Algebra: A Comprehensive Course
Algebra is primarily concerned with the study of mathematical symbols as well as the rules that operate such symbols. It is applied in most of the sub-domains within mathematics. Algebra makes use of letters to denote numerical values. Some of the major branches of algebra are elementary algebra and abstract algebra. Elementary algebra focuses on the study of variables and polynomials. Abstract algebra studies the abstraction such as groups, rings and fields as well as elementary equation solving. It is applied in the study of various fields such as algebraic topology and algebraic number theory. This book is compiled in such a manner, that it will provide in-depth knowledge about the theory and practice of algebra. Some of the diverse topics covered herein address the varied branches that fall under this category. Coherent flow of topics, student-friendly language and extensive use of examples make this book an invaluable source of knowledge.
Separation in Point-Free Topology
This book is the first systematic treatment of this area so far scattered in a vast number of articles. As in classical topology, concrete problems require restricting the (generalized point-free) spaces by various conditions playing the roles of classical separation axioms. These are typically formulated in the language of points; but in the point-free context one has either suitable translations, parallels, or satisfactory replacements. The interrelations of separation type conditions, their merits, advantages and disadvantages, and consequences are discussed. Highlights of the book include a treatment of the merits and consequences of subfitness, various approaches to the Hausdorff's axiom, and normality type axioms. Global treatment of the separation conditions put them in a new perspective, and, a.o., gave some of them unexpected importance. The text contains a lot of quite recent results; the reader will see the directions the area is taking, and may find inspirationfor her/his further work.The book will be of use for researchers already active in the area, but also for those interested in this growing field (sometimes even penetrating into some parts of theoretical computer science), for graduate and PhD students, and others. For the reader's convenience, the text is supplemented with an Appendix containing necessary background on posets, frames and locales.
Abstract Algebra
This is a high level introduction to abstract algebra which is aimed at readers whose interests lie in mathematics and the information and physical sciences. In addition to introducing the main concepts of modern algebra - groups, rings, modules and fields - the book contains numerous applications, which are intended to illustrate the concepts and to show the utility and relevance of algebra today. In particular applications to Polya coloring theory, latin squares, Steiner systems, error correcting codes and economics are described. There is ample material here for a two semester course in abstract algebra. Proofs of almost all results are given. The reader led through the proofs in gentle stages. There are more than 500 problems, of varying degrees of diffi culty. The book should be suitable for advanced undergraduate students in their fi nal year of study and for fi rst or second year graduate students at a university in Europe or North America. In this third edition three new chapters have been added: an introduction to the representation theory of fi nite groups, free groups and presentations of groups, an introduction to category theory.
Memory Dump Analysis Anthology, Volume 13
This reference volume consists of revised, edited, cross-referenced, and thematically organized selected articles from Software Diagnostics Institute (DumpAnalysis.org + TraceAnalysis.org) and Software Diagnostics Library (former Crash Dump Analysis blog, DumpAnalysis.org/blog) about software diagnostics, root cause analysis, debugging, crash and hang dump analysis, software trace and log analysis written in December 2019 - July 2020 for software engineers developing and maintaining products on Windows and Linux platforms, quality assurance engineers testing software, technical support, escalation and site reliability engineers dealing with complex software issues, security and vulnerability researchers, reverse engineers, malware and memory forensics analysts. This volume is fully cross-referenced with volumes 1 - 12 and features: - 9 new crash dump analysis patterns with selected downloadable example memory dumps; - 15 new software trace and log analysis patterns; - Introduction to diagnostic analysis gestures; - Introduction to the category-theoretic view of debugging; - Lists of recommended category theory, number theory, and cybersecurity books.
An Invitation to Abstract Algebra
Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics. To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois' achievement in understanding how we can--and cannot--represent the roots of polynomials. The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory. The presentation includes the following features: Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters. The text can be used for a one, two, or three-term course. Each new topic is motivated with a question. A collection of projects appears in Chapter 23. Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks--period. This book is offered as a manual to a new way of thinking. The author's aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.
