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Galois Theory of Linear Differential Equations by Marius Van der Put, Michael F. Singer

Posted by NTL at Thursday, April 30, 2009 , 0 Comments

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Book Details :
Title : Galois Theory of Linear Differential Equations
Author : Marius van der Put, Michael F. Singer
Hardcover : 478 pages
Publisher : Springer; 1 edition (March 10, 2003)
Language : English
ISBN-10 : 3540442286
ISBN-13 : 978-3540442288
Size : 3.03 MB.
Type : Pdf.
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Book Description :

Linear differential equations form the central topic of this volume, Galois theory being the unifying theme.
A large number of aspects are presented: algebraic theory especially differential Galois theory, formal theory, classification, algorithms to decide solvability in finite terms, monodromy and Hilbert's 21st problem, asymptotics and summability, the inverse problem and linear differential equations in positive characteristic. The appendices aim to help the reader with concepts used, from algebraic geometry, linear algebraic groups, sheaves, and tannakian categories that are used.
This volume will become a standard reference for all mathematicians in this area of mathematics, including graduate students.

This is a great book, which will hopefully become a classic in the subject of differential Galois theory. Although there is a large bibliography on the subject (which is very much en vogue today), it is almost exclusively composed of papers, and the specialist, as well as the novice, have long been missing an introductory book covering also specific and advanced research topics. This gap is filled by the volume under review, and more than satisfactorily.

At last, a thorough exposition, including most of the facets it presents nowadays, of this beautiful analogue of the Galois theory of field extensions … . The book is in fact already becoming a standard reference, not only for differential Galois theory proper, but also for the many areas which have accompanied its recent growth … . Any … student working in these areas will benefit from this book, which clearly belongs to all mathematical libraries.

This book, which is organized like a textbook with exercises … is a definitive account of the Galois theory of linear differential equations. … In sum, the book is a modern, comprehensive, and mostly self-contained account of the Galois theory of linear differential equations. It should be considered the standard reference in the field.

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Difference Equations and Inequalities: Theory, Methods, and Applications (Pure and Applied Mathematics) by Ravi P. Agarwal

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Book Details :
Title : Difference Equations and Inequalities: Theory, Methods, and Applications (Chapman & Hall/CRC Pure and Applied Mathematics)
Author : Ravi P. Agarwal
Hardcover : 1000 pages
Publisher : CRC Press; 2 edition (January 27, 2000)
Language : English
ISBN-10 : 0824790073
ISBN-13 : 978-0824790073
Size : 8.85 MB.
Type : Djvu.
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Book Description :

A study of difference equations and inequalities. This second edition offers real-world examples and uses of difference equations in probability theory, queuing and statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, quanta in radiation, genetics, economics, psychology, sociology, and other disciplines. It features 200 new problems, 400 additional references, and a new chapter on the qualitative properties of solutions of neutral difference equations.

This excellent monograph combines all aspects of the theory and methods of solutions of difference equations and their applications.

The exhaustive compilation and analytic representation of the various mathematical problems are indeed useful. . . .essential for the enrichment of knowledge in mathematics, physics and statistics.

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Functions of One Complex Variable by John B. Conway

Posted by NTL at Wednesday, April 29, 2009 , 0 Comments

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Book Details :
Title : Functions of One Complex Variable (Graduate Texts in Mathematics)
Author : John B. Conway
Paperback : 313 pages
Publisher : Springer; 1 edition (October 16, 1973)
Language : English
ISBN-10 : 0387900624
ISBN-13 : 978-0387900629
Size : 2.85 MB.
Type : Djvu.
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Book Description :

This book presents a basic introduction to complex analysis in both an interesting and a rigorous manner. It contains enough material for a full year's course, and the choice of material treated is reasonably standard and should be satisfactory for most first courses in complex analysis. The approach to each topic appears to be carefully thought out both as to mathematical treatment and pedagogical presentation, and the end result is a very satisfactory book for classroom use or self-study.

