Division Algebras:: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics (Mathematics and Its Applications)

Book Description

The four real division algebras (reals, complexes, quaternions and octonions) are the most obvious signposts to a rich and intricate realm of select and beautiful mathematical structures. Using the new tool of adjoint division algebras, with respect to which the division algebras themselves appear in the role of spinor spaces, some of these structures are developed, including parallelizable spheres, exceptional Lie groups, and triality. In the case of triality the use of adjoint octonions greatly simplifies its investigation. Motivating this work, however, is a strong conviction that the design of our physical reality arises from this select mathematical realm. A compelling case for that conviction is presented, a derivation of the standard model of leptons and quarks.
The book will be of particular interest to particle and high energy theorists, and to applied mathematicians.

Customer Reviews:

I am waiting for Dixon 's Octonians..........2007-02-10

I have not yet received my command. P. MERAT

Mathematics behind physics.......1997-09-29

This is an excellent book for those who want to study Hamilton's quaternions, and other algebraic structures, used in modern physics. Dixon believes that octonions and triality of Spin(8) are essential in understanding particle physics. This clear exposition contains many ideas which have gone unnoticed from other researchers. The book is a treasure trove for mathematical physicists. The author also compares the Cayley algebra of octonions to other algebraic systems used in physics: matrices and Clifford algebras, in particular the Dirac algebra.

A Java Library of Graph Algorithms and Optimization (Discrete Mathematics and Its Applications)

Average customer rating:

A very handy collection
Printout of Java programs
Useful problem-solving tool
A mere compendium of poorly written algorithms

A Java Library of Graph Algorithms and Optimization (Discrete Mathematics and Its Applications)
Hang T. Lau
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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An Inconvenient Truth

ASIN: 1584887184

Book Description

Because of its portability and platform-independence, Java is the ideal computer programming language to use when working on graph algorithms and other mathematical programming problems. Collecting some of the most popular graph algorithms and optimization procedures, A Java Library of Graph Algorithms and Optimization provides the source code for a library of Java programs that can be used to solve problems in graph theory and combinatorial optimization. Self-contained and largely independent, each topic starts with a problem description and an outline of the solution procedure, followed by its parameter list specification, source code, and a test example that illustrates the usage of the code. The book begins with a chapter on random graph generation that examines bipartite, regular, connected, Hamilton, and isomorphic graphs as well as spanning, labeled, and unlabeled rooted trees. It then discusses connectivity procedures, followed by a paths and cycles chapter that contains the Chinese postman and traveling salesman problems, Euler and Hamilton cycles, and shortest paths. The author proceeds to describe two test procedures involving planarity and graph isomorphism. Subsequent chapters deal with graph coloring, graph matching, network flow, and packing and covering, including the assignment, bottleneck assignment, quadratic assignment, multiple knapsack, set covering, and set partitioning problems. The final chapters explore linear, integer, and quadratic programming. The appendices provide references that offer further details of the algorithms and include the definitions of many graph theory terms used in the book.

Customer Reviews:

A very handy collection.......2007-04-15

There are many well-written textbooks that cover the theory
and algorithms on graphs and combinatorial optimization.
Very few provide the computer code for the methods. This
book offers an extensive collection of Java programs in
this area. Each program is self-contained and can be used
independently through parameter passing. The drawback of
the book is that the coding style is not object oriented,
and the programs would be difficult to maintain. The
description of the methods and their implementations is
terse. Hence the book is not intended as a learning text.
But the library of programs is a very convenient handy
device for students and researchers in locating solutions
to classroom didactic problems in graphs and optimization,
which apparently is the main objective of the book.