An Introduction to the Theory of Canonical Matrices
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Integral Quadratic Forms
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Complex Numbers and Conformal Mapping
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The Theory of Equations
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Joint Bulletin; no.7 (1921)
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An Introduction to the Theory of Groups
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Proceedings of the Bostonian Society, Annual Meeting; 1884
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Congruences in Number Theory
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Matrices
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The Complete Algebra
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Joint Bulletin; no.4-5 (1919)
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Skew Pbw Extensions
This monograph is devoted to a new class of non-commutative rings, skew Poincar矇-Birkhoff-Witt (PBW) extensions. Beginning with the basic definitions and ring-module theoretic/homological properties, it goes on to investigate finitely generated projective modules over skew PBW extensions from a matrix point of view. To make this theory constructive, the theory of Gr繹bner bases of left (right) ideals and modules for bijective skew PBW extensions is developed. For example, syzygies and the Ext and Tor modules over these rings are computed. Finally, applications to some key topics in the noncommutative algebraic geometry of quantum algebras are given, including an investigation of semi-graded Koszul algebras and semi-graded Artin-Schelter regular algebras, and the noncommutative Zariski cancellation problem. The book is addressed to researchers in noncommutative algebra and algebraic geometry as well as to graduate students and advanced undergraduate students.
Making Musical Time
This book is a comprehensive examination of the conception, perception, performance, and composition of time in music across time and culture. It surveys the literature of time in mathematics, philosophy, psychology, music theory, and somatic studies (medicine and disability studies) and looks ahead through original research in performance, composition, psychology, and education. It is the first monograph solely devoted to the theory of construction of musical time since Kramer in 1988, with new insights, mathematical precision, and an expansive global and historical context.The mathematical methods applied for the construction of musical time are totally new. They relate to category theory (projective limits) and the mathematical theory of gestures. These methods and results extend the music theory of time but also apply to the applied performative understanding of making music. In addition, it is the very first approach to a constructive theory of time, deduced from the recent theory of musical gestures and their categories.Making Musical Time is intended for a wide audience of scholars with interest in music. These include mathematicians, music theorists, (ethno)musicologists, music psychologists / educators / therapists, music performers, philosophers of music, audiologists, and acousticians.
Coarse Geometry of Topological Groups
This book provides a general framework for doing geometric group theory for many non-locally-compact topological transformation groups that arise in mathematical practice, including homeomorphism and diffeomorphism groups of manifolds, isometry groups of separable metric spaces and automorphism groups of countable structures. Using Roe's framework of coarse structures and spaces, the author defines a natural coarse geometric structure on all topological groups. This structure is accessible to investigation, especially in the case of Polish groups, and often has an explicit description, generalising well-known structures in familiar cases including finitely generated discrete groups, compactly generated locally compact groups and Banach spaces. In most cases, the coarse geometric structure is metrisable and may even be refined to a canonical quasimetric structure on the group. The book contains many worked examples and sufficient introductory material to be accessible to beginning graduate students. An appendix outlines several open problems in this young and rich theory.
Groups and Symmetry
This textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. This is the second book of the set of two books on groups theory. Topics on linear transformation and linear groups, group actions on sets, Sylow's theorem, simple groups, products of groups, normal series, free groups, platonic solids, Frieze and wallpaper symmetry groups and characters of groups have been discussed in depth. Covering all major topics, this book is targeted to advanced undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.
Algebra I All-In-One for Dummies
Solve for 'X' with this practical and easy guide to everything algebra A solid understanding of algebra is the key to unlocking other areas of math and science that rely on the concepts and skills that happen in a foundational Algebra class. Algebra I All-In-One For Dummies is the key! With it, you'll get everything you need to solve the mystery of Algebra I. This book proves that algebra is for everyone with straightforward, unit-based instruction, hundreds of examples and practice problems, and two quizzes for every chapter - one in the book and another (totally different!) online. From graph and word problems to the FOIL method and common algebra terminology, Algebra I All-In-One For Dummies walks you step-by-step through ALL the concepts you need to know to slay your Algebra I class. In this handy guide, you'll also: Receive instruction and tips on how to handle basic and intermediate algebraic tasks such as factoring and equation simplification Banish math anxiety forever by developing an intuitive understanding of how algebra works Get a handle on graphing problems and functions, as well as inequalities and word problems Algebra I All-In-One For Dummies is a must-read for Algebra students looking for an everything-in-one-book supplement to their coursework, as well as anyone hoping to brush up on their math before tackling a related subject, such as physics, chemistry, or a more advanced math topic.