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Functional Analysis: An Introduction by Yuli Eidelman, Vitali Milman, Antonis Tsolomit

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Book Details :
Title : Functional Analysis: An Introduction (Graduate Studies in Mathematics)
Author : Yuli Eidelman, Vitali Milman, Antonis Tsolomitis
Hardcover : 322 pages
Publisher : American Mathematical Society (November 23, 2004)
Language : English
ISBN-10 : 0821836463
ISBN-13 : 978-0821836460
Size : 0.5 MB.
Type : Pdf.
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Book Description :

This textbook provides an introduction to the methods and language of functional analysis, including Hilbert spaces, Fredholm theory for compact operators, and spectral theory of self-adjoint operators. It also presents the basic theorems and methods of abstract functional analysis and a few applications of these methods to Banach algebras and the theory of unbounded self-adjoint operators.
The text corresponds to material for two semester courses (Part I and Part II, respectively) and is essentially self-contained. Prerequisites for the first part are minimal amounts of linear algebra and calculus. For the second part, some knowledge of topology and measure theory is recommended. Each of the 11 chapters is followed by numerous exercises, with solutions given at the end of the book.

The text is ideal for a one-year course. It will also provide a sound basis for further study. It is suitable for graduate students and researchers interested in operator theory and functional analysis.

This is a gentle introduction to functional analysis that is clearly written and comes with detailed, elegant and effective proofs and well-chosen examples. ... This book is written with great care and with much sympathy to the reader. It is pleasant to read... It is simply a good book to learn the foundations of functional analysis.

Each chapter includes exercises, in total 195 of the them, all provided with solutions at the end of the book. The text is as self-contained as possible... The authors have taken special care to be brief and not to overload the students with the enormous amount of information available on the subject.


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Advanced Modern Algebra by Joseph J. Rotman

Posted by NTL at Tuesday, April 28, 2009 , 0 Comments

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Book Details :
Title : Advanced Modern Algebra
Author : Joseph J. Rotman
Hardcover : 1040 pages
Publisher : Prentice Hall (May 5, 2002)
Language : English
ISBN-10 : 0130878685
Size : 4.74 MB.
Type : Pdf.
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Book Description :

This book's organizing principle is the interplay between groups and rings, where “rings” includes the ideas of modules. It contains basic definitions, complete and clear theorems (the first with brief sketches of proofs), and gives attention to the topics of algebraic geometry, computers, homology, and representations. More than merely a succession of definition-theorem-proofs, this text put results and ideas in context so that students can appreciate why a certain topic is being studied, and where definitions originate. Chapter topics include groups; commutative rings; modules; principal ideal domains; algebras; cohomology and representations; and homological algebra. For individuals interested in a self-study guide to learning advanced algebra and its related topics.

Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowadays, everyone agrees that some knowledge of linear algebra, groups, and commutative rings is necessary, and these topics are introduced in undergraduate courses. We continue their study.

This book can be used as a text for the first year of graduate algebra, but it is much more than that. It can also serve more advanced graduate students wishing to learn topics on their own; while not reaching the frontiers, the book does provide a sense of the successes and methods arising in an area. Finally, this is a reference containing many of the standard theorems and definitions that users of algebra need to know. Thus, the book is not only an appetizer, but a hearty meal as well.

Let me now address readers and instructors who use the book as a text for a beginning graduate course. If I could assume that everyone had already read my book, A First Course in Abstract Algebra, then the prerequisites for this book would be plain. But this is not a realistic assumption; different undergraduate courses introducing abstract algebra abound, as do texts for these courses. For many, linear algebra concentrates on matrices and vector spaces over the real numbers, with an emphasis on computing solutions of linear systems of equations; other courses may treat vector spaces over arbitrary fields, as well as Jordan and rational canonical forms. Some courses discuss the Sylow theorems; some do not; some courses classify finite fields; some do not.

