Book Description

Jean Leray is one of the great French mathematicians of this century. His life's work divides into 3 major areas, reflected in these 3 volumes. Vol. 1, to which an Introduction in English has been contributed by A. Borel, covers Leray's seminal work in algebraic topology, where he created sheaf theory and discovered the spectral sequences. Vol. 2, with an introduction by P. Lax, covers fluid mechanics and PDE: Leray demonstrated the existence of the infinite-time extension of weak solutions of the Navier-Stokes equations; 60 years later this profound work has retained all its impact. Vol. 3, on the theory of several complex variables, has a long introduction by G. Henkin. Leray's work on the ramified Cauchy problem will stand for centuries alongside the Cauchy-Kovalevska theorem for the unramified case. ... Read more

Book Description

This book combining wavelets and the world of the spectrum focuses on recent developments in wavelet theory, emphasizing fundamental and relatively timeless techniques that have a geometric and spectral-theoretic flavor. The exposition is clearly motivated and unfolds systematically, aided by numerous graphics.

This self-contained book deals with important applications to signal processing, communications engineering, computer graphics algorithms, qubit algorithms and chaos theory, and is aimed at a broad readership of graduate students, practitioners, and researchers in applied mathematics and engineering. The book is also useful for other mathematicians with an interest in the interface between mathematics and communication theory. ... Read more

Reviews (6)

A masterpiece in wavelets
This is a very well written book on the theory of wavelets and its applications. The presentation is self-contained and may serve as an introduction for someone who wants to learn about this topic. It also is an excellent book for those who have an advanced graduate degree in applied mathematics since it demonstrates how to truly understand complex concepts. This book gives a general presentation of some recent developments in wavelet theory with an emphasis on techniques that have a geometric and spectral-theoretic flavor. It can be certainly used as a textbook for graduate students as well as as a reference book for the specialists and researchers. A very nice feature of the book are the tutorials at the beginning of each chapter and some sections, which serve as summaries of main ideas and guides through the background ideas and motivation. At the end of every chapter there are plenty of excercises of various level of difficulty. Overall, this is a great book and I recommend it highly to a motivated reader.

Distinguished link between math and computer science
This is an important mathematical reference written in excellent style. Wavelets have found applications in many areas of engineering and CS. The authors provide a detailed, rich and entertaining tour through this relatively young but important field for both math and CS/Eng. Connections are, e.g., made between advanced CS virtual-reality applications such as audio-systems processing, future applications such as quantum computing, and advanced math in functional analysis and operator theory.

A fresh look at wavelets
The Book by Bratteli and Jorgensen is a superb book on wavelet's theory. It is very well written and has new and a fresh point of view on the subject. Although there are several good books on wavelets, the book by Bratteli and Jorgensen covers an important niche that has not been covered before. In particular

1- The book covers the theory of wavelets from the point of view of operators and functional analysis and will appeal to a growing number of pure as well as applied mathematicians interested in the subject.

2- The writing of the book is very appealing: every chapter starts by a tutorial that gives motivation as well as intuition. It is then followed by a very clean mathematical development of the subject, together with many examples, figures, and applications from physics and engineering. A set of nice problems is provided at the end of each chapter. Thus this book can be used as a graduate textbook or for mathematical seminars in mathematics departments.

3- This book can even be used by experts in wavelet theory for learning about recent developments and new perspectives from operator theory and functional analysis.

I highly recommend this book.

an intriguing new wavelet book
This is a book about an important topic in applied mathematics by two authors with excellent credentials in both pure and applied areas. The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet index theorem, quantum computing, the ubiquitous C*-algebras O_n and, of course, spectral theory. The graphics are meticulously done.

I look forward to learning a lot from it.

An Intriguiging New Book
This is a book about an important topic in applied mathematics by two authors with excellent
credentials in both pure and applied areas.

The reader will find many intriguing threads connecting wavelets to other parts of mathematics, including a wavelet "index theorem", quantum computing, the ubiquitous Cuntz C*-algebras and, of course, spectral theory. The graphics are excellent. I look forward to learning a lot from it. ... Read more

Reviews (13)

Superb plane geometry primer for children's self-study
Easy to follow, logical, and methodical exposition of elementary classical plane geometry. Every term and concept is defined; therefore, it presupposes absolutely no prior knowledge (of either geometry or algebra). Yet, after four chapters it is possible for a child to knowledgeably explain Euclid's Fifth Postulate. The beauty of forms and patterns, the triumph of solving seemingly insoluble problems, the revelation of the power of logical thinking -- all serve to captivate my 9 year old daughter's attention. Most highly recommended for anyone who wishes to provide the means for a child to do his/her best original thinking. No cartoons, no shoot-em-up diversions, only the austere beaty of logic in an assimilable format for children. Every child in America should have the opportunity to go through this book with his/her parents. And every parent should have the oportunity of direct participation in the intellectual voyage of discovery of his/her child that this book affords.

Simplifies the subject beautifully.
Terrific tool for students of any age. My rising 9th grader is entering an advanced program which requires her to learn most of the basic theorems and postulates on her own over the summer. This book has worked beautifully for her. Even if you have one of those "I hate it if I don't understand it immediately" kind of kids, this book will work for them. I am recommending this to all of my friends with children about to take Geometry.

Geometry made easy and understandable!
This is an amazing book! It helped me personally. I had a 70 in my Geometry class (which was strange for me being a strait A student) and was studying for a quiz when my friend said i should check out her book and do some of the practice problems. The book explained the concepts in a new way and i could check my answers! I took the math quiz and got a 100%! Now im steadily on my way to become the strait A student i used to be! If you have trouble understanding geometry, or just want some back up help, then BUY THIS BOOK!

