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An Invitation to Abstract Algebra

By Steven J. Rosenberg
  • ISBN Code: : 1000516318
  • Publisher : CRC Press
  • Pages : 390
  • Category : Mathematics
  • Reads : 228
  • Book Compatibility : Pdf, ePub, Mobi, Kindle
  • Pdf : an-invitation-to-abstract-algebra.pdf

Book Excerpt :

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.

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Exam Prep for: The Heart of Mathematics; An Invitation to ...

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  • Publisher : N.a
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  • Pdf File: exam-prep-for.pdf

Book Short Summary:

Read and download full book Exam Prep for: The Heart of Mathematics; An Invitation to ...

An Invitation to Knot Theory

By Heather A. Dye
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 1315362384
  • Total of Pages : 256
  • Category : Mathematics
  • Members : 194
  • Pdf File: an-invitation-to-knot-theory.pdf

Book Short Summary:

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra. The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

An Invitation to Quantum Cohomology

By Joachim Kock,Israel Vainsencher
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 0817644954
  • Total of Pages : 162
  • Category : Mathematics
  • Members : 693
  • Pdf File: an-invitation-to-quantum-cohomology.pdf

Book Short Summary:

Elementary introduction to stable maps and quantum cohomology presents the problem of counting rational plane curves Viewpoint is mostly that of enumerative geometry Emphasis is on examples, heuristic discussions, and simple applications to best convey the intuition behind the subject Ideal for self-study, for a mini-course in quantum cohomology, or as a special topics text in a standard course in intersection theory

An Invitation to Morse Theory

By Liviu Nicolaescu
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 038749510X
  • Total of Pages : 242
  • Category : Mathematics
  • Members : 353
  • Pdf File: an-invitation-to-morse-theory.pdf

Book Short Summary:

This book offers readers a taste of the "unreasonable effectiveness" of Morse theory. It covers many of the most important topics in Morse theory along with applications. The book details topics such as Morse-Smale flows, min-max theory, moment maps and equivariant cohomology, and complex Morse theory. In addition, many examples, problems, and illustrations further enhance the value of this useful introduction to Morse Theory.

An Invitation to C*-Algebras

By W. Arveson
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 1461263719
  • Total of Pages : 108
  • Category : Mathematics
  • Members : 588
  • Pdf File: an-invitation-to-c-algebras.pdf

Book Short Summary:

This book gives an introduction to C*-algebras and their representations on Hilbert spaces. We have tried to present only what we believe are the most basic ideas, as simply and concretely as we could. So whenever it is convenient (and it usually is), Hilbert spaces become separable and C*-algebras become GCR. This practice probably creates an impression that nothing of value is known about other C*-algebras. Of course that is not true. But insofar as representations are con cerned, we can point to the empirical fact that to this day no one has given a concrete parametric description of even the irreducible representations of any C*-algebra which is not GCR. Indeed, there is metamathematical evidence which strongly suggests that no one ever will (see the discussion at the end of Section 3. 4). Occasionally, when the idea behind the proof of a general theorem is exposed very clearly in a special case, we prove only the special case and relegate generalizations to the exercises. In effect, we have systematically eschewed the Bourbaki tradition. We have also tried to take into account the interests of a variety of readers. For example, the multiplicity theory for normal operators is contained in Sections 2. 1 and 2. 2. (it would be desirable but not necessary to include Section 1. 1 as well), whereas someone interested in Borel structures could read Chapter 3 separately. Chapter I could be used as a bare-bones introduction to C*-algebras. Sections 2.

