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From the Preface: "This book was written for the active reader. The first part consists of problems, frequently preceded by definitions and motivation, and sometimes followed by corollaries and historical remarks... The second part, a very short one, consists of hints... The third part, the longest, consists of solutions: proofs, answers, or contructions, depending on the nature of the problem....

This is not an introduction to Hilbert space theory. Some knowledge of that subject is a prerequisite: at the very least, a study of the elements of Hilbert space theory should proceed concurrently with the reading of this book."

Customer Reviews (1)

Deserves 10 stars
This book should have been titled "A Hilbert Space Idea/Problem Book" as it not only challenges the reader to work out interesting problems in operator theory and the geometry of Hilbert space, but also motivates the essential ideas behind these fields. It is definitely a book that, even though out-of-print, will be referred to by many newcomers to operator theory and quantum physics. The insight one gains by the reading of this book is unequaled in any other books in existence on operator theory. It is becoming more rare as mathematics advances, to find books that attempt to explain the intuition behind the abstractions that are manifested in any area of mathematics. The problems in the book deal with both concrete examples and general theorems, and the reader should attempt to try and solve them without looking at the hints. The solutions found by the reader can then be compared with the author's, and some interesting differences will occur.

There are so many interesting discussions in this book that to list them all would probably entail listing everything in the book. The reader will find excellent discussions of the origin of normal operators on infinite dimensional Hilbert spaces as analogs to matrices on finite dimensional spaces; why the weak topology in infinite dimensions is not metrizable; the non-emptiness of the spectrum and why the spectral radius can be computed even though the spectrum cannot; the impossibility of isolated singular operators; the non-continuity of the spectrum: the existence of an operator with a large spectrum and the existence of operators with small spectra in every neighborhood of the large spectrum. The author then goes on to show that the spectrum is an upper semicontinuous function, thus preventing the existence of small spectra arbitrarily close to large spectra. This is an excellent discussion on the meaning and intuition behind semicontinuity; the result that every normal operator is unitarily equivalent to a multiplication and its equivalance to the spectral theorem. The author goes on to explain how one gives up the sigma-finiteness of the measure when doing this, and the origin of functional calculus; the difference between infinite and finite dimensions when attempting a polar decomposition for operators and its connection with partial isometries; the origin of compact operators and their connection with integral equations. The author shows how even the identity operator is not an integral operator on the space of square-integrable functions with Lebesgue measure.

In discussing the spectral theorem in chapter 13 the author statesmost profoundly: "In some contexts some authors choose to avoid a proof that uses the spectral theorem even if the alternative is longer and more involved. This sort of ritual circumlocution is common to many parts of mathematics; it is the fate of many big theorems to be more honored in evasion than in use. The reason is not just mathematical mischievousness. Often a long but 'elementary' proof gives more insight, and leads to more fruitful generalizations, than a short proof whose brevity is made possible by a powerful but overly specialized tool." In these few sentences the author has characterized the problem with current methods of teaching advanced mathematics. Too often the formalism masks the true meaning and intuitive motivation behind the mathematics. And even though mathematics is being applied to many different areas at an unprecedented rate, pure mathematics seems to be trapped in a local minimum, and I beleive this is due to the reluctance of authors to explain in detail the essentials of their ideas. This book is a perfect example of how mathematics can be taught that requires much thought and creativity on the part of students, without spoon-feeding them and thus encouraging a passive attitude to the learning of mathematics. I salute the author in his achievements in research and in teaching...one can only hope that his approach will be followed in all future works of mathematics. ... Read more

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foreword by Martin Davis and Hilary Putnam In 1900, the Germanmathematician David Hilbert put forth a list of 23 unsolved problemsthat he saw as being the greatest challenges for twentieth-centurymathematics. Hilbert's 10th problem, to find a method for decidingwhether a Diophantine equation has an integral solution, was solved byYuri Matiyasevich in 1970. Proving the undecidability of Hilbert's 10thproblem is clearly one of the great mathematical results of the century.This book presents the full, self-contained negative solution ofHilbert's 10th problem. In addition it contains a number of diverse,often striking applications of the technique developed for thatsolution, describes the many improvements and modifications of theoriginal proof since the problem was "unsolved" 20 years ago, and addsseveral new, previously unpublished proofs. More on this book... ... Read more

Customer Reviews (2)

Algorithm, Turing Machine, Turing Decidable, Solvability
Yuri V. Matiyasevich's "Hilbert's Tenth Problem" has two parts. "The first part, consisting of Chapters 1-5, presents the solution of Hilbert's Tenth Problem." The second part (Chapters 6-10) is "devoted to application."