Great Geometry Tool
I am teaching myself advanced math. I read the entire book, cover to cover, and did each problem in the book. I found it extrememly useful, and I now feel I have a very good grasp on geometry. There are some minor flaws, as already explained in other reviews, but overall, I would highly recommend this book.

It's not a just easy way. It's an ultimate way !!
This book is great! You may become a master of lines, points, angles, circles, triangles, polygons ... by reading this book from the first page to the last page sequentially.
The book contains some typos but those are not major errors.
Solid geometry part is somewhat disappointing but everything else is great !
I want to add one more property of proportions, if a/b=c/d then
(a+c)/(b+d)=a/b=c/d. ... Read more

Book Description

The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study. ... Read more

Reviews (8)

Objects and maps are everywhere
Excellent book for non-professional mathematicians, like me (I'm a software engineer), who wants to understand modern mathematics and apply its ideas in analysis of complex problems. Lots of pictures and diagrams (compared to terse wording in other mathematical books) really help to understand and master the subject. I think most of negative reviews come from professional mathematicians, but they don't need this book.

Very uneven, but still useful
As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics.

Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here.

There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together.

Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful.

Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth.

Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples. I think this will leave many readers less than satisfied, but realistically, the book would have been twice as long had they really delved deeper into examples (or they would have had to be very terse in the actual descriptions of category theory, which is the choice most authors writing for a more mathematically-inclined audience seem to make - e.g., _Mathematical Physics_ by Geroch (good book!) or _Basic Category Theory for Computer Scientists_ by Pierce).

If you are mathematically astute, you probably will find this book tedious. But if you are not a grad+ math major, then this book may well be worth the effort as a way to begin to learn a very profound and powerful set of tools and concepts.

Heavy Hitter Strikes Out
I sure hope Schanuel wrote this book and the publisher simply tacked on
Lawvere's name for marketing purposes. This text is a fantastic
example of why research mathematicians should not write for John Q.
Public. The random, pointless examples scattered throughout the book
remind me of the "word problems" that were so popular in high school
algebra texts written after the Chicago School hijacked the educational
textbook market.

After teasing the reader with examples of real mathematics, e.g.
Pick's Formula, the authors stop short of actually proving a theorem
and scurry back to their shelter of objects and arrows where they can
safely field trivial questions by ersatz students with politically
correct names.

Perhaps Category Theory is just not something that is accessible to the
general public? High school math teachers (I assume one intended
audience for the text) that can achieve even the slightest appreciation
of why Eilenberg and Mac Lane invented Category Theory are surely as
rare as rocking-horse poop.

What I would really like to see from someone as eminent as Lawvere write a
first year graduate level book that covers elementary set theory and/or
logic using Category Theory. Translating Model Theory and Topoi(1.) to
this level would be a good start. College math professors are really
the only people in a position to understand and transmit this beautiful
theory to aspiring mathematicians.

1. Model Theory and Topoi, Lecture Notes in Mathematics 445,
Springer-Verlag 1975

Keith A. Lewis ...

A retract in search of a section
There is a wonderful course in category theory for high school students, just begging to be excavated from this multi-layered book.
Please don't be put off by the disjointed and uneasy combination of materials that cluster around certain themes. You know you will have a lot of work to do when the same definition (of monomorphism) is presented both on page 52 and also on page 336.
With all the elementary themes covered in many varying ways, it would be best to consider this book as having been structured as a retract for which your job will be to construct the appropriate section.

A Good Introduction
As a first introduction to Categories, this book is well written, clever, simple and very clear. However, I was disappointed with it. From the notoriety of the authors and the, yes, cool illustrations I assumed it would be a gem. However, it fell short. I've been toying with Category Theory for a few years, and every time I try to get into a book on Categories I get stumped at the notions of Functors and Natural Transformations. This book, however, dealt with neither at length, despite the fact that Category Theory originated around the notion of Natural Transformations in the first place. (As I understand it at least.) That said, there are many very cool passages in the book, including a functional analysis of a Chinese restaurant and an elegent exposition of Brouwer's Fixed Point Theorem.

Still, for my purposes, I prefer Robert Goldblatt's "Topoi: The Categorical Analysis of Logig" and Michael Barr's "Category Theory for Computing Science". As both are intended for non Category Theorists, both build their presentations of Category Theory from sratch. Sadly, I think both are out of print. Not for the faint of heart, I'm told Saunders Mac Lane's "Categories for the Working Mathematician" is the classic. (It's on my wish list.) ... Read more

Book Description

As Chaos explained the science of disorder, Nexus reveals the new science of connection and the odd logic of six degrees of separation. How can geometry explain the puzzles of human behavior? In this incisive, insightful work Mark Buchanan presents the fundamental principles of the emerging field of "small worlds" theory—the idea that a hidden pattern is the key to how networks interact and exchange information, whether that network is the information highway or the firing of neurons in the brain. Mathematicians, physicists, computer scientists, and social scientists are working to decipher this complex organizational system, for it may yield a blueprint of dynamic interactions within our physical as well as social worlds. Highlighting groundbreaking research behind network theory, Buchanan documents mounting support for the small-worlds idea and demonstrates its multiple applications to diverse problems—whether explaining the volatile global economy or the Human Genome Project, the spread of infectious disease or ecological damage. Nexus is an exciting introduction to the hidden geometry that weaves our lives so inextricably together. 20 illustrations. ... Read more

Reviews (21)

Terrific snapshot of a hot new field!
I wasn't sure I would like this book when my brother bought it for me -- but I did! It covers a truly wide range of material. Extremely impressive. Amazingly, as the book shows, strong mathematical links seem to connect the workings of biological cells with the Internet, social networks and many other complex networks, even neural networks and the human brain. The writing is extremely clear and there is little chance of misunderstanding. This is one of those areas of "hyped" research that really lives up to the hype.

