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"‘Mathematicians, like Proust and everyone else, are at their best when writing about their first love’ … They are among the very best we have; and their best is very good indeed. … One approaches this book with high hopes. Happily, one is not disappointed.""…In paperback it might well have become a best seller. …read it."From The Mathematical Intelligencer"Mathematics is shaped by the consistent concerns and styles of powerful minds —three of which are represented here. Kac’s work is marked by deep commitment and breadth of inquiry."Rota is the easiest of these authors to read…a delight: witty and urbane, with a clear and interesting agenda and an astonishing intellectual range. To read him is to be a part of a pleasant and rewarding conversation."Jacob T. Schwartz attacks problems in…computer science, mathematical economics, computer and instructional … No matter how slippery the problem, Schwartz manages to capture a substantial chunk of it in his mathematical net."—Mathematics MagazineThis is a volume of essays and reviews that delightfully explore mathematics in all its moods — from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, Heidegger among others. ... Read more

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A carefully revised edition of the well-respected ODE text, whose unique treatment provides a smooth transition to critical understanding of proofs of basic theorems. First chapters present a rigorous treatment of background material; middle chapters deal in detail with systems of nonlinear differential equations; final chapters are devoted to the study of second-order linear differential equations. The power of the theory of ODE is illustrated throughout by deriving the properties of important special functions, such as Bessel functions, hypergeometric functions, and the more common orthogonal polynomials, from their defining differential equations and boundary conditions. Contains several hundred exercises. Prerequisite is a first course in ODE. ... Read more

Customer Reviews (5)

If you have a class which uses this book, drop it.
I can't believe that this is the fourth edition of this book. Considering the lack of organization and typos, I'd think it was an unpublished draft of a book. It's honestly the worst book I have ever used. I found it harder to learn anything from than rudin. It's unclear what knowledge is a prerequisite for the material here, as the authors seeem to do some sort of review of first order linear differential equations in the beginning but then go off on different tangents. Often terminology will be left undefined, especially in the exercises where it actually matters the most. One highlight of this happening is the chapter on limit cycles, where no formal definition "limit cycle" is actually given. Perhaps this wouldn't be too bad, but there are exercises asking to PROVE things about limit cycles. And I assure you that this is not at all an isolated incident.

My impression is that the authors attempted to write a terse, rigorous account of differential equations. Instead we get an unmotivated disorganized potpourri of topics whose significance is never made clear.

Worst math textbook I have ever had the misfortune of using.
This is a terrible textbook.After being forced to use it for a class, I can safely say that in every set of problems there are at least 5 errors.I'm not talking about small typos--these errors disrupt the entire problem.Often, the author screws up equations entirely in the problems and randomly changes what he has named each variable.Also, it appears as if the authors didn't bother to solve out the problems they were doing.Some of the problems lead to long and tedious computation: if you're in an Ordinary Differential Equations class, you should know how to take a derivative.There's no point in forcing people to solve a fifth order linear Differential Equation leading to a system of 5 equations, it just wastes time and serves no purpose in illuminating the ideas the authors were trying to teach (try taking the fourth derivative of 4 variations of c*e^((sqrt(2)/2)*x)*sin((sqrt(2)/2)*x) and solving for the constants and you'll see what I mean).The book is also painfully outdated (Look forward to computers that can calculate to 7 decimal places!!!) rendering entire sections of the book useless.Half of the formulas are wrong anyway and can't be trusted without verification from someone who is already knowledgeable about the subject.For a book that is so expensive, and in paperback no less, I expected more examples and at least a reasonable editor.Unless you're forced to buy this book, or enjoy reading an unedited copy that the authors clearly didn't even look at after they typed, I would highly recommend against wasting so much of your money.Overall, just a terrible terrible textbook all around and not worth even buying as a reference text.

About those mistakes...
Possibly the best, and certainly the most realistic, mathematics exam question is a statement that you are to prove, if it is true, or for which you're to provide a counterexample, if it's false.Since you're not told whether the statement is true, the first, and frequently hardest, step is to figure out whether you believe it, thus determining whether you next seek a proof or a counterexample.Extra credit is awarded, if you can elegantly amend the hypotheses of a false statement to produce one that's true and proceed to a proof, which can sometimes seem like an afterthought, once you've understood the situation.(For what it's worth, Birkhoff loved this type of question, when he taught from this book.)

