Series:

Pure and Applied Mathematics

2 total works

Cover of The Mathematics of Infinity
This title offers a balanced and clearly explained treatment of infinity in mathematics. The concept of infinity has fascinated and confused mankind for centuries with concepts and ideas that cause even seasoned mathematicians to wonder. For instance, the idea that a set is infinite if it is not a finite set is an elementary concept that jolts our common sense and imagination. The "Mathematics of Infinity: A guide to Great Ideas" uniquely explores how we can manipulate these ideas when our common sense rebels at the conclusions we are drawing. Writing with clear knowledge and affection for the subject, the author introduces and explores infinite sets, infinite cardinals, and ordinals, thus challenging the readers' intuitive beliefs about infinity. Requiring little mathematical training and a healthy curiosity, the book presents a user-friendly approach to ideas involving the infinite. Readers will discover the main ideas of infinite cardinals and ordinal numbers without experiencing in-depth mathematical rigor.
Classic arguments and illustrative examples are provided throughout the book and are accompanied by a gradual progression of sophisticated notions designed to stun your intuitive view of the world. With a thoughtful and balanced treatment of both concepts and theory, "The Mathematics of Infinity" focuses on the following topics: Sets and Functions; Images and Preimages of Functions; Hilbert's Infinite Hotel Cardinals and Ordinals; the Arithmetic of Cardinals and Ordinals; the Continuum Hypothesis Elementary; Number Theory; the Riemann Hypothesis; and, the Logic of Paradoxes. Recommended as recreational reading for the mathematically inquisitive or as supplemental reading for curious college students, the "Mathematics of Infinity: A Guide to Great Ideas" gently leads readers into the world of counterintuitive mathematics.

Cover of Direct Sum Decompositions of Torsion-Free Finite Rank Groups

With plenty of new material not found in other books, Direct Sum Decompositions of Torsion-Free Finite Rank Groups explores advanced topics in direct sum decompositions of abelian groups and their consequences. The book illustrates a new way of studying these groups while still honoring the rich history of unique direct sum decompositions of groups.

Offering a unified approach to theoretic concepts, this reference covers isomorphism, endomorphism, refinement, the Baer splitting property, Gabriel filters, and endomorphism modules. It shows how to effectively study a group G by considering finitely generated projective right End(G)-modules, the left End(G)-module G, and the ring E(G) = End(G)/N(End(G)). For instance, one of the naturally occurring properties considered is when E(G) is a commutative ring. Modern algebraic number theory provides results concerning the isomorphism of locally isomorphic rtffr groups, finitely faithful S-groups that are J-groups, and each rtffr L-group that is a J-group. The book concludes with useful appendices that contain background material and numerous examples.