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Physics and computer science genius Stephen Wolfram, whose Mathematica computer language launched a multimillion-dollar company, now sets his sights on a more daunting goal: understanding the universe. Wolfram lets the world see his work in A New Kind of Science, a gorgeous, 1,280-page tome more than a decade in the making. With patience, insight, and self-confidence to spare, Wolfram outlines a fundamental new way of modeling complex systems.

On the frontier of complexity science since he was a boy, Wolfram is achampion of cellular automata--256 "programs" governed by simplenonmathematical rules. He points out that even the most complexequations fail to accurately model biological systems, but the simplestcellular automata can produce results straight out of nature--treebranches, stream eddies, and leopard spots, for instance. The graphicsin A New Kind of Science show striking resemblance to thepatterns we see in nature every day.

Wolfram wrote the book in a distinct style meant to make it easy to read, even for nontechies; a basic familiarity with logic is helpful butnot essential. Readers will find themselves swept away by the elegantsimplicity of Wolfram's ideas and the accidental artistry of thecellular automaton models. Whether or not Wolfram's revolutionultimately gives us the keys to the universe, his new science isabsolutely awe-inspiring. --Therese Littleton ... Read more

Reviews (314)

Mind candy
A New Kind Of Science may come to be viewed as the Godel, Escher, Bach of our generation. It's full of challenging big ideas that touch on nearly every field of science and beyond. It's a brilliant and delightful read and makes wonderful mind candy.

The only problem is I don't believe any of it. Wolfram bases the entire opus on the complicated behavior of a few simple cellular automata (CAs). Curiously, he never discusses any of the cool things that originally got a lot of people so excited about CAs -- topics like adaptation on the edge of chaos, and genetic algorithm evolution of specific functions. Instead, the entire book is just about how it's sometimes possible to observe complex and unpredictable patterns. And he tries over and over to convince the reader of just how important that observation is for understanding the universe.

As a supposed harbinger of a major paradigm revolution, we can contrast it with Einstein's one-time dramatic new theory of the universe. While a lot of people didn't understand it, the theories of relativity gave quite a few very specific predictions that could be -- and were successfully -- tested by observation and experiment. I've now read through the entirety of A New Kind Of Science and I can't find any specific predictions that would show his worldview explains reality any better than conventional ideas.

The only prediction he gives us relating to his theories is that every field of science will ultimately be transformed by them, and he goes on to list many of those fields. As I have a doctorate in molecular evolution, I was particularly interested in his dismissal of Darwin's theory of evolution by natural selection -- one of the most firmly established theories in science. Wolfram claims that Darwinian evolution is not sufficient to produce complex adaptations. I'm loathe to criticize an intellectual of Wolfram's stature, but his understanding of evolutionary theory, at least insofar as is presented in this book, is not very sophisticated. At any rate, anyone wanting an authoritative explication of the power of natural selection to generate complex adaptations may refer to Richard Dawkins' The Blind Watchmaker. I wish Wolfram offered some sort of testable alternative, or evidence of any kind beyond an endless display of pictures of the output of his simple programs. While the output may match the complexity observed in nature, Wolfram never makes the case that they match the adaptivity or intelligence observed in nature.

Many of these pictures are indeed very pretty. But by the fourth or fifth hundred page his obsession with these automata becomes a bit tedious. And the outworldly conclusions he draws from observing their behavior will leave you bumfuzzled. For example: because his automata are discrete in space and in time he proposes (with no further justification) that the entire universe must be made up of discrete cells of space and time. Sounds great, but where's the evidence, and where are the testable hypotheses? He goes on to propose, again with no evidence other than the observed behavior of a select few of his automata, that the mysterious rules of the universe update only one discrete time cell at any given instant. Wolfram offers countless other extrapolations to the mechanisms of nature and structure of the universe, all similarly astounding and similarly unsupported.

As I read through this opus, and especially as I neared the end, I kept asking myself -- How is it possible for someone so brilliant to have spent so many years developing something so uncompelling? I came up with three possible explanations:

1) Wolfram has gone off the deep end. Just like Dr. Richard Daystrom of Star Trek's "The Ultimate Computer", the undisputed genius who goes mad trying to exceed his former glory. Perhaps Wolfram has been staring at his pretty pictures for so long his synapses can no longer make any other kind of connection.

2) Wolfram is perpetrating an elaborate hoax on the world, much like Dr. Alan Sokal's famous "Transgressing the Boundaries" paper, a parody of the academic humanities that the editors of Social Text were fooled into publishing. But Wolfram's physics flimflam is writ on an infinitely larger scale. Just to prove he's so much smarter than every one else, and just as a practical joke, he's trying to derail the entire scientific enterprise.

And finally,

3) I have become so entrenched in the practice and paradigms of traditional science that I am unable to grasp or appreciate the profundity of what's been laid before me in the simplest of terms.

Number three is always possible. And in fact it would be wonderful to bear witness to what he's calling the greatest discovery in the history of science, even if it does fly over my head at Mach 2. Wolfram is one of the smartest and most accomplished residents of the universe, and even though one of the basic tenets of the (traditional) scientific method is that the validity of a claim is judged independently of the stature and reputation of the one who proposes it, it's difficult not to give someone like Wolfram the benefit of the doubt -- no matter how much of a stretch.

All the same, I recommend this book to anyone who enjoys being intellectually stimulated and likes to think about big ideas. Even if he's wrong, I'm sure glad I read it.

Wolfram's "New" Science Simply Doesn't Work
As everyone who can read English and has been awake for the past six months knows, Stephen Wolfram has written how certain cellular automata have the "universal" property of being able to perform any calculation that can be performed on a computer or anything else. Wolfram generates thousands of lovely pictures that, he claims, are similar to those observed in many physical and biological systems. All these pictures are generated by simple rules and sometimes simple initial conditions. Yet, some show surprisingly complex and seemingly random behaviour.

So far so good. Wolfram's next contention is that the complexity found in what he calls Class 4 cellular automata cannot be exceeded by any physical, biological or computational process. Put more boldly, every physical, biological, psychological, financial, meteorolical and, no doubt, astrological feature of the universe that exhibits complexity is generated by some sort of cellular automaton with appropriate initial conditions.