Printout of Java programs.......2007-03-27

This is my third review; my two previous reviews have been removed. I have already notified Amazon about this fact

As I have stated, book is just a printout of Java program, without any explanation how program is doing what is doing, what are program limitations in terms of memory, time and complexity. Programming style is mostly Fortran IV like. Programs are without single line of comment and with non-intuitive variable names, what makes modificatios difficult or impossible. Book can be useful for somebody who needs "black box" library, doesn't need to understand programs and trusts the author that programs fave no flaws

Useful problem-solving tool.......2007-02-09

This library of ready-to-use programs is extremely useful. I have used the programs with very minimal effort in obtaining solutions to some graph optimization problems. Unfortunately the programs are not well documented; it would be a challenge to make modifications to the code. However, the library serves as an ideal black box tool in solving most of the pedagogical graph theory and optimization problems, especially well suited for users who are not of much concern for the underlying methodology and implementation.

A mere compendium of poorly written algorithms.......2007-02-07

There are so many problems with this book, it's hard to know where to begin. So I don't come across as all and only negative, I will first give it credit for gathering together, at least in name a large number of graph processing algorithms.

That said, here are the problems:

The book is just a catalog of graph algorithms with poorly done documentation and even worse actual code. To wit:

*Each algorithm is preceded by a very brief explanation of what it does and some of the issues involved. Suffice it to say that it's the sparsest and most minimal explanation imaginable; if you don't already understand the issues involved, you probably won't after reading the short paragraph or two that precedes each algorithm / method.

*There is but ONE class and every bit of functionality is contained in its own individual, single static method. This "design" causes not a few of the methods to literally run to a thousand and more lines and contain dozens and dozens of (cryptically named) member variables.

So for instance, if you are interested in planarity testing, there's a "method" called planarityTesting that takes four parameters and returns true or false.

All well and good until you actually look at that method and see declared 51 , that's fifty-one, member variables. Each of these variables has poorly chosen names like, "wkpathfind2" and "store2" and "store3" and of course "store4" and "sortptr1" and "sortptr2". I thought this tactic of vowel-conserving naming of variables went out with the 8 + 3 DOS naming convention. At any rate, the cryptic naming scheme combined with the lack of javadoc combine to render each variable's purpose completely opaque. This makes it all but impossible to relate the code to the underlying graph theory.

Then comes the code.

Imagine a thousand and more lines, literally page after page after page of streaming code, all one single method, manipulating these cryptic variables in virtually uncommented ways.

That is pretty much what you get with this book. One algorithm after another after another.

I would say the following:
1) the author codes as if from another time. There is NO object-oriented design to this code whatsoever. None. Zero. Zip.

2)The methods are hundreds or thousands of lines of what amounts to undocumented symbol manipulation. There is small chance to learn anything from this book with respect to relating the code to graph theory.

3) I can say that, having implemented many of the algorithms in this book myself prior to buying this book, the book has contributed nothing to my understanding and further, that already understanding the issues surrounding many of these methods, that is being a qualified reader, is NOT sufficient to allow the reader to follow and understand the algorithms.

4) If you only want to use the (static) methods to return a value or ascertain some property of a graph and you don't care to understand how it works or why it works, then perhaps you'll be happy with this book, but then , why not release the object code as blackbox library? If the code was never meant to be read, and there is no attempt at explaining graph theory as it relates to the code, then what of value is left for the reader?

5) Finally, if the purpose of the book is deliver a good "black-box" library, readers should know that the actual implementation of the graph "object" chosen in this book makes will make that problematic. The book uses an adjacency matrix to represent the graph, a well known data structure in graph theory. Unfortunately, this data structure has the following well-known problem: it is only suitable for the rare instance of dense graphs. The runtime performance and memory demands of this data structure make it unsuitable to any but very very small graphs. Most graphs are neither very very small nor very very dense, (as dense is defined in graph theory), and for that reason almost all graph drawing packages opt for a linked-list data structure to represent the graph.

