Book Description
Real Analysis is the third volume in the Princeton Lectures in Analysis, a series of four textbooks that aim to present, in an integrated manner, the core areas of analysis. Here the focus is on the development of measure and integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. This book reflects the objective of the series as a whole: to make plain the organic unity that exists between the various parts of the subject, and to illustrate the wide applicability of ideas of analysis to other fields of mathematics and science.
After setting forth the basic facts of measure theory, Lebesgue integration, and differentiation on Euclidian spaces, the authors move to the elements of Hilbert space, via the L2 theory. They next present basic illustrations of these concepts from Fourier analysis, partial differential equations, and complex analysis. The final part of the book introduces the reader to the fascinating subject of fractional-dimensional sets, including Hausdorff measure, self-replicating sets, space-filling curves, and Besicovitch sets. Each chapter has a series of exercises, from the relatively easy to the more complex, that are tied directly to the text. A substantial number of hints encourage the reader to take on even the more challenging exercises.
As with the other volumes in the series, Real Analysis is accessible to students interested in such diverse disciplines as mathematics, physics, engineering, and finance, at both the undergraduate and graduate levels.
Also available, the first two volumes in the Princeton Lectures in Analysis:
Customer Reviews:
great book.......2006-10-19
i found the first three chapters of this book very clear and well written. i'd strongly recommend it for someone looking to learn about analysis on the real line.
Good book for reading and as a graduate student.......2006-07-19
Easy to read. My university is using this book to get the graduate students ready for the real analysis qualifying exam. So go ahead and buy this book if you're planning to work on a PhD in mathematics. If you're not planning to work on a PhD in math, this is still a good book to read if you enjoy studying about the real line.
The book begins with measure theory, integration and differentiation. These are included in Chapters 1 to 3. Then in Chapters 4 and 5, we look into Hilbert spaces. This is similar to studying finite-dimensional inner-product spaces, but here, Hilbert space is infinite-dimensional. However, the analysis is very similar. If you know some linear algebra, it should feel like as if you have already read these two chapters.
Finally in Chapters 6 and 7, we see abstract measure theory, including Hausdorff measure, and we study fractals and self-similar sets. And this concludes the book.
Also recommend Walter Rudin's Real Analysis.
Suffers from all the flaws of a 1st edition.......2005-12-18
This book has a lot of problems. Several sections are poorly written/edited. Several important named theorems are not clearly labeled. Also some of the proofs contain typos or errors. The chapter on differentiation is particularly lacking. The chapter is poorly organized and presented. There is also a glaring TeX error in the chapter.
At Princeton this book is used as part of an undergraduate course, and it shows. This is not the ideal book for a graduate level course in real analysis(though I think it would be very well suited for advanced undergrads). Too much time is spent on Lebesgue measure and integration in the first 2 chapters, and abstract measure theory is not intoduced until chapter 6. Also the Monotone Class theorem is lacking from the chapter on abstract measure theory. Also, the book only touches on functional analysis in the two chapters on Hilbert spaces (where they assume all Hilbert spaces are separable).
On the other hand, the presentations of Lebesgue measure/integration and Hilbert spaces in the book are pretty good. The exercises and problems in teh book (when stated properly) are very good and instructive. Overall this book has a lot of potential to be very good, but seems to be suffering from a lack of revision. Hopefully these issues will be fixed in later editions.
Excellent sourse for graduate analysis.......2005-07-03
This book is the best book on real analysis I have ever studied. It does a wonderful job in bridging undergraduate level with graduate level analysis. I have not seen any book that makes measure and Lebesgue theory so easy to understand.
The books begins by defining what a "measure" is all about. And the description is so intuitive and geometrical that you would wonder why you weren't taught it this way before. The book then goes into Lebesgue theory and all of it suddenly becomes so easy.
The book has plenty of wonderful examples and a good set of over 30 problems per chapter.
Elias Stein (one of the authors) is a very renowned mathematician, and one need not worry about the accuracy of the proofs in the book--they are "bullet-proof", and at the same time succinct.
