Algebraic Statistics for Computational Biology

Book Description

The quantitative analysis of biological sequence data is based on methods from statistics coupled with efficient algorithms from computer science. Algebra provides a framework for unifying many of the seemingly disparate techniques used by computational biologists. This book offers an introduction to this mathematical framework and describes tools from computational algebra for designing new algorithms for exact, accurate results. These algorithms can be applied to biological problems such as aligning genomes, finding genes and constructing phylogenies. As the first book in the exciting and dynamic area, it will be welcomed as a text for self-study or for advanced undergraduate and beginning graduate courses.

Customer Reviews:

Algebraic Statistics for Computational Biology.......2007-09-21

This text has provided technical examples and learning material for a complicated mathematical/biological/computer science topic. The text is not for introductory purposes, but when used for graduate level course of algebraic statistics, it has proven useful in a variety of manners. The text offers a great overview of the field of algebraic statistics, along with several chapters showing applied examples of the field. This text is recommended for graduate to post graduate level students interested in the field of algebraic statistics.

A very interesting approach.......2005-11-04

This book is unique, in that is fuses together two subjects that may at first be thought of as having zero intersection. Indeed, computational biology and algebraic geometry are very different in their concepts, terminology, and goals, but thanks to the efforts of the editors (and their students), techniques from algebraic geometry are now being applied to problems in genomics, particularly sequence alignment, or at least to rephrase these problems in the language of algebraic geometry. The economy of thought obtained via the use of algebraic geometry though should be measured against the time required to master its subject matter. It might be rare to find a mathematician who is a specialist in algebraic geometry to also be interested in bioinformatics or biological sequence analysis, even though the editors of this book clearly are. Even more difficult would be to find a computational biologist or specialist in bioinformatics who is willing to learn algebraic geometry, which is a formidable subject, even if attention is restricted to a subfield called "computational" algebraic geometry. Those readers familiar with computational algebraic geometry will see that `algebraic statistics', the name that has been given to the subject matter of the book, can be thought of as the algebraic geometry of toric varieties. The models of algebraic statistics include those that are very familiar to researchers in genomic sequence analysis, namely hidden Markov models and graphical models. This reviewer only read the first three chapters of this book, and so the following commentary will be restricted to these.

The authors give an extremely clear description of how statistical considerations arise in genetic sequence analysis in chapter one. To motivate the subject by considering a fictional character called "DiaNA" who is producing sequence data by the throwing of tetrahedral dice, with each face labeled with the letters A, C, G, and T. The dice are not considered to be fair. The problem is to find the likelihood of observing a particular sequence of data. The probabilities of the individual letters are assumed to be functions of parameters ("theta" parameters) and the (log) likelihood function of these parameters must then be maximized. This is straightforwardly by taking partial derivatives. The authors mention `Grobner bases' as a tool for simplifying the resulting equations.

For the authors a general statistical model (for discrete data) is a family of probability distributions on a finite `state space', usually taken to be a string of positive integers of length M. A probability distribution on this state space is thought of as a point in a `probability simplex', the latter being a collection of vectors in M-dimensional Euclidean space (RM) whose entries sum to 1. An `algebraic statistical model' arises as the image of a polynomial map F from D-dimensional Euclidean space (RD) to RM. The model parameters are in RD and D is usually taken to be considerably less than M. A `linear model' is defined as one where each of the M coordinate polynomials is a linear function. In a `toric' or `log-linear' model the logarithms of the probabilities are linear functions in the logarithms of the parameters. The authors show in detail how to formulate maximum likelihood estimation for both of these models and prove that they both have a unique local maximum. Noted in this discussion is the concept of a (relatively open) polytope, which is very familiar in the theory of toric varieties where it is called the `moment map.'

