Book Description
College Geometry offers readers a deep understanding of the basic results in plane geometry and how they are used. Its unique coverage helps readers master Euclidean geometry, in preparation for non-Euclidean geometry. Focus on plane Euclidean geometry, reviewing high school level geometry and coverage of more advanced topics equips readers with a thorough understanding of Euclidean geometry, needed in order to understand non-Euclidean geometry. Coverage of Spherical Geometry in preparation for introduction of non-Euclidean geometry. A strong emphasis on proofs is provided, presented in various levels of difficulty and phrased in the manner of present-day mathematicians, helping the reader to focus more on learning to do proofs by keeping the material less abstract.For readers pursuing a career in mathematics. ... Read more

Book Description
This is one of the first monographs to deal with the metrictheory of spatial mappings and incorporates results in the theory ofquasi-conformal, quasi-isometric and other mappings.
The main subject is the study of the stability problem in Liouville'stheorem on conformal mappings in space, which is representative of anumber of problems on stability for transformation classes. To enablethis investigation a wide range of mathematical tools has beendeveloped which incorporate the calculus of variation, estimates fordifferential operators like Korn inequalities, properties of functionswith bounded mean oscillation, etc.
Results obtained by others researching similar topics are mentioned,and a survey is given of publications treating relevant questions orinvolving the technique proposed.
This volume will be of great value to graduate students andresearchers interested in geometric function theory.
... Read more

Product Description
This Elibron Classics edition is a facsimile reprint of a 1866 edition by Longmans, Green, and Co., London. ... Read more

Book Description
Sir Isaac Newton's philosophi Naturalis Principia Mathematica'(the Principia) contains a prose-style mixture of geometric and limit reasoning that has often been viewed as logically vague.
In A Combination of Geometry Theorem Proving and Nonstandard Analysis, Jacques Fleuriot presents a formalization of Lemmas and Propositions from the Principia using a combination of methods from geometry and nonstandard analysis. The mechanization of the procedures, which respects much of Newton's original reasoning, is developed within the theorem prover Isabelle. The application of this framework to the mechanization of elementary real analysis using nonstandard techniques is also discussed. ... Read more

Book Description
This book is a translation of Professor Wus seminal Chinese book of 1984 on Automated Geometric Theorem Proving. The translation was done by his former student Dongming Wang jointly with Xiaofan Jin so that authenticity is guaranteed. Meanwhile, automated geometric theorem proving based on Wus method of characteristic sets has become one of the fundamental, practically successful, methods in this area that has drastically enhanced the scope of what is computationally tractable in automated theorem proving. This book is a source book for students and researchers who want to study both the intuitive first ideas behind the method and the formal details together with many examples. ... Read more

Book Description
This book is a collection of essays centred around the subjectof mathematical mechanization. It tries to deal with mathematics in aconstructive and algorithmic manner so that reasoning becomesmechanical, automated and less laborious.
The book is divided into three parts. Part I concerns historicaldevelopments of mathematics mechanization, especially in ancientChina. Part II describes the underlying principles of polynomialequation-solving, with polynomial coefficients in fields restricted tothe case of characteristic 0. Based on the general principle, somemethods of solving such arbitrary polynomial systems may be found.This part also goes back to classical Chinese mathematics as well astreating modern works in this field. Finally, Part III containsapplications and examples.
Audience: This volume will be of interest to research andapplied mathematicians, computer scientists and historians inmathematics. ... Read more

Book Description
This book is intended for amateurs, students and teachers. The author presents partial results which could be obtained with exclusively elementary methods. The proofs are given in detail, with minimal prerequisites. An original feature are the ten interludes, devoted to important topics of elementary number theory, thus making the reading of this book self-contained. Their interest goes beyond Fermat's theorem. The Epilogue is a serious attempt to render accessible the strategy of the recent proof of Fermat's last theorem, a great mathematical feat. ... Read more

Customer Reviews (4)

Difficult book but great topic coverage
Solid coverage of proofs relating to Fermat's Last Theorem up to Kummer's Theory.You will find proofs for n=2, n=3, n=4, n=5, and n=7.Requires solid background in Algebraic Number Theory.For example, you should already have a good understanding of the Quadratic Law of Reciprocity, Quadratic Fields, and Congruences.If you don't, I recommend Elementary Number Theory for Congruences and the Quadratic Law of Reciprocity and Stark's An Introduction to Number Theory for Quadratic Fields.I would also recommend starting out with Edward's Book on Fermat's Last Theorem which includes detailed coverage of Kummer's Theory.