An Invitation to Unbounded Representations of ∗-Algebras on Hilbert Space

By Konrad Schmüdgen
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Nature
  • Book Code : 3030463664
  • Total of Pages : 381
  • Category : Mathematics
  • Members : 933
  • Pdf File: an-invitation-to-unbounded-representations-of-algebras-on-hilbert-space.pdf

Book Short Summary:

This textbook provides an introduction to representations of general ∗-algebras by unbounded operators on Hilbert space, a topic that naturally arises in quantum mechanics but has so far only been properly treated in advanced monographs aimed at researchers. The book covers both the general theory of unbounded representation theory on Hilbert space as well as representations of important special classes of ∗-algebra, such as the Weyl algebra and enveloping algebras associated to unitary representations of Lie groups. A broad scope of topics are treated in book form for the first time, including group graded ∗-algebras, the transition probability of states, Archimedean quadratic modules, noncommutative Positivstellensätze, induced representations, well-behaved representations and representations on rigged modules. Making advanced material accessible to graduate students, this book will appeal to students and researchers interested in advanced functional analysis and mathematical physics, and with many exercises it can be used for courses on the representation theory of Lie groups and its application to quantum physics. A rich selection of material and bibliographic notes also make it a valuable reference.

Quadratic Number Theory: An Invitation to Algebraic Methods in the Higher Arithmetic

By J. L. Lehman
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : American Mathematical Soc.
  • Book Code : 1470447371
  • Total of Pages : 394
  • Category : Algebraic fields
  • Members : 882
  • Pdf File: quadratic-number-theory.pdf

Book Short Summary:

Quadratic Number Theory is an introduction to algebraic number theory for readers with a moderate knowledge of elementary number theory and some familiarity with the terminology of abstract algebra. By restricting attention to questions about squares the author achieves the dual goals of making the presentation accessible to undergraduates and reflecting the historical roots of the subject. The representation of integers by quadratic forms is emphasized throughout the text. Lehman introduces an innovative notation for ideals of a quadratic domain that greatly facilitates computation and he uses this to particular effect. The text has an unusual focus on actual computation. This focus, and this notation, serve the author's historical purpose as well; ideals can be seen as number-like objects, as Kummer and Dedekind conceived of them. The notation can be adapted to quadratic forms and provides insight into the connection between quadratic forms and ideals. The computation of class groups and continued fraction representations are featured—the author's notation makes these computations particularly illuminating. Quadratic Number Theory, with its exceptionally clear prose, hundreds of exercises, and historical motivation, would make an excellent textbook for a second undergraduate course in number theory. The clarity of the exposition would also make it a terrific choice for independent reading. It will be exceptionally useful as a fruitful launching pad for undergraduate research projects in algebraic number theory.

Basic Modern Algebra with Applications

By Mahima Ranjan Adhikari,Avishek Adhikari
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 8132215990
  • Total of Pages : 637
  • Category : Mathematics
  • Members : 730
  • Pdf File: basic-modern-algebra-with-applications.pdf

Book Short Summary:

The book is primarily intended as a textbook on modern algebra for undergraduate mathematics students. It is also useful for those who are interested in supplementary reading at a higher level. The text is designed in such a way that it encourages independent thinking and motivates students towards further study. The book covers all major topics in group, ring, vector space and module theory that are usually contained in a standard modern algebra text. In addition, it studies semigroup, group action, Hopf's group, topological groups and Lie groups with their actions, applications of ring theory to algebraic geometry, and defines Zariski topology, as well as applications of module theory to structure theory of rings and homological algebra. Algebraic aspects of classical number theory and algebraic number theory are also discussed with an eye to developing modern cryptography. Topics on applications to algebraic topology, category theory, algebraic geometry, algebraic number theory, cryptography and theoretical computer science interlink the subject with different areas. Each chapter discusses individual topics, starting from the basics, with the help of illustrative examples. This comprehensive text with a broad variety of concepts, applications, examples, exercises and historical notes represents a valuable and unique resource.

An Invitation to Alexandrov Geometry

By Stephanie Alexander,Vitali Kapovitch,Anton Petrunin
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer
  • Book Code : 3030053121
  • Total of Pages : 88
  • Category : Mathematics
  • Members : 887
  • Pdf File: an-invitation-to-alexandrov-geometry.pdf

Book Short Summary:

Aimed toward graduate students and research mathematicians, with minimal prerequisites this book provides a fresh take on Alexandrov geometry and explains the importance of CAT(0) geometry in geometric group theory. Beginning with an overview of fundamentals, definitions, and conventions, this book quickly moves forward to discuss the Reshetnyak gluing theorem and applies it to the billiards problems. The Hadamard–Cartan globalization theorem is explored and applied to construct exotic aspherical manifolds.