The expectation maximization (EM) technique, used for models that are not linear or toric is also discussed for a class of algebraic statistical models that computational biologists will easily recognize as hidden Markov models. The convergence properties of the EM algorithm are discussed in the language of algebraic geometry in chapter 3 of the book. The closure of the set of critical points of the likelihood function is an algebraic variety in D-dimensional complex space and is referred to as the `likelihood variety'. The likelihood variety is computed by constructing a polynomial ring in M unknowns and D parameters, called the "big ring" by the authors, and s subring in D parameters called the "small ring." An ideal J of the big ring generated by a particular collection of M + D polynomials is introduced, so that the points of the variety V(J) are related to the critical points of the log-likelihood function. The `elimination ideal' I in the small ring is referred to as the likelihood ideal of the statistical model with respect to the data. The optimization problem is solved by computing V(I), intersecting it with the pre-image of the probability simplex, and then identifying the local maxima in this intersection. These constructions motivate the notion of the `maximum likelihood degree' (ML degree) of an algebraic statistical model, which is the number of complex critical points of the log-likelihood function. The ML degree, as defined by the authors, measures the algebraic complexity of the process of maximum likelihood estimation for an algebraic statistical model. In most cases of interest the ML degree will be a positive integer and will be equal to the degree of the (irreducible) polynomial in the coordinates of the maximum likelihood estimate.

Also embedded in the discussions of these chapters is the notion of `tropical' arithmetic and geometry. Tropical arithmetic arises in the discussion of computation and dynamical programming. The `tropical semiring' consists of the extended real numbers along with two operations, called `addition' and `multiplication'. Addition of two real numbers consists of taking the smaller of the two, and `multiplication' consists of adding them. The utility of tropical arithmetic lies in its ability to find the shortest path in a weighted directed graph. The authors show how to make algebraic varieties, and thus statistical models, `tropical.' The tropicalization G of a statistical model is linear on each cone in the normal fan of the Newton polytope of the statistical model. The evaluation of G corresponds to solving the dynamic programs for all possible observations.

A Computational Introduction to Number Theory and Algebra

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Really a treasure
The background you really need, clear and sweet

A Computational Introduction to Number Theory and Algebra
Victor Shoup
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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Foundations of Cryptography
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ASIN: 0521851548

Book Description

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

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Customer Reviews:

Really a treasure.......2006-03-30

I'm a student digging into the cryptology for an year. The more article I read, the more confusion I encounter because of my poor mathematical background. However, when I get this, I could find answer to my puzzles, and make an more explicit way to settle down my own idea.

The background you really need, clear and sweet.......2005-11-06

This book is a marvel. It is clear and concise yet thorough. The author is obviously a bit of an obsessive compulsive, he has found the shortest paths from the clearest definitions to the most important results, each given with the cleanest, most insight-inducing proofs ... the results (and definitions) he gives are the ones any student (practitioner!) of modern computer science (especially cryptology) *needs* to know -- having this book on your shelves (and its contents in your head) should be a requirement for any degree, at any level, in computer science.
[Caveat: I know the author and have read his book in draft form. I also required my students to get it and read it, in a computer science course I taught.]

Clifford Algebra: A Computational Tool for Physicists

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A gem!
Light, clear, and understandable.
Incredibly good!
An educational tool for Clifford algebras
A pedagogical gem.

Clifford Algebra: A Computational Tool for Physicists
John Snygg
Manufacturer: Oxford University Press, USA
ProductGroup: Book
Binding: Hardcover

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New Foundations for Classical Mechanics (Fundamental Theories of Physics)
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ASIN: 0195098242

Book Description

Clifford algebras have become an indispensable tool for physicists at the cutting edge of theoretical investigations. Applications in physics range from special relativity and the rotating top at one end of the spectrum, to general relativity and Dirac's equation for the electron at the other. Clifford algebras have also become a virtual necessity in some areas of physics, and their usefulness is expanding in other areas, such as algebraic manipulations involving Dirac matrices in quantum thermodynamics; Kaluza-Klein theories and dimensional renormalization theories; and the formation of superstring theories. This book, aimed at beginning graduate students in physics and math, introduces readers to the techniques of Clifford algebras.

Customer Reviews:

A gem!.......2001-05-09

I would rate this book as a gem! To calibrate that let me say that I think Weinreich's Geometrical Vectors and Foster and Nightingale's General Relativity are gems. Chapter 1 gives a beautiful, clear and concise introduction to Clifford Algebra in flat 3-space using Dirac's anti-commuting gamma matrices. If you have ever wondered about off-hand comments that rotations are double reflections and why half angles enter into this business this is the place to get enlightened. In an amusing series of photographs the author illustrates the 4-pi periodicity of certain objects. The object here is a copy of MTW's Gravitation - one of the more imaginative uses of this tome. As an example of the application of the CA results the chapter ends with a treatment of the spinning top without using Euler's equations for rigid body motion. If you have ever struggled through Goldstein's Classical Mechanics treatment of this problem, from Euler angles to infinitessimal rotations to d-Omega which is not a differential of a vector to dyadics to body diferentials and space differentials to Euler's equations, you will really appreciate Snygg's direct solution using CA. Sure, I know Goldstein's has to be a general treatment of solid body motion and thus more complex so he can treat more general problems, but it is good to find a more direct solution that is cristal clear and only a few pages long. This chapter is real little gem. Chapter 2 takes CA to Minkowsky 4-space rotations. Chapters 3 and 4 take you to flat n-dimensional spaces and curved subspaces embedded in them. Again beautiful explanations are presented of the meaning of tangent spaces, parallel transport and how the covariant derivative arises naturally in curved spaces. I had the silly hope that with Clifford numbers and their products all would be well and done. Unfortunately the exterior product wedges its nose under the tent flap and pretty soon the exterior derivative and its side-kick the co-differential operator soon follow it into the tent. All this is explained in Chapter 7. With Chapter 5 the learning curve steepens with the introduction of Fock-Ivanenko 2-vectors and the curvature 2-vector (or 2-form) and finally the curvature tensor. Chapter 6 solves the field equations for the Schwartzchild metric based on the F-I 2-vector approach. Chapter 8 on the Dirac equation is again an approach different than that found in the usual texts. Chapter 9 derives the Kerr metric, something you won't find in MTW published 8 to 10 years after Kerr's papers. Unfortunately the starting point is some obscure problem from an earlier chapter and Snygg does not provide the delightful physical insight of earlier examples. However, there is discussion at the of the chapter. While you might be able to solve the field equations for the Schartzchild metric on your own, once you know it can be done, I certainly would not be able to do so for the Kerr metric. Snygg takes you through step by step, none of them particularly difficult, but the sequence is certainly not something I would have found by myself. Chapter 10 I only skimmed, the index notation, with underscored and bracketed indices, becomes overloaded for my level of sophistication. Chapter 11 organizes all the matrix stuff together, again a beautiful, straightforward and clear presentation. Here is shown how to construct a matrix representation for the gammas. As you might expect, the book is a veritable beehive of sub- and superscripts over bars and carets Greek and Latin indices and full of gamma gymnastics. Even Pauli's less complementary comment on Dirac algebra comes to mind. The text has a few typos but blessedly few in the Clifford number and gamma indices. By the way, if you expect to find out how to do trace computations on gamma expressions you won't find it here. The explicit form of the gamma matrices is hardly ever mentioned until chapter 11 nor is it needed in the present context.

Light, clear, and understandable........2000-08-06

Snygg's book is a thoroughly delightful introduction to Clifford Algebra and its applications in physics. It is detailed, readable, and at times even humorous... but always clear and educational. Snygg presents Clifford Algebra above all as a practical tool, rather than as the ultimate algebraic representation of spatial geometry. This gives a refreshing alternative to Hestenes' writings which, although quite good, can at times be philosophically pedantic and difficult to connect with standard theory.

Incredibly good!.......1998-02-14

From a letter to the author. John, I have to write you to tell you what a wonderful book you wrote. I still can't believe how good it is. Yesterday I was waiting for a television show to begin in ten minutes and I picked your book up while I sat front of the TV set. When I finally looked up 45 minutes later, I had missed the show! In 35 years as an algebraic topologist, I have tried to learn various things about Clifford algebras because of their role in K-theory and in the Atiyah- Singer Index theorem, and more recently because of the Seiberg-Witten equations. With only mediocre intensity, mostly browsing, I have had little success. In the month since I met you and bought your book, I have browsed through it while occupied with several other competing projects. In the process I have internalized the classification of Clifford algebras, learned how physicists use Dirac's equation, what they are doing when they talk about gauge theory, understood Hodge duality much better and so the codifferential operator. And I still have only browsed through a small portion of the text. I think we mathematicians should study your book to learn how to improve our own levels of exposition. Sincerely, Daniel Henry Gottlieb

An educational tool for Clifford algebras.......1997-09-29

The author writes in the Introduction: "Much of Clifford algebra is quite simple. If this fact were generally recognized, Clifford algebra would be more widely used as a computational tool." However, a few applications discussed in the book may require some physics usually covered in the first year of graduate school.