From a personal point of view, I especially enjoyed the final chapters on economics and social capital. Something really seems to be emerging here -- a deep link between social patterns and natural patterns in the physical world

Six Degrees of seperation.
Actually, I bought this book with the intention of reading about genetics algorithms although I was pleasantly surprised with the out come of the book.

The book is about how our large world is small and what seems chaotic is actually an organized small network.

The author starts with how networks in nature relate to networks in technology. A very strong case for "6 degrees of separation" for our society and "19 degrees of (link) separation" for the Internet. The rest of the book explains with historical examples how scientists were able to prove the networking concepts through human decision and thought process.

I gave this book 4 star because I did not think that the conclusion had the continuity of the other chapters. I would recommend this book to all individuals who would be interested in reading and understanding the connections and influences of nature in our "connected" world.

Have fun understanding that you closer then you think to the person next door.

Good introduction to a broad subject
The author makes a strong case that many diverse phenomena can be modelled in very similar ways. This book can be summarized as a very brief introduction to network models, followed by numerous examples from the real world.

The level of mathematical sophistication needed to comprehend the matterial is minimal. I do not believe there are any equations in the entire book. There are many easily understood graphs and a few percentages.

The basic concept of the networks is very easy to explain and to understand. The applications are the interesting part. Thoughout the pages are clear and interesting examples that make you want to turn the page to see what is coming next. In my case I often found myself thinking how I would have approached the problem and more importantly what problems could this have been applied to. Any book that can do that is a good one in my book!

Like many good books, this one leaves more questions unanswered than it answers. The subject area is a generic one that allows it's self to be applied in many many different fields. The question becomes not is this model of the world valid but rather how can it be applied.

This was a quick read, certain to change my views on how the world works.

It's a small world after all.
I just finished reading Nexus right after I finished Steven Johnson's book, Emergence. Both are great, quick reads. The ideas are fascinating and build upon chaos theory that James Gleick gives a history of in Chaos, which is the last book I read that addressed topics such as complexity. It's a great thrill to receive journalistic reports on what has happened in the small-worlds theory and gaining a cursory understanding of its current and future applications. I also just started reading Harold Morowitz's The Emergence of Everything, which is interesting in its subject matter while the writing is much more austere than in Emergence and Nexus. I look forward to reading everything I can on the small-worlds, complexity theory-type popular science books.

Networks of sex partners and the Net-Are they really related
The surprising answer is yes. I picked this book up after reading Steven Strogatz's Sync which mentions a great deal about the science of networks. Buchanan explains how networks exist everywhere - the net, the web, the power grid, our circle of friends, our sex partners - and that they are in fact very similar to one another.

The phrase "six degrees of separation" comes from the fact that two randomly chosen people, A and B, will on average be connected by six social links. A knows C who knows D who knows E who knows F who knows G who finally knows B. Considering the world has over 6 billion people, an average separation of 6 seems unbelievable small, but the explanation of this incredible phenomenon lies in the makeup of our social network. Our close friends know each other but our cluster of friends has weak ties to other clusters through acquaintances, people we really don't know that well - that's why when one is looking for a job, it's better to tell an acquaintance rather than a friend so that our inquiry can jump to other clusters. Our social network is essentially highly clustered but enough links exist between these clusters to allow us to jump from ourselves to any other person through just an average of six links. Buchanan shows us how this kind of network exists everywhere as mentioned above although he distinguishes between egalitarian networks where clusters are roughly the same size and aristocratic networks such as the WWW where gigantic hubs like Amazon.com exist that link to millions of websites.

One of the most interesting chapters in the book deals with sexual networks. It turns out that in the network of sex partners, certain people have a great many more links than the average person in the network. Buchanan explains how the structure of the sexual network actually accounts for the rapid spread of HIV. The virus spread quickly because the hubs in the network spread it to their numerous partners. In fact, it turns out that a significant percentage of the inital HIV cases had a sexual relationship with one particular flight attendant.

As I wrote in my review for Strogatz's Sync, we are entering an era of science where disparate fields of study are being linked because many phenomena that we used to regard as unrelated now appear to have very similar underlying bases. It is exciting to read books like Nexus because it illustrates this point. You should definitely read this book if your are interested in the science of networks and want to know how so many different phenomena are being explained by the same underlying principles. ... Read more

Book Description

This is an introduction to the theory and applications of modern geometry. It differs from other books in its field in its emphasis on applications and its discussion of Special Relativity as a major example of a non-Euclidean geometry. Besides Special Relativity, it covers two other important ares of non-Euclidean goemetry: spherical geometry (used in navigation and astronomy) and projective geometry (used in art). In addition, it reviews many useful topics from Euclidean geometry, emphasizing transformations, and includes a chapter on conics and planetary orbits. Applications are stressed throughout the book. Every topic is motivated by an application and many additional applications are given in the exercises. The book would be an excellent introduction to higher geometry for those students, especially prospective mathematics and teachers, who need to know how geometry is used in addition to its formal theory. ... Read more

Reviews (1)

Excellent elementary introduction to modern geometry
I am a Ph.D student in the field of symplectic geometry and topology. This book introduces the foundations of modern geometry in a beautiful and a very clear way,and I am saying this having some experience with geometry and topology books. If you are a skilled high school student or an under graduate student for mathematics or related area,this is a good book to start with in understanding what is modern geometry. The level of the book is about undergraduate level using very elementary notions. The content of the book is: Euclidean geometry and its logical foundations(so one could understand the motivation of the other geometries), Sphirical geometry,conic sections,Projective geometry,and the ending chapter is about the geometrical foundations of special relativity. The approach is not theorem-proof style but rather a more intuitive approch!. This is a recommended book. ... Read more

Book Description

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry.Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry.This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered it the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century.This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric Theorem proving.