This is solidly the worst book on this subject I have yet encountered. Amazon offers a number of alternative books, including the fine Graph Algorithms, Third Edition by Robert Sedgewick and Michael Schidlowsky, a book I have no connection with whatsoever and two authors who are otherwise unknown to me. Bundle of Algorithms in Java, Third Edition (Parts 1-5): Fundamentals, Data Structures, Sorting, Searching, and Graph Algorithms, Third Edition

Elliptic Curves: Number Theory and Cryptography (Discrete Mathematics and Its Applications)

Average customer rating:

Washington Elliptic Curves
Solid intermediate introduction to elliptic curves
A clear, concise introduction to elliptic curves
It might be a good book for a mathematic student but not a good one for an engineering student.
Excellent

Elliptic Curves: Number Theory and Cryptography (Discrete Mathematics and Its Applications)
Lawrence C. Washington
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 1584883650

Book Description

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.

Customer Reviews:

Washington Elliptic Curves.......2007-01-12

I bought this book as a follow-up to working my way through "Introduction to Cryptography with Coding Theory" (by the same author together Wade Trappe) (which I strongly recommend as well). I was not disappointed - Washington covers a difficult but important topic in a masterly fashion which should be accessible to anyone with a serious interest in elliptic curve cryptography. It successfully follows a middle road between the standard, but rather abstract texts on number theory and those which give details of algorithms but few proofs. There are ample examples and enjoyable exercises. Strongly recommended.

Solid intermediate introduction to elliptic curves.......2006-06-12

I compare this book to Rational Points on Elliptic Curves (RP) by Tate and Silverman, and The Arithmetic of Ellipitic Curves (AEC) by Silverman.

RP is definitely for junior and senior undergraduates interested in elliptic curves. With modest knowledge of real and complex analysis (calculus and some complex calculus), RP introduces the concept of elliptic curves and presents many interesting results. Unfortunately, a lot of hand waving goes on, i.e., many results are merely stated, instead of proved.

AEC is definitely for graduate students who have all ready taken the graduate algebra and geometry sequences. A lot of high powered mathematics is used in this text to get at the heart of elliptic curves.

Washington's book falls right in between these two books. He assumes knowledge of some analysis and algebra (particulary abelian groups), then develops much of what else is needed. Some hand waving exists (mainly for some of the high powered projective geometry needed to fully understand the geometry of elliptic curves) in this book, but this does not detract from the understanding of the additive group on elliptic curves, the primary focus of the book.

For those with a basic handle on real analysis and group theory, this book can easily be used for self-teaching.

A clear, concise introduction to elliptic curves.......2006-02-20

I used this book as my main resource when writing my undergraduate dissertation on elliptic curve group structure. Although once I wanted to have a more in-depth look into any particular subject I had to chase up the references, this book made an excellent starting point. This book is a solid, clear introduction to the subject, which can be easily understood even by maths undergrads in the later years of their study (though if you're not a mathematician you may find it hard going!!) I found it be the clearest textbook on elliptic curves I came across, especially as it doesn't assume any background knowledge of algebraic geometry.

It might be a good book for a mathematic student but not a good one for an engineering student........2005-09-06

It might be a good book for a mathematic student but not a good one for an engineering student. There are too many mathematic jargons with very limited explanations. Many notations just take for granted that the readers have already known them. It is very hard for people who have limited math background. Moreover, there are so many editorial errors in the current version. I would suggest that the author put a mathematical symbol/sign index at the end of the book and make it easier for the readers to look for their meanings.

Excellent.......2003-07-19

Anyone who writes a book on elliptic curves will never do a bad job, for these objects are so beautiful that it would be a sacrilege to do otherwise. Those who study elliptic curves fall under their spell, not only because of their beauty, but also because of their many applications: the spinning top in mechanics, cryptography, exactly solved models in statistical mechanics, precession of the Mercury perihelion in general relativity, the proof of Fermat's Last (Wiles) Theorem, control theory, and string theory, to name a few. This book is an excellent treatment of ECs and would be good for a graduate student starting out in the field. The author gives many concrete examples of the main theorems, and helpful exercises are found at the end of each chapter.