Great selection of material, difficult book
I find that this is a great book if you are an instructor or have a solid background in algebraic number theory.If you are unfamiliar with the Legendre Symbol, Gaussian integers, or the Law of Quadratic Reciprocity, you may wish to start out with a book such as Elementary Number Theory.If your are familiar with Algebraic Number Theory and wish to study in detail the Fermat Last Theorem proofs up to Kummer's Theory, this is a great book.I would recommend starting out with Edward's Book (Fermat's Last Theorem), for analysis of Euclid's proof of N=3.I found this very useful as an example of applications of Gaussian integers and Eisenstein integers.Ribenboim is one of the top experts about Fermat's Last Theorem and he is to praised for putting these beautiful proofs down.Even so, I would recommend purchasing other books to help explain this one.I found Stark's book very helpful in understanding Quadratic Fields.

"Amateur" mathematicians, that is !
If, like me, you were fascinated to hear that Fermat's so-called "last theorem" had been proven in 1995, then read Simon Singh or Amir Aczel's books popularizing the proof in outline, you probably wanted something more.

If, like me, you are a person who took some math in college, enjoys recreational mathematics books of the Douglas Hofstadter and Ian Stewart genre, and even sometimes picks up the odd number-theory book, you might consider yourself an "amateur."

If...if... this might seem like the book for you. I'd suggest that its not.

The mathematics in this book and its level of presentation was simply impenetrable by me. Not slow going... "no" going. That's frustrating to admit, but in a way fine, since it affirms of my admiration at a distance of the work that professional mathematicians do. I have seen many cited who state that Wiles' proof is simply beyond the ken of even 95% of working mathematicians. I believe this book must really be intended to serve some fraction of that group.Perhaps within the fold of mathematics these would consider themselves "amateurs". My two stars are offered only for them.

The book is simply not for the "lay" amateur. And Ribenboim's titling of it suggests that he does not even know that this lower caste, containing those of us who enjoy recreational mathematics and would describe ourselves as "amateurs", even exists. We know we exist as something mathematically distinct from the general population by the simple fact of the universally raised eyebrows that confront any mention of our interest in mathematics. Nevertheless, like any other species in a niche, we will have to continue to feed on a sparse supply of intellectual sustenance and learn to avoid the over-rich and indigestible fare of the higher forms.

Finally, if you haven't read Singh or Aczel I'd offer the former 5 stars and the latter 3 but recommend both. A truly fascinating story.

An excellant work, good for any serious study of FLT.
I am a math instructor and graduate student at PVAMU, and am working on a thesis detailing the history of attempts to prove and the Wiles proof of FLT. The text was easily readable and the proofs were very well done. I wasable to follow the logic and math of all the presented proofs very well. ... Read more

Book Description

Students will find all the information covered in the standard textbooks--and more--explained clearly and concisely in this powerful study tool. Unusually detailed, it elucidates all the most difficult-to-grasp concepts that class studies and texts sometimes gloss over. The hundreds of problems with fully explained solutions illuminate important points and teach students sound problem-solving skills. Ideal, also, for independent study.

Customer Reviews (3)

Differential Geometry review
I have found this to be an excellent addition to my library.

Good as a basic textbook and a source of solve problems
This book is intended to assist upper level undergraduate and graduate students in their understanding of differential geometry, which is the study of geometry using calculus.Usually students study differential geometry in reference to its use in relativity. I personally have a rather oddball application for the subject - modeling of curved geometry for computer graphics applications. The fundamental concepts are presented for curves and surfaces in three-dimensional Euclidean space to add to the intuitive nature of the material.
The book presumes very little in the way of background and thus starts out with the basic theory of vectors and vector calculus of a single variable in the first two chapters. The following three chapters discuss the concept and theory of curves in three dimensions including selected topics in the theory of contact.
Great care is given to the definition of a surface so that the reader has a firm foundation in preparation for further study in modern differential geometry. Thus, there is some background material in analysis and in point set topology in Euclidean spaces presented in chapters 6 and 7. The definition of a surface is detailed in chapter eight. Chapters 9 and 10 are devoted to the theory of the non-intrinsic geometry of a surface. This includes an introduction to tensor methods and selected topics in the global geometry of surfaces. The last chapter of the outline presents the basic theory of the intrinsic geometry of surfaces in three-dimensional Euclidean space.
Exercises are primarily in the form of proofs, and there are plenty of worked examples. Since the examples are kept to no more than three dimensions, the outline contains plenty of good instructive diagrams that illustrate key concepts. This Schaum's outline has quite a bit of instruction in it past the bare required minimum, but you might still want a good primary textbook. My personal favorite is Pressley's "Elementary Differential Geometry". Overall I find this to be a very good outline and source of solved problems on the subject and I highly recommend it.

