment of fundamental concepts of homogeneous finite Markov chain theory that offers examples of applicable models.
The text advances to studies of two basic types of homogeneous finite Markov chains: absorbing and...

The most effective way to study any branch of mathematics is to tackle its problems. This wide-ranging anthology offers a straightforward approach, with 431 challenging problems in all phases of group theory, from elementary to the most advanced.
The problems are arranged in eleven chapters: subgroups, permutation groups, automorphisms and finitely generated Abelian groups, normal series, commutators and derived series, solvable and nilpotent groups, the group ring and monomial representations, Frattini subgroup, factorization, linear groups, and representations and characters. Each...

For many years, this elementary treatise on advanced Euclidean geometry has been the standard textbook in this area of classical mathematics; no other book has covered the subject quite as well. It explores the geometry of the triangle and the circle, concentrating on extensions of Euclidean theory, and examining in detail many relatively recent theorems. Several hundred theorems and corollaries are formulated and proved completely; numerous others remain unproved, to be used by students as exercises.
The author makes liberal use of circular inversion, the theory of pole and polar, and...

This popular text was created for a one-year undergraduate course or beginning graduate course in partial differential equations, including the elementary theory of complex variables. It employs a framework in which the general properties of partial differential equations, such as characteristics, domains of independence, and maximum principles, can be clearly seen. The only prerequisite is a good course in calculus.
Incorporating many of the techniques of applied mathematics, the book also contains most of the concepts of rigorous analysis usually found in a course in advanced calculus....

This concise text offers an introduction to discrete mathematics for undergraduate students in computer science and mathematics. Mathematics educators consider it vital that their students be exposed to a course in discrete methods that introduces them to combinatorial mathematics and to algebraic and logical structures focusing on the interplay between computer science and mathematics. The present volume emphasizes combinatorics, graph theory with applications to some stand network optimization problems, and algorithms to solve these problems.
Chapters 0-3 cover fundamental operations...