The author begins the book with two neat problems that motivate well the subject of elliptic curves: the pyramid of cannonballs and the right triangle problem, i.e. which integers can occur as areas of right triangles with integer sides? He then immediately begins the elementary theory of ECs in chapter 2. The treatment is pretty standard, although he proves Pascal's and Pappus's theorems using the associativity of the group operation on ECs, which is not usually done in books on ECs. Also somewhat non-standard this early in the game is the discussion of reduction of ECs modulo various primes, and the subsequent definitions of additive, split multiplicative, and non-split multiplicative reduction.

The study of torsion points is done in chapter 3 with the Weil pairing on the n-torsion of an EC taking center stage. A fairly short chapter, the author delays the proof of the properties of the Weil pairing until chapter 11, where it is done with divisors.

Chapter 4 deals with elliptic curves over finite fields, and is one of the most important in the book from the standpoint of cryptographic applications of ECs. Hasse's theorem, giving the bounds for the group of points on an EC over a finite field, is proven in detail. The Frobenius endomorphism is introduced, and a proof of Schoof's algorithm for computing the number of points on ECs over a finite field is given a detailed treatment. There are many symbolic computational software packages in both the open and commerical realm which will do the counting straightforwardly, and anyone interested in cryptography will need to be familiar with some of these. Supersingular curves in characteristic p are introduced, and the author gives a good discussion of the reason why they are named as such.

The discrete logarithm problem, a topic also very important for cryptographic applications, is discussed in chapter 5. The chapter beings with the index calculus, and, recognizing that it does not apply to general groups, the Pohlig-Hellman, baby step-giant step method, and Pollards rho and lambda methods are discussed in details. The author then shows that for supersingular and "anomalous" curves, that the discrete logarithm problem can be reduced to an easier discrete logarithm problem. Along the way, two important concepts are introduced: the p-adic valuation, and the Tate-Lichtenbaum pairing, the latter of which is related to the Weil pairing, but applies to situations where the Weil pairing does not.

Elliptic curve cryptography is then discussed in chapter 6, and the treatment is fairly thorough. The author shows to what extent the Decision Diffie-Hellman problem can be solved using the Weil pairing. He also shows how to represent a message on an elliptic curve, satisfying early on any reader's curiosity on just how this is done. The El Gamal and ECDSA are compared in terms of their computational efficiency. An EC generalization of RSA is also discussed in some detail, along with a cryptosystem based on the Weil pairing. Chapter 7 then gives other applications of ECs, such as factoring and primality testing.

Chapter 8 marks the beginning of the "heavy artillery" in the theory of ECs, for here the author begins the discussion of elliptic curves over the rational numbers, which can be viewed as an example of Diophantine geometry. The famous Mordell-Weil theorem is proved, and as a sign that one is definitely in the arena of modern mathematics, the proof is given in terms of Galois cohomology, which is an abstraction of the Fermat method of descent. The reader gets a taste of height functions, and via some good examples, gets insight into why the rank of the EC is so difficult to compute. A neat example is given of a nontrivial Shafarevich-Tate group.

I did not read the chapters 9, 10, or 11 on ECs over the complex numbers, complex multiplication, and divisors, so I will omit their review. Chapter 12 introduces the famous zeta functions, and their use in obtaining arithmetic information about an EC. Zeta functions motivate the definition of an L-function of an EC, these being tremendously important in modern developments in the theory of ECs, such as the Swinnerton-Dyer and Birch conjecture, the latter of which is motivated rather nicely in this chapter.

Book Description

This book is devoted to mean-square and weak approximations of solutions of stochastic differential equations (SDE). These approximations represent two fundamental aspects in the contemporary theory of SDE. Firstly, the construction of numerical methods for such systems is important as the solutions provided serve as characteristics for a number of mathematical physics problems. Secondly, the employment of probability representations together with a Monte Carlo method allows us to reduce the solution of complex multidimensional problems of mathematical physics to the integration of stochastic equations.
Along with a general theory of numerical integrations of such systems, both in the mean-square and the weak sense, a number of concrete and sufficiently constructive numerical schemes are considered. Various applications and particularly the approximate calculation of Wiener integrals are also dealt with.
This book is of interest to graduate students in the mathematical, physical and engineering sciences, and to specialists whose work involves differential equations, mathematical physics, numerical mathematics, the theory of random processes, estimation and control theory.