Such a statement cannot, of course, be proved in any acceptable way. To compensate, Wolfram gives us many examples of phenomena whose random behaviour resembles those of cellular automata. He is most convincing with his pictures of real seashells and arguments about turbulence in fluids (I especially liked his wafting smoke in the air anology.) He is less persuasive when he argues that evolution has nothing to do with maximizing anything and everything to do with generated patterns, some of which survive. When he talks about the analogy between Class 4 cellular automata and human cognition, he is downright silly.

Yet this is all irrelevant. Wolfram is scathing in the inability of mathematics to solve anything but the simplest physical problems. Thus Newton could tell us how to calculate the orbit of a planet around a star but neither he nor any of his successors could come up with a reasonable mathematical model for turbulence. And no one has even attempted a mathematical model of evolution.

But describing the disease is easier than prescribing a cure. Suppose that Wolfram is correct and that every meaningful physical and biological process is generated by an ongoing cellular automoton--or something equivalent. Then we could understand how we got where we are and predict where we will go. All we need is to discover the underlying rules and initial conditions for each system we wish to model. But therein lies the rub.

Wolfram argues persuasively that the systems generated by Class 4 cellular automata are irreducible. This means that there is no shorthand method for calculating future behaviour. The only thing we can do is go through the iteration millions, billions, gazillions of times and observe the outcomes at each step. Since the behaviour is random, knowing where you are at any step doesn't help you to predict where you will be at a future step.

The inverse problem is far more intractable. It is practically impossible to determine the underlying rules and initial conditions of a cellular automoton by looking at the deterministic pattern that it generated--especially if the pattern is complex and random (the only case of interest). But that's the whole point. Even if we knew with certainty that some complex process was generated by a cellular automoton with simple rules, it would still be impossible to describe its past behaviour or predict its future because we could never find the rule and starting conditions.

So, at its most profound level, even if Wolfram's new science is correct, it fails at doing two of the most fundamental things that science is supposed to do: telling us how we got where we are and making predictions about future behaviour.

In the final analysis, Wolfram's book is brilliant and well worth reading. But its new ideas may prove to be as useful as those in astrology.

Wolfram is not as revolutionary as ME
Stephen Wolfram is making a lot of noise about his "new kind of science," and the revolutionary impact it will have. But Wolfram is wrong, because his theories are not as revolutionary as my own. If you really want to know about the "new kind of science" you will find it in my work, not his.

Let me explain why I can so confidently make this statement. First, the issue of computers. While Stephen is right that computers will play an important role in the new kind of science, he failed to realize that his computer is not as good as mine. He has a pathetic, outdated model manufactured more than two months ago, while I buy a new computer each time a new breakthrough is made in micrprocessor speed and memory. Thus, my computer models are better than Wolfram's.

Also, Wolfram does not and cannot account for the amazing scientific discoveries made by ME using MY computers. I am a recognized pioneer in the field of quantum mitosis, and my studies of the statistical entropic confabulations of subcognitive querktons have become benchmarks in modern science. But I don't remember Wolfram ever calling ME to discuss these important breakthroughs.

Looks like it's back to the drawing board, Steve. Your supposedly all-encompassing new science may draw admiring glances for a week or two, but MY ideas are better. Right now, I'm even working on a grand unified theory of cognitive assonance which will allow me to objectively determine the degree to which Wolfram's ideas are inferior to my own, and finally get the crab grass out of my lawn. Who's your daddy, Wolfy?

Thought-provoking book
Given that you bothered to read these reviews, you probably should buy the book.

Wolfram's either on to something or he's not. I don't know and don't care. His writing style is tedious and annoyingly arrogant, but again, who cares?

His assertions about science are bold and unconventional. If he's right about something, maybe there will be some benefit to mankind (or even better, to me ;-) ) If not, so I wasted a few bucks. It's not the first time. Won't be the last....

Bottom line: Thinking Wolfram's ideas through is fun, whether you agree, disagree, or have no opinion. If you disagree, you'll feel good that you're smarter than a "physics and computer science genius". Otherwise, you'll feel that you have thought about some intriguing possibilities.

Single mindedness is bad even if it's a brilliant mind
The lack of proper reference to those minds that have studied and articulated complexity and cellular automata gives the impression that Wolfram solely thought up and now presents the ideas. No great ideas are exclusive. The book appears to be a self generated hoo rah for himself, oh and a little bit of computer generated plot. After lugging this monster around for way too long I finally just dropped it to door stop status and now it serves a much higher purpose than its content deserves. ... Read more

Book Description

Lebesgue Integration on Euclidean Space contains a concrete, intuitive, and patient derivation of Lebesgue measure and integration on Rn. Throughout the text, many exercises are incorporated, enabling students to apply new ideas immediately.Jones strives to present a slow introduction to Lebesgue integration by dealing with n-dimensional spaces from the outset.In addition, the text provides students a through treatment of Fourier analysis, while holistically preparing students to become "workers" in real analysis. ... Read more

Reviews (3)

an excellent introductory text
As someone who wasn't a math major but who has been trying to get up to speed on lebesgue measure and integration, I found this book to be truly accessible. Unlike other "introductory" texts (such as Kopp's "Measure, Integral and Probability") I could follow the reasoning in this book without much difficulty.

The only criticism I have of the book has to do with the first chapter. Its purpose is to provide background mathematical material and given the author's clear ability to explain difficult concepts, I wish that it covered that material in greater detail.

For others who may be looking to build a foundational understanding of this material but who may not be mathematicians, I'd also recommend Pitt's "Measure and Integration for Use" (1985) or his "Integration, Measure and Probability" (1963) (both out of print but fairly easy to find). Those books, along with Jones', are well-used items in my library.

High Praise for Jones
"Lebesgue Integration on Euclidean Space" is a nearly ideal introduction to Lebesgue measure, integration, and differentiation. Though he omits some crucial theory, such as Egorov's Theorem, Jones strengthens his book by offereing as examples subjects that others leave as exercises. The best example of this is his section on L^p spaces for 0 < p < 1.

The book's greatest strength, however, is its readability. Whereas Royden gives no hint as to how much work is needed between steps, Jones highlights important steps in proofs, not just the important proofs. It is this motivated style that makes his book useful.

