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The Diophantine Frobenius Problem
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198568209.jpg" alt="The Diophantine Frobenius Problem"/><br/></td><td><dl><dt>Author:</dt><dd>Jorge L. Ramírez Alfonsín</dd><dt>ISBN:</dt><dd>9780198568209</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198568209.001.0001</dd><dt>Published in print:</dt><dd>2005</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>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.</p>Jorge L. Ramírez Alfonsín2007-09-01The p-adic Simpson Correspondence (AM-193)
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691170282.jpg" alt="The p-adic Simpson Correspondence (AM-193)"/><br/></td><td><dl><dt>Author:</dt><dd>Ahmed Abbes, Michel Gros, Takeshi Tsuji</dd><dt>ISBN:</dt><dd>9780691170282</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Algebra</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691170282.001.0001</dd><dt>Published in print:</dt><dd>2016</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches. It mainly focuses on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored.</p>Ahmed Abbes, Michel Gros, and Takeshi Tsuji2017-10-19Category Theory
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198568612.jpg" alt="Category Theory"/><br/></td><td><dl><dt>Author:</dt><dd>Steve Awodey</dd><dt>ISBN:</dt><dd>9780198568612</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Algebra</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198568612.001.0001</dd><dt>Published in print:</dt><dd>2006</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>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.</p>Steve Awodey2007-09-01Hilbert Modular Forms and Iwasawa Theory
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198571025.jpg" alt="Hilbert Modular Forms and Iwasawa Theory"/><br/></td><td><dl><dt>Author:</dt><dd>Haruzo Hida</dd><dt>ISBN:</dt><dd>9780198571025</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Algebra</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198571025.001.0001</dd><dt>Published in print:</dt><dd>2006</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>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).</p>Haruzo Hida2007-09-01Krylov Subspace Methods
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199655410.jpg" alt="Krylov Subspace MethodsPrinciples and Analysis"/><br/></td><td><dl><dt>Author:</dt><dd>Jörg Liesen, Zdenek Strakos</dd><dt>ISBN:</dt><dd>9780199655410</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Applied Mathematics, Algebra</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199655410.001.0001</dd><dt>Published in print:</dt><dd>2012</dd><dt>Published Online:</dt><dd>2013-01-24</dd></dl></td></tr></table><p>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.</p>Jörg Liesen and Zdenek Strakos2013-01-24Cyclic Modules and the Structure of Rings
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199664511.jpg" alt="Cyclic Modules and the Structure of Rings"/><br/></td><td><dl><dt>Author:</dt><dd>S.K. Jain, Ashish K. Srivastava, Askar A. Tuganbaev</dd><dt>ISBN:</dt><dd>9780199664511</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Algebra</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199664511.001.0001</dd><dt>Published in print:</dt><dd>2012</dd><dt>Published Online:</dt><dd>2013-01-24</dd></dl></td></tr></table><p>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.</p>S.K. Jain, Ashish K. Srivastava, and Askar A. Tuganbaev2013-01-24Topics in Quaternion Linear Algebra
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780691161853.jpg" alt="Topics in Quaternion Linear Algebra"/><br/></td><td><dl><dt>Author:</dt><dd>Leiba Rodman</dd><dt>ISBN:</dt><dd>9780691161853</dd><dt>Publisher:</dt><dd>Princeton University Press</dd><dt>Subjects:</dt><dd>Mathematics, Algebra</dd><dt>DOI:</dt><dd>10.23943/princeton/9780691161853.001.0001</dd><dt>Published in print:</dt><dd>2014</dd><dt>Published Online:</dt><dd>2017-10-19</dd></dl></td></tr></table><p>Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.</p>Leiba Rodman2017-10-19