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Modulo I
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Book Synopsis Modular Algorithms in Symbolic Summation and Symbolic Integration by : Jürgen Gerhard
Download or read book Modular Algorithms in Symbolic Summation and Symbolic Integration written by Jürgen Gerhard and published by Springer. This book was released on 2004-11-12 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work brings together two streams in computer algebra: symbolic integration and summation on the one hand, and fast algorithmics on the other hand. In many algorithmically oriented areas of computer science, theanalysisof- gorithms–placedintothe limelightbyDonKnuth’stalkat the 1970ICM –provides a crystal-clear criterion for success. The researcher who designs an algorithmthat is faster (asymptotically, in the worst case) than any previous method receives instant grati?cation: her result will be recognized as valuable. Alas, the downside is that such results come along quite infrequently, despite our best efforts. An alternative evaluation method is to run a new algorithm on examples; this has its obvious problems, but is sometimes the best we can do. George Collins, one of the fathers of computer algebra and a great experimenter,wrote in 1969: “I think this demonstrates again that a simple analysis is often more revealing than a ream of empirical data (although both are important). ” Within computer algebra, some areas have traditionally followed the former methodology, notably some parts of polynomial algebra and linear algebra. Other areas, such as polynomial system solving, have not yet been amenable to this - proach. The usual “input size” parameters of computer science seem inadequate, and although some natural “geometric” parameters have been identi?ed (solution dimension, regularity), not all (potential) major progress can be expressed in this framework. Symbolic integration and summation have been in a similar state.
Book Synopsis Ideals, Varieties, and Algorithms by : David Cox
Download or read book Ideals, Varieties, and Algorithms written by David Cox and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 549 pages. Available in PDF, EPUB and Kindle. Book excerpt: Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. The book bases its discussion of algorithms on a generalisation of the division algorithm for polynomials in one variable that was only discovered in the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century. This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric theorem proving. In preparing this new edition, the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem.
Book Synopsis Abstract Algebra by : Jonathan K. Hodge
Download or read book Abstract Algebra written by Jonathan K. Hodge and published by CRC Press. This book was released on 2013-12-21 with total page 593 pages. Available in PDF, EPUB and Kindle. Book excerpt: Emphasizing active learning, this text not only teaches abstract algebra but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The book can be used in both rings-first and groups-first abstract algebra courses. Numerous activities, examples, and exercises illustrate the definitions, theorems, and concepts. Each chapter also discusses the connections among topics in ring theory and group theory, helping students see the relationships between the two main types of algebraic objects studied throughout the text.
Book Synopsis Ideals, Varieties, and Algorithms by : David A Cox
Download or read book Ideals, Varieties, and Algorithms written by David A Cox and published by Springer Science & Business Media. This book was released on 2008-07-31 with total page 565 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book details the heart and soul of modern commutative and algebraic geometry. It covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory. In addition to enhancing the text of the second edition, with over 200 pages reflecting changes to enhance clarity and correctness, this third edition of Ideals, Varieties and Algorithms includes: a significantly updated section on Maple; updated information on AXIOM, CoCoA, Macaulay 2, Magma, Mathematica and SINGULAR; and presents a shorter proof of the Extension Theorem.
Book Synopsis Algebra Interactive! by : Arjeh M. Cohen
Download or read book Algebra Interactive! written by Arjeh M. Cohen and published by Springer Science & Business Media. This book was released on 1999-08-25 with total page 174 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first interactive course covering first and second year algebra. Starting from such fundamental topics as integers and divisions, modular arithmetic and polynomials the content extends to rings, fields and permutation groups. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the theory while also contributing interactive calculators for computing with integers, polynomials and permutations. The computer algebra system GAP is integrated throughout, allowing the calculation and manipulation of mathematical objects. In addition, collections for Mathematica notebooks and Maple worksheets review the algorithms presented. Multiple choice exercises provide users with instant feedback, while facilities for monitoring students and a bulletin board complete this digital course.