An Invitation to Arithmetic Geometry

By Dino Lorenzini
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : American Mathematical Soc.
  • Book Code : 0821802674
  • Total of Pages : 397
  • Category : Arithmetical algebraic geometry
  • Members : 370
  • Pdf File: an-invitation-to-arithmetic-geometry.pdf

Book Short Summary:

Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject ... a highly welcome addition to the existing literature. --Zentralblatt MATH The interaction between number theory and algebraic geometry has been especially fruitful. In this volume, the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the one-dimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes, which will aid the reader who goes to the next level of this rich subject.

An Invitation To Algebraic Numbers And Algebraic Functions

By Franz Halter-Koch
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  • Publisher : CRC Press
  • Book Code : 0429014678
  • Total of Pages : 580
  • Category : Mathematics
  • Members : 548
  • Pdf File: an-invitation-to-algebraic-numbers-and-algebraic-functions.pdf

Book Short Summary:

The author offers a thorough presentation of the classical theory of algebraic numbers and algebraic functions which both in its conception and in many details differs from the current literature on the subject. The basic features are: Field-theoretic preliminaries and a detailed presentation of Dedekind’s ideal theory including non-principal orders and various types of class groups; the classical theory of algebraic number fields with a focus on quadratic, cubic and cyclotomic fields; basics of the analytic theory including the prime ideal theorem, density results and the determination of the arithmetic by the class group; a thorough presentation of valuation theory including the theory of difference, discriminants, and higher ramification. The theory of function fields is based on the ideal and valuation theory developed before; it presents the Riemann-Roch theorem on the basis of Weil differentials and highlights in detail the connection with classical differentials. The theory of congruence zeta functions and a proof of the Hasse-Weil theorem represent the culminating point of the volume. The volume is accessible with a basic knowledge in algebra and elementary number theory. It empowers the reader to follow the advanced number-theoretic literature, and is a solid basis for the study of the forthcoming volume on the foundations and main results of class field theory. Key features: • A thorough presentation of the theory of Algebraic Numbers and Algebraic Functions on an ideal and valuation-theoretic basis. • Several of the topics both in the number field and in the function field case were not presented before in this context. • Despite presenting many advanced topics, the text is easily readable. Franz Halter-Koch is professor emeritus at the university of Graz. He is the author of “Ideal Systems” (Marcel Dekker,1998), “Quadratic Irrationals” (CRC, 2013), and a co-author of “Non-Unique Factorizations” (CRC 2006).

A Project-Based Guide to Undergraduate Research in Mathematics

By Pamela E. Harris,Erik Insko,Aaron Wootton
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Nature
  • Book Code : 3030378535
  • Total of Pages : 324
  • Category : Mathematics
  • Members : 920
  • Pdf File: a-project-based-guide-to-undergraduate-research-in-mathematics.pdf

Book Short Summary:

This volume provides accessible and self-contained research problems designed for undergraduate student projects, and simultaneously promotes the development of sustainable undergraduate research programs. The chapters in this work span a variety of topical areas of pure and applied mathematics and mathematics education. Each chapter gives a self-contained introduction on a research topic with an emphasis on the specific tools and knowledge needed to create and maintain fruitful research programs for undergraduates. Some of the topics discussed include:• Disease modeling• Tropical curves and surfaces• Numerical semigroups• Mathematics EducationThis volume will primarily appeal to undergraduate students interested in pursuing research projects and faculty members seeking to mentor them. It may also aid students and faculty participating in independent studies and capstone projects.