Jones is so careful in his construction of the theory that differentiation does not appear until Chapter 15, and specific results for R^1 come only in Chapter 16. But the wait is worth it.

While Jones has written a great introduction, the book cannot be used for more advanced courses. As the title suggests, the discussion is restricted to Euclidean spaces. In addition, his direct jump to measure on R^n and the use of "special rectangles" therein make the development incongruous with other books. But what is sacrificed in depth is made up for in breadth, with Jones hinting at how the theory is used in other branches of math. There's even an entire chapter devoted to the Gamma function!

As a student, I have found Jones's book more instructive on basic theory than Royden, Rudin, and Wheeden & Zygmund. I highly recommend it as a first-semester introduction to Lebesgue theory or as a source of clean, fundamental presentations of proofs.

treasure trove of mathematical technique
This book is a treasure trove of mathematical technique. It covers topics that are relevant to many broad areas of real and functional analysis including signal processing and approximation theory. The author takes the time not only to prove the results, but also to construct the proofs so that the technique is made explicit to the reader. The author also motivates definitions by breaking them into the successively more complicated pieces so as to build intuition in the reader.

I especially recommend this book to anyone who lacks formal training in mathematics or wishes to develop mathematical technique in the areas of real and functional analysis. ... Read more

Book Description

Of the many different approaches to solving partial differential equations numerically, this book studies difference methods. Written for the beginning graduate student in applied mathematics and engineering, this text offers a means of coming out of a course with a large number of methods that provide both theoretical knowledge and numerical experience. The reader will learn that numerical experimentation is a part of the subject of numerical solution of partial differential equations, and will be shown some uses and taught some techniques of numerical experimentation.

Prerequisites suggested for using this book in a course might include at least one semester of partial differential equations and some programming capability. The author stresses the use of technology throughout the text allowing the student to utilize it as much as possible. The use of graphics for both illustration and analysis is emphasized, and algebraic manipulators are used when convenient.

This is the first volume of a two-part book. The second part is entitled Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations. ... Read more

Reviews (2)

Good, practical book for FDM applied to PDE
This is a book that approximates the solution of parabolic, first order hyperbolic and systems of partial differential equations using standard finite difference schemes (FDM). The theory and practice of FDM is discussed in detail and numerous practical examples (heat equation, convection-diffusion) in one and two space variables are given. In particular, Alternating Direction Implicit (ADI) methods are the standard means of solving PDE in 2 and 3 dimensions.
In almost all cases model problems are taken in order to show how the schemes work for initial value problems, initial boundary value problem with Dirichlet and Neumann boundary conditions.
This book is a *must* for those in science, engineering and quantitative financial analysis. It digs into the nitty-gritty of mapping a PDE to a FDM scheme while taking nasty boundary conditions into consideration. The resulting algorithms are documented are are easily programmed in C++ or other language.
The book does not cover topics that are also important: operator splitting (Marchuk/Janenko), non-constant coefficient PDEs, nonlinearities. Finally, the book uses von Neumann analysis as a means of proving stability (getting a bit long in the tooth). There are more robust methods that use monotone schemes, M-matrices and the maximum principle. You should consult other specialised references.
This is Volume I of a two-volume set (Volume II deals with Conversation Laws and first-order hyperbolic as well as Elliptic problems.

(...)

Numerical Partial Differential Equations
Thomas wrote a good book on a quite specialized subject. Although finite difference schemes have been traditionally viewed as a game field for physicists, they are given today much more commercial attention as financial option market evolves. Those who seek standard numerical recipes are advised to read this book. You will enjoy it (easy reading) and learn. But the book may not satisfy quests of a more rigorous readership. It abuses the Fourier method in stability analysis while considering only PDEs with constant coefficients. The bibliographical work has not been done at all. In addition, the cover does not state that this is the first book of two. I'd also advise to read G.Marchuk "Methods of Numerical Mathematics" (Springer, 1982) where a more general approach for stability of numerical schemes is developed. ... Read more

Book Description

This classic of the mathematical literature forms a comprehensive study of the inequalities used throughout mathematics. First published in 1934, it presents clearly and lucidly both the statement and proof of all the standard inequalities of analysis.The authors were well-known for their powers of exposition and made this subject accessible to a wide audience of mathematicians. ... Read more

Reviews (2)

What ?? Only one review (so far) for this classic?
A well written, classic text written by three larger than life math legends (Hardy, Littlewood, Polya). This is the definitive and monumental text on inequality -- the little bent lines you see in all fields of mathematics -- from analysis to number theory to topology to combinatorics (i would say all fields of science but I am not a math apologist, mind you, so I would stay with those things I know something about).

This is one of the most important books of the century.
The authors are among the greatest mathematicians of this century. The contents of this book form the basis for countless applications of mathematics in economics, finance, engineering and physical sciences. ... Read more

Book Description

This book gives an introduction to the discrete wavelet transform and some of its applications. It is based on a novel approach to discrete wavelets called lifting. The first part is a completely elementary introduction to the subject, and the prerequisites for this part are knowledge of basic calculus and linear algebra. The second part requires some knowledge of Fourier series and digital signal analysis. The connections between lifting and filter theory are presented and the wavelet packet transforms are defined. The time-frequency plane is used for interpretation of signals. The problems with finite length signals are treated in detail. MATLAB is used as the computational environment for examples and implementation of transforms. The book is well suited for undergraduate mathematics and electrical engineering students and engineers in industry. ... Read more

Reviews (5)

The non-mathematician-friendly introduction to wavelets
While most wavelets books are terribly difficult from a mathematical viewpoint, this book introduces the reader to wavelets from the practical viewpoint. It introduces the procedure to compute the wavelet coefficients without speaking about mathematical functions. If you don't care about functional spaces, then go to this one. I really think this book is the most friendly introduction to wavelets for people who don't care about maths.

Users of wavelets.
The users of wavelets include engineers who design filters. The wavelet lifting technique is a powerful tool, and extraordinarily practical. The authors present it very nicely, and the book is an attractive source where anyone can pick up the essentials.

