This is a concise introductory textbook for a one semester course in the history and philosophy of mathematics. It is written for mathematics majors, philosophy students, history of science students and secondary school mathematics teachers. The only prerequisite is a solid command of pre-calculus mathematics. It is shorter than the standard textbooks in that area and thus more accessible to students who have trouble coping with vast amounts of reading. Furthermore, there are many detailed explanations of the important mathematical procedures actually used by famous mathematicians, giving more mathematically talented students a greater opportunity to learn the history and philosophy by way of problem solving. Several important philosophical topics are pursued throughout the text, giving the student an opportunity to come to a full and consistent knowledge of their development. These topics include infinity, the nature of motion, and Platonism. This book offers, in fewer pages, a deep penetration into the key mathematical and philosophical aspects of the history of mathematics.

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.

In Mathematical Mountaintops, John Casti brilliantly recreates the solutions to the five greatest mathematical problems of all time: The Four-Color Map Problem, Fermat's Last Theorem, The Continuum Hypothesis, Kepler's Conjecture, and Hilbert's Tenth Problem. Casti recounts these mathematical quests with great verve. In retelling the story of Hilbert's Tenth Problem, for instance, he sweeps from Britain to New York to Leningrad and introduces us to such luminaries as Alan Turing, before turning to the young Soviet researcher who credited his breakthrough to a 700-year-old Italian problem about rabbits. He describes how Fermat's Last Theorem tantalized generations of scientists, who tried for three centuries to answer it, and relates how the final solution was greeted with the unprecedented front-page headlines, prize money, and international celebration--before a flaw (soon resolved) turned up. Casti's account of the struggle to solve Kepler's Conjecture wittily reveals how the "proof of the obvious" sometimes eludes us for centuries. And his discussion of The Continuum Hypothesis movingly portrays the tragic figure of Georg Cantor, the troubled genius who created the first truly original mathematics since the Greeks, yet died insane in an institution. Casti closes with a preview of the "Magnificent Seven"--the greatest unsolved mathematical mysteries, each of which carries a million-dollar bounty from the Clay Mathematics Institute--including the Poincare Conjecture, the Yang-Mills Existence and Mass Gap (why physicists can't isolate quarks), and the Reimann Hypothesis ("the granddaddy of all mathematical mysteries"). Mathematical Mountaintops is a brilliant account of mathematicians in action--seeking hidden patterns and structures, forging elegant chains of reasoning--as they struggle with problems that challenged the greatest minds for decades, if not centuries.

A dynamical system is called isochronous if it features in its phase space an open, fully-dimensional region where all its solutions are periodic in all its degrees of freedom with the same, fixed period. Recently a simple transformation has been introduced, applicable to quite a large class of dynamical systems, that yields autonomous systems which are isochronous. This justifies the notion that isochronous systems are not rare. In this book the procedure to manufacture isochronous systems is reviewed, and many examples of such systems are provided. Examples include many-body problems characterized by Newtonian equations of motion in spaces of one or more dimensions, Hamiltonian systems, and also nonlinear evolution equations (PDEs).

In recent years, mathematicians have detailed simpler proofs of known theorems, have identified new applications of the method of averaging, and have obtained many new results of these applications. Encompassing these novel aspects, Method of Averaging of the Infinite Interval: Theory and Applications rigorously explains the modern theory of the method of averaging and provides a solid understanding of the results obtained when applying this theory. The book starts with the less complicated theory of averaging linear differential equations (LDEs), focusing on almost periodic functions. It describes stability theory and Shtokalo's method, and examines various applications, including parametric resonance and the construction of asymptotics. After establishing this foundation, the author goes on to explore nonlinear equations. He studies standard form systems in which the right-hand side of a system is proportional to a small parameter and proves theorems similar to Banfi's theorem. The final chapters are devoted to systems with a rapidly rotating phase. Covering an important asymptotic method of differential equations, this book provides a thorough understanding of the method of averaging theory and its resulting applications.

This book is devoted to the study of scalar and asymptotic scalar derivatives and their applications to some problems in nonlinear analysis, Riemannian geometry and applied mathematics. The theoretical results are developed in particular with respect to the study of complementarity problems, monotonicity of nonlinear mappings and the non-gradient type monotonicity on Riemannian manifolds. Scalar and Asymptotic Derivatives: Theory and Applications also presents the material in relation to Euclidean spaces, Hilbert spaces, Banach spaces, Riemannian manifolds, and Hadamard manifolds.

Combinatorial mathematicians and statisticians have made a wide range of contributions to the development of block designs, and this book brings together much of that work. The designs developed for a specific problem are used in a variety of different settings. Applications include controlled sampling, randomized response, validation and valuation studies, intercropping experiments, brand cross-effect designs, lotto and tournaments. The intra- and inter- block, nonparametric and covariance analysis are discussed for general block designs, and the concepts of connectedness, orthogonality, and all types of balances in designs are carefully summarized. Readers are also introduced to the designs currently playing a prominent role in the field: alpha designs, trend-free designs, balanced treatment-control designs, nearest neighbor designs, and nested designs. This book provides the important background results required by researchers in block designs and related areas and prepares them for more complex research on the subject.