Introduction To Linear Algebra

By Mark J. DeBonis
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 100054169X
  • Total of Pages : 434
  • Category : Mathematics
  • Members : 818
  • Pdf File: introduction-to-linear-algebra.pdf

Book Short Summary:

Introduction to Linear Algebra: Computation, Application, and Theory is designed for students who have never been exposed to the topics in a linear algebra course. The text is filled with interesting and diverse application sections but is also a theoretical text which aims to train students to do succinct computation in a knowledgeable way. After completing the course with this text, the student will not only know the best and shortest way to do linear algebraic computations but will also know why such computations are both effective and successful. Features: Includes cutting edge applications in machine learning and data analytics Suitable as a primary text for undergraduates studying linear algebra Requires very little in the way of pre-requisites

An Invitation to the Rogers-Ramanujan Identities

By Andrew V. Sills
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 1498745261
  • Total of Pages : 234
  • Category : Mathematics
  • Members : 796
  • Pdf File: an-invitation-to-the-rogers-ramanujan-identities.pdf

Book Short Summary:

The Rogers--Ramanujan identities are a pair of infinite series—infinite product identities that were first discovered in 1894. Over the past several decades these identities, and identities of similar type, have found applications in number theory, combinatorics, Lie algebra and vertex operator algebra theory, physics (especially statistical mechanics), and computer science (especially algorithmic proof theory). Presented in a coherant and clear way, this will be the first book entirely devoted to the Rogers—Ramanujan identities and will include related historical material that is unavailable elsewhere.

Advanced Number Theory

By Harvey Cohn
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Courier Corporation
  • Book Code : 0486149242
  • Total of Pages : 288
  • Category : Mathematics
  • Members : 460
  • Pdf File: advanced-number-theory.pdf

Book Short Summary:

Eminent mathematician/teacher approaches algebraic number theory from historical standpoint. Demonstrates how concepts, definitions, and theories have evolved during last two centuries. Features over 200 problems and specific theorems. Includes numerous graphs and tables.

Bridging Algebra, Geometry, and Topology

By Denis Ibadula,Willem Veys
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer
  • Book Code : 3319091867
  • Total of Pages : 289
  • Category : Mathematics
  • Members : 246
  • Pdf File: bridging-algebra-geometry-and-topology.pdf

Book Short Summary:

Algebra, geometry and topology cover a variety of different, but intimately related research fields in modern mathematics. This book focuses on specific aspects of this interaction. The present volume contains refereed papers which were presented at the International Conference “Experimental and Theoretical Methods in Algebra, Geometry and Topology”, held in Eforie Nord (near Constanta), Romania, during 20-25 June 2013. The conference was devoted to the 60th anniversary of the distinguished Romanian mathematicians Alexandru Dimca and Ştefan Papadima. The selected papers consist of original research work and a survey paper. They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra. The papers are written by well-known experts from different fields of mathematics, affiliated to universities from all over the word, they cover a broad range of topics and explore the research frontiers of a wide variety of contemporary problems of modern mathematics.

An Invitation to Knot Theory

By Heather A. Dye
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 1498701655
  • Total of Pages : 256
  • Category : Mathematics
  • Members : 596
  • Pdf File: an-invitation-to-knot-theory.pdf

Book Short Summary:

The Only Undergraduate Textbook to Teach Both Classical and Virtual Knot Theory An Invitation to Knot Theory: Virtual and Classical gives advanced undergraduate students a gentle introduction to the field of virtual knot theory and mathematical research. It provides the foundation for students to research knot theory and read journal articles on their own. Each chapter includes numerous examples, problems, projects, and suggested readings from research papers. The proofs are written as simply as possible using combinatorial approaches, equivalence classes, and linear algebra. The text begins with an introduction to virtual knots and counted invariants. It then covers the normalized f-polynomial (Jones polynomial) and other skein invariants before discussing algebraic invariants, such as the quandle and biquandle. The book concludes with two applications of virtual knots: textiles and quantum computation.