Algebraic Number Theory (Crc Press Series on Discrete Mathematics and Its Applications)

Average customer rating:

A Mature and Concrete Introduction to Algebraic Number Theory
Worst textbook ever!
BUY THIS BOOK!

Algebraic Number Theory (Crc Press Series on Discrete Mathematics and Its Applications)
Richard A. Mollin
Manufacturer: CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 0849339898

Book Description

From its history as an elegant but abstract area of mathematics, algebraic number theory now takes its place as a useful and accessible study with important real-world practicality. Unique among algebraic number theory texts, this important work offers a wealth of applications to cryptography, including factoring, primality-testing, and public-key cryptosystems. A follow-up to Dr. Mollin's popular Fundamental Number Theory with Applications, Algebraic Number Theory provides a global approach to the subject that selectively avoids local theory. Instead, it carefully leads the student through each topic from the level of the algebraic integer, to the arithmetic of number fields, to ideal theory, and closes with reciprocity laws. In each chapter the author includes a section on a cryptographic application of the ideas presented, effectively demonstrating the pragmatic side of theory. In this way Algebraic Number Theory provides a comprehensible yet thorough treatment of the material. Written for upper-level undergraduate and graduate courses in algebraic number theory, this one-of-a-kind text brings the subject matter to life with historical background and real-world practicality. It easily serves as the basis for a range of courses, from bare-bones algebraic number theory, to a course rich with cryptography applications, to a course using the basic theory to prove Fermat's Last Theorem for regular primes. Its offering of over 430 exercises with odd-numbered solutions provided in the back of the book and, even-numbered solutions available a separate manual makes this the ideal text for both students and instructors.

Customer Reviews:

A Mature and Concrete Introduction to Algebraic Number Theory.......2006-09-08

I used this book in a one on one course in algebraic number theory in my fourth year of college. We finished just before ramification.

My background at the time included a year of undergraduate linear algebra, a year of undergraduate abstract algebra, a semester of intro. graduate algebra, intro. Galois theory, and intro. commutative algebra. The only things I used were Galois theory, my second abstract algebra course, and my second linear algebra course. Commutative algebra helped, but wasn't necessary in that abstract setting.

Organization: The book is very well organized with helpful appendices on abstract algebra basics (Groups, Rings, Fields) and Galois theory. The first chapter is slow-paced and provides a strong historical background for the material. A reviewer below suggested that there were "logical leaps" in the text--I never found such stuff, and I am always very picky about details. The author uses easy propositions that are assigned for homework sometimes, but they're mostly straightforward.

Exercises: They range from straightforward to quite thought-invoking... I remember one in particular, a starred problem, in which I had to use three "tricks" to solve.

Content: I like this book a lot. It's not super abstract on the level of Lang, but has hints of great generality throughout, and it's not some trivial algebraic number theory full of history, anecdotes, useless junk book with "Fermat's Last Theorem" misleadingly stated in the title somewhere. This book has a lot of stuff on applications to cryptography. The book covers things including

- Euclidean domains and unique factorization,
- special cases of FLT,
- Dirichlet's Unit Theorem,
- geometry of numbers,
- ideal class group,
- ramification,
- basics of class field theory,
- reciprocity laws.

It's a really nice all-around introduction. You need to be mature enough to read this book--the problems require that the reader is familiar with the relevent math. I was really impressed with the organization of this book--with topics like these, it's hard to have a nice balance of generality with concrete and useful results. This is for people not ready to appreciate Lang's book.