I say this type of question is realistic because you find yourself in the same place, when trying to solve a real problem.For example, suppose you know that a certain proposition is true, but some calculations you're doing would be made easier, if its converse were also true.Your first step isn't (or shouldn't be!) to jump in and try to prove the converse.Rather, you ask yourself whether the converse makes sense--i.e., would its truth contradict some other statement that you know to be true, or do other things you know suggest its truth?If it passes this test, you proceed to construct a simple example in which the converse holds, then try to make the example a little more realistic and probably harder, because you've weakened its hypotheses.Only after constructing a few examples whose hypotheses don't seem artificially strong do you finally try to prove the converse.(If your examples are good ones, they may suggest how a proof works but unfortunately not always, especially if the most likely proof is nonconstructive.They will in any case give you an idea of what the hypotheses and any auxiliary conditions should look like.)

My point is that in good exam questions and real problems, you almost never know whether what you're trying to prove is actually true, despite the best evidence you've accumulated. Mathematical experience entails time spent trying to prove things later realized to be false--or, almost as-bad, true but uninteresting--because that's how we learn in any logically rigorous subject.Regrettably, knowledge absent the real frustration of framing appropriate questions is beginner's luxury soon left behind in the practice of pure or applied science.Indeed, the balance-of-frustration, as it were, not infrequently tilts toward posing the right questions, walking the maze sometimes being less problematic than mapping it in the first place.

To my mind this is the best high-end ODE book around, assuming you've already worked your way through a book like Braun that emphasizes calculation.Among the more applied texts its main competitor is Carrier and Pearson, but that text is in many ways idiosyncratic because of the authors' very Socratic approach.If your bent is a bit more theoretical, at this level there are Arnold; Devaney, Hirsch, and Smale; and its predecessor, Hirsch and Smale, which Amazon incorrectly attributes to Smale alone.But the geometric (dynamical-system) approach of these books is sufficiently different from B&R that they're complements, rather than substitutes.Pretty much the same is true of Hurewicz's superb older text.At an advanced level Coddington and Levinson remains the best reference, for my purposes at least, despite its age.Arnold's advanced text is also excellent but again has a geometric emphasis, as its title suggests.(Full citations are in Listmania; see my Amazon profile.I should stress that I'm no doubt ignoring many fine favorites, a few of which appear at the end of my citations, either through unfamiliarity, or because I haven't actually used them.)

The errors in B&R obviously shouldn't be there.But having been warned, you'll learn a great deal, if you think of them as practice for problem-solving by making sure you actually believe a statement before you try to prove it.We do this in some measure all the time, of course, but the errors in B&R mean that the reader should sometimes make explicit what's usually an implicit step.For example, think through the statement's implications, try to construct counterexamples, and so forth.If you conclude that it''s wrong, ask how the statement's hypotheses need to be modified to make it work.

To wit, my fellow reviewer did finally conclude that an infinite sequence ofSturm-Liouville eigenfunctions is bounded, and his bringing this up in his review suggests that he's never forgotten it.Thus, for the thoughtful the exercise was successful, if not quite in the way intended, and even if "frustrating [for] a student," as our learned friend complains.Catching just a few errors in this way will make you so gloriously self-confident that you'll start probing the questions and problems in every math book you use, which is a very good thing.

As the saying goes, "When life gives you lemons [especially the juicy ones, like this book], make lemonade!"

A WORD ABOUT COUNTEREXAMPLES.Talented analysts, like mathematicians in other fields, tend to keep a a relatively small number of very carefully constructed counterexamples in mind and try to adapt them to situations that arise as a step toward their understanding.One of my professors, a mathematician of some note (not Birkhoff or Rota but easily in their league), had a counterexample he called the "wandering, shrinking interval" that wrecked many incorrect conjectures, much to his delight.It takes astonishingly few of these counterexamples to be really effective--a dozen, say, may be on the high side, if the counterexamples are well-chosen and evolve over time--and each analyst seems to have his or her own list, with a much larger, less subtle, core of interesting pathologies that all good analysts seem to know.

The subtleties ultimately derive from an oral tradition that's handed down from professor to student, which is the real purpose of advanced instruction at any level.However, a number of books talk about the well-known core in various areas of mathematics--e.g., analysis, topology, topological vector spaces (functional analysis), differential equations, and probability.They can get you started in constructing your own counterexamples, not least by getting you into the conversations where the more sophisticated concepts are discussed; this review's Listmania page on my profile has pointers.

POSTSCRIPT (added 18 Mar 2009). When I read this review after almost a year, I'm struck that there's no errata in later printings of the Fourth Edition.This edition first appeared in January 1989; Birkhoff died in November 1996; and Rota in April 1999.It seems highly likely that the authors became aware of many errors before their deaths, colleagues and students being what they are, so there probably was ample opportunity to issue sucha sheet, if they'd so desired.