An Introduction to Abstract Mathematics

By Robert J. Bond,William J. Keane
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Waveland Press
  • Book Code : 1478608056
  • Total of Pages : 323
  • Category : Mathematics
  • Members : 915
  • Pdf File: an-introduction-to-abstract-mathematics.pdf

Book Short Summary:

Bond and Keane explicate the elements of logical, mathematical argument to elucidate the meaning and importance of mathematical rigor. With definitions of concepts at their disposal, students learn the rules of logical inference, read and understand proofs of theorems, and write their own proofs all while becoming familiar with the grammar of mathematics and its style. In addition, they will develop an appreciation of the different methods of proof (contradiction, induction), the value of a proof, and the beauty of an elegant argument. The authors emphasize that mathematics is an ongoing, vibrant disciplineits long, fascinating history continually intersects with territory still uncharted and questions still in need of answers. The authors extensive background in teaching mathematics shines through in this balanced, explicit, and engaging text, designed as a primer for higher- level mathematics courses. They elegantly demonstrate process and application and recognize the byproducts of both the achievements and the missteps of past thinkers. Chapters 1-5 introduce the fundamentals of abstract mathematics and chapters 6-8 apply the ideas and techniques, placing the earlier material in a real context. Readers interest is continually piqued by the use of clear explanations, practical examples, discussion and discovery exercises, and historical comments.

Algebraic and Discrete Mathematical Methods for Modern Biology

By Raina Robeva
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Academic Press
  • Book Code : 0128012714
  • Total of Pages : 382
  • Category : Mathematics
  • Members : 496
  • Pdf File: algebraic-and-discrete-mathematical-methods-for-modern-biology.pdf

Book Short Summary:

Written by experts in both mathematics and biology, Algebraic and Discrete Mathematical Methods for Modern Biology offers a bridge between math and biology, providing a framework for simulating, analyzing, predicting, and modulating the behavior of complex biological systems. Each chapter begins with a question from modern biology, followed by the description of certain mathematical methods and theory appropriate in the search of answers. Every topic provides a fast-track pathway through the problem by presenting the biological foundation, covering the relevant mathematical theory, and highlighting connections between them. Many of the projects and exercises embedded in each chapter utilize specialized software, providing students with much-needed familiarity and experience with computing applications, critical components of the "modern biology" skill set. This book is appropriate for mathematics courses such as finite mathematics, discrete structures, linear algebra, abstract/modern algebra, graph theory, probability, bioinformatics, statistics, biostatistics, and modeling, as well as for biology courses such as genetics, cell and molecular biology, biochemistry, ecology, and evolution. Examines significant questions in modern biology and their mathematical treatments Presents important mathematical concepts and tools in the context of essential biology Features material of interest to students in both mathematics and biology Presents chapters in modular format so coverage need not follow the Table of Contents Introduces projects appropriate for undergraduate research Utilizes freely accessible software for visualization, simulation, and analysis in modern biology Requires no calculus as a prerequisite Provides a complete Solutions Manual Features a companion website with supplementary resources

Introduction to Algebraic Geometry

By Serge Lang
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Courier Dover Publications
  • Book Code : 048683980X
  • Total of Pages : 272
  • Category : Mathematics
  • Members : 574
  • Pdf File: introduction-to-algebraic-geometry.pdf

Book Short Summary:

Rapid, concise, self-contained introduction assumes only familiarity with elementary algebra. Subjects include algebraic varieties; products, projections, and correspondences; normal varieties; differential forms; theory of simple points; algebraic groups; more. 1958 edition.

Category Theory in Context

By Emily Riehl
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Courier Dover Publications
  • Book Code : 0486820807
  • Total of Pages : 272
  • Category : Mathematics
  • Members : 185
  • Pdf File: category-theory-in-context.pdf

Book Short Summary:

Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition.

A Concrete Approach to Abstract Algebra

By W. W. Sawyer
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Courier Dover Publications
  • Book Code : 0486833313
  • Total of Pages : 240
  • Category : Mathematics
  • Members : 662
  • Pdf File: a-concrete-approach-to-abstract-algebra.pdf

Book Short Summary:

Brief, clear, and well written, this introductory treatment bridges the gap between traditional and modern algebra. Includes exercises with complete solutions. The only prerequisite is high school-level algebra. 1959 edition.