Very good book on how to do wavelet based signal processing
This is one of the books that shows you how to implement wavelets, it does so in a good way. Some of the sentences, although, could have been written better. It is not a book for 'dummies'. It does require you to think hard. Lifting is very well explained, as are the implementation issues that prop up when you try to work with finite data sets.

Example codes in Matlab, given at the end helped me to understand a lot. Wonderful book for implementation, provided that you work hard and think hard. This book does require you to be familiar with Z transform to understand how lifting is linked to Filter Banks. So be warned..

The best wavelet book I've found so far
There is a vast mathematics literature on wavelets.
Much of this literature deals with wavelets from
a theoretical point of view. Although books like
Strang and Nguyen's "Wavelets and Filter Banks" are
well written math books, their emphasis is on mathematics,
not application. For example, not until Chapter 8, half
way through the book, do Strang and Nguyen discuss the
mathematics for dealing with a finite, rather than an
infinite data set.

"Ripples in Mathematics" is the seventh book on wavelets that
I've worked with. So far it is the best. The concentration
is on applying wavelet techniques. The book approaches
wavelets through a relatively new technique developed
by Wim Sweldens and others called "the Lifting Scheme". The
lifting scheme provides a structure for wavelets that is
easier to understand. Lifting scheme wavelets also have the
elegant feature that the transform and the inverse transform
are mirrors of each other.

The authors of "Ripples in Mathematics" keep the mathematics
level at a relatively introductory level (e.g., relative
to some of the other wavelet books).

"Ripples in Mathematics" provides the first explaination of
wavelet packets that I have understood. Even better they
discuss the actual implementation of the wavelet packet
algorithm. They also provide a chapter that covers wavelets
applied to finite data sets in a clear non-theoretical fashion
(I found this much more approachable than Strang and Nguyen).

The perfect wavelet book for me has not yet been written,
so I have given this book only four stars. I think of
my perfect book on wavelets as "Wavelets for Dumb Engineers".
This book has been written for Fourier analysis and classical
signal processing (see Richard Lyons' outstanding book
"Understanding Digital Signal Processing").

There is a difference in point of view between mathematicans
and most software and hardware engineers. Our concern is
how the technique can be applied. "Ripples in Mathematics"
provides the necessary material to implement the algorithms,
but you will have to put in some work reading this book and
writing the software (or software models for a hardware
implementation). There are no application hints of the kind
that Lyons provides for applying the Fourier transform. Unlike
Lyons the authors are mathematicians, not practicing digital
signal processing engineers.

Discrete Wavelet Transform for Digital Systems Weenies
If you're interested in the Discrete Wavelet Transform, have I got a book for you! "Ripples In Mathematics" is a complete discussion of the topic. It is chockablock with exercises and example MATLAB code. There are also URLs pointing to keen stuff available for free out in Internetland.

Technical books - particularly math books - usually mean some extra sleepy time for me. Not so with this book. The math is clear and to the point, with examples worked through very nicely. Great illustrations. There's a chapter devoted to different applications. It's just really well put together.

I'm new to wavelet filter design, but this book lays out all the background material I need to do a hardware implementation. ... Read more

Book Description

THIS IS BOTH PROMO COPY AND BACK COVER COPY!!!!! This book provides an introduction to functional analysis and treats in detail its application to boundary-value problems and finite elements. The book is intended for use by senior undergraduate and graduate students in mathematics, the physical sciences and engineering, who may not have been exposed to the conventional prerequisites for a course in functional analysis, such as real analysis. Mature researchers wishing to learn the basic ideas of functional analysis would also find the text useful. The text is distinguished by the fact that abstract concepts are motivated and illustrated wherever possible. Readers of this book can expect to obtain a good grounding in those aspects of functional analysis which are most relevant to a proper understanding and appreciation of the mathematical aspects of boundary-value problems and the finite element method. ... Read more

Reviews (2)

superb introduction to Functional analysis
Daya Reddy has produced a new edition of an older text which now has someextra material and has improved the explanations of the earlier edition aswell as include many useful diagrams.This is an excellent introduction toFunctional Analysis avoiding the often seen theorem/proof approach whichoverwhelms students who are not literate in the mathematics. All conceptsare very well explained and demonstrated, the progression is slow andcareful so as not to disallusion students who are not pure mathematicians.Even for mathematicians this is a good, gentle introduction. When I firstsaw the earlier edition in the library I was captivated by the style andease of development so I emailed Professor Reddy and asked about a newedition, the preprint of which he sent me to peruse, I then later boughtthe new one as it came out. The beauty and power of functional analysis isalso evident

Well worth buying so you don't get lost in the none toosimple mathematics of functional analysis.

The most readablefunctional analysis book for engineers.
This book is essentially the second edition of his now out of print book titled "Functional Analysis and Boundary-Value Problems: An Introductory Treatment". I like this bood very much. It is not verydemanding on mathematical procedures, but lays out the nessary concepts ina very logical progression for Engineers and none math majors, though somemath people may not like it. The examples and graphs are very revealing;they illustrates mathematical concepts very well, makes them easier tounderstand.

I first noticed the power of Functional Analysis when theprofessor proved that you'll never get a converged finite element resultsaround a point load, no matter how you refine the mesh. The odd things wasthat he didn't even mention shape functions and stuff; just used someproperties of spaces and operators. Though not on the same scale, thatreminded me the spirit of Andrew Wiles's proof of Fermat's last theorem.This book easy starters in to appreciate the power and beauty ofmathematics. ... Read more

Book Description

User-friendly--yet rigorous--in approach,this introduction to analysis meets readers where they areby providing extra support for those who like a slower, less detailedapproach, but not getting in the way of those who want a quickerpace and deeper focus. It uses analogy and geometry to motivate andexplain the theory, and precedes many complicated proofs with a"Strategy" which motivates the proof, shows why it was chosen, and whyit should work. Examples follow many theorems, showing why eachhypothesis is needed, allowing readers to remember the hypotheses byrecalling the examples.Proofs are presented in complete detail, witheach step carefully documented, and proofs are linked together in away that teaches readers to think ahead. Physical interpretations areused to examine some concepts from a second or third point of view. Includes over 200 worked examples and over 600 exercises.Provides extensive coverage of multidimensionalanalysis. ... Read more

Reviews (5)

Wonderful book on Analysis
One of the best and clearest book on analysis of all times. Modern introduction. I would like to have a solutions manual so it would be perfect for self study.