If you are struggling with W. Rudin's book on Analysis, this book is a MUST for you.
Book Description
Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes.
Customer Reviews:
A fast introduction to mathematics in physics.......2006-01-02
The book does not assume prior knowledge of the topics covered. However, the reader will find use of prior knowledge in algebra, in particular group theory, and topology. Compared to texts, such as Arfken Weber, Mathematical Methods for Physics, A Course in Modern Mathematical Physics is different, and emphasis is on proof and theory. The text is reasonably rigorous and build around stating theorems, giving the proofs and lemmas with occasional examples. The style is not the strictest, although making the text more reader friendly, it is easy to get confused with which assumptions have been made, and the direction of the proof. Sometimes only the "if" part is proven.
Students familiar with algebra will notice that the emphasis is on group theory, interestingly the concept of ideals is left mostly untouched. For more on representation theory a good reference is Groups Representations and Physics by H.F. Jones where solutions to some of the exercises can be found, and examples of the use of the fundamental orthogonality theorem applied to characters of represenations.
The first 6 chapters are relatively straight forward, but in chapter 7 Tensors the text becomes much more advanced and difficult. Chapter 10 on topology offers some lighter material but the reader should be careful, these consepts are to re-appear in the discussion of differential geometry, differentiable forms, integration on manifolds and curvature. These are not the most simple subjects and it is clear that they deserve entire courses of their own.
The book has insight and makes many good remarks. However, chapter 15 on Differential Geometry is perhaps too brief considering the importance of understanding this material, which is applied in the chapters thereinafter. The book is suitable for second to third year student in theoretical physics.
Jumping over the Gap.......2005-12-30
Most physicists avoid mathematical formalism, the book attacks this by exposing mathematical structures, the best approach I've ever experience. After reading the first chapter of this books I can assure is a must for everyone lacking mathematical formation undergraduate or graduate.
It surely jumps over this technical gap experienced by most physics opening the gate for advanced books an mathematical thinking with physic intuition.
Unfortunately is very expensive, i hope i could have it some day.
A serious, wide spectrum introduction to modern mathematical physics.......2005-10-10
This book covers almost every subject one needs to begin a serious graduate study in mathematical and/or theoretical physics. The language is clear, objective and the concepts are presented in a well organized and logical order. This book can be regarded as a solid preparation for further reading such as the works of Reed/Simon, Bratteli/Robinson or Nakahara.
Not a review, only a little more information.......2004-12-11
Since I don't yet have this book, I cannot review it; however, I have found the contents of this book on the publisher's web site in case it would help anyone decide to purchase it or not.
Contents
Preface
1. Sets and structures
2. Groups
3. Vector spaces
4. Linear operators and matrices
5. Inner product spaces
6. Algebras
7. Tensors
8. Exterior algebra
9. Special relativity
10. Topology
11. Measure theory and integration
12. Distributions
13. Hilbert space
14. Quantum theory
15. Differential geometry
16. Differentiable forms
17. Integration on manifolds
18. Connections and curvature
19. Lie groups and lie algebras
I will return at a later date to properly review it in case I need to change the rating I gave it.
Book Description
Updated to reflect the latest Graduate Management Admission Tests as it is given in the computer-adaptive format, this manual presents a diagnostic test and five full-length practice tests. Answers and explanations are given for all questions. The manual's extensive subject review sections focus on all six GMAT testing areas: essay writing, reading comprehension, sentence correction, critical reasoning, problem solving, and data sufficiency. The accompanying CD-ROM simulates actual test-taking conditions. It presents a computer-adaptive exam with automatic scoring.
Customer Reviews:
Great Reference Guide .......2007-09-10
This book really helped me to prepare for the gmat. The cd was somewhat useful. The prep test were similar to that of the real gmat, however the explanations for the answers were not that clear.