Linear Algebra

By Larry E. Knop
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 1584887834
  • Total of Pages : 752
  • Category : Mathematics
  • Members : 460
  • Pdf File: linear-algebra.pdf

Book Short Summary:

Linear Algebra: A First Course with Applications explores the fundamental ideas of linear algebra, including vector spaces, subspaces, basis, span, linear independence, linear transformation, eigenvalues, and eigenvectors, as well as a variety of applications, from inventories to graphics to Google’s PageRank. Unlike other texts on the subject, this classroom-tested book gives students enough time to absorb the material by focusing on vector spaces early on and using computational sections as numerical interludes. It offers introductions to MapleTM, MATLAB®, and TI-83 Plus for calculating matrix inverses, determinants, eigenvalues, and eigenvectors. Moving from the specific to the general, the author raises questions, provides motivation, and discusses strategy before presenting answers. Discussions of motivation and strategy include content and context to help students learn.

Abstract Linear Algebra

By Morton L. Curtis
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 1441987649
  • Total of Pages : 168
  • Category : Mathematics
  • Members : 164
  • Pdf File: abstract-linear-algebra.pdf

Book Short Summary:

Intended for a first course on the subject, this text begins from scratch and develops the standard topics of Linear Algebra. Its progresses simply towards its ultimate goal, the Theorem of Hurwitz, which argues that the only normed algebras over the real numbers are the real numbers, the complex numbers, the quaternions, and the octonions. The book stresses the complete logical development of the subject.

A Royal Road to Algebraic Geometry

By Audun Holme
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 9783642192258
  • Total of Pages : 366
  • Category : Mathematics
  • Members : 572
  • Pdf File: a-royal-road-to-algebraic-geometry.pdf

Book Short Summary:

This book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements. Euclid is said to have answered: “There is no royal road to geometry!” The book starts by explaining this enigmatic answer, the aim of the book being to argue that indeed, in some sense there is a royal road to algebraic geometry. From a point of departure in algebraic curves, the exposition moves on to the present shape of the field, culminating with Alexander Grothendieck’s theory of schemes. Contemporary homological tools are explained. The reader will follow a directed path leading up to the main elements of modern algebraic geometry. When the road is completed, the reader is empowered to start navigating in this immense field, and to open up the door to a wonderful field of research. The greatest scientific experience of a lifetime!

Abstract Algebra

By Stephen Lovett
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 1000605523
  • Total of Pages : 569
  • Category : Mathematics
  • Members : 858
  • Pdf File: abstract-algebra.pdf

Book Short Summary:

When a student of mathematics studies abstract algebra, he or she inevitably faces questions in the vein of, "What is abstract algebra" or "What makes it abstract?" Algebra, in its broadest sense, describes a way of thinking about classes of sets equipped with binary operations. In high school algebra, a student explores properties of operations (+, −, ×, and ÷) on real numbers. Abstract algebra studies properties of operations without specifying what types of number or object we work with. Any theorem established in the abstract context holds not only for real numbers but for every possible algebraic structure that has operations with the stated properties. This textbook intends to serve as a first course in abstract algebra. The selection of topics serves both of the common trends in such a course: a balanced introduction to groups, rings, and fields; or a course that primarily emphasizes group theory. The writing style is student-centered, conscientiously motivating definitions and offering many illustrative examples. Various sections or sometimes just examples or exercises introduce applications to geometry, number theory, cryptography and many other areas. This book offers a unique feature in the lists of projects at the end of each section. the author does not view projects as just something extra or cute, but rather an opportunity for a student to work on and demonstrate their potential for open-ended investigation. The projects ideas come in two flavors: investigative or expository. The investigative projects briefly present a topic and posed open-ended questions that invite the student to explore the topic, asking and to trying to answer their own questions. Expository projects invite the student to explore a topic with algebraic content or pertain to a particular mathematician’s work through responsible research. The exercises challenge the student to prove new results using the theorems presented in the text. The student then becomes an active participant in the development of the field.