Differential Geometry - A Schaum's Outline Series
As with all of the Schaum's Outline Series, this book is particularly useful if the readers intent is to gain a working knowledge of the subject. The subject of Differential Geometry is no exception. Dr. Lipschultz hasdone an excellent job of communicating the essential aspects ofdifferential geometry to the reader. The book assumes a fairly low level ofmathematical ability having calculus as the primary prerequisite. From thishumble beginning, Dr. Lipschultz takes the reader through the necessarydiscussions of vector functions, curvature, fundamental forms, and tensoranalysis. Given the theoretical nature of the subject,Dr. Lipschultz hasincluded most of the theorems and associated proofs necessary for a generalunderstanding of the subject. However, this book is not a substitute for aserious study of differential geometry. In addition most of the problemsare limited to two dimensional surfaces and this reader would have enjoyeda more adventurous investigation of higher dimensional spaces. Like allSchaum's series, the text is chock full of problems and their solution. Irecommend this book for anyone interested in quickly gaining a workingknowledge of the subject. ... Read more

Book Description

A number of important results in combinatorics, discrete geometry, and theoretical computer science have been proved using algebraic topology. While the results are quite famous, their proofs are not so widely understood. They are scattered in research papers or outlined in surveys, and they often use topological notions not commonly known among combinatorialists or computer scientists.

Customer Reviews (1)

Insight Pedagogy and Honesty
While allowing you to learn new concepts and definitions in a classical sense , this book shows what mathematics is about. It makes connections and gives you insight.
If you think that mathematics is made out of connections and is an "Art of variations" you will be rewarded. Those interested in the "frontier" between discrete and continuous will be interested too.
Definitely pedagogical and moreover intellectually honest.
... Read more

Book Description
...reflects the exciting developments in number theory during the past two decades. ... Read more

Customer Reviews (4)

tough problems => good for the student
The motivation of explaining Fermat's Last Theorem is a nice device by which Stewart takes you on a tour of algebraic number theory. Things like rings of integers, Abelian groups, Minkowski's Theorem and Kummer's Theorem arise fluidly and naturally out of the presentation.

The inclusion of problem sets in each chapter also enlivens its appeal to a student. Typically, the first problems in each set are easy. But later problems can be quite formidable, and really give a good mental workout of the salient issues just covered in the chapter.

Very clear introduction to Algebraic Number Theory
This book is a very clear intoductino to ANT.It is a good first step for many reasons.One: it stays with algebraic number fields that are extensions of Q, the rational numbers.You get a good feel for the subject.When you go to more advanced books Q is replaced by other fields (P-adic, function fields, finite fields,..).
Two: He assumes very little and writes very clearly
Three: You only needs to read his Galois theory book for the prerequisite
Four: His book is what is usually left for the reader to do as an excersize in more advanced books.

thoughts from an amateur
good overview of algebraic number theory as it applies to FLT, however not exactly pitched at beginners.you'll want to have a grounding in abstract algebra & linear algebra at the minimum.still, even if you don't, you can get a good sense of the "big picture" and a high-level understanding of the advances in mathematics that were directly or indirectly related to attempts to solve FLT.overall a fascinating read if you're a math geek who wants something a little deeper than Simon Singh's pop treatment of Wiles' proof.

Lucid introduction
Lucid and clear introduction to algebraic number theory, in style very much like the author's other book on Galois theory. Very elementary though, doesn't cover any analytic method, nor gives even a taste of class field theory, besides the problem set is less than challenging. But the book serves its purpose well, strongly recommended for beginners. ... Read more

Book Description
This book is an introduction to algebraic number theory via the famous problem of "Fermat's Last Theorem." The exposition follows the historical development of the problem, beginning with the work of Fermat and ending with Kummer's theory of "ideal" factorization, by means of which the theorem is proved for all prime exponents less than 37. The more elementary topics, such as Euler's proof of the impossibilty of x+y=z, are treated in an elementary way, and new concepts and techniques are introduced only after having been motivated by specific problems. The book also covers in detail the application of Kummer's ideal theory to quadratic integers and relates this theory to Gauss' theory of binary quadratic forms, an interesting and important connection that is not explored in any other book. ... Read more

Customer Reviews (3)

Old school algebraic number theory with heavy Kummer bias
Algebraic number theory eventually metamorphosed into a sub-discipline of modern algebra, which makes a genetic approach both pointless and very interesting at the same time. Edwards makes the bold choice of dealing almost exclusively with Kummer and stopping before Dedekind. Kummer's theory is introduced by focusing on Fermat's Last Theorem. As Edwards confirms, this cross-section of history is on the whole artificial--Fermat's Last Theorem was never the main driving force; not for Kummer, nor for anyone else--but it fits its purpose quite well, and besides, Edwards only adheres to it for about half the book. Kummer-Edwards's style has a heavily computational emphasis. Edwards defends this aspect fiercely. Perhaps feeling that the authority of Kummer is not enough to convince us of the virtues of excessive computations, Edwards trumps us with a Gauss quotation (p. 81) and we must throw up our hands.