Worst textbook ever!.......2006-03-26

I am a graduate student specializing in Ring Theory and I have to tell you this is the absolute worst book I have ever had. Not only does the author make these humongous jumps in each section, he also has massive logical gaps. There are plenty of errors in the text starting right from the first section. You could easily spend a whole year deciphering (with a massive headache) the first chapter. The author is definetly wrong in assuming that all you need is a basic undergraduate number theory class and basic abstract algebra. You could have 2 comprehensive years of graduate modern algebra and still not be ready for the massive logical gaps in the book. Sure if we were all Galois and Eulers, the book would be easier, but I'd bet they'd even be scratching their heads often enough. My advice is stay away from the text at all cost. You'll regret paying the outrageous price for a text that is worth firepaper.

BUY THIS BOOK!.......2003-04-26

I learned a tremendous amount about Algebraic Number Theory from this excellent source. I have looked at other books that just skim the topics. This one covers them in depth and even has applications to cryptography (the author shuld have put that in the title). Even more advanced topics such as the higher reciprocity laws are covered with rigorous detail and extreme clarity. I read the AMS review for this book by Charles Parry and it is right on! This book should replace the old standards such as Janusz's and Marcus' books for instance. I'd say that this is a gem to be enjoyed.

Elementary Number Theory and Its Applications

Average customer rating:

Omits obvious visual and intuitive ways of looking at concepts.
Excellent introduction to a vast, fascinating subject
Good foundational text for Cryptography
Very helpful resource
Very helpful for the Learning Undergraduate

Elementary Number Theory and Its Applications
Kenneth H. Rosen
Manufacturer: Longman Higher Education
ProductGroup: Book
Binding: Paperback

ASIN: 0201119587

Book Description

This latest edition of Kenneth Rosen's widely used Elementary Number Theory and Its Applications enhances the flexibility and depth of previous editions while preserving their strengths. Rosen effortlessly blends classic theory with contemporary applications. New examples, additional applications and increased cryptology coverage are also included. The book has also been accuracy-checked to ensure the quality of the content. A diverse group of exercises are presented to help develop skills. Also included are computer projects. The book contains updated and increased coverage of Cryptography and new sections on Möbius Inversion and solving Polynomial Congruences. Historical content has also been enhanced to show the history for the modern material. For those interested in number theory.

Customer Reviews:

Omits obvious visual and intuitive ways of looking at concepts........2006-10-23

This book is wordy, but less clear than I think it could be. It moves slowly, yet omits helpful ways of looking at concepts in the interest of being elementary. Also, it does not pave the way for future study of the material, nor does it pave the way for later study of connections to algebra, analysis, or combinatorics.

This book overlooks intuitive, visual ways of representing basic concepts. For example, lattice/Hasse diagrams helped me understand divisibility and GCD's, but this book does not even mention this way of looking at divisibility. Congruence relations can be visualized in a number of ways, but this book only treats them using basic algebra. A book that moves as slowly and is as elementary as this one should really explore these sorts of visual presentations of concepts.

Since this book is elementary, it does not explore any connections to groups, rings and ideals, fields, or lattices, and I think this is a shame because these structures make number theory make more sense. I think the book would do better to introduce a few of these structures in a very basic way. The book also passes up the opportunity to introduce generating functions, a critical and fairly elementary topic in number theory, which are only touched in one exercise. The first few chapters of Wilf's Generatingfunctionology and Newman's "Analytic Number Theory" show that generating functions can be presented at an elementary level.

There are blurbs of history interspersed throughout the text, and I like the idea, but the history focuses almost exclusively on biographical information, with a tiny bit of history of famous problems and conjectures. There is little discussion of how the core mathematical ideas in the book were discovered and evolved over time. This book would do well to cut out the biographies and replace them with richer discussion of the historical development of the subject itself.

I like the idea of a number theory book that focuses on applications, but this book does so at the expense of other things: its treatment of truly fascinating topics (such as continued fractions) is so weak that I do not think it's worth the trade-off. The book does nothing to pave the way towards the study of either analytic or algebraic number theory. The Zeta function only gets a token mention.