Poorly Written and Edited.......2007-08-08
After using numerous other GMAT preparation tools from Kaplan, GMAC, and elsewhere, the quality of this product is below subpar. Numerous typographical and logical errors are present in the text, including:
-Questions on reading comprehension sections that were not printed (page 18)
-Duplicate answer choices for a question (page 71)
-An apparenty inability to number questions from 1 to 41 (page 45)
Given the lack of attention to detail along with a number of questionable justifications for answers, I would not recommend this book to anybody. Go with Kaplan, Kaplan800, or Princeton Review instead.
Error-ridden and buggy.......2007-05-06
My husband has been out of school for over 25 years, when he decided he wanted to take a Masters degree, so we needed a really good test prep product. The book is just fine at explaining concepts and ideas, and providing test strategies. But the CD is deeply disappointing. The questions are full of errors, many of them simply a matter of poor proofreading (in one, the greater-than sign is reversed into a less-than sign, making the entire question unsolvable). My husband was two questions away from the end of one of the practice tests on the CD when it froze up. Am I too picky in thinking a test-prep product should have ZERO errors, and software that works consistently? I don't think so!
An O.K. Prep Guide.......2007-03-18
This book is not the best, but too bad either. I ended up NOT using the CD because I didn't want to take the chance of having it screwing up my computer. Now, I am not saying that the CD is defective because I don't know. I am just saying that I was a bit paranoid :) I really should've just bought the edition without the CD. The writing style of the book is good. My beef is mostly with the math review section and some of the testing hints. It has about 90-95% of the GMAT math topics. A few topics are mentioned for the first time in the answer section for the sample tests at the back of the book. There are also some typos in the percentage/fraction conversion tables in the math review section. My biggest beef is that the author forgot to update the testing techniques because he said to "work on famalier topics first" for the verbal section. Hmmm, this advice would've worked on paper-based GMAT, but NOT on the CAT version. The strategy for data sufficiency and the logic behind reading comprehension and critical reasoning are extremely helpful though. Contrary to what some people said about the previous edition of the book, I don't think there is any wrong answer for the questions IN the book (again, I don't know anything about the questions on the CD). Finally, the grammar review section is good and bad at the same time. It is helpful to know the correct grammar for general purpose, but the topics covered in the book are sort of "off" as far as GMAT questions are concerned.
Explantions and buggy software a problem.......2007-03-03
The book was decent with many practice problems, but the explanations were sometimes too complicated and not structured enough for you to truly understand it. The CD is buggy, occasionally it stop and all the answer are lost, which is a major pain. Also, the answers are not saved, so you cannot go back to the ones you got wrong later on. I do have not used any other books, so i am not sure how it compares.
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The Hilbert-Huang Transform and Its Applications (Interdisciplinary Mathematical Sciences)
Manufacturer: World Scientific Publishing Company
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The Hilbert-Huang Transform in Engineering
ASIN: 9812563768 |
Book Description
The Hilbert-Huang Transform (HHT) represents a desperate attempt to break the suffocating hold on the field of data analysis by the twin assumptions of linearity and stationarity. Unlike spectrograms, wavelet analysis, or the Wigner-Ville Distribution, HHT is truly a time-frequency analysis, but it does not require an a priori functional basis and, therefore, the convolution computation of frequency. The method provides a magnifying glass to examine the data, and also offers a different view of data from nonlinear processes, with the results no longer shackled by spurious harmonics — the artifacts of imposing a linearity property on a nonlinear system or of limiting by the uncertainty principle, and a consequence of Fourier transform pairs in data analysis. This is the first HHT book containing papers covering a wide variety of interests. The chapters are divided into mathematical aspects and applications, with the applications further grouped into geophysics, structural safety and visualization.
Customer Reviews:
High quality.......2007-03-09
This is a high quality introduction to what is known as the Hilbert-Huang transform. The chapters are well-prepared and nicely ordered. The examples are valuable for those who might consider using this new technique. The relevant software is not provided.
Book Description
Since the first volume of this work came out in Germany in 1937, this book, together with its first volume, has remained standard in the field. Courant and Hilbert's treatment restores the historically deep connections between physical intuition and mathematical development, providing the reader with a unified approach to mathematical physics. The present volume represents Richard Courant's final revision of 1961.