In preparing a new edition of "Ideals, Varieties and Algorithms" the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple, Mathematica and REDUCE. ... Read more

Reviews (6)

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

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

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

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

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

Book Description

Benoit Mandelbrot discovered what is now called the M-Set in the early seventies and coined the term ‘fractal’ to describe the geometry behind it. The power and the beauty of fractals were only capable of being seen with the advent of computers, which become psychedelic windows on the infinite when using simple fractal equations.

In 1992 Nigel Lesmoir-Gordon made the TV documentary, The Colors of Infinity about the Mandelbrot Set and fractals, which has since been seen right round the world. Nigel’s enthusiasm brought together a dream team of contributors for the film who all now contribute to the book tracking how fractals have developed since the film was made. Sir Arthur C Clarke presented the film and in the book gives a lucidly simple account of the mathematics of the M-Set. Benoit Mandelbrot, the Belgian mathematician explains how it began. Professor Michael Barnsley, the computer graphics researcher who developed fractal image compression technology, explains the applications of the breakthroughs. Professor Ian Stewart, author of Does God Play Dice? adds his insights into the beautifully simple equation that gives birth to fractals.

Two of the most interesting applications of fractal geometry, reflected by the two new contributors to the book, are to the Internet and to the Stock Market. Dr Gary Flake, Chief Technology Officer at Overture, the leading provider of commercial search on the Internet and just taken over by Yahoo for 1.6 billion dollars, discusses the profoundly fractal nature of the Web in his article: The Self-ish Web. Robert Prechter Jr is President of Elliott Wave International, Inc. and founder of the Socionomics Institute. His latest title is Socionomics: The Science of History and Social Prediction (2003). He writes about how fractals can help us understand the oscillations of stock markets.

In the back of the book is a DVD of the original documentary with soundtrack by David Gilmour of Pink Floyd PLUS a 30-minute fractal animation to the music of members of Quintessence. ... Read more

Book Description

You, too, can understand geometry—— just ask Dr. Math ® !

Are things starting to get tougher in geometry class? Don't panic. Dr. Math—the popular online math resource—is here to help you figure out even the trickiest of your geometry problems.

Students just like you have been turning to Dr. Math for years asking questions about math problems, and the math doctors at The Math Forum have helped them find the answers with lots of clear explanations and helpful hints. Now, with Dr. Math Presents More Geometry, you'll learn just what it takes to succeed in this subject. You'll find the answers to dozens of real questions from students in a typical geometry class. You'll also find plenty of hints and shortcuts for using coordinate geometry, finding angle relationships, and working with circles. Pretty soon, everything from the Pythagorean theorem to logic and proofs will make more sense. Plus, you'll get plenty of tips for working with all kinds of real-life problems.

You won't find a better explanation of high school geometry anywhere! ... Read more

Book Description

Reviews (9)

Agree -- one of the best; elegant; beautiful
I had been seeking a book on differential geometry for self-study, as a preface to learning general relativity. A seasoned mathematics friend recommended Kreyszig.

So, I waded in, and patiently made my way through every page of the first six chapters, working the problems along the way, at a pace of a few pages per day. Now that the journey is behind me, I can say that I appreciated this book. It compares favorably to some other texts I had tried reading, with less success.

I realize that the author's approach is an old-style classical one, with a reliance on specific coordinate systems and transformations between coordinate systems. To work the problems requires a fair amount of paper and pencil work. Nonetheless, this approach worked well for me. On those occasions when my reading bogged down, inevitably there was a good reason. If I went back carefully, re-read and pondered, doodled on paper, and tried to visualize what Kreyszig was describing, it always worked! The light would soon go on, usually with a pleasurable sense of discovery.

I went back to re-read certain sections of the book to refresh my memory, and realized how elegant the writing is. Crystal clear, right to the heart, and always trustworthy. Everything follows in a gentle persuasive way; there are no jarring leaps or gaps.

Additionally, I had a nice sense of the different flavor brought to the field by the French geometers who made many of the key advances around the turn of the 19th-20th century.

Finally, the summary of key results and equations at the end is very smart and helpful.

Since finishing Kreysig, I did find it helpful to push on and try to grasp these same ideas from the standpoint of one-forms and the coordinate-free approach to tensors. But I'm not sorry I came at the subject this way first.

I do recommend this book, and think that a beginner needs only a moderate amount of stamina and patience here.

A postscript -- the book is also beautiful. I like that in a math book.

One of the best
Good, thorough, self-contained. Spend a little more time on the first three chapters, and the understanding will follow. What can I say? Just rush out and get it!

highly recommended
This is a wonderfully well written book. If you have a good background in calculus and analytic geometry, you will have no problems with understanding most of the book. (If you don't, you shouldn't be studying differential geometry anyway.) The last couple of chapters are more difficult. Make sure to do the problems after each chapter; they are very well designed to enhance your understanding, and as a huge bonus, their solutions can be found at the end of the book. Forget about those books with a fancy hard cover and cost ten times as much. Buy this book and enjoy!