Rings and Categories of Modules

By Frank W. Anderson,Kent R. Fuller
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 1461244188
  • Total of Pages : 378
  • Category : Mathematics
  • Members : 262
  • Pdf File: rings-and-categories-of-modules.pdf

Book Short Summary:

This book is intended to provide a reasonably self-contained account of a major portion of the general theory of rings and modules suitable as a text for introductory and more advanced graduate courses. We assume the famil iarity with rings usually acquired in standard undergraduate algebra courses. Our general approach is categorical rather than arithmetical. The continuing theme of the text is the study of the relationship between the one-sided ideal structure that a ring may possess and the behavior of its categories of modules. Following a brief outline of set-theoretic and categorical foundations, the text begins with the basic definitions and properties of rings, modules and homomorphisms and ranges through comprehensive treatments of direct sums, finiteness conditions, the Wedderburn-Artin Theorem, the Jacobson radical, the hom and tensor functions, Morita equivalence and duality, de composition theory of injective and projective modules, and semi perfect and perfect rings. In this second edition we have included a chapter containing many of the classical results on artinian rings that have hdped to form the foundation for much of the contemporary research on the representation theory of artinian rings and finite dimensional algebras. Both to illustrate the text and to extend it we have included a substantial number of exercises covering a wide spectrum of difficulty. There are, of course" many important areas of ring and module theory that the text does not touch upon.

An Introduction to Complex Analysis and the Laplace Transform

By Vladimir Eiderman
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 100051112X
  • Total of Pages : 392
  • Category : Mathematics
  • Members : 557
  • Pdf File: an-introduction-to-complex-analysis-and-the-laplace-transform.pdf

Book Short Summary:

The aim of this comparatively short textbook is a sufficiently full exposition of the fundamentals of the theory of functions of a complex variable to prepare the student for various applications. Several important applications in physics and engineering are considered in the book. This thorough presentation includes all theorems (with a few exceptions) presented with proofs. No previous exposure to complex numbers is assumed. The textbook can be used in one-semester or two-semester courses. In one respect this book is larger than usual, namely in the number of detailed solutions of typical problems. This, together with various problems, makes the book useful both for self- study and for the instructor as well. A specific point of the book is the inclusion of the Laplace transform. These two topics are closely related. Concepts in complex analysis are needed to formulate and prove basic theorems in Laplace transforms, such as the inverse Laplace transform formula. Methods of complex analysis provide solutions for problems involving Laplace transforms. Complex numbers lend clarity and completion to some areas of classical analysis. These numbers found important applications not only in the mathematical theory, but in the mathematical descriptions of processes in physics and engineering.

Directions For Mathematics Research Experience For Undergraduates

By Rubinstein Yanir A,Peterson Mark A
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : World Scientific
  • Book Code : 9814630330
  • Total of Pages : 252
  • Category : Mathematics
  • Members : 187
  • Pdf File: directions-for-mathematics-research-experience-for-undergraduates.pdf

Book Short Summary:

“The US National Science Foundation (NSF) Research Experiences for Undergraduates (REU) program in mathematics is now 25 years old, and it is a good time to think about what it has achieved, how it has changed, and where this idea will go next.”This was the premise of the conference held at Mt. Holyoke College during 21-22 June, 2013, and this circle of ideas is brought forward in this volume. The conference brought together diverse points of view, from NSF administrators, leaders of university-wide honors programs, to faculty who had led REUs, recent PhDs who are expected to lead them soon, and students currently in an REU themselves. The conversation was so varied that it justifies a book-length attempt to capture all that was suggested, reported, and said. Among the contributors are Ravi Vakil (Stanford), Haynes Miller (MIT), and Carlos Castillo-Chavez (Arizona, President's Obama Committee on the National Medal of Science 2010-2012).This book should serve not only as a collection of speakers' notes, but also as a source book for anyone interested in teaching mathematics and in the possibility of incorporating research-like experiences in mathematics classes at any level, as well as designing research experiences for undergraduates outside of the classroom.