The Absolute Galois Group of a Semi-Local Field
This book is devoted to the structure of the absolute Galois groups of certain algebraic extensions of the field of rational numbers. Its main result, a theorem proved by the authors and Florian Pop in 2012, describes the absolute Galois group of distinguished semi-local algebraic (and other) extensions of the rational numbers as free products of the free profinite group on countably many generators and local Galois groups. This is an instance of a positive answer to the generalized inverse problem of Galois theory. Adopting both an arithmetic and probabilistic approach, the book carefully sets out the preliminary material needed to prove the main theorem and its supporting results. In addition, it includes a description of Melnikov's construction of free products of profinite groups and, for the first time in book form, an account of a generalization of the theory of free products of profinite groups and their subgroups. The book will be of interest to researchers in field arithmetic, Galois theory and profinite groups.
Structure and Regularity of Group Actions on One-Manifolds
This book presents the theory of optimal and critical regularities of groups of diffeomorphisms, from the classical work of Denjoy and Herman, up through recent advances. Beginning with an investigation of regularity phenomena for single diffeomorphisms, the book goes on to describes a circle of ideas surrounding Filipkiewicz's Theorem, which recovers the smooth structure of a manifold from its full diffeomorphism group. Topics covered include the simplicity of homeomorphism groups, differentiability of continuous Lie group actions, smooth conjugation of diffeomorphism groups, and the reconstruction of spaces from group actions. Various classical and modern tools are developed for controlling the dynamics of general finitely generated group actions on one-dimensional manifolds, subject to regularity bounds, including material on Thompson's group F, nilpotent groups, right-angled Artin groups, chain groups, finitely generated groups with prescribed critical regularities, and applications to foliation theory and the study of mapping class groups. The book will be of interest to researchers in geometric group theory.
Algebra for High School
This book provides students with a tool to improve their knowledge in preparation to learning Mathematics. This book is intended to be a quick review of the Linear Equations and Inequalities. The book follows the main precalculus concepts needed in order to understand High School Algebra. It starts with the Cartesian System, then Straight lines, Linear Equations, and Linear Inequalities. It explains concepts as distance between points, slope of a line, solving linear one step and multi step equations, parallel and perpendicular lines, solving one step and multistep inequalities. This is the first book on the High School Algebra.
A First Course in Group Theory
Preliminaries Notions.- Symmetries of Shapes.- Binary Operations.- Cyclic Groups.- Inverse Functions and Permutations.- Group of Arithmetical Functions.- Matrix Groups.- Translation and Scaling Matrices.- Cosets of Subgroups and Lagrange's Theorem.- Normal Subgroups and Factor Groups.- Some Special Subgroups.- Commutators and Derived Subgroups.- Maximal Subgroups.- Group Homomorphisms.- Homomorphisms and Their Properties.- Cayley's Theorem.- Another View of Linear Groups.
Homological Theory of Representations
Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.
Direct and Inverse Finite-Dimensional Spectral Problems on Graphs
Considering that the motion of strings with finitely many masses on them is described by difference equations, this book presents the spectral theory of such problems on finite graphs of strings. The direct problem of finding the eigenvalues as well as the inverse problem of finding strings with a prescribed spectrum are considered. This monograph gives a comprehensive and self-contained account on the subject, thereby also generalizing known results. The interplay between the representation of rational functions and their zeros and poles is at the center of the methods used. The book also unravels connections between finite dimensional and infinite dimensional spectral problems on graphs, and between self-adjoint and non-self-adjoint finite-dimensional problems. This book is addressed to researchers in spectral theory of differential and difference equations as well as physicists and engineers who may apply the presented results and methods to their research.
Annual Report / Museum of Comparative Zoology; 2014-2015
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The Pharmacological Assay of Digitalis by Different Methods
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Introduction to Logic and Sets
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Sets, Sequences, and Mappings
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Elements of the Topology of Plane Sets of Points
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Algebraic Formulae [microform]
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The Teacher's Hand-book of Algebra [microform]
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The Theory Of Ruled Surfaces
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The Theory Of Ruled Surfaces
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Equations and Graphs; a Handbook of Algebraic Techniques
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Sets, Sequences, and Mappings
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The Young Algebraist's Companion
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A Study of Ergodicity and Redundancy Based on Inter-symbol Correlation of Finite Range Ergodicity and Redundancy.
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Matrix Iterative Analysis. --; 3
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The Continuum and Other Types of Serial Order; With an Introduction to Cantor's Transfinite Numbers. --
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Elements of Algebra
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A Treatise on the Theory of Determinants and Their Applications in Analysis and Geometry
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Elements of Vector Analysis
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Epidemiology, Old and New
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Matrix Iterative Analysis
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Modern Algebra. In Part a Development From Lectures by E. Artin and E. Noether; Translated From the 2d Rev. German Ed. by Fred Blum, With Revisions and Additions by the Author; 1
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