Customer Reviews:
state of the art in the 50's.......2006-10-17
Beware the german edition is the prewar edition, and the english edition is a
complete overhaul to include modern material at the time. This is not really a
textbook, but a sourcebook of "issues" in partial differential equations. Some
problems have been forgotten now due to the focussing on real, linear equations
and weak solutions (as described, for example, in the books by Hormander). It is
a gem for who wants some taste of the full breadth of theory of PDE's.
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- GOOD for control theory theory
- A Tantalizing Introduction to Hilbert Space
- Very Clear,short and useful
- An unusually readable book on Hilbert space
- Concise and accurate introduction to Hilbert space
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An Introduction to Hilbert Space
N. Young
Manufacturer: Cambridge University Press
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Theory of Linear Operators in Hilbert Space
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Integral Equations
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Introduction to Hilbert Spaces with Applications
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A Course in Functional Analysis
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A Hilbert Space Problem Book (Graduate Texts in Mathematics)
ASIN: 0521337178 |
Book Description
This textbook is an introduction to the theory of Hilbert spaces and its applications. The notion of a Hilbert space is a central idea in functional analysis and can be used in numerous branches of pure and applied mathematics. Dr. Young stresses these applications particularly for the solution of partial differential equations in mathematical physics and to the approximation of functions in complex analysis. Some basic familiarity with real analysis, linear algebra and metric spaces is assumed, but otherwise the book is self-contained. The book is based on courses given at the University of Glasgow and contains numerous examples and exercises (many with solutions). The book will make an excellent first course in Hilbert space theory at either undergraduate or graduate level and will also be of interest to electrical engineers and physicists, particularly those involved in control theory and filter design.
Customer Reviews:
GOOD for control theory theory.......2002-10-02
This book is good to any control engineer who wants to know the background theory of optimization and robust control, but read read an analysis book first.
A Tantalizing Introduction to Hilbert Space.......2002-02-27
Young has done an admirable job at presenting some really beautiful and useful aspects of Hilbert spaces in a manner comprehendable for advanced undergraduates. After reading the book and reflecting on the experience, I'm somewhat amazed at the amount of nice ideas that were presented in such a compact text. The book cannot be compared with more rigorous and comprehensive texts such as Rudin, but you still get all the fundamentals of Hilbert space plus some wonderful applications.
I must strongly disagree with the reader from Sao Paolo who says that chapters 12 and 13 are poorly motivated. These chapters are crucial for the final theorem of the book in chapter 16. Parrott's Theorem in chapter 12 is the key to the foundational Nehari's theorem of chapter 15. Chapter 13 explores Hardy spaces which are the setting place for the major theorem of Adamyan, Arov, and Krein in chapter 16. In fact, I found the movement of ideas from chapter 12 to chapter 16 to be marvelously compelling. These chapters have extreme importance for theoretically oriented control engineers.
Only a modicum of real and complex analysis is necessary to understand the book. Knowledge of measure theory is not required.
Very Clear,short and useful.......2000-10-20
The first eleven chapters are an excellent introduction to functional analysis . Both Hilbert and Banach spaces are introduced carefully. Then there are two short chapters on orthogonal expansions and classical fourier series and then linear operators are studied. From the point of view of a person who is interested in applications to physics and engineering one can say that the book is well motivated mainly because is so compact and because of the many notes on applications. Chapters nine , ten and eleven on Green's functions and eigenfunctions expansions are extremely good. Chapters twelve and thirteen are poorly motivated from the point of view of applications.Finally chapters fourteen to sixteen try to exhibit the applications to complex analysis of operator theory and be helpfull to eletrical engineers.I think the book fails in this. So the ten first chapters of the book are excellent . The remaining less so
An unusually readable book on Hilbert space.......2000-06-12
An unusually readable book on Hilbert space. Very clean notation and very detailed proofs. There are also numerous diagrams. There are also answers to selected problems, but no detailed solutions. If you own one book on Hilbert space, or even functional analysis, this should be it. The author takes great pains to illustrate the ideas involved, not just pound out the theorems.
