Knot Theory (Mathematical Association of America Textbooks, Series Number 24)

As a survey of the basics of knot theory, this book is as good as it gets. The opening chapter is a history of knot theory, which is followed by a chapter on the mathematical definition of knots. The remainder of the book is a series of descriptions of knots, how they are represented, classified and the mathematical machinery used to transform them. Very little in the way of deep mathematical knowledge is needed to understand the presentation, one of the most important requirements is the ability to think in spatial terms. Exercises are given at the end of each section although no solutions are provided. Many areas of mathematics began as an abstract theory and after some time, applications are found. Knot theory is an element of this set; one of the applications is that it can be used to describe how proteins fold. A protein is a long chain of connected amino acids, but its' ability to be biochemically active is based on the structure that it folds into after construction. This book is a lively understandable introduction to this fascinating field; it is suitable for self-study or a special topics class in the area of knots. Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots. Livingston guides you through a general survey of the topic showing how to use the techniques of linear algebra to address some sophisticated problems, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject#x97;the Conway, Jones, and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology "This is a revised, updated, and augmented edition of a classic Carus monograph with a new chapter on integration and its applications. Earlier editions covered sets, metric spaces, continuous functions, and differentiable functions. To that, this edition adds sections on measurable sets and functions and the Lebesgue and Stieltjes integrals. The book retains the informal chatty style of the previous editions. It presents a variety of interesting topics, many of which are not commonly encountered in undergraduate textbooks, such as the existence of continuous everywhere-oscillating functions; two functions having equal derivatives, yet not differing by a constant; application of Stieltjes integration to the speed of convergence of infinite series. For readers with a background in calculus, the book is suitable either for self-study or for supplemental reading in a course on advanced calculus or real analysis. Students of mathematics will find here the sense of wonder that was associated with the subject in its early days"--Publisher description Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject - the Conway, Jones and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology. Inequalities from Complex Analysis is a careful, friendly exposition of some rather interesting mathematics. The author begins by defining the complex number field; he gives a novel presentation of some standard mathematical analysis in the early chapters. The development culminates with some results from recent research literature. The book provides complete yet comprehensible proofs as well as some surprising consequences of the results. One unifying theme is a complex variables analogue of Hilbert's seventeenth problem. Numerous examples, exercises and discussions of geometric reasoning aid the reader. The book is accessible to undergraduate mathematicians, as well as physicists and engineers This book is an introduction to the ergodic theory behind common number expansions, for instance decimal expansions, continued fractions and many others. The questions studied are dynamical as well as number theoretic in nature, and the answers are obtained with the help of ergodic theory. What it means to be ergodic and the basic ideas behind ergodic theory are explained along the way. The book is aimed at introducing students with sufficient background knowledge in real analysis to a 'dynamical way of thinking'. The subjects covered vary from the classical to recent research which should increase the appeal of this book to researchers working in the field Principal ideas of classical function theory Basic notions of differential geometry Curvature and applications Some new invariant metrics Introduction to the Bergman Theory A glimpse of several complex variables. In 1877 P.G. Tait published the first in a series of papers addressing the enumeration of knots. Karma Dajani, Cor Kraaikamp. Includes Bibliographical References (p. 179-185) And Index. Mathematical analysis requires a thorough study of inequalities.