Book Synopsis Introduction to Cardinal Arithmetic by : Michael Holz
Download or read book Introduction to Cardinal Arithmetic written by Michael Holz and published by Springer Science & Business Media. This book was released on 2009-11-23 with total page 309 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to modern cardinal arithmetic, developed in the frame of the axioms of Zermelo-Fraenkel set theory together with the axiom of choice. It splits into three parts. Part one, which is contained in Chapter 1, describes the classical cardinal arithmetic due to Bernstein, Cantor, Hausdorff, Konig, and Tarski. The results were found in the years between 1870 and 1930. Part two, which is Chapter 2, characterizes the development of cardinal arith metic in the seventies, which was led by Galvin, Hajnal, and Silver. The third part, contained in Chapters 3 to 9, presents the fundamental investigations in pcf-theory which has been developed by S. Shelah to answer the questions left open in the seventies. All theorems presented in Chapter 3 and Chapters 5 to 9 are due to Shelah, unless otherwise stated. We are greatly indebted to all those set theorists whose work we have tried to expound. Concerning the literature we owe very much to S. Shelah's book [Sh5] and to the article by M. R. Burke and M. Magidor [BM] which also initiated our students' interest for Shelah's pcf-theory.
Download or read book Lifting Modules written by John Clark and published by Springer Science & Business Media. This book was released on 2008-08-17 with total page 403 pages. Available in PDF, EPUB and Kindle. Book excerpt: Extending modules are generalizations of injective modules and, dually, lifting modules generalize projective supplemented modules. This duality exhibits a certain asymmetry. While the theory of extending modules is well documented in monographs and text books, the purpose of this monograph is to provide a thorough study of supplements and projectivity conditions needed to investigate classes of modules related to lifting modules.
Book Synopsis Modern Computer Algebra by : Joachim von zur Gathen
Download or read book Modern Computer Algebra written by Joachim von zur Gathen and published by Cambridge University Press. This book was released on 2013-04-25 with total page 811 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany one- or two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.
Download or read book Gröbner Bases written by Takayuki Hibi and published by Springer Science & Business Media. This book was released on 2014-01-07 with total page 488 pages. Available in PDF, EPUB and Kindle. Book excerpt: The idea of the Gröbner basis first appeared in a 1927 paper by F. S. Macaulay, who succeeded in creating a combinatorial characterization of the Hilbert functions of homogeneous ideals of the polynomial ring. Later, the modern definition of the Gröbner basis was independently introduced by Heisuke Hironaka in 1964 and Bruno Buchberger in 1965. However, after the discovery of the notion of the Gröbner basis by Hironaka and Buchberger, it was not actively pursued for 20 years. A breakthrough was made in the mid-1980s by David Bayer and Michael Stillman, who created the Macaulay computer algebra system with the help of the Gröbner basis. Since then, rapid development on the Gröbner basis has been achieved by many researchers, including Bernd Sturmfels. This book serves as a standard bible of the Gröbner basis, for which the harmony of theory, application, and computation are indispensable. It provides all the fundamentals for graduate students to learn the ABC’s of the Gröbner basis, requiring no special knowledge to understand those basic points. Starting from the introductory performance of the Gröbner basis (Chapter 1), a trip around mathematical software follows (Chapter 2). Then comes a deep discussion of how to compute the Gröbner basis (Chapter 3). These three chapters may be regarded as the first act of a mathematical play. The second act opens with topics on algebraic statistics (Chapter 4), a fascinating research area where the Gröbner basis of a toric ideal is a fundamental tool of the Markov chain Monte Carlo method. Moreover, the Gröbner basis of a toric ideal has had a great influence on the study of convex polytopes (Chapter 5). In addition, the Gröbner basis of the ring of differential operators gives effective algorithms on holonomic functions (Chapter 6). The third act (Chapter 7) is a collection of concrete examples and problems for Chapters 4, 5 and 6 emphasizing computation by using various software systems.
Download or read book Gröbner Bases written by Thomas Becker and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 587 pages. Available in PDF, EPUB and Kindle. Book excerpt: The origins of the mathematics in this book date back more than two thou sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu clid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbek istan and worked in Baghdad at the court of Harun al-Rashid's son. The word "algorithm" is actually a westernization of al-Khowarizmi's name, while "algebra" derives from "al-jabr," a term that appears in the title of his book Kitab al-jabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and al gorithms lasted roughly up to the beginning of this century; until then, the primary goal of algebra was the design of constructive methods for solving equations by means of symbolic transformations. During the second half of the nineteenth century, a new line of thought began to enter algebra from the realm of geometry, where it had been successful since Euclid's time, namely, the axiomatic method.