Concise and accurate introduction to Hilbert space.......2000-01-14
I found this book a concise, well written and accurate introduction to linear algebra. Although some fellow students told me they found it too dry, I had no problem with that.
Book Description
This remarkable book has endured as a true masterpiece of mathematical exposition. There are few mathematics books that are still so widely read and continue to have so much to offer--after more than half a century! The book is overflowing with mathematical ideas, which are always explained clearly and elegantly, and above all, with penetrating insight. It is a joy to read, both for beginners and experienced mathematicians.
"Hilbert and Cohn-Vossen" is full of interesting facts, many of which you wish you had known before, or had wondered where they could be found. The book begins with examples of the simplest curves and surfaces, including thread constructions of certain quadrics and other surfaces. The chapter on regular systems of points leads to the crystallographic groups and the regular polyhedra in $\mathbb{R}^3$. In this chapter, they also discuss plane lattices. By considering unit lattices, and throwing in a small amount of number theory when necessary, they effortlessly derive Leibniz's series: $\pi/4 = 1 - 1/3 + 1/5 - 1/7 + - \ldots$. In the section on lattices in three and more dimensions, the authors consider sphere-packing problems, including the famous Kepler problem.
One of the most remarkable chapters is "Projective Configurations". In a short introductory section, Hilbert and Cohn-Vossen give perhaps the most concise and lucid description of why a general geometer would care about projective geometry and why such an ostensibly plain setup is truly rich in structure and ideas. Here, we see regular polyhedra again, from a different perspective. One of the high points of the chapter is the discussion of Schlafli's Double-Six, which leads to the description of the 27 lines on the general smooth cubic surface. As is true throughout the book, the magnificent drawings in this chapter immeasurably help the reader.
A particularly intriguing section in the chapter on differential geometry is Eleven Properties of the Sphere. Which eleven properties of such a ubiquitous mathematical object caught their discerning eye and why? Many mathematicians are familiar with the plaster models of surfaces found in many mathematics departments. The book includes pictures of some of the models that are found in the Göttingen collection. Furthermore, the mysterious lines that mark these surfaces are finally explained!
The chapter on kinematics includes a nice discussion of linkages and the geometry of configurations of points and rods that are connected and, perhaps, constrained in some way. This topic in geometry has become increasingly important in recent times, especially in applications to robotics. This is another example of a simple situation that leads to a rich geometry.
It would be hard to overestimate the continuing influence Hilbert-Cohn-Vossen's book has had on mathematicians of this century. It surely belongs in the "pantheon" of great mathematics books.
Customer Reviews:
Many beautiful things.......2007-01-12
This is a marvellous book. I will illustrate by one sample from each chapter (except chapter 1 on "the simplest curves and surfaces" which is the least exciting chapter). Chapter 2 on "regular system of points" contains a beautiful derivation of Leibnitz' series pi/4=1-1/3+1/5-1/7+... If we draw a large circle centred at the origin then of course a good measure of its area is the number of integer points it contains. Now, for any such point, x^2+y^2 is an integer less than r^2. So the number of such points can be obtained by going through all integers less than r^2 and counting how many times it can be written as the sum of two squares. But this is a classical problem in number theory and the solution is known. So this number theoretic result essentially tells us the area of a large circle, so it implies an expression for pi, namely Leibnitz' series. Chapter 3 is on projective geometry. We go through many projective configurations that are not seen very often today, but still the classics are the best, such as Desargues' theorem. If we have a triangular pyramid and cut it with two planes to get two triangles then the three points of intersection of the extensions of corresponding sides will or course be on a line (the intersection of the two planes), which is the three-dimensional Desargues' theorem. But by projecting the triangles onto one of the walls of the pyramid we get two projectively related plane triangles and the theorem holds for them also. All we have to do to prove the plane Desargues' theorem is to prove that all such configurations can be obtained in his way (i.e. that one can always erect an appropriate pyramid based on two projectively related plane triangles) which is practically obvious. Chapter 4 is on differential geometry. The fundamental concept of differential geometry is curvature, which