They may have concluded, however, that corrections are superfluous, because a careful student of mathematics--the type that they tried to reach, and of which there's never been any shortage in Cambridge--would proceed as I've suggested without further ado.Indeed, an instructor can see the uncertainty as beneficial, as it is in the exam questions I've described.There's no knowing at this point, of course, but I would find it quite plausible that they reasoned in this way. --PJES

Full of mistakes
This text is based upon lecture notes (written either by Rota and then bound by Birkhoff or vice versa) and it is VERY apparant that the authors spent little time on checking for mistakes.It seems as though every fourth problem has a typographical error or is completely incorrect.This textbook does not take an intuitive approach to the subject and the many errors gives me reason to rate it low.

Nothing is more frustrating to a student than to have a text make a claim, ask you to prove it, and then you find out the claim is false.This is not a unique occurance.Steer clear of this textbook.

Good text but watch for errors
This is a good text for a course in ODE after the computational course is completed.This book covers many topics in ODE, but be careful and watchfor errors; especially near the end of the text.Most of the errors comein the excercises.For instance: you are asked to show that "everyfinite sequence of eigenfunctions of a S-L system is bounded".Duh! This is trivial until one realizes that every infinite sequence is alsobounded. ... Read more

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The essays in this volume pay tribute to Gian-Carlo Rota on the occasion of his 64th birthday. The breadth and depth of Rota's interests, research and influence are reflected in such areas as: combinatorics, invariant theory, geometry, algebraic topology, and representation theory. ... Read more

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Written by two of Gian-Carlo Rota's former students, this book is based on notes from his courses and on personal discussions with him. Topics include sets and valuations, partially ordered sets, distributive lattices, partitions and entropy, matching theory, free matrices, doubly stochastic matrices, Moebius functions, chains and antichains, Sperner theory, commuting equivalence relations and linear lattices, modular and geometric lattices, valuation rings, generating functions, umbral calculus, symmetric functions, Baxter algebras, unimodality of sequences, and location of zeros of polynomials. Many exercises and research problems are included, and unexplored areas of possible research are discussed.This book should be on the shelf of all students and researchers in combinatorics and related areas. ... Read more

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Gian-Carlo Rota was one of those rare mathematicians who made major contributions to several areas of mathematics. Presented in the first part of this volume are reprints of his papers in analysis, which were written at the beginning of his career. These papers on differential equations, operator theory, ergodic theory, and other subjects have a continuing and pervasive influence. Reprints of his papers on convexity and probability theory are presented in the second part of the work. These were written towards the end of his career and contain many ideas that have yet to be fully developed. Comprehensive commentaries are included in every chapter. These survey articles detail work inspired by Rota's papers and also include discussions of many unsolved problems.

As is customary with Rota's writings, the papers included in the volume---some published here for the first time---contain many fresh and unexpected ideas for further research. Thus, this volume will be of interest to both experts and beginners in the above-mentioned fields.

Contributors: J. Dhombres, P.L. Duren, W.N. Everitt, D.A. Klain, J.P.S. Kung, A. Ramsay, M.M. Rao, J. Rovnyak, H.H. Schaefer, B. Schultze, J.T. Schwartz, N. Starr, G. Strang, D.C. Torney, R. Zaharopol, A. Zettl, X.-D. Zhang ... Read more

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It has been clear within the last ten years that combinatorial geometry, together with its order-theoretic counterpart, the geometric lattice, can serve to catalyze the whole field of combinatorial theory, and a major aim of this book, now available in a preliminary edition, is to present the theory in a form accessible to mathematicians working in disparate subjects.

Earlier studies have been one-sided or restricted in their point of view; they were motivated primarily by the desire to extend the classical theory of graphs, or were lattice-theoretic approaches confined to axiomatics and algebraic dependence. These approaches largely ignored the original geometric motivations.

The present work brings all these aspects together in order to emphasize the many-sidedness of combinatorial geometry, and to point up the unifying role it may well play in current developments in combinatorics and its applications.

The book defines the axiomatics of combinatorial geometry, describes a variety of geometrical examples, and discusses the notion of a strong map between geometries. In addition, there is a brief presentation of coordinatization theory and a sketch of two important lines of future work, the "critical problem" and matching theory. The full chapter titles are given below.