Hilbert's Tenth Problem is about the "determination of the solvability of a Diophantine equation." To be specific, the problem asked for "devise a process ... which ... can ... [determine in] a finite number of operations whether the equation is solvable in ... integers." David Hilbert posted the problem in 1900. "Today ... the words `devise a process' ... mean `find an algorithm.' When Hilbert's Problem was posed, there was no ... rigorous ... notion of algorithm ... [Until 1930s] Kurt Godel, Alonzo Church, Alan Turing, and other logicians provided a rigorous formulation ... of computability; [then] ... it [is] possible to establish algorithmic insolvability ... "

The problem was considered solved by Yuri Matiyasevich in 1970. In short, Matiyasevich proved the Martin Davis's conjecture. The readers will find Matiyasevich's "Hilbert's Tenth Problem: What can we do with Diophantine equations?" helpful. Martin Davis's conjecture states that a set is Diophantine if and only if it is list-able. There is a classical result in the computability theory: there exists an un-decidable list-able set. The un-decidability of the set implies that there is no algorithm to determine [the] values of the parameters [of] the Diophantine representation [so that the representation] has a solution.

On the other hand, the material on the book is more technical. "... we can reformulate Hilbert's Tenth Problem in the following ... way: is the set of codes of all solvable Diophantine equations ... Turing decidable? ... the complement of [the set of codes] is not Diophantine. This implies that [the set] is not Turing decidable. In other words, it is impossible to construct a Turing machine that ... will halt after a finite number of steps in state q2 [yes] or q3 [no], depending on whether the equation ... is or is not solvable."

In terms of application, "we can construct a Diophantine equation whose un-solvability is equivalent to the Riemann Hypothesis." Similar utilization can be applied to number theory, calculus, and game theory problems. But we have no obvious way to restate the twin prime conjecture ... as the problem of the solvability or un-solvability of a particular Diophantine equation."

Masterful and elucidating on many levels
Hilbert's 10th problem was solved, as well explicated in this book, but many of the ramifications of this solution were very unexpected and almost surprising beyond belief!

This book is not easy, but it is also not hard in the way if many advanced mathematical texts. The authors have done a great service by presenting proofs well within the range of non-experts with a general college level of mathematical sophistication. They are truly to be congratulated for this unique and priceless contribution to mathematical literature. No one had any idea of the rich results that would ensue on the solution to this seemingly simple to state problem, and the not so surprising result that the answer was in the negative. If you like mathematics, you will find many delightful and surprising results presented here in a way very comprehensible to those willing to work through these proofs designed for the most general audience possible.

... Read more

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In May 1974, the American Mathematical Society sponsored a special symposium on the mathematical consequences of the Hilbert problems, held at Northern Illinois University, DeKalb, Illinois. The central concern of the symposium was to focus upon areas of importance in contemporary mathematical research which can be seen as descended in some way from the ideas and tendencies put forward by Hilbert in his speech at the International Congress of Mathematicians in Paris in 1900. The Organizing Committee's basic objective was to obtain as broad a representation of significant mathematical research as possible within the general constraint of relevance to the Hilbert problems. The Committee consisted of P. R. Bateman (secretary), F. E. Browder (chairman), R. C. Buck, D. Lewis, and D. Zelinsky.

The volume contains the proceedings of that symposium and includes papers corresponding to all the invited addresses with one exception. It contains as well the address of Professor B. Stanpacchia that could not be delivered at the symposium because of health problems. The volume includes photographs of the speakers (by the courtesy of Paul Halmos), and a translation of the text of the Hilbert Problems as published in the Bulletin of the American Mathematical Society of 1903. The papers are published in the order of the problems to which they are filiated, and not in the alphabetical order of their authors.