Not for beginer
Not suitable for beginer if you try to study yourself, esecially from chapter IV on the prove is not quite straightforward.

By far the best intro to classical differential geometry
This is the sort of math book that you pick up, get something to drink, sit on the couch and read through as you would read a novel. I dont know if its possible to write a simpler or clearer treatment on differential geomerty. But be warned that it is still only "classical". Tensros are treated as objects that tranform in a certain way, rather than studied as general multilinear functions. However, after reading this book, any book on tensors is a breeeze to go through. Well worth having, especially considering the price. ... Read more

Book Description

This well-accepted introduction to computational geometry is a textbook for high-level undergraduate and low-level graduate courses. The focus is on algorithms and hence the book is well suited for students in computer science and engineering. Motivation is provided from the application areas: all solutions and techniques from computational geometry are related to particular applications in robotics, graphics, CAD/CAM, and geographic information systems. For students this motivation will be especially welcome. Modern insights in computational geometry are used to provide solutions that are both efficient and easy to understand and implement. All the basic techniques and topics from computational geometry, as well as several more advanced topics, are covered. The book is largely self-contained and can be used for self-study by anyone with a basic background in algorithms. In the second edition, besides revisions to the first edition, a number of new exercises have been added. ... Read more

Reviews (11)

The best computational geometry book!
I also completely disagree with the one-star review below. The "Dutch book" is the clearest, most complete, most up-to-date, best designed, best illustrated computational geometry textbook out there. Some of the material may be a bit advanced for undergraduates (and for those people I would recommend Joe O'Rourke's excellent "Computational Geometry in C"), but for graduate students and other researchers who want to learn computational geometry, this book is absolutely essential.

This is an algorithms textbook, though, not a textbook full of code. You will not find compilable code in the author's favorite programming language du jour -- this may be what the first reviewer meant by "desperately needed details". What you will find is clear, correct, well-motivated explanations of the underlying algorithms, data structures, and mathematics.

The book does have a few faults. The motivating examples are often forced ("mixing things" for convex hulls??). The authors deliberately chose to show only one algorithm for each problem they consider, and occasionally the algorithm they chose is not the simplest or most efficient. But these are minor points.

If you're going to buy just one computational geometry book, this is the one to get.

Interesting read, excellent theory, no code
This book serves as a survey of computational geometry algorithms. The explanations are very readable. The authors have taken special care to prove algorithm correctness and time complexity bounds.

Although I have yet to actually implement one of the algorithms in the book directly, I was exposed to a number of general techniques which I have used, such as randomized techniques to eliminate pathological worst-case performance problems, and various space partitioning techniques.

The algorithms are all presented in pseudocode, unfortunately, which is the reason for only 4 out of 5 stars. Also, some important details are omitted which make a few of their algorithms practically useless (although they are interesting theoritically). For example, there is an algorithm for pathfinding and collision avoidance for a translating (but not ROTATING!) robot.

If you're lookin for a computational geometry bible, this isn't it. But there are certainly some gems in this book and it is a very interesting read.

Good Introduction but look elsewhere for detailed reference
Pro:
(1) Each chapter begins with a practical example. For example, the chapter computing intersections of lines starts with a discussion of a map-making application that goes into enough detail to see how the algorithms they present would be useful. This is a considerable step up from the common practice in algorithms literature of motivation by way of vaguely mentioning some related field (i.e. "These string matching algorithms are useful in computational biology"). This book does a much better job of motivating the material it presents, but if you're primarily interested in the abstract problem, these sections can be skipped.

(2) Each chapter is relatively self-contained. Feel free to skip ahead to subjects that interest you.

(3) Surprisingly readable. Unlike most technical material, one can read an entire chapter in a single sitting without missing much. Generally, each chapter will develop a single algorithm for a single kind of problem.

(4) It's very up to date. This second edition is less than two years old, it includes some new results in the field.

Con:
(1) Algorithms are only given in pseudocode. The emphasis is on describing algorithms and data structures clearly and completely. If you're looking for a "cookbook" with code to copy and paste into an application, perhaps O'Rourke's "Computational Geometry in C" would be a better choice.

(2) There are many important advanced results that are not discussed in the main text. An obvious example is the first chapter, which describes a well-known convex hull algorithm that takes O(n log n) time but algorithms that are faster for most inputs are mentioned only in the "Notes and Comments" at the end of the chapter. Someone interested in lots of gory details would be well-served to combine this book with Boissonnat and Yvinec's more detailed and mathematical "Algorithmic Geometry".

Extremely well written
Algorithm books are often quite hard to understand, but this is not the case with this book. The information is very compact so it is a slow read but due to the high quality of the text this is only an advantage. You are never left wondering what the authors might have meant with a certain statement.

The book focuses solely on theory, so it presents no real source code (only pseudo-code) which I think is good thing since that would otherwise have polluted the clarity of the explanations.

Many of the topics it covers has been a help to me as a programmer. Can be recommended for anyone interested in computation geometry - but it requires some computer science maturity so I don't recommend it unless you have a bachelor's degree in C.S. or something similar.

Jacob Marner, M.Sc.