Book Synopsis Orthomorphism Graphs of Groups by : Anthony B. Evans
Download or read book Orthomorphism Graphs of Groups written by Anthony B. Evans and published by Springer. This book was released on 2006-11-15 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is about orthomorphisms and complete mappings of groups, and related constructions of orthogonal latin squares. It brings together, for the first time in book form, many of the results in this area. The aim of this book is to lay the foundations for a theory of orthomorphism graphsof groups, and to encourage research in this area. To this end, many directions for future research are suggested. The material in this book should be accessible to any graduate student who has taken courses in algebra (group theory and field theory). It will mainly be useful in research on combinatorial design theory, group theory and field theory.
Book Synopsis Introduction to Elliptic Curves and Modular Forms by : N. Koblitz
Download or read book Introduction to Elliptic Curves and Modular Forms written by N. Koblitz and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 258 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses, thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work.
Download or read book Algebra I written by Aleksej I. Kostrikin and published by Springer. This book was released on 2013-12-01 with total page 265 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Basic Notions of Algebra by : Igor R. Shafarevich
Download or read book Basic Notions of Algebra written by Igor R. Shafarevich and published by Springer Science & Business Media. This book was released on 2005-08-15 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: Wholeheartedly recommended to every student and user of mathematics, this is an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields studied in every university maths course, through Lie groups to cohomology and category theory, the author shows how the origins of each concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Required reading for mathematicians, from beginners to experts.
Download or read book Module Theory written by Alberto Facchini and published by Springer Science & Business Media. This book was released on 2012-02-05 with total page 296 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents topics in module theory and ring theory: some, such as Goldie dimension and semiperfect rings are now considered classical and others more specialized, such as dual Goldie dimension, semilocal endomorphism rings, serial rings and modules.
Download or read book Computer Algebra written by R. Albrecht and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: The journal Computing has established a series of supplement volumes the fourth of which appears this year. Its purpose is to provide a coherent presentation of a new topic in a single volume. The previous subjects were Computer Arithmetic 1977, Fundamentals of Numerical Computation 1980, and Parallel Processes and Related Automata 1981; the topic of this 1982 Supplementum to Computing is Computer Algebra. This subject, which emerged in the early nineteen sixties, has also been referred to as "symbolic and algebraic computation" or "formula manipulation". Algebraic algorithms have been receiving increasing interest as a result of the recognition of the central role of algorithms in computer science. They can be easily specified in a formal and rigorous way and provide solutions to problems known and studied for a long time. Whereas traditional algebra is concerned with constructive methods, computer algebra is furthermore interested in efficiency, in implementation, and in hardware and software aspects of the algorithms. It develops that in deciding effectiveness and determining efficiency of algebraic methods many other tools - recursion theory, logic, analysis and combinatorics, for example - are necessary. In the beginning of the use of computers for symbolic algebra it soon became apparent that the straightforward textbook methods were often very inefficient. Instead of turning to numerical approximation methods, computer algebra studies systematically the sources of the inefficiency and searches for alternative algebraic methods to improve or even replace the algorithms.
Book Synopsis The Future of Software Engineering by : Sebastian Nanz
Download or read book The Future of Software Engineering written by Sebastian Nanz and published by Springer Science & Business Media. This book was released on 2010-10-20 with total page 191 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on defining the achievements of software engineering in the past decades and showcasing visions for the future. It features a collection of articles by some of the most prominent researchers and technologists who have shaped the field: Barry Boehm, Manfred Broy, Patrick Cousot, Erich Gamma, Yuri Gurevich, Tony Hoare, Michael A. Jackson, Rustan Leino, David L. Parnas, Dieter Rombach, Joseph Sifakis, Niklaus Wirth, Pamela Zave, and Andreas Zeller. The contributed articles reflect the authors‘ individual views on what constitutes the most important issues facing software development. Both research- and technology-oriented contributions are included. The book provides at the same time a record of a symposium held at ETH Zurich on the occasion of Bertrand Meyer‘s 60th birthday.