To accommodate readers having different backgrounds, the first three chapters contain many familiar results, with many proofs merely sketched. The first chapter contains the fundamental theorem of arithmetic, congruences, De Moivre's theorem, roots of unity, cyclotomic polynomials, and some standard notions of set theory, such as equivalence relations and verification of the group axioms for symmetric groups. The next two chapters contain both familiar and unfamiliar material. "New" results, that is, results rarely taught in a first course, have complete proofs, while proofs of "old" results are usually sketched. In more detail, Chapter 2 is an introduction to group theory, reviewing permutations, Lagrange's theorem, quotient groups, the isomorphism theorems, and groups acting on sets. Chapter 3 is an introduction to commutative rings, reviewing domains, fraction fields, polynomial rings in one variable, quotient rings, isomorphism theorems, irreducible polynomials, finite fields, and some linear algebra over arbitrary fields. Readers may use "older" portions of these chapters to refresh their memory of this material (and also to see my notational choices); on the other hand, these chapters can also serve as a guide for learning what may have been omitted from an earlier course (complete proofs can be found in A First Course in Abstract Algebra). This format gives more freedom to an instructor, for there is a variety of choices for the starting point of a course of lectures, depending on what best fits the backgrounds of the students in a class. I expect that most instructors would begin a course somewhere in the middle of Chapter 2 and, afterwards, would continue from some point in the middle of Chapter 3. Finally, this format is convenient for the author, because it allows me to refer back to these earlier results in the midst of a discussion or a proof. Proofs in subsequent chapters are complete and are not sketched.

I have tried to write clear and complete proofs, omitting only those parts that are truly routine; thus, it is not necessary for an instructor to expound every detail in lectures, for students should be able to read the text.

When I was a student, Birkhoff and Mac Lane's A Survey of Modern Algebra was the text for my first algebra course, and van der Waerden's Modern Algebra was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first appearance: Birkhoff and Mac Lane's book first appeared in 1941, and van der Waerden's book first appeared in 1930. There are today major directions that either did not exist over 60 years ago, or that were not then recognized to be so important. These new directions involve algebraic geometry, computers; homology, and representations (A Survey of Modern Algebra has been rewritten as Mac Lane-Birkhoff, Algebra, Macmillan, New York, 1967, and this version introduces categorical methods; category theory emerged from algebraic topology, but was then used by Grothendieck to revolutionize algebraic geometry).

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Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V.I. Arnold by V.B. Alekseev

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Book Details :
Title : Abel's Theorem in Problems and Solutions: Based on the lectures of Professor V.I. Arnold (The Kluwer International Series in Engineering & Computer Science)
Author : V. B. Alekseev, Francesca Aicardi
Hardcover : 269 pages
Publisher : Springer; 1 edition (May 31, 2004)
Language : English
ISBN-10 : 1402021860
ISBN-13 : 978-1402021862
Size : 0.9 MB.
Type : Pdf.
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Book Description :

Do formulas exist for the solution to algebraical equations in one variable of any degree like the formulas for quadratic equations? The main aim of this book is to give new geometrical proof of Abel's theorem, as proposed by Professor V.I. Arnold. The theorem states that for general algebraical equations of a degree higher than 4, there are no formulas representing roots of these equations in terms of coefficients with only arithmetic operations and radicals.

A secondary, and more important aim of this book, is to acquaint the reader with two very important branches of modern mathematics: group theory and theory of functions of a complex variable.

This book also has the added bonus of an extensive appendix devoted to the differential Galois theory, written by Professor A.G. Khovanskii.

As this text has been written assuming no specialist prior knowledge and is composed of definitions, examples, problems and solutions, it is suitable for self-study or teaching students of mathematics, from high school to graduate.

This very special and brilliant text has been written for bright non-specialists in mathematics, but it leads the reader up to topical research problems in the field, and that in a masterly manner. The book is absolutely self-contained, in its own particular fashion, and it is therefore perfectly suited for self-study, ranging from advanced high school to graduate level. No doubt, the thorough and serious working with this outstanding text could turn very beginners into creative almost-experts in the field.