Clear and concise
The book is well written and easy to understand. An ideal book for someone planning to apply computation geometry for real-life problems. This is not a definitive book for computational geometry, but does give you good examples and ideas. Could do with more references to figures. There is scope for expansion of this book to include more detailed case studies and more pseudo code examples ... Read more

Book Description

Exposition of fourth dimension, concepts of relativity as Flatland characters continue adventures. Popular, easily followed yet accurate, profound. Topics include curved space time as a higher dimension, special relativity and shape of space-time. Accessible to layman but also of interest to specialist. 141 illustrations.
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Reviews (12)

Amazing concepts of space and time !!!
An excellent introduction to concepts of space and time in modern physics, including non-Euclidean geometry - the geometry of the curved spaces. Minimal background in mathematics is requested and multiple diagrams help a better understanding of the most difficult passages. The book is so interesting that I finished it in 5 days !!!

Solid Intro to Special Relativity and Non-Euclidean Geometry
In his own introduction the author, Mr. R. Rucker, states, "My goal has been to present an intuitive picture of the curved space-time we call home. There are a number of excellent introductions to the separate topics treated here, but there has been no prior weaving of them into a sustained visual account. I looked for a book like this for many years- and finding none, I wrote it." His dedication has been rewarded, as the text is one of the finer introductory books on the curvature of space time and special relativity.

The 'book like this' as the author calls it, walks the reader through several visual explanations that allow a solid mathematical and graphical explanation of modern physics. This isn't always a simple explanation, but there is a certain reward to struggling with the concepts before understanding them. In particular, Chapter 4 on time as a higher dimension makes the entire book worth reading, with many fascinating examples and a host of thought-provoking examples, such as "Schrodinger's Cat."

This is a very interesting book which would be of use to anyone who wishes to push just a little bit further than the typical popular physics text. For those who wish to push even further to solidify their knowledge, there are even questions at the end of each chapter. I highly recommend this book.

an extra dimension
This book is mainly concerned with exactly what the title says and I have been searching for a book like this for a very long time. because for one, it provides a very detailed explanation of topics that are intersting in the realm of physics. Such as the fourth dimension. it is very visual and explains things in a way that I can understand. I also like this book because it doesn't spend half the book telling you about which scientist hated the other scientist, Or the entire biography of Dr. Planck before they tell me what the planck length is

Good but Missing a Few Things
I haven't completely read this book, but I've read several like it. I want to point out some things that other reviewers haven't touched on. There is no index to the Dover edition. Maybe the original one had an index. That automatically knocks off one star in any book rating I give. It has some pretty sturdy exercises at the end of each chapter. There are no answers in the book. That's OK though. One can get some additional sense of the subject by looking at the questions. There is a very good annotated bibliography at the end of the book. It is not tied into page numbers, but I get the feeling the order of the list and their reference in the book are in the same order. There's good and bad news about the list. He makes many of these books sound very appealing, but many are long out of print. Rucker's book was produced around 1975.

There are times when I wish the author would have pressed a little harder one some seemingly simple points. Maybe by giving an alternative view. For example, early on in the book he talks about a flatlander being inside a balloon as he expands the balloon from the inside. Suddenly the flatlander is on the outside. Maybe it's me, but how that happens is not clear. I've found other such passages. However, a studious reader will find the topics interesting. The price is certainly right.

Weird in all the right ways
I really enjoy Rudy Rucker's nonfiction, and some of his fiction too (_White Light_ is great). He's very good at presenting mind-blowingly cool ideas in accessible expository prose, and he knows _just_ when to throw in the bombs.

This particular book is published by Dover, and it's not one of their usual reprints; it was _originally_ published by Dover. (In 1977, but the geometry of spacetime hasn't changed much since then.) It's an exploration of just what the title says: the geometry of the four-dimensional spacetime that the theory of relativity says is Really Out There.

Well, this is a good book on the subject, but you can get others (although one of the best -- Cornelius Lanczos's delightful _Space Through the Ages_ -- has long been out of print). What's coolest about this one is that Rudy Rucker wrote it.

Which means you get those little bombs thrown in at all the right places. Of course Rucker gives you what any competent mathematician will give you -- a sound introductory presentation of the mathematics of 4D spacetime and relativity theory, which are weird enough if you haven't encountered them before (and maybe even if you have) -- but he doesn't stop there. You also get an argument that the apparent passage of time is an illusion, and a little speculation about how this might tie in with the Many-Worlds Interpretation of quantum mechanics. And even that isn't all: you get a suggestion that it's possible to _develop a spacetime consciousness_ via some sort of meditation techniques or mystical insight, together with an entry in the annotated bibliography referring you (cautiously) to Robert A. Monroe's _Journeys Out of the Body_, whose experiments Rucker himself has tried.

It's like Raymond Smullyan on acid, if you know what I mean. But honest, it really does make sense. And it really will knock your mind loose from your brain even without the use of chemical aids.

This is the sort of thing Rucker does best. He does it in _Infinity and the Mind_, too (with which this volume has a little bit of overlap, but you won't care). Check out that book as well, along with _White Light_. Mathematical hippie mysticism just doesn't get any better. ... Read more

Reviews (24)

good resource for geometry teachers
This book is appropriate for highly motivated middle school students whe have studied algebra in the seventh grade. The text uses a guided exploration approach to discovering the facts of geometry. For students who take any pleasure in math, this book can be fun. It is attuned to the 13 year old mind. Proof is introduced in a systematic way in the last two chapters.

The teacher must compensate for the fact that the book is not self-contained and is not useful as a reference (no glossary for example). Students must develop a notebook to organize what they learn and for future reference. Students who are left on their own to "construct" their knowledge of geometry through group activities and reflection could find it very tedious. The teacher has to be both a "sage on the stage" as well as a "guide on the side".