An additional unusual feature of the volume is the article entitled "Problems of present day mathematics" which appears immediately after the text of Hilbert's article. The development of this material was initiated by Jean Dieudonné through correspondence with a nummber of mathematicians throughout the world. The resulting problems, as well as others obtained by the editor, appear in the form in which they were suggested. ... Read more

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Positivity is one of the most basic mathematical concepts, involved in many areas of mathematics (analysis, real algebraic geometry, functional analysis, etc.). The main objective of the book is to give useful characterizations of polynomials. Beyond basic knowledge in algebra, only valuation theory as explained in the appendix is needed.

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In a memorable address given at the International Congress of Mathematicians in Paris in 1900, David Hilbert, perhaps the most respected mathematician of his time, developed a blueprint for mathematical research in the new century.

The collections of problems he presented in that address has become a guiding inspiration to many mathematicians, and those who have succeeded in solving or advancing their solutions form an Honors Class among research mathematicians.

With the support of many of the major players in the field, the author has written an engaging account of the achievements of this Honors Class, covering mathematical substance and biographical aspects. ... Read more

Customer Reviews (12)

great mathematicians who contributed to solutions of Hibert's 23 unsolved problems
The Honors Class is the collection of mathematicians that individually or in collaboration solved or partially solved at least one of Hilbert's 23 problems. Yandell does a great of gathering up the historical information so that we have an up-to-date account of the progress on each problem and even how some problems evolved because of their vague or incorrect original proposal.
This is a popular math book and is accessible to the nonmathematician such as the fine books by Casti on mathematicians and mathematical developments. It is also similar to Singh's book on Fermat.

I think the historical research and accounting of the mathematics deserves 5 stars. I am a little unsure about how well the technical mathematics is conveyed to the layperson however. Admittedly, this is a very difficult task as much of the mathematics is very abstract, especially the early chapters on the foundational questions. The number theory, geometry and even some of the abstract algebra problems are easier to explain and Yandell does a fine job with them.

As a mathematician who studied algebra, analysis and even some symbolic logic as an undergraduate and graduate student, I still had a hard time feeling that I got the essence of the mathematics associated with some of these problems. Yandell's discussion is at times detailed but is necessarily sketchy on some of the mathematics. This works for me sometimes but not so well at other times. I think it would be much harder for a novice, but I guess it depends on the depth of understanding one is looking for.

I have always found the work of Cantor mysterious and so the ealry chapters that cover Godel and Cohen's amazing results are not the most enlightening for me. I had learned about the axiom of choice in my real analysis classes and was told something about the undecidability of it and its equivalence to the continuum hypothesis but have never really seen the connection or gotten much insight. The material on Paul Cohen is interesting to me because I attended Stanford in the 1970s when he was the buzz of the campus. A younger and less accomplished mathematician compared to many of his famous colleagues in Stanford's prestigious mathematics department, he still was revered because he solved one of Hilbert's problems. Still I am no closer to understanding symbolic logic and the method of deciding whether or not a proposition can be deduced from a set of axioms or can exist independently of the axiom system.

I got hooked on the book with the chapter on the tenth problem. This problem seemed more easily understandable and it was very interesting to see how the many players work together and separately to attack the problem including the very interesting Julia Robinson who was a key player in the middle of alll this.

The lives of these mathematicians, in some cases their suffering and insanity (similar to Nash) is very interesting and entertaining. There is too much here to handle in one reading.

I think this is a book I will go back to again and again. I am interested in reading more on Kolmogorov and want to try to understand some of the abstract algebra and number theory questions in more detail. There is a great deal of commonality in many of the stories. A large number of the members of the Honors Class were from Germany and fled during World War II. Many also traveled through or spent great portions of their career at Princeton University (some at the Institute for Advanced Study).

The book is thorough and gives an account of all the unsolved problems as well providing the insights of the mathematicians who have made attempts at them.

too much chat, not enough math
I was hoping for something more like The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time, an explanation of the problems and how they were solved, at a reasonably accessible level.This book appears to be just chat about the funny people who become math professors.Sort of interesting, but not what I wanted.

A wonderful book!
As a career scientist for over 50 years, I am versed in mathematics but not exactly a mathematician.I bought it to become familiar with Hilbert's problems.I quickly realized that Yandell's book was more about the attempters and solvers than about the problems.Yet the problems are described too, in considerable and certainly sufficient detail.