This book is usable as a textbook or for self-study, but it is not outstanding for either of these goals, nor is it useful as a reference for advanced students. Studying from this book alone won't help one develop a good sense of intuition in number theory. Stillwell's thin book is about as easy to follow as this one, and yet it slowly introduces ideals and other algebraic concepts by the end of the book. My favorite book on number theory is Apostol's "Analytic Number Theory". It is considerably more advanced than this one, but I think that students with a strong background will actually find it easier to learn from. I have not yet found a truly elementary book on number theory that I liked; I think students would be better off to first acquire enough background and mathematical maturity to dive right into some of these more advanced texts, than to spend their time working through a book like this.

Excellent introduction to a vast, fascinating subject.......2006-05-16

This book is awesome. Tons of material covered, but at a decent pace and with good examples. All the reader needs is a good working knowledge of how to read and construct proofs, and the time and patience to get through the material. The exercises range from the computational to the theoretical, from the routine to the extremely challenging. Many of the examples and exercises are intriguing, well-known results. The author touches on a large amount of subject matter and has many references for those interested in further reading. He makes no use of any of the methods that come from the main branches of Mathematics, namely Algebra, Analysis, etc. (though he mentions a few famous results), but he also mentions that he will not be using these methods in the beginning. It starts off a bit easy but gets moderately challenging. This book will never leave my shelf, and has sparked an interest inside of me that shows no signs of burning out. Again, an excellent book.

Good foundational text for Cryptography.......2005-05-17

I bought this book as a supplemental study guide - my course was a mix of Number Theory, Abstact Algebra, Finite Fields and Probablility (pre-requisites for Cryptography study). The professor used lectures with hand-written supplements as primary study media.

I found this text to be well-written, each concept was dealt with separately and concisely, then later assimilated with others to form broader ideas. Even though the text is directed at Number Theory, many of the other related concepts I studied were mentioned or covered in this text.

This book will stay on my bookshelf for quite a while.

Very helpful resource.......2005-03-05

I'm currently taking a Discrete Mathematics class that is a stew of number theory, abstract algebra and probability.

This books has been very helpful as a reference as my professor teaches the course from his own lecture materials that often do not provide enough examples or coherent explanations.

The chapters on cryptography add a welcome flavor to the material.

Very helpful for the Learning Undergraduate.......2004-11-20

This books gives an excellent background to elementary number theory. I've found this book gives lots of examples, theorems that are helpful with interesting excercises. What I do find frustrating is that here and there the methods given are innefficient compared to what my proffesor teaches in class. As well some of the proofs are written as one giant run on sentance, and written line by line. This is VERY hard on the eyes and totally annoys me.

Sums of Squares of Integers (Discrete Mathematics and Its Applications)

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Sums of Squares of Integers (Discrete Mathematics and Its Applications)
Carlos J. Moreno , and Jr., Samuel S. Wagstaff
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 1584884568

Book Description

Sums of Squares of Integers covers topics in combinatorial number theory as they relate to counting representations of integers as sums of a certain number of squares. The book introduces a stimulating area of number theory where research continues to proliferate. It is a book of "firsts" - namely it is the first book to combine Liouville's elementary methods with the analytic methods of modular functions to study the representation of integers as sums of squares. It is the first book to tell how to compute the number of representations of an integer n as the sum of s squares of integers for any s and n. It is also the first book to give a proof of Szemeredi's theorem, and is the first number theory book to discuss how the modern theory of modular forms complements and clarifies the classical fundamental results about sums of squares. The book presents several existing, yet still interesting and instructive, examples of modular forms. Two chapters develop useful properties of the Bernoulli numbers and illustrate arithmetic progressions, proving the theorems of van der Waerden, Roth, and Szemeredi. The book also explains applications of the theory to three problems that lie outside of number theory in the areas of cryptanalysis, microwave radiation, and diamond cutting. The text is complemented by the inclusion of over one hundred exercises to test the reader's understanding.