The author starts by Clifford algebras of the 3-dimensional Euclidean space and the 4-dimensional Minkowski space-time. He discusses the Maxwell equations in flat space, the Dirac equation for the electron, the Dirac operator, spherical harmonics, and curved space-times with Schwarzschild and Kerr metrics. The book ends with matrix representation and classification of Clifford algebras of real non-degenerate quadratic spaces and an appendix on Lorentz transformations.

A pedagogical gem........1996-10-11

The author is my brother. You may doubt the credibility of my commentary, but check it out for yourself. Based on an early draft, I predict it will be a pedagogical gem. Style and content make anything John Snygg writes a pleasure to read. Snygg is a hidden treasure. As of October 1996, "Clifford Algebra: A Computational Tool for Physicists" was not out.

Computational Methods for Representations of Groups and Algebras (Progress in Mathematics)

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Computational Methods for Representations of Groups and Algebras (Progress in Mathematics)

Manufacturer: Birkhäuser Basel
ProductGroup: Book
Binding: Hardcover

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ASIN: 3764360631

Book Description

This book presents material from three survey lectures and 14 additional invited lectures given at the Euroconference "Computational Methods for Representations of Groups and Algebras" held at Essen University in April 1997. The purpose of this meeting was to provide a survey of general theoretical and computational methods and recent advances in the representation theory of groups and algebras. Furthermore, new applications of the computational methods in linear algebra to the revision of the classification of finite simple sporadic groups are presented. Computational tools (including high-performance computations on supercomputers) have become increasingly important for classification problems. They are also inevitable for the construction of projective resolutions of finitely generated modules over finite-dimensional algebras and the study of group cohomology and rings of invariants.

A major part of this book is devoted to a survey of algorithms for computing special examples in the study of Grothendieck groups, quadratic forms and derived categories of finite-dimensional algebras. Open questions on Lie algebras, Bruhat orders, Coxeter groups and Kazhdan Lusztig polynomials are investigated with the aid of computer programs. The representation theory of finite groups and finite-dimensional algebras are linked by the condensation technique. The contents of this book provide an overview on the present state of the art, for graduate students and researchers in mathematics, computer science and physics.

Book Description

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry. 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 algorithms to answer questions such as those posed above are an important part of algebraic geometry. This book bases its discussion of algorithms on a generalization 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 a new edition of Ideals, Varieties and Algorithms the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple , Mathematica and REDUCE.

Customer Reviews:

Symbolic computation.......2003-08-29

This book explains and illustrates the algorithms used by symbolic math packages such as Mathematica, Maple, CoCoA, MatLab, MuPAD,... to solve problems involving polynomials in many variables, and along the way teaches the elements of real algebraic geometry-- most mathematics texts concentrate on the complex-variable version. It is not just for undergraduates; electrical engineers, for instance, should see it. Lots of pictures!

Easiest introduction to Algebraic Geometry.......2003-04-23

This is the easiest introduction to algebraic geometry and commutative algebra, the authors had done a great job in writing a book that assume very little from the readers. To learn some algebraic geometry, you can either start with this book, or you can spend a year to read a lot of background materials in algebra and then go to a Graduate Text like Harris' book. Of course, if you want to be an expert in algebra, you eventually need a lot of background, what this book can help you is to offer you a quick start, much quicker than you would ever imagine.

Straightforward and lucidly written.......2002-04-09

Having just finished using this text in the course of an undergraduate seminar, I can attest to the fact that the authors' style is outstanding - they are able to synthesize an enormous amount of material in this volume and present it in a manner that is highly accessible to almost all students of mathematics. The presentation of important theorems (for example, Hilbert's Nullstellensatz and Basis Theorem) along with just the right amount of copncrete examples makes for a book of superb quality. All-around, I highly recommend this volume to anyone who has an interest in learning about Algebraic Geometry.

Good book.......2001-05-27

I don't have the second edition of this book but did read the first, and the authors do a fine job of introducing the reader to the computational side of algebraic geometry. I will forego a chapter by chapter review therefore, but no doubt the second edition (which I do not own) is as well-written as the first. I would recommend it to anyone interested in the many applications of algebraic geometry and to those who need to understand how to compute things in algebraic geometry. The good thing about this book is that it gives a concrete flavor to a highly abstract subject. Algebraic geometry, through its applications to coding theory, cryptography, and computer graphics, is fast becoming the subject to learn. It is no longer just an esoteric, high-brow subject but one that is taking on major importance in the information age. Even without applications though it is a fascinating subject, and readers will get a taste of this in this book.