Contents: 1. Introduction. 2. Geometrics and Geometric Lattices. 3. Six Classical Examples. 4. Span, Bases, Bonds, Dependence, and Circuits. 5. Cryptomorphic Versions of Geometry. 6. Simplicial Geometries. 7. Semimodular Functions. 8. A Glimpse of Matching Theory. 9. Maps. 10. The Extension Theorem. 11. Orthogonality. 12. Factorization of Relatively Complemented Lattices. 13. Factorization of Geometries. 14. Connected Sets. 15. Representation. 16. The Critical Problem. 17. Bibliography. ... Read more

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Essays explore the emerging modalities and potential impact of this technological revolution. Provides a clear explanation of massively parallel computing. Paper. DLC: Parallel processing (Electronic computers) ... Read more

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This is the third volume of the collected papers of George Pólya, one of the most influential mathematicians and teachers of our time. Volumes I (Singularities of Analytic Functions) and II (Location of Zeros) were published in 1974.

Volume III contains 58 papers spanning Pólya's career (the earliest is from 1913, the latest from 1976) and covering a wide range of subjects in mathematical analysis and mathematical physics. The commentaries on these papers attest to the fertility and continued importance of Pólya's ideas in current mathematics.

This volume is the twenty-second in the series Mathematicians of Our Time, edited by Gian-Carlo Rota. ... Read more

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The present work represents a unique undertaking in scientific publishing to honor Nick Metropolis, who passed away in October, 1999. Nick was the last survivor of the Manhattan Project that began during World War II in Los Alamos and later became the Los Alamos National Laboratory. In this volume, some of the leading scientists and humanists of our time have contributed essays related to their respective disciplines exploring various aspects of future developments in science, technology, and society. Speculations on the future developments of science and society, philosophy, national security, nuclear power, pure and applied mathematics, physics and biology, particle physics, computing, information science, among many others, are included.

Contributors include:

H. Agnew * R. Ashenhurst * K. Baclawski * G. Baker * N. Balazs * J.A. Freed * R. Hamming * M. Hawrylycz * O. Judd * D. Kleitman * M. Krieger * N. Krikorian * P. Lax * J.D. Louck * T. Puck * M. Raju * R. Richtmyer * J. Schwartz * R. Sokolowski * E. Teller * M. Waterman ... Read more

Customer Reviews (1)

Foundations of Rota's thinking on polynomials, etc.
This book is an anthology of Rota's papers, all but two with co-authors, and one paper by another author.The title essay, "Finite Operator Calculus" by Rota, Kahaner, and Odlyzko, was published in 1973.It is the founding paper of the modern (as opposed to 19th-century) umbral calculus.It inaugurated Rota's general theory of special polynomial sequences.A later paper, _The_Umbral_Calculus_, by Steven Roman and Gian-Carlo Rota, published in Advances in Mathematics, volume 31, pages 95-188, in 1978, approaches the subject from a different direction and makes it clearer in what sense the theory of polynomial sequences is the same thing as the 19th-century umbral calculus. That older umbral calculus is essentially a notational trick by which subscripts that index sequences are treated as if they were exponents.The paper includes a fairly long list of research problems in combinatorics and probability, perhaps many of which have yet to be addressed.Application to cumulants of compound Poisson processes is barely mentioned in a remark that seemed cryptic to me.He returned to that topic not long before his death in a joint paper with Jianhong Shen, but seemed to be barely beginning to figure out how to think about the problem.

An earlier paper of Ronald Mullin and G.-C. Rota, _Theory_of_Binomial_Enumeration_, anticipates this theory of special polynomial sequences, and, as the title suggests, is concerned with its application to combinatorial enumeration.

Not everyone agrees that the modern umbral calculus can be very fruitful.

This book begins not with the title paper, but with Rota's first paper on combinatorics, _The_Number_of_Partitions_of_a_Set_, published in 1964.It is in that paper that Rota first applied linear functionals on spaces of polynomials to combinatorics.It is remarkable that this paper never explicitly mentions that the number of partitions of a set of size n is the nth moment of a Poisson distribution with expected value 1, since it mentions it _implicitly_ repeatedly.Rota uses a lemma that is formally very similar to the "Robbins lemma" of empirical Bayes methods in statistics, without appearing to be aware of any connection.

This book also contains the only joint paper Rota wrote with Richard Stanley (author of _Enumerative_Combinatorics_), on basic theory of generating functions; Peter Doubilet is also a co-author.