The worst book I have never seen before
I have studied this book for a semester. My professor chose this book because he thinks it easy for us to understand analysis. He professed that every student must learn Math step by step and whether the books are good or bad, any student could not understand concepts on Euclidean n space or metric space unless the concepts on R, the set of real numbers, are comprehended first.

However, this book is a counterexample. Chapter 1 to 7 discuss theorems on R including concepts of sequence, continuity, uniform contunuity, integral, convergence, series.... There is nothing about metric space or vector space until chapter 8. How could a book of analysis not mention these spaces in half of its content? I cite my professor's point of view. any student could not understand concepts on Euclidean n space or metric space unless the concepts on R, the set of real numbers, are comprehended first. It is a specious argument. Whether a student can understand or not depends on that the book chosen is easy for the student or not . It does not depend on what is learn first or second. Obviously, this book is difficult to read and understand not only for me but also for the reader who ever read the book, Elementary classical analysis 2nd edition by Jerrold E. Marsden.

I here give an example. In page 191 is theorem 7.15, Weierstrass M test. The proof uses uniform Cauchy criterion of series of function but it is not mentioned in the previous passage. Looking for it in the preceding pages, uniform Cauchy criterion of sequence of function is found in page 187 which is far away from the page of Weierstrass M test. It means that the reader can't connect the concepts used in the proof of Weierstrass M test. The book by Marsden does perfectly in this point. In page 244 in Marsden's book, Weierstrass M test follows the uniform Cauchy criterion of sequence of function immediately. Uniform Cauchy criterion of series of function is also mentioned between them. The way that Marsden wrote the book makes me connect concepts and feel easy to comprehend.

Marsden's book explains why Weierstrass M test is introduced in the passage preceding to the passage of Weierstrass M test. It also gives an intuitive interpretation to let the reader have insight into the theorem and its proof in a natural point of view, not merely a syntatic point of view like Wade's book. It applies to every theorem in Marsden's book. I could not understand the proof of Weierstrass M test in Wade's book after I read it 10 times, but I comprehended it immediately and derived the proof by myself after I read Marsden's book.

I can't comprehend Math only in syntax. I comprehend Math intuitively by reason. And I think every Math learner does in this way. A proof like the proof of Weierstrass M test can be comprehended as an a priori analytic judgement or an a priori synthetic judgement. It is said analytic because it uses Uniform Cauchy criterion. It is said synthetic because the reader can think of some examples of the theroem which are consistent with the phenomena in the real world. Thus the concepts and the theorems have objective references which can be found in our experience of phenomena. A Math learner can't merely check the validity of the process of a proof syntatically and then says he comprehends the theorem.

A Great Textbook for mathematics lover
It has a lot of problems and concise anwsers and hints. It is a very good book to challeng your mind and push you to think more. I've read Apostol's, Rudin's and Courant John's Analysis textbook. Wade's is even easier and also clear.

THE BOOK before graduate level real variable theory !
I have taken a two-semester course of senior level Advanced (honors) Calculus with this as text. It is a great way to learn real analysis - covering rigorous single variable analysis(first 4 chapters) in one semester - immediately followed in the subsequent semester by multivariable theory (next 4-5 chapters). The author introduces Point Set Topology not before the 5th chapter, as a tool for multivariable analysis. The 10th chapter includes an introduction to calculus on manifolds.

This book needs more revision
I have studied this book in a two semester advanced calculus basis and the reason my teacher chose the book was because it is the only book she could find that touches both singl and multivariable functions calculus. This book is said by many teachers of mine to be a very dry book and here by dry I mean that it just gives you the theory of the concept not the applications which are very important in the study of mathematics (or atleast to attract people to mathematics) I think this book is still allright at least to complete the collections of mathematics books in one's library. ... Read more

Book Description

A powerful XML-based markup language for publishing mathematics on the Web, MathML makes it possible to develop Web-based applications for displaying, searching, indexing, archiving, and evaluating mathematical content.

The MathML Handbook is the definitive resource for this exciting new Web technology. The book begins with an overview of MathML and its practical applications. It then moves on to a clear exposition of the basic concepts of MathML, including XML syntax, presentation markup, content markup, and combined markup. Additional chapters deal with advanced topics such as using CSS and XSLT style sheets to control the display of MathML, tools for converting TeX equations to MathML, and using JavaScript with MathML to create dynamic math web sites featuring interactive equations. The book will be useful to anyone interested in working with mathematics on the Web, such as scientists, engineers, Web developers, publishers, teachers and students.

KEY FEATURES
* Provides a detailed tutorial introduction to MathML, with many examples to illustrate the syntax.

* Reviews the commercial and free software available for working with MathML, including browsers, plug-ins, equation editors, and mathematical software

* Explains how to convert TeX documents to XHTML+MathML

* Illustrates the use of CSS and XSLT to render MathML for display

* Includes examples and templates for creating dynamic web sites with interactive mathematical content

* CD includes demo software for viewing and authoring MathML

ON THE CD!
* Amaya 6.2 (Full Version)
- W3C

* EzMath 1.1 (Full Version)
- Dave Raggett

* Mathematica 4.2 (30-Day Trial Version)
- Wolfram Research, Inc.

* MathPlayer 1.0 (Full Version)

* MathType 5.1 (30-day Trial version)

* WebEQ 3.0 (30-Day Trial Version)
- Design Science, Inc.

FONTS
* Mathematica - Mathematica Fonts in TrueType Format
* TeX - Computer Modern and AMS fonts in Type 1 Format

Plus all the Source Code for Examples from the Book ... Read more

Book Description

This volume offers a gentle introduction to proof in a concrete setting, the introduction of abstract concepts only after a careful study of important examples, and the gradual increase of the level of sophistication as one progresses through the book. It offers an extensive set of exercises that help to build proof writing skills. In addition, chapter introductions give motivation and historical context while tying the subject matter in with the broader picture. ... Read more

Reviews (3)

Buy this book!!
Not only is the best book I have seen on Abstract Algebra, this is the best mathematics book I own. I have used it as a suppliment while studying, in research, and in teaching. It is clear and readable. The authors also have a wonderful web site with scores of resources on the subject.