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A Course of Modern Analysis by E. T. Whittaker, G. N. Watson

Posted by NTL at Monday, April 27, 2009 , 0 Comments

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Book Details :
Title : A Course of Modern Analysis (Cambridge Mathematical Library)
Author : E. T. Whittaker, G. N. Watson
Paperback : 620 pages
Publisher : Cambridge University Press; 4 edition (September 13, 1996)
Language : English
ISBN-10 : 0521588073
ISBN-13 : 978-0521588072
Size : 0.9 MB.
Type : Pdf.
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Book Description :

This classic text has entered and held the field as the standard book on the applications of analysis to the transcendental functions. The authors explain the methods of modern analysis in the first part of the book and then proceed to a detailed discussion of the transcendental function, unhampered by the necessity of continually proving new theorems for special applications. In this way the authors have succeeded in being rigorous without imposing on the reader the mass of detail that so often tends to make a rigorous demonstration tedious. Researchers and students will find this book as valuable as ever.

This classic text is known to and used by thousands of mathematicians and students of mathematics thorughout the world. It gives an introduction to the general theory of infinite processes and of analytic functions together with an account of the principle transcendental functions.

Whittaker and Watson has entered and held the field as the standard book of reference in English on the applications of analysis to the transcendental functions. This end has been successfully achieved by following the sensible course of explaining the methods of modern analysis in the first part of the book and then proceeding to a detailed discussion of the transcendental function, unhampered by the necessity of continually proving new theorems for special applications. In this way the authors have succeeded in being rigorous without imposing on the reader the mass of detail which so often tends to make a rigorous demonstration tedious.

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Problems and Solutions in Theoretical and Mathematical Physics : Advanced Level by Willi-Hans Steeb

Posted by NTL at Sunday, April 26, 2009 , 0 Comments

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Book Details :
Title : Problems and Solutions in Theoretical and Mathematical Physics : Advanced Level
Author : Willi-Hans Steeb
Paperback : 240 pages
Publisher : World Scientific Pub Co Inc (December 1996)
Language : English
ISBN-10 : 9810229445
ISBN-13 : 978-9810229443
Size : 4.0 MB.
Type : Pdf.
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Book Description :

The purpose of Problems and Solutions in Theoretical and Mathematical Physics is to supply a collection of problems together with their detailed solution which will prove to be valuable to students as well as to research workers in the fields of mathematics, physics, engineering and other sciences. The topics range in difficulty from elementary to advanced. Almost all problems are solved in detail and most of the problems are self-contained. All relevant definitions are given. Students can learn important principles and strategies required for problem solving. Teachers will also find this text useful as a supplement, since important concepts and techniques are developed in the problems. The material was tested in the author's lectures given around the world.
Problems and Solutions in Theoretical and Mathematical Physics is divided into two volumes. Volume I presents the introductory problems for undergraduate and advanced undergraduate students. In volume II, the more advanced problems, together with their detailed solutions are collected, to meet the needs of graduate students and researchers. Problems included cover most of the new fields in theoretical and mathematical physics such as Lax representation. Bocklund transformation, soliton equations, Lie algebra valued differential forms, Hirota technique, Painleve test, the Bethe ansatz, the Yang-Baxter relation, chaos, fractals, complexity, etc.

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Mathematical Statistics: Exercises and Solutions by Jun Shao

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Book Details :
Title : Mathematical Statistics: Exercises and Solutions
Author : Jun Shao
Paperback : 388 pages
Publisher : Springer; 1 edition (June 30, 2005)
Language : English
ISBN-10 : 0387249702
ISBN-13 : 978-0387249704
Size : 2.19 MB.
Type : Pdf.
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Book Description :