The book would not be appropriate at the high school level unless aggressively supplemented with a systematic treatment of Euclidean synthetic geometry. Some high school students would find its comic book style childish and unappealing.

I would recommend the book to math teachers and those studying to teach math as a rich source of ideas, activities and problems.

I HATE THIS BOOK!!!!
OMG! This book was so complicated to understand. They expect you to know everything. Because of this mindless crap, I failed Geometry and had to take the regular one. And the worst part is that they don't include enough examples to explain how to work the problem. I wouldn't recommend this God forsaken piece of crap to anyone who wants to do good in Geometry.

A terrible textbook
Whoever wrote this book has a great idea in having students actually working things out to find solutions, but to expect students to discover in an hour what geniuses discovered after many years is unrealistic. There are little to no examples, no glossary, and the hints are useless in helping the student understand the concept. All in all, a very poor book.

This Book SUCKS!
I don't know who would waste their money on such a horrilbe text book. I'm currently using this book right now for Geometry and I never experince such a bad textbook. This book is hard to understand and it's also usless to use this book if you are a "learn it yourself type of person." I rate this book an "F" and I wouldn't recommended this textbook to a genius, that's how horrible it is!

Terrible text
I have taught HS and college level Math for many years. I have never encountered a less user-friendly text. It is only with complete arrogance that an instructor or district would adopt this text. No examples to speak of, no odd-numbered answers in the back, no glossary (!), hints that don't help. How would anyone say that this is a good way to teach? Instead it sets up students for failure. If you are looking for a supplemental resource instead of a text, keep looking. There is no value to this text as a supplement since there is nothing that is self-explanatory. ... Read more

Book Description

Recent advances in computing and algorithms make it easier to do many classical problems in algebra. Suitable for graduate students, this book brings advanced algebra to life with many examples. The first three chapters provide an introduction to commutative algebra and connections to geometry. The remainder of the book focuses on three active areas of contemporary algebra: homological algebra; algebraic combinatorics and algebraic topology; and algebraic geometry. ... Read more

Book Description

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications through the study of interesting examples and the development of computational tools. Coverage ranges from analytic to geometric. Treats basic techniques and results of complex manifold theory, focusing on results applicable to projective varieties, and includes discussion of the theory of Riemann surfaces and algebraic curves, algebraic surfaces and the quadric line complex as well as special topics in complex manifolds. ... Read more

Reviews (7)

A review from a graduate student
If you are a graduate student in mathematics or related fields and you are interested in learning algebraic geometry in the Griffiths-Harris way, then I suggest before buying this book to have a good background in the following:

1. Complex Analysis
2. Differential Geometry and calculus on manifolds
3. Homology-Cohomology Theory
4. Undergraduate Algebraic Geometry

Do not expect chapter 0, "Foundational Material", to be the place where you are supposed to build your "foundation". You can try the books of Michael Spivak, David A. Cox, Fangyang Zheng, among other books for foundational material but not chapter 0.

However, if you have most of the above-mentioned foundational material, then this book is good in presenting complex manifolds for example in chapter 0 section 2 and also in presenting (complex) holomorphic vector bundles, as well as many other things.

So, in summary, I would say a good book but not for students trying to learn the basics in algebraic geometry.

algebraic geometry: the real stuff
The book is beautifully written and easy to read, with emphasis on geometric picture instead of abstract nonsense. By far the best introduction to algebraic geometry for string theorists.

Work of Art
This is an amazing book with an amazing subject (complex algebraic geometry). Every section presents something interesting and wonderful. I've only read chapters 0 (Complex manifolds, Hodge theory), 1 (Divisors & line bundles, vanishing theorems, embeddings), and 2 (Riemann surfaces). I had had a bad experience with alg geom before this book. Required reading for mathematicians in complex manifolds, algebraic geometry, or string theorists. There are some very trivial typos scattered, but nothing problematic in the least (like capital lambda instead of a big wedge, or indices). If you read the book carefully you will get a lot out of it.

Absolutely indispensable
This book is fabulous - it is an indispensable reference for complex algebraic geometry. It is very clearly written and ideas are always motivated by examples and problems. Moreover, if you want to learn modern algebraic geometry, it's imperative to learn the classical case (over the complexes - which in practice is easier to work in) in order to understand the generalisations a la Grothendieck.

Will never collect dust
Once thought to be highly esoteric and useless by those interested in applications, algebraic geometry has literally taken the world by storm. Indeed, coding theory, cryptography, steganography, computer graphics, control theory, and artificial intelligence are just a few of the areas that are now making heavy use of algebraic geometry.

This book would probably be one the most useful one for those interested in applications, for it is an overview of algebraic geometry from the complex analytic point of view, and complex analysis is a subject that most engineers and scientists have had to learn at some point in their careers. But one must not think that this book is entirely concrete in its content. There are many places where the authors discuss concepts that are very abstract, particularly the discussion of sheaf theory, and this might make its reading difficult. The complex analytic point of view however is the best way of learning the material from a practical point of view, and mastery of this book will pave the way for indulging oneself in its many applications.

Algebraic geometry is an exciting subject, but one must master some background material before beginning a study of it. This is done in the initial part of the book (Part 0), wherein the reader will find an overview of harmonic analysis (potential theory) and Kahler geometry in the context of compact complex manifolds. Readers first encountering Kahler geometry should just view it as a generalization of Euclidean geometry in a complex setting. Indeed, the so-called Kahler condition is nothing other than an approximation of the Euclidean metric to order 2 at each point.