Transition to Advanced Mathematics

By Danilo R. Diedrichs,Stephen Lovett
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : CRC Press
  • Book Code : 1000581861
  • Total of Pages : 520
  • Category : Mathematics
  • Members : 865
  • Pdf File: transition-to-advanced-mathematics.pdf

Book Short Summary:

This unique and contemporary text not only offers an introduction to proofs with a view towards algebra and analysis, a standard fare for a transition course, but also presents practical skills for upper-level mathematics coursework and exposes undergraduate students to the context and culture of contemporary mathematics. The authors implement the practice recommended by the Committee on the Undergraduate Program in Mathematics (CUPM) curriculum guide, that a modern mathematics program should include cognitive goals and offer a broad perspective of the discipline. Part I offers: An introduction to logic and set theory. Proof methods as a vehicle leading to topics useful for analysis, topology, algebra, and probability. Many illustrated examples, often drawing on what students already know, that minimize conversation about "doing proofs." An appendix that provides an annotated rubric with feedback codes for assessing proof writing. Part II presents the context and culture aspects of the transition experience, including: 21st century mathematics, including the current mathematical culture, vocations, and careers. History and philosophical issues in mathematics. Approaching, reading, and learning from journal articles and other primary sources. Mathematical writing and typesetting in LaTeX. Together, these Parts provide a complete introduction to modern mathematics, both in content and practice. Table of Contents Part I - Introduction to Proofs Logic and Sets Arguments and Proofs Functions Properties of the Integers Counting and Combinatorial Arguments Relations Part II - Culture, History, Reading, and Writing Mathematical Culture, Vocation, and Careers History and Philosophy of Mathematics Reading and Researching Mathematics Writing and Presenting Mathematics Appendix A. Rubric for Assessing Proofs Appendix B. Index of Theorems and Definitions from Calculus and Linear Algebra Bibliography Index Biographies Danilo R. Diedrichs is an Associate Professor of Mathematics at Wheaton College in Illinois. Raised and educated in Switzerland, he holds a PhD in applied mathematical and computational sciences from the University of Iowa, as well as a master’s degree in civil engineering from the Ecole Polytechnique Fédérale in Lausanne, Switzerland. His research interests are in dynamical systems modeling applied to biology, ecology, and epidemiology. Stephen Lovett is a Professor of Mathematics at Wheaton College in Illinois. He holds a PhD in representation theory from Northeastern University. His other books include Abstract Algebra: Structures and Applications (2015), Differential Geometry of Curves and Surfaces, with Tom Banchoff (2016), and Differential Geometry of Manifolds (2019).

Arithmetics

By Marc Hindry
  • File : Pdf, ePub, Mobi, Kindle
  • Publisher : Springer Science & Business Media
  • Book Code : 1447121317
  • Total of Pages : 322
  • Category : Mathematics
  • Members : 531
  • Pdf File: arithmetics.pdf

Book Short Summary:

Number theory is a branch of mathematics which draws its vitality from a rich historical background. It is also traditionally nourished through interactions with other areas of research, such as algebra, algebraic geometry, topology, complex analysis and harmonic analysis. More recently, it has made a spectacular appearance in the field of theoretical computer science and in questions of communication, cryptography and error-correcting codes. Providing an elementary introduction to the central topics in number theory, this book spans multiple areas of research. The first part corresponds to an advanced undergraduate course. All of the statements given in this part are of course accompanied by their proofs, with perhaps the exception of some results appearing at the end of the chapters. A copious list of exercises, of varying difficulty, are also included here. The second part is of a higher level and is relevant for the first year of graduate school. It contains an introduction to elliptic curves and a chapter entitled “Developments and Open Problems”, which introduces and brings together various themes oriented toward ongoing mathematical research. Given the multifaceted nature of number theory, the primary aims of this book are to: - provide an overview of the various forms of mathematics useful for studying numbers - demonstrate the necessity of deep and classical themes such as Gauss sums - highlight the role that arithmetic plays in modern applied mathematics - include recent proofs such as the polynomial primality algorithm - approach subjects of contemporary research such as elliptic curves - illustrate the beauty of arithmetic The prerequisites for this text are undergraduate level algebra and a little topology of Rn. It will be of use to undergraduates, graduates and phd students, and may also appeal to professional mathematicians as a reference text.