Carefully develops proof writing skills
This excellent book was my textbook for 2 semesters of senior level abstract algebra. The unique feature of this book is that elementary number theory, equivalence relations, and permutations are carefully introduced at the beginning. Other books launch right into groups and then have to make long digressions to cover these topics. Comparing this book to the best-selling Contemporary Abstract Algebra by Joseph Gallian, I like that Gallian's book adds many applications which students will find interesting. However, Beachy and Blair's book puts a greater emphasis on developing student's ability to do proofs. The book also incorporates more number theory than many other texts. Answers to selected problems are included, so I recommend this book for self study as well as a textbook for any undergraduate abstract algebra course.

to slow
In trying to teach students algebra, I tried to use this book to teach them from, but I found that all the concepts were introduced at what is seemingly high school level. I think that a moderate high school student (with some curiosity) could teach himself the basics of algebra with this book. As an undergraduate text, though, it is way too slow and way too elementary. ... Read more

Book Description

Handbook of Analysis and Its Foundations is a self-contained and unified handbook on mathematical analysis and its foundations. Intended as a self-study guide for advanced undergraduates and beginning graduatestudents in mathematics and a reference for more advanced mathematicians, this highly readable book provides broader coverage than competing texts in the area. Handbook of Analysis and Its Foundations provides an introduction to a wide range of topics, including: algebra; topology; normed spaces; integration theory; topological vector spaces; and differential equations. The author effectively demonstrates the relationships between these topics and includes a few chapters on set theory and logic to explain the lack of examples for classical pathological objects whose existence proofs are not constructive. More complete than any other book on the subject, students will find this to be an invaluable handbook. For more information on this book, see http://math.vanderbilt.edu/

Key Features
* Covers some hard-to-find results including:
* Bessagas and Meyers converses of the Contraction Fixed Point Theorem
* Redefinition of subnets by Aarnes and Andenaes
* Ghermans characterization of topological convergences
* Neumanns nonlinear Closed Graph Theorem
* van Maarens geometry-free version of Sperners Lemma
* Includes a few advanced topics in functional analysis
* Features all areas of the foundations of analysis except geometry
Combines material usually found in many different sources, making this unified treatment more convenient for the user
* Has its own webpage: http://math.vanderbilt.edu/ ... Read more

Reviews (1)

From an amateur's point of view.
I've always wanted to know more math. Finally, I've discovered a big book that makes the tough stuff accessible without sacrificing detail or over-simplifying the hard concepts. For someone with a little bit of college math - calculus and maybe a few more courses - this is it. Analysis, abstract algebra, set theory, topology .... It's all here despite the focus on foundations. And it's organized in a way that makes it possible to follow the thread of a topic throughout the book without reading everything from the beginning. A CDROM version (the one I'm using) not only allows the sort of random access that makes surfing the book convenient but also presents the text in either HTML or PDF format. Open it up in a Java-enabled browser and you can play with the fonts until it suits your reading style. Ideally, for the reader, the CDROM might have been bundled with the book.

Books geared to self instruction need to enlist the readers' aid in educating themselves. Good teachers have that knack. It is a gift to be able to make complicated ideas understandable by building up gradually from simple and familiar concepts to ones that are unusual, obscure, or even initially incomprehensible. Eric Schecter must be a good teacher because in reading his book I am learning more math than I ever thought I would or could. I need to leave questions about the quality of the math itself in depth or scope to the judgment of the professionals but the text appears to make few concessions of substance to us amateurs. Armchair or would-be mathematicians as well as teachers and researchers might find this book a place in their personal libraries. ... Read more

Book Description

For over a hundred years, the theory of water waves has been a source of intriguing and often difficult mathematical problems. Virtually every classical mathematical technique appears somewhere within its confines. Beginning with the introduction of the appropriate equations of fluid mechanics, the opening chapters of this text consider the classical problems in linear and nonlinear water-wave theory. This sets the stage for a study of more modern aspects, problems that give rise to soliton-type equations. The book closes with an introduction to the effects of viscosity. All the mathematical developments are presented in the most straightforward manner, with worked examples and simple cases carefully explained. Exercises, further reading, and historical notes on some of the important characters in the field round off the book and make this an ideal text for a beginning graduate course on water waves. ... Read more

Book Description

From the reviews: "Volume 1 covers a basic course in real analysis of one variable and Fourier series. It is well-illustrated, well-motivated and very well-provided with a multitude of unusually useful and accessible exercises. (...) There are three aspects of Courant and John in which it outshines (some) contemporaries: (i) the extensive historical references, (ii) the chapter on numerical methods, and (iii) the two chapters on physics and geometry. The exercises in Courant and John are put together purposefully, and either look numerically interesting, or are intuitively significant, or lead to applications. It is the best text known to the reviewer for anyone trying to make an analysis course less abstract. (...)" The Mathematical Gazette (75.1991.471 ... Read more

Reviews (8)

simply the Best Calculus Book
An intuitive, rigorous and a beautifully conceptual approach to calculus is what distinguishes this book from the thousands of run-of-the-mill "Calculus I" textbooks published every year.

This is not surprising because 1) Courant and John were both important German-born mathematicians, both schooled in that great mathematical mecca, Gottingen, both making fundamental contributions to many classical branches of pure and applied mathematics. Courant is an especially important mathematician since he not only studied under the greats Minkowski and Hilbert - even serving as the latter's assistant - but founded the Courant Institute of Mathematical Sciences in New York, modelled on the Gottingen Mathematical Institute. 2) That typical German thoroughness and emphasis on the mastery of the "fundamental concepts", so dear to German textbooks, is evident in all sections of the book, particularly in the introductory material on the number continuum, functions, continuity etc.

The exercises at the end of chapters are substantial and excellent, and help to develop proof skills in students as well as a subtle mathematical intuition.

Mathematics is best learnt by studying books written by important mathematicians. Classic books like these should always serve to prove the truth of Abel's dictum that to master mathematics one should 'study the masters and not the pupils'.