The authors choose to introduce algebraic varieties in a projective space setting in chapter 1, i.e. they are the set of complex zeros of homogeneous polynomials in projective space. The absence of a global holomorphic function for a compact complex manifold motivates a study of meromorphic functions and divisors. Divisors are introduced as formal sums of irreducible analytic hypersurfaces, but they are related to the defining functions for these hypersurfaces also, via the poles and zeros of meromorphic functions. For the mathematical purist, a "sheafified" version of divisors is also outlined. Divisors and line bundles are basically "linear" tools used to investigate complex varieties through their representation as complex submanifolds of projective space. In addition, various approaches are used to study codimension-one subvarieties, such as the results of Kodaira and Spencer. Although the famous Kodaira vanishing theorem is clothed in the language of Cech cohomology, this cohomology is represented by harmonic forms, thus making its understanding more accessible. The authors also show explicitly to what extent an algebraic variety can be thought of as a compact complex manifold via the Kodaira embedding theorem. Projective space of course is not the most complicated of constructions, as readers familiar with the theory of vector bundles will know. Grassmannians are an example of this, and they are introduced and discussed in the book as generalizations of projective space. And, just as in the ordinary theory of vector bundles, the authors show how to use Grassmannians to act as universal bundles for holomorphic vector bundles.

The presence of meromorphic functions will alert the astute reader as to the role of Riemann surfaces in the study of complex algebraic varieties. Indeed, in chapter 2, the authors cast many classical complex analytic results to modern ones, and they prove the famous Riemann-Roch theorem, which essentially counts the number of meromorphic functions on a Riemann surface of genus g. The theory of Abelian varieties is outlined, and the reader gets a taste of "Italian" algebraic geometry but done in the rigorous setting of Plucker formulas and coordinates.

Chapter 3 is a summary of some of the other methodologies and techniques used to study general analytic varieties, the first of these being the theory of currents, i.e differential forms with distribution coefficients. It is perhaps not surprising to see this applied here, given that it can handle both the smooth and piecewise smooth chains simultaneously. The currents are associated to analytic varieties and allow a definition of their intersection numbers and a proof that they are positive. The all-important Chern classes are introduced here, and it is shown that the Chern classes of a holomorphic vector bundle over an algebraic variety are fundamental classes of algebraic cycles. Most importantly the authors introduce spectral sequences, a topic that is usually formidable for newcomers to algebraic geometry.

The study of surfaces is studied in chapter 4, with the differences between its study and the theory of curves (Riemann surfaces) emphasized. The reader gets a first crack at the notion of a rational map, and the birational classification of surfaces is shown. Intuitively, one expects that the classification of surfaces would be easy if it were not for "singular points", and this is born out in the use of blowing up singularities in this chapter. Rational surfaces are characterized using Noether's lemma, and a rather detailed discussion is given of surfaces that are not rational, giving the reader more examples of rigorous "Italian" geometry. ... Read more

Book Description

Each chapter devoted to single type of problem with accompanying commentary and set of practice problems. Amateur puzzlists, students of mathematics and geometry will enjoy this rare opportunity to match wits with civilization’s great mathematicians and witness the invention of modern mathematics.
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Reviews (3)

Very Good for the interested reader
Although not reasonable for most high school students, this book does bring to light the background of modern geometry through its treatment of impossible and hard to solve problems.

Excellent background reading for a teacher!

Classic problems of geometry made simple
The principles of geometry are both elegant and timeless. One cannot help but understand why the Greeks considered it a pinnacle of intellectual achievement. This book sets down several of the classic problems, explaining them in such a clear way that it is easy to forget that it sometimes took centuries of work before the problem was resolved. Of course, those who first toiled on the problems did not have some of the additional machinery that we so take for granted today.
The problems covered are trisecting an angle, squaring the circle, constructing regular polygons and constructing a cube whose volume is twice that of a given cube. The background needed to understand the problems and solutions is nothing more than a solid grounding in basic algebra and trigonometry. Calculus is mentioned, but not used. Problem sets are included in all chapters and solutions to all are in the back of the book. They are well-posed, solidly reinforcing the points made in the text.
This is a book that covers several thousand years of progress in geometry in a little over one hundred pages. It is done well and it can be used as a supplement in any course in geometry, from high school to college.

Well done..
Bold has a gem of a book here. It's only a little bit over a hundred pages, but it's packed full of the great geometry problems that occupied the minds of the world's greatest thinkers for the past 2000 years.

The title describes the book perfectly. These really are "Famous Problems from Geometry" and he does indeed explain how to solve them.

The book has four major sections/chapters. He discusses in detail the three problems from antiquity (one section each): squaring a circle, doubling a cube, and trisecting an angle. Furthermore, he spends significant time with constructions of regular polygons (the fourth section) - which ones can be constructed and why. He also discusses which ones cannot be constructed and why.

The reader will be expected to understand concepts from Modern Algebra, particularly the concept of a Field. While Bold does spend time explaining what a Field is, his definition is quick and is assumed to be more of a refresher for someone who has already learned about them. Bold also has a section on Complex Numbers where he derives one of the formulas used later in the book. Again - this section is assumed to be a refresher on Complex Numbers. High School Geometry or Algebra students would have significant trouble understanding his explanations and proofs.

Bold provides problems for the reader to work along the way. These are problems that logically lead to the proof of the problem being studied. The problems are good. As a third year college student majoring in mathematics, I found the explanations/solutions to be sometimes