Chapter 1 surveys Fermat's number theory. Chapter 2 deals with Euler's proof of the n=3 case of Fermat's Last Theorem, which is (erroneously) based on unique factorisation in Z[sqrt(-3)] and thus contains the fundamental idea of algebraic number theory. Still, progress towards Fermat's Last Theorem during the next ninety years is quite pitiful (chapter 3). The stage is set for our hero: Kummer, who developed a theory of factorisation for cyclotomic integers. One may of course not trust unique factorisation to hold here, but Kummer has a marvellous idea: the concept of "ideal" prime factors--curious ghost entities that save unique factorisation in many cases (chapter 4); enough to prove Fermat's Last Theorem for "regular" prime exponents (chapter 5). Telling whether a given prime is regular involves computing the corresponding class number, which is done analytically by means of an appropriate analog of the zeta function (chapter 6). Now, for all of this there is an analogous theory with quadratic integers in place of cyclotomic integers (cf. Euler above). Since it was not important for Fermat's Last Theorem, Edwards skipped past it before, but now we plunge into this theory and the allied theory of quadratic forms (chapters 7-9) to see how Kummer's theory helps elucidate some aspects of it, especially Gauss's notoriously complicated theory of quadratic forms.

great book
This is a great book.If you want to learn algebraic number theory from a very example/computational oriented book, then this is the book you want.it really has a lot of stuff in it.all other graduate books are theory without examples or motivation.this book is the exact opposite.the only drawback is that it doesn't use any modern algebra, but you can figure out how to shorten the arguments with algebra if you wanted to.

Read this if you're seriously interested in math.
There was a great burst of excitement, and several popular books, when Andrew Wiles proved "Fermat's last theorem". The popular books are fine, but they don't address the deepest issue: among all the many long-standing unsolved problems in number theory that are easy to state but resistant to solution, why did "Fermat's last theorem" attract the efforts of so many top-flight mathematicians: Euler, Sophie Germain, Kummer, and many others? The problem itself has no useful application or extension, and as stated seems like just another piece of obstinate trivia. So why is it mathematically interesting?

The answer, of course, is that attacks on the problem revealed deep and important connections between elementary number theory and various other branches of mathematics, such as the theory of rings. Thus, as so often in mathematics, the importance of the problem lies in where it leads the mind, rather than in the problem itself. Harold M. Edwards' book

is a minor classic of exposition, showing how the instincts of top-flight research mathematicians lead them to fruitful work from a seemingly unimportant starting point. I'm only sorry that Professor Edwards seems never to have completed the second volume he had hoped to write.

Thus book deserves to be read by a much larger audience than it has gotten; in particular, I believe every graduate student in math who hopes to do good research, regardless of specialty, would benefit from reading it. Beyond that, any mathematically inclined reader with a modicum of training in math, is likely to find this a fascinating book. ... Read more

Book Description
Noncommutative Geometry and Cayley-smooth Orders explains the theory of Cayley-smooth orders in central simple algebras over function fields of varieties. In particular, the book describes the étale local structure of such orders as well as their central singularities and finite dimensional representations.After an introduction to partial desingularizations of commutative singularities from noncommutative algebras, the book presents the invariant theoretic description of orders and their centers. It proceeds to introduce étale topology and its use in noncommutative algebra as well as to collect the necessary material on representations of quivers. The subsequent chapters explain the étale local structure of a Cayley-smooth order in a semisimple representation, classify the associated central singularity to smooth equivalence, describe the nullcone of these marked quiver representations, and relate them to the study of all isomorphism classes of n-dimensional representations of a Cayley-smooth order. The final chapters study Quillen-smooth algebras via their finite dimensional representations.Noncommutative Geometry and Cayley-smooth Orders provides a gentle introduction to one of mathematics' and physics' hottest topics. ... Read more