What was ultimately fascinating was the web Yandell weaves throughout the book.Those famous mathematicians and their colleagues, their personal lives, those famous problems, and all integrated so cohesively.

When I started reading I knew I was in for a long adventure.In fact, it took me over a year to read - of course, only an hour or so every few days.What extended it was the temptation to go back and reread, again and again.Finally, a week ago, I turned the last page.With great reluctance I put it on my bookshelf.I had a strong urge to start all over again from the very beginning, and I knew if I succumbed I was in for another year with it.

As I reflect, partly it was the subject - those difficult problems in such vastly different fields.Partly it was those mathematicians - many of them already heroes of mine.But mostly it was Yandell's skill in putting together this riveting accounting.His love of and fascination with mathematics, and his desire to share his romanticism with others, comes through so clearly.It is sad that he died, at the young age of 53, a scant two years after writing this book.Of a heart attack and multiple sclerosis.What a tragic loss.He was a gem.

Even if you are only mildly interested in mathematics, its history and personalities, you will absolutely love this accounting of it.

Feels like a dutiful summary
First, my background: I am not a mathematician, but an academic with fair knowledge of college math and even some advanced materials. I do greatly enjoy reading books about mathematics and mathematicians.

This book is obviously a work of great effort by the author. My difficulty probably came from the work's ambitious premise: offering mathematical and biographical history of Hilbert's problems. There are simply too many ideas and persons (some well known, others a bit obscure to lay reader like myself) to cover in one book.The author dutifully and honestly gives references to his sources. My impression is that the author collected as much material as he could about each problem and solvers, and tried to squeeze the information as compactly as possible into the pages.

The result: the narrative is very methodically told - explanation of the problem, some necessary ideas introduced, who the major solvers were, then a short biography of each solver, when and where they were born, who their parents were, where they went to school, who they married and so on; then another cycle begins. Halfway into the book, I began to get bored.

I can imagine mathematicians enjoying a quick review of and glimpses into their discipline and heroes, but lay readers much beware. I recommend lay readers to check out a few pagescarefully online or at your local library to see whether you like it. I certainly didn't hate it but did not like it as much as I expected.

A great work
Due to rapid development of mathemtics in the last century, now one cannot master all subfects of mathematics. This is also true for those historians. Most of the boods of " History of Mathematics " end in the beginning of 20th century. So we know very little about the conteporary mathematicians. This book can be described as a gap for it. After readiming this book, not only you have a knowledge about the life of the great mathemaitcians, you also get the period in World War II how Nazis forced those mathematicians out of Germany and the reason why U. S. A. is now the leading centre of mathematics. ... Read more

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In the late sixties Matiyasevich, building on the work of Davis, Putnam and Robinson, showed that there was no algorithm to determine whether a polynomial equation in several variables and with integer coefficients has integer solutions. Hilbert gave finding such an algorithm as problem number ten on a list he presented at an international congress of mathematicians in 1900. Thus the problem, which has become known as Hilbert's Tenth Problem, was shown to be unsolvable. This book presents an account of results extending Hilbert's Tenth Problem to integrally closed subrings of global fields including, in the function field case, the fields themselves. While written from the point of view of Algebraic Number Theory, the book includes chapters on Mazur's conjectures on topology of rational points and Poonen's elliptic curve method for constructing a Diophatine model of rational integers over a 'very large' subring of the field of rational numbers. ... Read more

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Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials.

This monograph consists of three parts:- the abstract theory of Hilbert spaces, leading up to the spectral theory of unbounded self-adjoined operators;- the application to linear Hamiltonian systems, giving the details of the spectral resolution;- further applications such as to orthogonal polynomials and Sobolev differential operators. Written in textbook style this up-to-date volume is geared towards graduate and postgraduate students and researchers interested in boundary value problems of linear differential equations or in orthogonal polynomials ... Read more

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This self-contained treatment of bounded linear operators on a Hilbert space provides an examination of the theory from a problem-solving viewpoint.Each chapter interweaves theoretical results with a number of problems, ranging from simple yet instructive exercises to open questions at the forefront of current research; complete solutions to all stated problems are provided.