is a number that indicates how curved a surface is at a given point. It may be defined as follows. We draw a little circle around the point on the surface and consider all the normals to the surface at these points. Take these normals and put them with their origin at the center of a sphere; then they will sweep out a section of the surface of the sphere. The curvature is the ratio of the area enclosed on the surface and that on the sphere as the circle is taken infinitesimally small. This quantity is seen to be invariant under bending by triangulating the surface; then the the circles are polygons with fixed angles and the theorem follows from the fact that the area of a spherical triangle is determined by its angles (proof omitted here; see any Stillwell geometry book for Harriot's beautiful proof (a.k.a. "Euler's proof")). Now, there are two fundamentally different types of points. Either the surface bends in the same direction in every direction, as on a sphere, or it bends in different directions like a saddle. In the first case the boundary on the sphere traced out by the normals has the same orientation as the boundary on the surface; in the second case the orientation is reversed. So, using signed area, the second type of points have negative curvature. A typical surface will have areas of positive curvature and areas of negative curvature and in between there will be lines of zero curvature. An absolutely wonderful, although perhaps not entirely successful, application of this concept is Klein's Apollo Belvidere hypothesis that the curves of zero curvature on a human face determine beauty. Chapter 5 on kinematics contains a determination of the curve that "we may observe ... every day in cups and tin cans when the light shines on them", i.e. the coffee cup caustic. With the sun at x=-infinity, the radius that makes an angle theta with the x-axis will point to a point where the angle of reflection is also theta. Consider a concentric circle of half the radius, and another circle with the other half of the radius as its diameter. The arc cut out of the inner circle by the radius and the x-axis is equal to the arc cut out of the outer circle by the radius and the reflected ray (arc with central angle theta in the big circle = arc with central angle 2*theta in the small cirlce). The shape of the caustic follows by rolling the outer circle on the inner. The reflected light rays are tangent to this curve since they are perpendicular to the line connecting the generating point with the center of motion (intersection of the two circles). From chapter 6 on topology one nice result is that any continuous mapping of a disc onto itself has a fixed point. For suppose it did not. Then any point in the circle can be connected with its image by an arrow. Now consider the point on the boundary. The arrow direction varies continuously as we walk once around the circle, and it end up where it started so it must have made an integer number of revolutions. But there is also a tangent at each point, and the tangent of course make one revolution as we walk once around. The arrows always point to some point in the disc so they could never point in a direction parallel to the tangent so the arrows in fact have to make one revolution also (they would have to be parallel to the tangent for a moment to overtake it, and if they stood still they would be parallel to the tangent "at six o'clock" so to speak). But if we consider the same situation for a concentric circle inside the disc then it too must have arrows making one revolution because the number of revolutions can not make jumps since the new circle is obtained by continuous shrinking of the circumference circle. But as we shrink this circle to infinitesimal radius then all its arrows point in the same direction, so they don't make one revolution, so we have a contradiction. One sees similarly that a continuous mapping of the sphere onto itself also has a fixed point. Since the projective plane is the sphere with diametrically opposite points identified this proves that any projective transformation has a fixed point.
Don't expect to find it "easy.".......2006-12-24
I agree that this book, co-authored by the co-greatest mathematician of the first quarter of the twentieth century, is a masterpiece to be treasured and kept in print, as other reviewers have stated.
However: The Preface states: "This book was written to bring about a greater enjoyment of mathematics, by making it easier for the reader to penetrate to the essence of mathematics without having to weight himself down under a laborious course of studies."
All I can say is that if you read this and find it "easy," then you have terrific mathematical talent! Yes, the drawings and the intuitive descriptions are helpful, but much of the book is so obscure that I have been told that one of the world's leading geometers is working on an annotated edition explaining what the authors were talking about. On topics which I had already studied elsewhere, I found the presentation illuminating.