The best book on the topic.......2001-01-26

I learned the basics of Groebner bases from this book and its the best introductory book on this topic. Authors have explained all concepts with the help of examples which makes it readable for people from other fields also. It also talks about applications of Groebner bases to other fields. The book gives lot of exercises which help in understanding the contents more. I recommend that if you wish to learn Algebraic Geometry and Groebner bases then this is the book to start with.

Applied Functional Analysis (Crc Series in Computational Mechanics and Applied Analysis)

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Discussions are a bit convoluted.
Mediocre
A very good first book on functional analysis.
Excellent book for engineers seeking to learn mathematics

Applied Functional Analysis (Crc Series in Computational Mechanics and Applied Analysis)
J. Tinsley Oden , and Leszek Demkowicz
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 084932551X

Book Description

Functional analysis-the study of the properties of mathematical functions-is widely used in modern scientific and engineering disciplines, particularly in mathematical modeling and computer simulation. Applied Functional Analysis, the only textbook of its kind, is designed specifically for the graduate student in engineering and science who has little or no training in advanced mathematics. Comprehensive and easy-to-understand, this innovative textbook progresses from the essentials of preparatory mathematics to sophisticated functional analysis. This self-contained presentation requires few mathematical prerequisites and provides students with the fundamental concepts and theorems essential to mathematical analysis and modeling. Applied Functional Analysis combines various principles of mathematics, computer science, engineering, and science, laying the foundation for further specialty work in partial differential equations, approximation theory, numerical mathematics, control theory, mathematical physics, and related subjects. This new treatment of a classic subject outfits engineering and science majors with a graduate-level mathematics standing, otherwise accessible only through regular mathematics studies.

Customer Reviews:

Discussions are a bit convoluted........2004-12-01

I took Functional Analysis from professor Demkowicz. Actually the course is a misnomer, since you learn very little functional analysis and quite a bit more about set theory, Lebesgue measure theory, and topology (through Ch. 4 in the book). While I feel that the book is very meticulously written, it tries to cover in too much detail everything starting from the most basic laws of logic. The proofs and explanations are concise and clear but for my taste I also appreciate a bit of "plain English" explanation before tackling a proof, so I at least have some idea of what is going on. Only get this book if you want an *extremely* detailed development of basic theory and don't care much about applications.

Mediocre.......2001-09-13

Yet another book that resulted out of Class Notes ! Don't waste your money buying this book if you want to learn applied functional analysis. This book is meant solely for those students who have registered for Leszek's Class at UT, Austin.

A very good first book on functional analysis........2000-06-12

A very good first book on functional analysis. The book starts with preliminary set theory, logic, functions, and concepts of abstract algebra and moves on to the big convergence theorems in the Lebesgue integration and onto topological and metric spaces. The book ends with basic theory of Hilbert space. Each section concludes with exercises. The author takes great pains to include a lot of detail in the proofs. I would recommend this book to an advanced undergraduate or beginning graduate student.

Excellent book for engineers seeking to learn mathematics.......1997-10-25

Written by two experts in the area of computational and applied mathematics, this book is ideal for first/second year graduate students in engineering who wish to use concepts from functional analysis in their work. On one hand, the treatment is mathematically precise and yet the authors use their extensive engineering experience to present examples that are highly intuitive.

The introduction is greatly self-contained and serves as an appetizer for further chapters. Having worked in matrix methods for a while, I found the chapter on linear algebra especially interesting and informative. It explains the underlying concepts behind results that are generally taken for granted otherwise. The next few chapters consist of material that is, in general, not easy to explain and yet the style of the authors significantly simplifies the flow of the book. The topics covered in these chapters are at a level of abstraction higher than that encountered in engineering mathematics. This is evident, for example in the chapter on Lebesgue Integration. The heavy notation that is used sometimes makes it necessary to pay close attention while reading but this is perhaps a small price to pay for precision. The chapters on Banach and Hilbert spaces should be of special interest to people who wish to study the solution of variational boundary value problems.