You must have this.
My review of the first volume pretty much applies here as well. How many *calculus* texts have an introduction to complex variables, and the theory of analytic functions? This is the only one I've ever seen, and I don't think anyone else could make it more enriching than Courant. Useful material on vector calculus, the theory of matrices, and even introductory material on the *calculus of variations* (something we usually don't see at *all* in the undergrad curriculum) is included. It is refreshing to have an instructor like Courant, who doesn't assume we can't follow higher mathematical roads, but also doesn't sit at the other end of the spectrum, just waving a wand and "poof, here is the result".

Courant also published a standard reference work (also two volumes, I believe) on Mathematical Physics. While the level of mathematics required is post-grad, I was still able to read sizeable sections of it without getting lost.

We can only hope Dover decides to publish Courant's works one day, to make them a little more affordable. But still, you can buy both volumes of Courant's intro to calculus for about the same price as a modern calculus text that waters down the material, and on top of that, provides inadequate explanation for the material it does cover.

Superior as an introductory calculus text!
I don't use the word "superior" lightly, but this book definitely warrants it. Courant was a first rate teacher and mathematician, and his brilliance shows in his exposition. The main obstacle to some readers may be that Courant does not follow the "cookbook calculus" approach that seems so rampant today, but actually bothers to prove his results. He does, however, reserve most of the more difficult proofs for the appendices at the end of the chapter, which is most appreciated. The result is an exciting read, yet rigorous. The reader is very well prepared for future courses in mathematical analysis, and even has a leg up on real analysis. While Courant's insistence on proof does mean that the student needs to have a basic grounding in proof methods, this is usually a standard part of the undergraduate curriclum. Anyone with a background in symbolic logic will instantly be able to follow the proof methods, and most discrete math courses have a section on proofs. In any event, ignorance of proof methods will not detract much from the book's value. Courant rightly recognizes that calculus should be taught in a logical, yet rigorous presentation from the beginning. The absence of this in modern texts mean that students learn how to manipulate formulas, but have no idea what makes the results they are assuming true. The "mechanics" of calculus and analysis, the most crucial thing to be learn, is missed. In particular, I enjoyed his presentation of integration *before* differentiation, which goes against the grain of basic calc texts, yet is historically and pedagogically correct. Integration actually paves the way for differentiation, and gives more motivation for the FTC. In addition, most texts on real analysis work in that order anyway, as an understanding of Lebesgue measure and integration is crucial to understanding the process of differentiation. In addition, I don't think I have ever before or since seen such a careful explanation of the theory of the logarithm or exponential functions. Again, the presentation makes it work, as just introducing the "exponential function", then a little later, the "log function" as the "inverse" of the exponential function is, to put it mildly, artificial and distasteful. The natural progression from the definite integral definition of the logarithm to the exponential function is displayed in its full glory.

In short, Courant manages to present some of the most crucial results of calculus and basic analysis without boring the reader to tears with arcane details, or worse, leaving the reader hanging on important theorems and ideas. This is a balance only a great mathematician could strike, and it is clear why this book remains a classic after almost 60 years.

Note: The second volume of this work covers the multivariable portion of calculus, and will be more difficult to follow without prior exposure to the subject. However, the introductions to the theory of matrices and the calculus of variations are very readable, and it is recommended that the reader take the time to peruse them. Also, don't miss the material on special functions, lightly touched on in the first volume, but explained in fuller detail in the second.

A very good course in Calculus and Analysis
Courant and John have done an excellent job in making me a Calculus-and-Analysis fan. The book is very well written, well motivated and easy to understand - even for me, as an absolute beginner. This doesn't mean it hasn't got any difficult parts, but because the books is so well written, it becomes even more challenging. One of the best books on Calculus and Analysis, even on Mathematics in general. It makes you want to throw your poor high school textbooks in the waste bin.

A good book about analysis
This book is maybe a bit old in style, but you can't deny its worth. As a student of electronics I wanted a complement to my ordinary book (swedish). I don't give this book 5 stars because all problems in the book are proofs. If you don't mind this, go for it. ... Read more

Book Description

This powerful problem-solver gives you 3,000 problems in calculus, fully solved step-by-step! From Schaum’s, the originator of the solved-problem guide, and students’ favorite with over 30 million study guides sold—this timesaver helps you master every type of calculus problem that you will face in your homework and on your tests, from inequalities to differential equations. Work the problems yourself, then check the answers, or go directly to the answers you need with a complete index. Compatible with any classroom text, Schaum’s 3000 Solved Problems in Calculus is so complete it’s the perfect tool for graduate or professional exam review! ... Read more

Reviews (10)

good concept but book falls short
This book sounds like a great idea, and to a student considering whether to get this or a more traditional book, I would recommend a book like The Complete Idiot's Guide to Calculus by W. Michael Kelley.

You will probably have plenty of problems at your disposal in a calc class, so ones from Schaums don't necessarily correlate to what the professor is teaching. Furthermore there are many steps, but not always enough, and no explanations. I think it would be better to have 1000 problems with discussion rather then 3000 crammed in tiny print with bad formatting. That's the final qualm with this book, it is printed on poor quality paper and the ink is thick and the print is small and really crammed in there. Not what you need when there are so many sub and super scripts.

Recommend for Students Needing Help - No Brainer
The best way to learn calculus is to do problems. I was a professor for 8 years and it was a technique that I learned as a student at MIT and later passed on to my students.

Do these problems to improve your scores on tests or just to pass if you are having a problem. To succeed in calculus do problems, problems, and more problems. Here is a book with 3000 problems. That will keep the student busy.

Recommend buy. It is cheap and effective.

Jack in Toronto

A good supplement
This is a very good book, the contents are cleanly and easily accessible, actually the second page. The problems are plenty and of all levels. This is my first supplement for calculus and I must say the book should be used after you do textbook problems or if you can't do basic textbook problems, it's a good help to get you started. There are two downfalls to the books: 1) Basic elementary proofs take up a page or two in some very short sections, these things can EASILY be found in the appendix of a textbook. 2) The book shows steps, but may sometimes think you understood a step, this can cause hell and take time to figure out, though rare, but still annoying. 3) Lack of explanation, this only happend to me in one section the Related Rates section, the beginning was easy but from the middle it was hell. But I still gave this book a 5 because it is a very good supplment

More like Crapulus...
It was funny, but not ha-ha funny...