This book consists of four hundred exercises in mathematical statistics and their solutions, over 95% of which are in the author's Mathematical Statistics, Second Edition (Springer, 2003). For students preparing for work on a Ph.D. degree in statistics and instructors of mathematical statistics courses, this useful book provides solutions to train students for their research ability in mathematical statistics and presents many additional results and examples that complement any text in mathematical statistics. To develop problem-solving skills, two solutions and/or notes of brief discussions accompany a few exercises. The exercises are grouped into seven chapters with titles matching those in the author's Mathematical Statistics. On the other hand, the book is stand-alone because exercises and solutions are comprehensible independently of their source, and notation and terminology are explained in the front of the book. Readers are assumed to have a good knowledge in advanced calculus. A course in real analysis or measure theory is highly recommended. If this book is used with a statistics textbook that does not include probability theory, then knowledge in measure-theoretic probability theory is required.

Mathematical Statistics: Exercises and Solutions collection consists of 400 exercises and their solutions. Most of the exercises (over 95%) are introduced in the cited textbook. … The collection is a stand-alone book. It is written very rigorously and solutions are presented in detail. It can be recommended as a source of solved problems for teachers and students of advanced mathematical statistics.

The exercise book accompanies an excellent text book for a graduate course in mathematical statistics ... and is an excellent support for the book. The exercise book is excellent by itself also: a challenging collection of problems in mathematical statistics. … Using both however, the textbook and the exercise book, the mathematical statistician will enjoy the wealth of interesting problems.

About author :

Jun Shao is Professor of Statistics at the University of Wisconsin, Madison.

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Problems in Applied Mathematics by Murray S. Klamkin

Posted by NTL at Saturday, April 25, 2009 , 0 Comments

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Book Details :
Title : Problems in Applied Mathematics
Author : Murray S. Klamkin
Paperback : 614 pages
Publisher : Society for Industrial Mathematics (January 1, 1987)
Language : English
ISBN-10 : 0898712599
ISBN-13 : 978-0898712599
Size : 22.05 MB.
Type : Pdf.
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Book Description :

People in all walks of life - and perhaps mathematicians especially - delight in working on problems for the sheer pleasure of meeting a challenge. The problem section of SIAM Review has always provided such a challenge for mathematicians. The section was started to offer classroom instructors and their students as well as other interested problemists, a set of problems -solved or unsolved - illustrating various applications of mathematics. In many cases the unsolved problems were eventually solved. Problems in Applied Mathematics is a compilation of 380 of SIAM Review's most interesting problems dating back to the journal's inception in 1959. The problems are classified into 22 broad categories including Series, Special Functions, Integrals, Polynomials, Probability, Combinatorics, Matrices and Determinants, Optimization, Inequalities, Ordinary Differential Equations, Boundary Value Problems, Asymptotics and Approximations, Mechanics, Graph Theory, and Geometry.

This outstanding collection of instructive problems is organized into twenty-two broad sections such as probability, combinatorics, series, special functions, and so forth. Most of the problems include motivational material describing how the problem arose; the solutions, some of which approach the status of short papers, are accessible to a wide audience (say at the level of the American Mathematical Monthly). Solutions generally include references to further literature and pertinent comments by Klamkin. There is something in this volume for everyone, and is a welcome addition to the problems literature.

About author :

Murray S. Klamkin is Professor Emeritus of Mathematics at the University of Alberta, and has served as Problem Editor for SIAM Review since it was first published in 1959. Professor Klamkin received the 1988 Award for Distinguished Service from the MAA for "his contributions to the realm of problem solving, for his inspiring influence on young problem solvers, and for his many other contributions to mathematics." He has chaired the Advisory Board of the Applied Mathematics Division of the National Bureau of Standards, and has authored three books, approximately 165 articles, and many problems and/or solutions.

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Problem Solving Through Problems by Loren C. Larson

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Book Details :
Title : Problem Solving Through Problems
Author : Loren C Larson
Paperback : 363 pages
Publisher : Springer; 1st edition (1983)
Language : English
ISBN-10 : 0387961712
ISBN-13 : 978-0387961712
Size : 2.37 MB.
Type : Djvu.
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Book Description :

This is a practical anthology of some of the best elementary problems in different branches of mathematics. Arranged by subject, the problems highlight the most common problem-solving techniques encountered in undergraduate mathematics. This book teaches the important principles and broad strategies for coping with the experience of solving problems. It has been found very helpful for students preparing for the Putnam exam.