Finite Fields (Encyclopedia of Mathematics and its Applications)

Average customer rating:

An excellent and thorough reference
2-nd edition = 1-st edition
Book for professional mathematician

Finite Fields (Encyclopedia of Mathematics and its Applications)
Rudolf Lidl , and Harald Niederreiter
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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ASIN: 0521392314

Book Description

The theory of finite fields is a branch of algebra with diverse applications in such areas as combinatorics, coding theory and the mathematical study of switching circuits. This updated second edition is devoted entirely to the theory of finite fields, and it provides comprehensive coverage of the literature. Bibliographical notes at the end of each chapter give a historical survey of the development of the subject. Worked examples and lists of exercises throughout the book make it useful as a text for advanced level courses for students of algebra.

Customer Reviews:

An excellent and thorough reference.......2007-07-06

This book is an excellent reference on finite fields. Chapter 1 covers the necessary background material at the right level for someone with a good knowledge of mathematics, but little algebra. Chapter 2 covers the basics of finite fields and chapter 3 goes into more detail on polynomials. The later chapters treat special topics and some applications. The proofs are clear, not over detailed and not overly terse either. If you just want to know the basics then there may be more in this book than you will ever need, but if you want a thorough reference it is recommended.

2-nd edition = 1-st edition.......2001-02-13

Undoubtedly an excellent book, but comprehensive ... only up to 1983, as this is essentially an IDENTICAL reprint by CUP of the original 1983 book by Addison-Wesley. Don't be misled by the 1997 edition date; second edition = first edition, not a single result from 1983 - 1997.

Book for professional mathematician.......2001-01-06

This is the best book on Galois (Finite) Fields, written by Austrian and Australian professors can be recomended as the most complete reference in the field. It is amazing how helpful it has been to me since time I wrote my MS diploma. Though an instance I was using was borrowed from library, I'd recomend everybody, who is related to deescrete math to have one.

Spline Functions and Multivariate Interpolations (Mathematics and Its Applications)

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Spline Functions and Multivariate Interpolations (Mathematics and Its Applications)
B. Bojanov , H. Hakopian , and B. Sahakian
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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All Titles | Qualifying Textbooks - Fall 2007 | Stores | Books

Computers & Internet | Qualifying Textbooks - Fall 2007 | Stores | Books

Professional | Qualifying Textbooks - Fall 2007 | Stores | Books

Science | Qualifying Textbooks - Fall 2007 | Stores | Books
ASIN: 0792322290

Book Description

This volume provides a comprehensive introduction to the theory of spline functions. Emphasis is given to new developments, such as the general Birkhoff-type interpolation, the extremal properties of splines, their prominent role in the optimal recovery of functions, and multivariate interpolation by polynomials and splines.
The book has thirteen chapters dealing, respectively, with interpolation by algebraic polynomials, the space of splines, B-splines, interpolation by spline functions, natural spline functions, perfect splines, monosplines, periodic splines, multivariate B-splines and truncated powers, multivariate spline functions and divided differences, box splines, multivariate mean value interpolation, multivariate polynomial interpolations arising by hyperplanes, and multivariate pointwise interpolation.
Some of the results described are presented as exercises and hints are given for their solution.
For researchers and graduate students whose work involves approximation theory.

Diophantine Analysis (Discrete Mathematics and Its Applications)

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Diophantine Analysis (Discrete Mathematics and Its Applications)
Jorn Steuding
Manufacturer: Chapman & Hall/CRC
ProductGroup: Book
Binding: Hardcover

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ASIN: 1584884827

Book Description

While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications. He also explores modern topics rarely addressed in other texts, including the abc conjecture, the polynomial Pell equation, and the irrationality of the zeta function and touches on topics and applications related to discrete mathematics, such as factoring methods for large integers. Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field. A detailed appendix supplies the necessary background material, more than 200 exercises reinforce the concepts, and engaging historical notes bring the subject to life.

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