Book Description

The origins of computation group theory (CGT) date back to the late 19th and early 20th centuries. Since then, the field has flourished, particularly during the past 30 to 40 years, and today it remains a lively and active branch of mathematics. The Handbook of Computational Group Theory offers the first complete treatment of all the fundamental methods and algorithms in CGT presented at a level accessible even to advanced undergraduate students. It develops the theory of algorithms in full detail and highlights the connections between the different aspects of CGT and other areas of computer algebra. While acknowledging the importance of the complexity analysis of CGT algorithms, the authors' primary focus is on algorithms that perform well in practice rather than on those with the best theoretical complexity. Throughout the book, applications of all the key topics and algorithms to areas both within and outside of mathematics demonstrate how CGT fits into the wider world of mathematics and science. The authors include detailed pseudocode for all of the fundamental algorithms, and provide detailed worked examples that bring the theorems and algorithms to life.

Customer Reviews:

Important Text on CGT.......2005-05-12

Handbook of Computational Group Theory by Derek F. Holt (Discrete Mathematics and Its Applications: Chapman & Hall/CRC) is about computational group theory, which we shall frequently abbreviate to CGT. The origins of this lively and active branch of mathematics can he traced back to the nineteenth and early twentieth centuries, but it has been flourishing particularly during the past 30 to 40 years. The aim of this book is to provide as complete a treatment as possible of all of the fundamental methods and algorithms in CGT, without straying above a level suitable for a beginning postgraduate student.
The most basic algorithms in CGT tend to be representation specific; that is, there are separate methods for groups given as permutation or matrix groups, groups defined by means of polycyclic presentations, and groups that are defined using a general finite presentation. The author has devoted separate chapters to algorithms that apply to groups in these different types of repre¬sentations, but there are other chapters that cover important methods involving more than one type. For example, Chapter 6 is about finding presentations of permutation groups and the connections between coset enumeration and methods for finding the order of a finite permutation group.
There is also included a chapter (Chapter 11) on the increasing number of precomputed stored libraries and databases of groups, character tables, etc. that are now publicly available. They have been playing a major rôle in CGT in recent years, both as an invaluable resource for the general mathematical public, and as components for use in some advanced algorithms in CGT. The library of all finite groups of order up to 2000 (except for order 1024) has proved to be particularly popular with the wider community.
It is inevitable that our choice of topics and treatment of the individual topics will reflect the authors' personal expertise and preferences to some extent. On the positive side, the final two chapters of the book cover appli¬cations of string-rewriting techniques to CGT (which is, however, treated in much greater detail, and the application of finite state automata to the computation of automatic structures of finitely presented groups. On the other hand, there may be some topics for which our treatment is more superficial than it would ideally be.
One such area is the complexity analysis of the algorithms of CGT. During the 1980s and 1990s some, for the most part friendly and respectful, rivalry developed between those whose research in CGT was principally directed to-wards producing better performance of their code, and those who were more interested in proving theoretical results concerning the complexity of the al¬gorithms. This study of complexity began with the work of Eugene Luks, who established a connection in his 1982 article between permutation group algorithms and the problem of testing two finite graphs for isomorphism. Our emphasis in this book will be more geared towards algorithms that per-form well in practice, rather than those with the best theoretical complexity. Fortunately, Seress' book includes a very thorough treatment of com¬plexity issues, and so we can safely refer the interested reader there. In any case, as machines become faster, computer memories larger, and bigger and bigger groups come within the range of practical computation, it is becom¬ing more and more the case that those algorithms with the more favourable complexity will also run faster when implemented.
The important topic of computational group representation theory and computations with group characters is perhaps not treated as thoroughly as it might be in this book. Some of the basic material is covered in Chapter 7, but there is unfortunately no specialized book on this topic.
One of the most active areas of research in CGT at the present time, both from the viewpoint of complexity and of practical performance, is the development of effective methods for computing with large finite groups of matrices. Much of this material is beyond the scope of this book. It is, in any case, developing and changing too rapidly to make it sensible to attempt to cover it properly here. Some pointers to the literature will of course be provided, mainly in Section 7.8.
Yet another topic that is beyond the scope of this book, but which is of increasing importance in CGT, is computational Lie theory. This includes computations with Coxeter groups, reflection groups, and groups of Lie type and their representations. It also connects with computations in Lie algebras, which is an area of independent importance. The article by Cohen, Murray, and Taylor provides a possible starting point for the interested reader.
The author firmly believes that the correct way to present a mathematical algorithm is by means of pseudocode, since a textual description will generally lack precision, and will usually involve rather vague instructions like "carry on in a similar manner". So we have included pseudocode for all of the most basic algorithms, and it is only for the more advanced procedures that we have occasionally lapsed into sketchy summaries. We are very grateful to Thomas Cormen who has made his LATEX package `clrscode' for displaying algorithms publicly available. This was used by him and his coauthors in the well-known textbook on algorithms.
Although working through all but the most trivial examples with procedures that are intended to be run on a computer can be very tedious, the author attempted to include illustrative examples for as many algorithms as is practical.
At the end of each chapter, or sometimes section, the reader's attention directed to some applications of the techniques developed in that chapter either to other areas of mathematics or to other sciences. It is generally difficult to do this effectively. Although there are many important and interesting applications of CGT around, the most significant of them will typically use methods of CGT as only one of many components, and so it not possible to do them full justice without venturing a long way outside of the main topic of the book.
The author assumes that the reader is familiar with group theory up to an advanced undergraduate level, and has a basic knowledge of other topics in algebra, such as ring and field theory. Chapter 2 includes a more or less complete survey of the required background material in group theory, but we shall assume that at least most of the topics reviewed will be already familiar to readers. Chapter 7 assumes some basic knowledge of group representation theory, such as the equivalence between matrix representations of a group G over a field K and KG-modules, but it is interesting to note that many of the most fundamental algorithms in the area, such as the `Meataxe', use only rather basic linear algebra.