This is a very welcome addition. The main message of the book is that the only way to learn to solve problems is to solve problems! I found this book very helpful. I am quite sure the book will be in constant use and I have no hesitation in recommending it.

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Problems in Analytic Number Theory by Maruti Ram Murty

Posted by NTL at Friday, April 24, 2009 , 0 Comments

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Book Details :
Title : Problems in Analytic Number Theory (Graduate Texts in Mathematics / Readings in Mathematics)
Author : Maruti Ram Murty
Hardcover : 452 pages
Publisher : Springer; 1 edition (December 21, 2000)
Language : English
ISBN-10 : 0387951431
ISBN-13 : 978-0387951430
Size : 2.35 MB.
Type : Djvu.
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Book Description :

This book gives a problem-solving approach to the difficult subject of analytic number theory. It is primarily aimed at graduate students and senior undergraduates. The goal is to provide a rapid introduction to analytic methods and the ways in which they are used to study the distribution of prime numbers. The book also includes an introduction to p-adic analytic methods. It is ideal for a first course in analytic number theory. The new edition has been completely rewritten, errors have been corrected, and there is a new chapter on equidistribution.

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The Mathematical Century: The 30 Greatest Problems of the Last 100 Years by Piergiorgio Odifreddi

Posted by NTL at Thursday, April 23, 2009 , 0 Comments

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Book Details :
Title : The Mathematical Century: The 30 Greatest Problems of the Last 100 Years
Author : Piergiorgio Odifreddi, Arturo Sangalli, Freeman Dyson
Hardcover : 224 pages
Publisher : Princeton University Press (March 8, 2004)
Language : English
ISBN-10 : 069109294X
ISBN-13 : 978-0691092942
Size : 1.54 MB.
Type : Djvu.
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Book Description :

The twentieth century was a time of unprecedented development in mathematics, as well as in all sciences: more theorems were proved and results found in a hundred years than in all of previous history. In The Mathematical Century, Piergiorgio Odifreddi distills this unwieldy mass of knowledge into a fascinating and authoritative overview of the subject. He concentrates on thirty highlights of pure and applied mathematics. Each tells the story of an exciting problem, from its historical origins to its modern solution, in lively prose free of technical details.

Odifreddi opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four most important open mathematical problems of the twenty-first century. In presenting the thirty problems at the heart of the book he devotes equal attention to pure and applied mathematics, with applications ranging from physics and computer science to biology and economics. Special attention is dedicated to the famous "23 problems" outlined by David Hilbert in his address to the International Congress of Mathematicians in 1900 as a research program for the new century, and to the work of the winners of the Fields Medal, the equivalent of a Nobel prize in mathematics.

This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics.

"The Mathematical Century is both popular and scholarly. Piergiorgio Odifreddi clearly and accurately covers many important mathematical problems and the contributions that leading mathematicians have made to their solutions. Offering a personal but very balanced perspective, his book is one that amateur and professional alike can learn from.

This is an astonishingly readable, succinct, and wonderful account of twentieth-century mathematics! It is a great book for mathematics majors, students in liberal-arts courses in mathematics, and the general public. I am amazed at how easily the author has set out the achievements in a broad array of mathematical fields. The writing appears effortless.

About author :

Piergiorgio Odifreddi is Professor of Mathematical Logic at the University of Turin and has been a visiting professor at Cornell University for many years. He is the author of the textbook "Classical Recursion Theory". He is also a regular contributor to the Italian daily "La Repubblica".
Freeman Dyson, Professor Emeritus of Physics at the Institute for Advanced Study, is the author of several books, including "Disturbing the Universe".

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