Written in a motivating and rigorous style, the text covers much of the classical theory: it begins with the basics of invariant subspaces, linear operators, convergence, shifts, and decompositions, and then proceeds to hyponormal operators, spectral properties, and paranormal and quasireducible operators.The book concludes with a detailed presentation of the Lomonosov Theorem on nontrivial hyperinvariant subspaces for compact operators.

Some knowledge of elementary functional analysis and a familiarity with the basics of operator theory are all that is required. While this problem-solving approach to the study of Hilbert space operators is primarily aimed at graduate students, it will benefit researchers and working scientists as well, given the far-reaching applications of the subject to pure and applied mathematics, physics, engineering, economics, and statistics. ... Read more

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This volume is a collection of papers presented at a special session on integrable systems and Riemann-Hilbert problems. The goal of the meeting was to foster new research by bringing together experts from different areas. Their contributions to the volume provide a useful portrait of the breadth and depth of integrable systems.

Topics covered include discrete Painlevé equations, integrable nonlinear partial differential equations, random matrix theory, Bose-Einstein condensation, spectral and inverse spectral theory, and last passage percolation models. In most of these articles, the Riemann-Hilbert problem approach plays a central role, which is powerful both analytically and algebraically.

The book is intended for graduate students and researchers interested in integrable systems and its applications. ... Read more

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The famous twelfth Hilbert problem calls for holomorphic functions in several variables with properties analogous to the exponential function and the elliptic modular function with a view to the explicit construction of (Hilbert) class fields by means of special values. The lecture notes present those functions living on the two-dimensional complex unit ball. In the course of their construction, the reader is introduced to work with complex multiplication, moduli fields, moduli space of curves, surface uniformizations, Gauss-Manin connection, Jacobian varieties, Torelli's theorem, Picard modular forms, Theta functions, class fields and transcen- dental values in an effective manner. ... Read more

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This book is the result of a meeting that took place at the University of Ghent (Belgium) on the relations between Hilbert's tenth problem, arithmetic, and algebraic geometry. Included are written articles detailing the lectures that were given as well as contributed papers on current topics of interest.

The following areas are addressed: an historical overview of Hilbert's tenth problem, Hilbert's tenth problem for various rings and fields, model theory and local-global principles, including relations between model theory and algebraic groups and analytic geometry, conjectures in arithmetic geometry and the structure of diophantine sets, for example with Mazur's conjecture, Lang's conjecture, and Bücchi's problem, and results on the complexity of diophantine geometry, highlighting the relation to the theory of computation.

The volume allows the reader to learn and compare different approaches (arithmetical, geometrical, topological, model-theoretical, and computational) to the general structural analysis of the set of solutions of polynomial equations. It would make a nice contribution to graduate and advanced graduate courses on logic, algebraic geometry, and number theory. ... Read more

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This book presents global actions of arbitrary Lie groups onlarge classes of generalised functions by using a novel parametricapproach. This new method extends and completes earlier results of theauthor and collaborators, in which global Lie group actions ongeneralised functions were only defined in the case of projectable orfibre-preserving Lie group actions.
The parametric method opens the possibility of dealing with vastlylarger classes of Lie semigroup actions which still transformsolutions into solutions. These Lie semigroups can contain arbitrarynoninvertible smooth mappings. Thus, they cannot be subsemigroups ofLie groups.
Audience: This volume is addressed to graduate students andresearchers involved in solving linear and nonlinear partialdifferential equations, and in particular, in dealing with the Liegroup symmetries of their classical or generalised solutions. ... Read more

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Text discusses the qualitative investigation of two-dimensional polynomial dynamical systems, and is aimed at solving Hilbert's Sixteenth Problem on the maximum number and relative position of limit cycles. The volume number facing the title page is erroneously listed as Volume 559. ... Read more

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In a coherent, exhaustive and progressive way, this book presents the tools for studying local bifurcations of limit cycles in families of planar vector fields. A systematic introduction is given to such methods as division of an analytic family of functions in its ideal of coefficients, and asymptotic expansion of non-differentiable return maps and desingularisation. The exposition moves from classical analytic geometric methods applied to regular limit periodic sets to more recent tools for singular limit sets. The methods can be applied to theoretical problems such as Hilbert's 16th problem, but also for the purpose of establishing bifurcation diagrams of specific families as well as explicit computations. ... Read more