The Development of the Number Field Sieve

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The Development of the Number Field Sieve

Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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Prime Numbers: A Computational Perspective

ASIN: 3540570136

Book Description

The number field sieve is an algorithm for finding the prime factors of large integers. It depends on algebraic number theory. Proposed by John Pollard in 1988, the method was used in 1990 to factor the ninth Fermat number, a 155-digit integer. The algorithm is most suited to numbers of a special form, but there is a promising variant that applies in general. This volume contains six research papers that describe the operation of the number field sieve, from both theoretical and practical perspectives. Pollard's original manuscript is included. In addition, there is an annotated bibliography of directly related literature.

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Excellent reference book
Great, but tough to use
Loaded with Content
There are several Bronshteins
Clasic

Handbook of Mathematics
I.N. Bronshtein , K.A. Semendyayev , G. Musiol , H. Muehlig , and H. Mühlig
Manufacturer: Springer
ProductGroup: Book
Binding: Turtleback

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Handbook of Physics
Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review
Handbook of Mathematics and Computational Science
Oxford Users' Guide to Mathematics
Mathematics Handbook for Science and Engineering

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Fuzzy and Rough Techniques in Medical Diagnosis and Medication (Studies in Fuzziness and Soft Computing)
Advances in Discrete Tomography and Its Applications (Applied and Numerical Harmonic Analysis)

ASIN: 3540434917

Book Description

This guide book to mathematics contains in handbook form the fundamental working knowledge of mathematics which is needed as an everyday guide for working scientists and engineers, as well as for students. Easy to understand, and convenient to use, this guide book gives concisely the information necessary to evaluate most problems which occur in concrete applications. For the 4th edition, the concept of the book has been completely re-arranged. The new emphasis is on those fields of mathematics that became more important for the formulation and modeling of technical and natural processes, namely Numerical Mathematics, Probability Theory and Statistics, as well as Information Processing.

Customer Reviews:

Excellent reference book.......2007-09-19

I highly recommend the Handbook of Mathematics. It is an excellent resource for every engineering student and professional engineer.

Great, but tough to use.......2007-04-25

This handbook contains more material than I find in any other single source that I happen to have. But I don't use it as frequently as I use the analogous CRC handbook, or MathWorld and Wikipedia on the Web, or the ancient NBS handbook. Why not? Because one uses a handbook, not as a textbook, but as a source for things one should know, but don't (or perhaps once knew but have forgotten). So one wants to find the thing one is looking for, refresh or extend one's memory, and then put the handbook aside and go back to the problem one is trying to solve. I have trouble locating what I'm looking for in Bronshtein and Semendayev, and when I find it, I often find that I have to look up things elsewhere in the volume to get all of whatever it is I was looking for. So I try my other sources first, and if they don't answer my question, I pick up this book, resignedly, and expect to spend hours rather than minutes getting whatever it is I want to know. There is nothing wrong with that; indeed, this book often supplies me with answers to questions I can't find answered elsewhere.