Got me through Calculus 2!
This book is a fabulous reference to problems I had to work through. I was failing calculus 2 up until the final. I decided to get my buns in gear and buy a few books to help me along. I purchased this book along with The Idiots Guide to Calculus and, no lie, I passed the class with a C instead of the F I was receiving. Very Helpful I'd recommend it to any Calc student in need of help! ... Read more

Book Description

The new edition of this highly acclaimed textbook contains several major additions, including more than four hundred new exercises (with hints and answers).To match the mathematical preparation of current senior college and university entrants, the authors have included a preliminary chapter covering areas such as polynomial equations, trigonometric identities, coordinate geometry, partial fractions, binomial expansions, induction, and the proof of necessary and sufficient conditions.Elsewhere, matrix decompositions, nearly-singular matrices and non-square sets of linear equations are treated in detail.The presentation of probability has been reorganized and greatly extended, and includes all physically important distributions.New topics covered in a separate statistics chapter include estimator efficiency, distributions of samples, t- and F- tests for comparing means and variances, applications of the chi-squared distribution, and maximum likelihood and least-squares fitting.In other chapters the following topics have been added:linear recurrence relations, curvature, envelopes, curve-sketching, and more refined numerical methods. ... Read more

Book Description

A new, revised edition of a yet unrivaled work on frequency domain analysis

Long recognized for his unique focus on frequency domain methods for the analysis of time series data as well as for his applied, easy-to-understand approach, Peter Bloomfield brings his well-known 1976 work thoroughly up to date. With a minimum of mathematics and an engaging, highly rewarding style, Bloomfield provides in-depth discussions of harmonic regression, harmonic analysis, complex demodulation, and spectrum analysis. All methods are clearly illustrated using examples of specific data sets, while ample exercises acquaint readers with Fourier analysis and its applications. The Second Edition:
* Devotes an entire chapter to complex demodulation
* Treats harmonic regression in two separate chapters
* Features a more succinct discussion of the fast Fourier transform
* Uses S-PLUS commands (replacing FORTRAN) to accommodate programming needs and graphic flexibility
* Includes Web addresses for all time series data used in the examples

An invaluable reference for statisticians seeking to expand their understanding of frequency domain methods, Fourier Analysis of Time Series, Second Edition also provides easy access to sophisticated statistical tools for scientists and professionals in such areas as atmospheric science, oceanography, climatology, and biology. ... Read more

Reviews (1)

nice beginning treatment of frequency domain time series
As a time series analyst who studied time series in graduate school when the first edition of Bloomfield's book came out, I really appreciated it. The frequency approach to time series is not natural because we tend to look at the series to see how it changes (fluctuates) over time. This makes the time domain approach very natural. It is only when one sees periodicity in the series and can relate the series to a summation of sine and cosine waves that the frequency domain ideas get appreciated.

Bloomfield introduces Fourier series in a gentle way, making it clear. Concepts are introduced as needed to deal with specific aspects of real time series data. I particularly liked his treatment of complex demodulation, which could be a very difficult concept to grasp if not presented carefully. But Bloomfield explains it well and provides another way to look at the famous Wolfer sunspot data using an approach based on complex demodulation (assuming a slowly changing period for the series). ... Read more

Book Description

In the late 1950s, many of the more refined aspects of Fourier analysis were transferred from their original settings (the unit circle, the integers, the real line) to arbitrary locally compact abelian (LCA) groups. Rudin's book, published in 1962, was the first to give a systematic account of these developments and has come to be regarded as a classic in the field. The basic facts concerning Fourier analysis and the structure of LCA groups are proved in the opening chapters, in order to make the treatment relatively self-contained. ... Read more

Book Description

This new edition of a successful textbook for undergraduate students in physics, computer science and electrical engineering describes important contemporary ideas in practical science and information technology at an understandable level, illustrated with worked examples and copious diagrams. The field is covered broadly rather than in depth, and includes references to more extended works on various topics. This new edition is slightly expanded, and includes additional new material in the applications sections. ... Read more

Book Description

This is a modern introduction to the analytic techniques used in the investigation of zeta-function. Riemann introduced this function in connection with his study of prime numbers, and from this has developed the subject of analytic number theory. Since then, many other classes of "zeta-function" have been introduced and they are now some of the most intensively studied objects in number theory.Professor Patterson has emphasized central ideas of broad application, avoiding technical results and the customary function-theoretic approach. ... Read more

Reviews (1)

An excellent resource for those interested in Riemann's
Zeta function. This book contains a lot of application, theory, and, to my surprise, several practice problems at the end of each section to maximize the learning experience. The chapters are concise and the mathematics is relatively easy follow for those with some experience in special functions.
That, however, is the major thing to note: one needs some experience with special functions in order to find this material accessible. (Obviously, right? Otherwise one wouldn't be buying this book! However, much of this material is beyond the grasp of the average mathematics student that stopped at a bachelor's degree.)
Although this book is called an introduction, I don't think that view is entirely appropriate. The material is quite extensive, and the historocity of the zeta function and its development were kept to a minimum. The precursors to the zeta function and its development by Euler and Riemann (especially Euler's original proof to the Basel Problem) are fantastic. If you're interested in Euler's role in the development, I would look to Dunham's Euler: The Master of Us All, and for Riemann, one should turn to Edwads' Riemann's Zeta Function to read Riemann's original paper.
If you're looking for depth, conciseness, and a broad view of Riemann's zeta function, this book should suit your purposes. If you want a more historical view, I would suggest either of the other books I've mentioned, and not this one. ... Read more

Book Description

This is a corrected third printing of the first part of the text Differential Equations: A Dynamical Systems Approach written by John Hubbard and Beverly West. The authors' main emphasis in this book is on ordinary differential equations. The book is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. Traditional courses on differential equations focus on techniques leading to solutions. Yet most differential equations do not admit solutions which can be written in elementary terms. The authors have taken the view that a differential equations defines functions; the object of the theory is to understand the behavior of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods. The companion software, MacMath, is designed to bring these notions to life. ... Read more