Mathematics

This book is a text and reference book on Category Theory, a branch of abstract algebra. The book contains clear definitions of the essential concepts, which are illuminated with numerous accessible examples. It provides full proofs of all the important propositions and theorems, and aims to make the basic ideas, theorems, and methods of Category Theory understandable. Although it assumes few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; and monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided.

This book provides an up-to-date account of the literature on the subject of determining the structure of rings over which cyclic modules or proper cyclic modules have a finiteness condition or a homological property. The finiteness conditions and homological properties are closely interrelated in the sense that either hypothesis induces the other in some form. The main objective behind writing this volume is the absence of a book that contains most of the relevant material on the subject. Since before the last half century, numerous authors including Armendariz, Beidar, Camillo, Chatters, Clark, Cohen, Cozzens, Faith, Farkas, Fisher, Goodearl, Gómez Pardo, Guil Asensio, Hajarnavis, Huynh, Jain, Kohler, Levy, López-Permouth, Mohamed, Ornstein, Osofsky, Singh, Skornyakov, Smith, Tuganbaev, and Wisbauer have investigated rings whose factor rings or factor modules have a finiteness condition or a homological property. They made important contributions leading to new directions and questions that have been listed at the end of each chapter for the benefit of future researchers. The bibliography has more than 200 references and is not claimed to be exhaustive.

During the early part of the last century, F. G. Frobenius raised, in his lectures, the following problem (called the Diophantine Frobenius Problem FP): given relatively prime positive integers a1, . . . , an, find the largest natural number (called the Frobenius number and denoted by g(a1, . . . , an)) that is not representable as a nonnegative integer combination of a1, . . . , an. It turned out that the knowledge of g(a1, . . . , an) has been extremely useful to investigate many different problems. A number of methods, from several areas of mathematics, have been used in the hope of finding a formula giving the Frobenius number and algorithms to calculate it. The main intention of this book is to highlight such ‘methods, ideas, viewpoints, and applications’ for as wide an audience as possible. This book aims to provide a comprehensive exposition of what is known today on FP.

The 1995 work by Wiles and Taylor-Wiles opened up a whole new technique in algebraic number theory and, a decade on, the waves caused by this incredibly important work are still being felt. This book describes a generalization of their techniques to Hilbert modular forms (towards the proof of the celebrated ‘R=T’ theorem) and applications of the theorem that have been found. Applications include a proof of the torsion of the adjoint Selmer group (over a totally real field F and over the Iwasawa tower of F) and an explicit formula of the L-invariant of the arithmetic p-adic adjoint L-functions. This implies the torsion of the classical anticyclotomic Iwasawa module of a CM field over the Iwasawa algebra. When specialized to an elliptic Tate curve over F by the L-invariant formula, the invariant of the adjoint square of the curve has exactly the same expression as the one in the conjecture of Mazur-Tate-Teitelbaum (which is for the standard L-function of the elliptic curve and is now a theorem of Greenberg-Stevens).

This book offers a detailed treatment of the mathematical theory of Krylov subspace methods with focus on solving systems of linear algebraic equations. Starting from the idea of projections, Krylov subspace methods are characterised by their orthogonality and minimisation properties. Projections onto highly nonlinear Krylov subspaces can be linked with the underlying problem of moments, and therefore Krylov subspace methods can be viewed as matching moments model reduction. This allows enlightening reformulations of questions from matrix computations into the language of orthogonal polynomials, Gauss–Christoffel quadrature, continued fractions, and, more generally, of Vorobyev method of moments. Using the concept of cyclic invariant subspaces conditions are studied that allow generation of orthogonal Krylov subspace bases via short recurrences. The results motivate the practically important distinction between Hermitian and non-Hermitian problems. Finally, the book thoroughly addresses the computational cost while using Krylov subspace methods. The investigation includes effects of finite precision arithmetic and focuses on the method of conjugate gradients (CG) and generalised minimal residuals (GMRES) as major examples. The book emphasises that algebraic computations must always be considered in the context of solving real-world problems, where the mathematical modelling, discretisation, and computation cannot be separated from each other. Moreover, the book underlines the importance of the historical context and it demonstrates that knowledge of early developments can play an important role in understanding and resolving very recent computational problems. Many extensive historical notes are therefore included as an inherent part of the text. The book ends with formulating some omitted issues and challenges which need to be addressed in future work. The book is intended as a research monograph which can be used in a wide scope of graduate courses on related subjects. It can be beneficial also for readers interested in the history of mathematics.