In case the reader of this review attributes my difficulties with this book to a lack of mathematical background, I'll remark that my academic training, very many years ago, was in math, so I find that I can follow the discussion in this book of any particular thing I look up; it's just a slow process for me. I'm not surprised that it's a favorite in Germany (and in Europe more generally); Europeans in their mathematical training are expected to deal with tough subjects by dogged persistence, and probably feel more comfortable with this style than I do, given my US background. So, overall, it's a book I couldn't do without, but hate having to spend time in.

Loaded with Content.......2005-02-25

This book is exactly what the title says it is; a handbook of mathematical techniques and formulas for scientists and engineers. It is more a handbook than a book on mathematics and assumes a prior knowledge on the subjects covered. Readers of this english version of the Bronshtein should take note that it is a "raw" translation of the german version and so some discussion may not do justice to the theory. This in no way takes away from the fact it is an exceptional book and you'd be hard pressed to find any other book with more mathematical content.

There are several Bronshteins.......2004-10-25

Bronstein's "Taschenbuch der Mathematik" is a longtime favorite among german science and engineering students. English language readers should be aware however, that there are numerous different editions of this book. Not only were the german editions constantly enlarged and reworked, but there were also two publishers of the same book, one in East Germany (Teubner Verlag), one in West Germany (Harri Deutsch Verlag). Today both of these publishers sell a "Taschenbuch der Mathematik" based on the original Bronstein, yet they are completely different books. The english edition by Springer Verlag advertised above is based on the current Harri Deutsch edition. An english translation of the Teubner edition is now available as the "Oxford Users' Guide to Mathematics" from Oxford University Press. It is mostly considered to be the better 'Bronstein' (even though Teubner and OUP have dropped his name because the new edition was completely rewritten by E. Zeidler).

Clasic.......2004-03-22

For me it was a remarkable fact, how few of the people in western countries have heard about this handbook. In the east this is probably the most popular mathematical handbook ever. I dont know any eastern european scientist in a field of mathematics, engineering and physics who don't have it. The main reason is it's extensivity and usefulness. In Croatian print it has about 1000 pages in very small format covering all possible parts of applied mathematics up to special functions/markov chains/complex integration/vector algebra. It's "allways in backpack book" and my hot recomandation to any technical scientist.

Krylov Solvers for Linear Algebraic Systems, Volume 11: Krylov Solvers (Studies in Computational Mathematics)

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Krylov Solvers for Linear Algebraic Systems, Volume 11: Krylov Solvers (Studies in Computational Mathematics)
Charles George Broyden , and Maria Teresa Vespucci
Manufacturer: Elsevier Science
ProductGroup: Book
Binding: Hardcover

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ASIN: 0444514740

Book Description

The first four chapters of this book give a comprehensive and unified theory of the Krylov methods. Many of these are shown to be particular examples of
the block conjugate-gradient algorithm and it is this observation that
permits the unification of the theory. The two major sub-classes of those
methods, the Lanczos and the Hestenes-Stiefel, are developed in parallel as
natural generalisations of the Orthodir (GCR) and Orthomin algorithms. These
are themselves based on Arnoldi's algorithm and a generalised Gram-Schmidt
algorithm and their properties, in particular their stability properties,
are determined by the two matrices that define the block conjugate-gradient
algorithm. These are the matrix of coefficients and the preconditioning
matrix.

In Chapter 5 the"transpose-free" algorithms based on the conjugate-gradient squared algorithm are presented while Chapter 6 examines the various ways in which the QMR technique has been exploited. Look-ahead methods and general block methods are dealt with in Chapters 7 and 8 while Chapter 9 is devoted to error analysis of two basic algorithms.

In Chapter 10 the results of numerical testing of the more important algorithms in their basic forms (i.e. without look-ahead or preconditioning) are presented and these are related to the structure of the algorithms and the general theory. Graphs illustrating the performances of various algorithm/problem combinations are given via a CD-ROM.

Chapter 11, by far the longest, gives a survey of preconditioning techniques. These range from the old idea of polynomial preconditioning via SOR and ILU preconditioning to methods like SpAI, AInv and the multigrid methods that were developed specifically for use with parallel computers. Chapter 12 is devoted to dual algorithms like Orthores and the reverse algorithms of Hegedus. Finally certain ancillary matters like reduction to Hessenberg form, Chebychev polynomials and the companion matrix are described in a series of appendices.

· comprehensive and unified approach
· up-to-date chapter on preconditioners
· complete theory of stability
· includes dual and reverse methods
· comparison of algorithms on CD-ROM
· objective assessment of algorithms

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