Book Description
This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. The purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi- stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. Contributors to this volume include: B. Conrad, H. Darmon, E. de Shalit, B. de Smit, F. Diamond, S.J. Edixhoven, G. Frey, S. Gelbart, K. Kramer, H.W. Lenstra, Jr., B. Mazur, K. Ribet, D.E. Rohrlich, M. Rosen, K. Rubin, R. Schoof, A. Silverberg, J.H. Silverman, P. Stevenhagen, G. Stevens, J. Tate, J. Tilouine, and L. Washington. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, universal deformation rings, Hecke algebras, complete intersections and more, as the reader is led step-by-step through Wiles' proof. In recognition of the historical significance of Fermat's Last Theorem, the volume concludes by looking both forward and backward in time, reflecting on the history of the problem, while placing Wiles' theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this volume to be an indispensable ... Read more

Customer Reviews (3)

Yet another application of elliptic curves...
The successful proof of Fermat's Last Theorem by Andrew Wiles was probably the most widely publicized mathematical result of the 20th century. And once again, among their numerous other applications, elliptic curves are employed in the proof. The book is a compilation of articles written by first-class mathematicians, the reading of which will give one a thorough understanding of the proof, along with an overview of some very interesting topics in number theory and algebraic geometry. The reader who undertakes an understanding of the proof already no doubt has a substantial amount of 'mathematical maturity', and no review, no matter how complete, would influence greatly such a reader. Suffice it to say then that this book is excellent, and even a reader interested solely in elliptic curves and modular forms could benefit greatly from the reading of this book.

Great?!?!
This book might be good if you like number theory. But if you're an analyst who hates number theory or a brick-layer, then this book is probably not meant for you. I hope you found this review helpful. Have a nice day.

Highly recommended
This item is very instructively, not only for "real"mathematicians. Of course, sometimes it's very difficult to"read". It gives me pleasure to own the proof of FLT. ... Read more

Book Description
The manifolds investigated in this monograph are generalizations of (XX)-rank one locally symmetric spaces. In the first part of the book the author develops spectral theory for the differential Laplacian operator associated to the so-called generalized Dirac operators on manifolds with cusps of rank one. This includes the case of spinor Laplacians on (XX)-rank one locally symmetric spaces. The time-dependent approach to scattering theory is taken to derive the main results about the spectral resolution of these operators. The second part of the book deals with the derivation of an index formula for generalized Dirac operators on manifolds with cusps of rank one. This index formula is used to prove a conjecture of Hirzebruch concerning the relation of signature defects of cusps of Hilbert modular varieties and special values of L-series. This book is intended for readers working in the field of automorphic forms and analysis on non-compact Riemannian manifolds, and assumes a knowledge of PDE, scattering theory and harmonic analysis on semisimple Lie groups. ... Read more

Book Description
The Novikov conjecture is the single most important unsolved problem in the topology of high-dimensional non-simply connected manifolds.These two volumes give a snapshot of the status of work on the Novikov conjecture and related topics from many points of view: geometric topology, homotopy theory, algebra, geometry, and analysis. Volume 1 contains a detailed historical survey and bibliography of the Novikov conjecture and of related subsequent developments, including an annotated reprint (both in the original Russian and in English translation) of Novikov's original 1970 statement of his conjecture; an annotated problem list; the texts of several important unpublished classic papers by Milnor, Browder, and Kasparov; and research/survey papers on the Novikov conjecture by Ferry/Weinberger, Gromov, Mishchenko, Quinn, Ranicki, and Rosenberg.Volume 2 contains fundamental long research papers by G. Carlsson on "Bounded K-theory and the assembly map in algebraic K-theory" and by S. Ferry and E. Pedersen on "Epsilon surgery theory"; and shorter research and survey papers on various topics related to the Novikov conjecture, by Bekka, Cherix, Valette, Eichhorn, and others. These volumes will appeal to researchers interested in learning more about this intriguing area. ... Read more

Product Description
Students will find this text useful for these reasons: 1) learning each rule, formula and principle. 2) learning each set of solved problems. 3) learning each set of supplementary problems. 4) integrating the learning of plane geometry. 5) Learning geometry through self-study. 6) extending plane geometry into solid geometry. CONTENTS: 1) Lines, angles and triangles 2) Methods of Proof 3) Congruent Triangles 4) Parallel Lines, Distances and Angle Sums 5) Parallelograms, Trapezoids, Medians and Midpoints 6) Circles 7) Similarity 8) Trigonometry 9) Areas 10) Regular Polygons and the Circle 11) Locus 12) Coordinate Geometry 13) Inequalities and Indirect Reasoning 14) Improvement of Reasoning 15) Constructions 16) Proofs of Required Theorems 17) Extending Plane Geometry into Solid Geometry. Also includes formulas for reference; table of trigonometric functions; table of squares and square roots; answers to supplementary problems. ... Read more