The one paper appearing here in which Rota is not identified as one of the authors is by Curtis Greene, on valuation rings of lattices. ... Read more

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This is a volume of essays and reviews that delightfuly explore mathematics in all its moods - from the light and the witty, and humorous to serious, rational, and cerebral. Topics include: logic, combinatorics, statistics, economics, artifical intellligence, computer science, and applications of mathematics broadly. You will also find history and philosophy covered, including discussion of the work of Ulam, Kant, and Heidegger among others. ... Read more

Customer Reviews (1)

An enlightening book on mathematics
This is an unusual book on the discipline of mathematics. Don't look here for the usual fluffy collages of fractals, chaos theory and hagiographies.

What this book offers is a collection of essays on the state and nature of mathematics by three top mathematicians. They're honestly worried about the schism between math and science, interested in the effect the computer could have on math, critical about the mathematical establishment. They're completely honest, and their criticisms are incisive and well reasoned. This book is food for thought, and not only about math.

The authors differ in their writing styles. While all three have good essays, Rota is a fantastic writer, and his mixture of serious thought, pith and gossip is completely refreshing. You should also read (before this book) Rota's "Indiscrete Thoughts". ... Read more

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Here is the first modern introduction to geometric probability, also known as integral geometry, presented at an elementary level, requiring little more than first-year graduate mathematics. Klein and Rota present the theory of intrinsic volumes due to Hadwiger, McMullen, Santaló and others, along with a complete and elementary proof of Hadwiger's characterization theorem of invariant measures in Euclidean n-space. They develop the theory of the Euler characteristic from an integral-geometric point of view. The authors then prove the fundamental theorem of integral geometry, namely, the kinematic formula. Finally, the analogies between invariant measures on polyconvex sets and measures on order ideals of finite partially ordered sets are investigated. The relationship between convex geometry and enumerative combinatorics motivates much of the presentation. Every chapter concludes with a list of unsolved problems. ... Read more

Customer Reviews (1)

a beautiful book
Everyone with any interest in probability or combinatorics should take alook at this book, and at least read Chapter 1, on the Buffon needleproblem.It gives a beautiful conceptual solution, quite different fromthe more well-known solution using integrals to get conditionalprobabilities. I find it hard to imagine anyone reading Chapter 1 and notdeciding to read the entire book.

I heard Rota lecture on this material,and the book has much the same feeling as his lectures: it is clear,elegant, and concise, full of illuminating examples.Relatively littlebackground is required, and it should be easily accessible to beginninggraduate students (or undergraduates with unusually strong backgrounds). ... Read more

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This volume surveys the development of combinatorics since 1930 by presenting in chronological order the fundamental results of the subject proved in over five decades of original papers by: T. van Aardenne-Ehrenfest.- R.L. Brooks.- N.G. de Bruijn.- G.F. Clements.- H.H. Crapo.- R.P. Dilworth.- J. Edmonds.- P. Erdös.- L.R. Ford, Jr.- D.R. Fulkerson.- D. Gale.- L. Geissinger.- I.J. Good.- R.L. Graham.- A.W. Hales.- P. Hall.- P.R. Halmos.- R.I. Jewett.- I. Kaplansky.- P.W. Kasteleyn.- G. Katona.- D.J. Kleitman.- K. Leeb.- B. Lindström.- L. Lovász.- D. Lubell.- C. St. J.A. Nash-Williams.- G. Pólya.-R. Rado.- F.P. Ramsey.- G.-C. Rota.- B.L. Rothschild.- H.J. Ryser.- C. Schensted.- M.P. Schützenberger.- R.P. Stanley.- G. Szekeres.- W.T. Tutte.- H.E. Vaughan.- H. Whitney.

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Word count: 923. ... Read more

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This book brings the reader to the frontiers of research in some topics in superalgebras and symbolic method in invariant theory. Superalgebras are algebras containing positively-signed and negatively-signed variables. One of the book's major results is an extension of the standard basis theorem to superalgebras. This extension requires a rethinking of some basic concepts of linear algebra, such as matrices and coordinate systems, and may lead to an extension of the entire apparatus of linear algebra to ``signed'' modules. The authors also present the symbolic method for the invariant theory of symmetric and of skew-symmetric tensors. In both cases, the invariants are obtained from the symbolic representation by applying what the authors call the umbral operator. This operator can be used to systematically develop anticommutative analogs of concepts of algebraic geometry, and such results may ultimately turn out to be the main byproduct of this investigation. While it will be of special interest to mathematicians and physicists doing research in superalgebras, invariant theory, straightening algorithms, Young bitableaux, and Grassmann's calculus of extension, the book starts from basic principles and should therefore be accessible to those who have completed the standard graduate level courses in algebra and/or combinatorics. ... Read more