I still recommend this book.
Beautiful, Rewarding, and Deep........2003-07-21
I have some 47 books in the geometry section of my shelves. If I had to discard 40 of these, Geometry and the Imagination would be among the 7 remaining.
Geometry is the study of relationships between shapes, and this book helps you see how shapes fit together. Ultimately, you must make the connections in your mind using your mind's eye. The illustrations and text help you make these connections. This is a book that requires effort and delivers rewards.
A glimpse of mathematics as Hilbert saw it.......2001-11-09
The leading mathematician of the 20th century, David Hilbert liked to quote "an old French mathematician" saying "A mathematical theory should not be considered complete until you have made it so clear that you can explain it to the first man you meet on the street". By that standard, this book by Hilbert was the first to complete several branches of geometry: for example, plane projective geometry and projective duality, regular polyhedra in 4 dimensions, elliptic and hyperbolic non-Euclidean geometries, topology of surfaces, curves in space, Gaussian curvature of surfaces (esp. that fact that you cannot bend a sphere without stretching some part of it, but you can if there is just one hole however small), and how lattices in the plane relate to number theory.
It is beautiful geometry, beautifully described. Besides the relatively recent topics he handles classics like conic sections, ruled surfaces, crystal groups, and 3 dimensional polyhedra. In line with Hilbert's thinking, the results and the descriptions are beautiful because they are so clear.
More than that, this book is an accessible look at how Hilbert saw mathematics. In the preface he denounces "the superstition that mathematics is but a continuation ... of juggling with numbers". Ironically, some people today will tell you Hilbert thought math was precisely juggling with formal symbols. That is a misunderstanding of Hilbert's logical strategy of "formalism" which he created to avoid various criticisms of set theory. This book is the only written work where Hilbert actually applied that strategy by dividing proofs up into intuitive and infinitary/set-theoretic parts. Alongside many thoroughly intuitive proofs, Hilbert gives several extensively intuitive proofs which also require detailed calculation with the infinite sets of real of complex numbers. In those cases Hilbert says "we would use analysis to show ..." and then he wraps up the proof without actually giving the analytic part.
If you find it terribly easy to absorb Hilbert's THEORY OF ALGEBRAIC NUMBER FIELDS and also Hilbert and Courant METHODS OF MATHEMATICAL PHYSICS, then of course you'll get a fuller idea of his math by reading them--but only if you find it very easy. Hilbert did. And that ease is a part of how he saw the subject. I do not mean he found the results easily but he easily grasped them once found. And you'll have to read both, and a lot more, to see the sweep of his view. For Hilbert the lectures in GEOMETRY AND THE IMAGINATION were among the crowns of his career. He showed the wide scope of geometry and finally completed the proofs of recent, advanced results from all around it. He made them so clear he could explain them to you or me.
A Book to Put under Your Pillow.......2000-10-20
There might be less than 10 mathematics books in the world that I am glad to put under my pillow when I go to sleep. And this book is one of the top three.
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Fourier, Hadamard, and Hilbert Transforms in Chemistry
Alan Marshall
Manufacturer: Springer
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Quantum Logic in Algebraic Approach (Fundamental Theories of Physics)
Miklós Rédei
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ASIN: 0792349032 |
Book Description
This book is the first to present quantum logic in relation to von Neumann algebra theory. Based on developing quantum logic in terms of operator algebras, the book reconstructs and reevaluates the Birkhoff-von Neumann concept of quantum logic. It also covers recent results such as the violation of Bell's inequality in relativistic quantum field theory, the logical independence of von Neumann lattices and the status of the common cause principle in quantum field theory. Other topics treated include the theory of quantum conditional and statistical inference, an operator algebraic treatment of the hidden variable problem and the semantic approach to physical theories.
Audience: This volume will be of interest to mathematicians, physicists, mathematical physicists and historians and philosophers of science involved in interpretational problems of quantum mechanics and quantum field theory.
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