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"Algebra, Universal."
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Congruence Lattices of Ideals in Categories and (Partial) Semigroups
This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations,
diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain
normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several
specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions;
Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations
are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid
categories.
eBook
Groups, Algebras and Identities
This volume contains the proceedings of the Research Workshop of the Israel Science Foundation on Groups, Algebras and Identities, held from March 20-24, 2016, at Bar-Ilan University and The Hebrew University of Jerusalem, Israel, in honor of Boris Plotkin's 90th birthday.The papers in this volume cover various topics of universal algebra, universal algebraic geometry, logic geometry, and algebraic logic, as well as applications of universal algebra to computer science, geometric ring theory, small cancellation theory, and Boolean algebras.
eBook
Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
An operator C on a Hilbert space \\mathcal H dilates to an operator T on a Hilbert space \\mathcal K if there is an isometry V:\\mathcal H\\to \\mathcal K such that C= V^* TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor \\vartheta (d), expressed as a ratio of \\Gamma functions for d even, of all d\\times d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space.
eBook
Unknown Quantity
Prime Obsession taught us not to be afraid to put the math in a math book. Unknown Quantity heeds the lesson well. So grab your graphing calculators, slip out the slide rules, and buckle up! John Derbyshire is introducing us to algebra through the ages-and it promises to be just what his die-hard fans have been waiting for. \"Here is the story of algebra.\" With this deceptively simple introduction, we begin our journey. Flanked by formulae, shadowed by roots and radicals, escorted by an expert who navigates unerringly on our behalf, we are guaranteed safe passage through even the most treacherous mathematical terrain. Our first encounter with algebraic arithmetic takes us back 38 centuries to the time of Abraham and Isaac, Jacob and Joseph, Ur and Haran, Sodom and Gomorrah. Moving deftly from Abel's proof to the higher levels of abstraction developed by Galois, we are eventually introduced to what algebraists have been focusing on during the last century. As we travel through the ages, it becomes apparent that the invention of algebra was more than the start of a specific discipline of mathematics-it was also the birth of a new way of thinking that clarified both basic numeric concepts as well as our perception of the world around us. Algebraists broke new ground when they discarded the simple search for solutions to equations and concentrated instead on abstract groups. This dramatic shift in thinking revolutionized mathematics. Written for those among us who are unencumbered by a fear of formulae, Unknown Quantity delivers on its promise to present a history of algebra. Astonishing in its bold presentation of the math and graced with narrative authority, our journey through the world of algebra is at once intellectually satisfying and pleasantly challenging.
eBook
Optimization algorithms on matrix manifolds
Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra.
eBook
Noncommutative rings and their applications : International Conference on Noncommutative Rings and Their Applications, July 1-4, 2013, Université d'Artois, Lens, France
This volume contains the Proceedings of an International Conference on Noncommutative Rings and Their Applications, held July 1-4, 2013, at the Universite d'Artois, Lens, France. It presents recent developments in the theories of noncommutative rings and modules over such rings as well as applications of these to coding theory, enveloping algebras, and Leavitt path algebras.Material from the course ``Foundations of Algebraic Coding Theory``, given by Steven Dougherty, is included and provides the reader with the history and background of coding theory as well as the interplay between coding theory and algebra. In module theory, many new results related to (almost) injective modules, injective hulls and automorphism-invariant modules are presented. Broad generalizations of classical projective covers are studied and category theory is used to describe the structure of some modules. In some papers related to more classical ring theory such as quasi duo rings or clean elements, new points of view on classical conjectures and standard open problems are given. Descriptions of codes over local commutative Frobenius rings are discussed, and a list of open problems in coding theory is presented within their context.
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