Abstract Algebra 2 Osu

First Annual OSU Workshop on Parallel Computing

Abstract Algebra and Famous Impossibilities

This textbook develops the abstract algebra necessary to prove the impossibility of four famous mathematical feats: squaring the circle, trisecting the angle, doubling the cube, and solving quintic equations. All the relevant concepts about fields are introduced concretely, with the geometrical questions providing motivation for the algebraic concepts. By focusing on problems that are as easy to approach as they were fiendishly difficult to resolve, the authors provide a uniquely accessible introduction to the power of abstraction. Beginning with a brief account of the history of these fabled problems, the book goes on to present the theory of fields, polynomials, field extensions, and irreducible polynomials. Straightedge and compass constructions establish the standards for constructability, and offer a glimpse into why squaring, doubling, and trisecting appeared so tractable to professional and amateur mathematicians alike. However, the connection between geometry and algebra allows the reader to bypass two millennia of failed geometric attempts, arriving at the elegant algebraic conclusion that such constructions are impossible. From here, focus turns to a challenging problem within algebra itself: finding a general formula for solving a quintic polynomial. The proof of the impossibility of this task is presented using Abel’s original approach. Abstract Algebra and Famous Impossibilities illustrates the enormous power of algebraic abstraction by exploring several notable historical triumphs. This new edition adds the fourth impossibility: solving general quintic equations. Students and instructors alike will appreciate the illuminating examples, conversational commentary, and engaging exercises that accompany each section. A first course in linear algebra is assumed, along with a basic familiarity with integral calculus.

Parallel Processing and Applied Mathematics

This two-volume-set (LNCS 8384 and 8385) constitutes the refereed proceedings of the 10th International Conference of Parallel Processing and Applied Mathematics, PPAM 2013, held in Warsaw, Poland, in September 2013. The 143 revised full papers presented in both volumes were carefully reviewed and selected from numerous submissions. The papers cover important fields of parallel/distributed/cloud computing and applied mathematics, such as numerical algorithms and parallel scientific computing; parallel non-numerical algorithms; tools and environments for parallel/distributed/cloud computing; applications of parallel computing; applied mathematics, evolutionary computing and metaheuristics.