No comments about Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications.

5 comments about A Friendly Guide to Wavelets.

1. I bought this book when I was particularly interested in continuous wavelet transforms. I found myself flying through the exposition. I felt confident about the knowledge I was acquiring and I was quickly able to apply it. Although I come at this book as a mathematician, I think that it is ideal for engineers and physical scientists who usually have far better grounding in signal processing and related issues in Fourier analysis than do mathematicians. I have recommended this book to students, friends and colleagues with high praise.



2. It is not a friendly guide to Wavelets. However, chapters 1 one to six are excelent. Chapter 7, Multiresolution Analysis, the heart of the matter is hard to understand. So, you get the flavor of Wavelets but not learn to use it.



3. I really did find this lovely book reader friendly! The author has succeeded in communicating the exciting subject of wavelets, and their many applications, to students and more advanced readers alike. He realized that different communities, math, computer science, engineering, and physics,--
   that they have variations in their emphasis, their terminology,
   and their thinking. When authors speak to the various user groups, and do it well,-- like in this case, the result is a cucess. This friendly book is!



4. Some reviewers have commented that this is not such a "friendly"
   guide if you are not friends with upper division or graduate
   level mathematics. There is some truth to this. Unlike the
   pragmatic approach taken in "Ripples in Mathematics" this is
   a mathematical coverage of wavelets.

   For me the value in this book is that it provides a clear
   introduction to the notation and theory behind wavelets. This
   book provides the tools you need to understand the wavelet
   literature better. If you are a software engineer searching
   for a quick guide to wavelet algorithms, this book may disappoint.




5. A classic text *but* look at the graph on the cover very carefully because it is the only one you'll get until you reach the last chapters of the book. In terms of the presentation of the basic theorems and equations, the text is excellent ... BUT ... there is precious little to guide a student to an intuitive and practical understanding of the theory. In practice, one looks at the graphical representations of wavelets quite often -- just like the ones used for Fourier analysis (can you imagine your first Fourier analysis class with no graphs of convolution, impulses, etc? Wavelets *are* Fourier analysis taken a step further in order to handle time varying systems. Graphs are essential to the uninitiated.) In fact, it is helpful to think of wavelets (i.e., wavelet packet decompositions) as a Fourier spectrum with an additional time axis added so you can see how the spectrum changes over time. The "wavelet" method simply optimizes the resolution used at different frequencies. True, it is a little more complicated than that, but it is easy to lose sight of the simple elegance of the topic when so many equations are flung at you without intuitive context. The topic could be presented MUCH better, so don't worry that it takes a long time to get through this material. Read other basic wavelet texts first and then *definitely* come back to this one. You will appreciate its otherwise *excellent* presentation much more and the principles will be easier to put into actual practice rather than mere academic conjecture.

5 comments about Everything and More: A Compact History of Infinity (Great Discoveries).

1. It was Isaac Asimov who once pointed out that to be a great science fiction writer, you must first be a great writer, but the converse doesn't necessarily hold true: you can be a great writer but a poor science fiction writer. The same is true for other genres as well as David Foster Wallace's Everything and More illustrates: he may be a great novelist (though I can't even be positive of that, as I've never read any of his books prior to this one), but he is a mediocre science/math writer.
   
   What is Everything and More about? It is a history of the mathematic concept "infinity". From ancient times, the concept of infinity was troublesome and often worked around. Paradoxes such as Achilles and the Tortoise demonstrated the seeming contradictions of the infinite; for example, this ancient paradox pointed out that to go from point A to point B, you must first go to the halfway point, but to get to that point C, you need to first go to the halfway point between A and C, and so on, ad infinitum. Since there is always another midpoint standing between you and your destination, you can never reach it, but, as anyone who has walked from A to B knows, this seeming impossibility is really possible.
   
   Infinity would be a concept more or less ignored or danced around until the development of calculus made it essential. Even then, for a while, infinity (and the related concept, the infinitesimal) was a shaky idea. Yes, calculus worked, but the foundation it was built on was of uncertain strength. It would take the work of Cantor to finally give infinity its strong theoretical basis; indeed, Cantor is the hero of Everything and More, though he really only appears in the end to clean things up.
   
   Wallace is something of a literary writer, which is not a quality that really fits a math history. He is an occasionally witty and generally wordy writer who is often clever but more often too clever. His constant asides and footnotes are distracting and diminish the clarity that this subject requires. He enjoys abbreviations to the point of annoyance. In addition, a book of this type demands a table of contents or at least an index, but neither are provided.
   
   Reading Everything and More is like going diamond mining. You know there is a gem somewhere, but you need to a lot of work to get to it. In the end, I don't think it is worth the effort. Wallace may be a good writer in other contexts, and certainly this is an interesting concept, but he is not the right man for the job. If you want to see what good math or science writing is like, read Martin Gardener, Isaac Asimov, Carl Sagan, Brian Greene or a dozen others; if you want to see how not to write on such subjects, Everything and More is an ideal example.



2. this book offers no recommendation for what mathematical principles a reader should be familiar with before starting it but any claim of it being accessable to an average reader would be misleading.
   if seems not only like no attempt was made to relate most of what is being described to any commonsense foundation, but that it was academically overwritten into a code that even someone who already knew all the information contained in the book would have trouble following. in my ironic experience the "emergency glossary" definitions themselves contain more undefined or ambiguous terms than any other part of the text.



3. I suppose this might just be his style of writing but I just can't stand it. Having read 9 other math related books over the past month, this was a huge disappointment. He uses all sorts of acronyms and idiosyncrasies that just go too far. I got half way through it and then decided to skim seeing if I could find anything that caught my eye. Thinking maybe his discussion of the Continuum Hypothesis should be good, I read that. Of course, he misstated it, confusing which equality was known and which was hypothesized. This doesn't seem huge, but its just silly that in a book about infinity, DFW states one of the most important undecidable hypotheses in all of math incorrectly and actually presents something that is easily provable (c=2^N0). Why not just one star? He did get me to read 100 pages...



4. I (and many of my professional scientist colleagues) thought Gleick's "Chaos" was one of the worst books ever written on math - so confusing and uninstructive it called the whole subject into question. So it is not surprising Gleick praises this book: it is worse than "Chaos". The grammar, punctuation, and style are so tangled I found myself rereading passage after passage to sort out Wallace's meaning. He uses dozens of obscure, undefined, unusual, and unobvious abbreviations, with the index to them lost in the text, and no index at all to the book as a whole, which is very negligent for a technical work. There is no organization into chapters, just numbered sections which do not coincide with any natural divisions in the material. "Stream-of-consciousness" writing may do for Joyce (though he was not known for lucidity), but it is hopeless for presenting technical material. Many of Wallace's explanations explain nothing: "Fourier Series is vital to understanding transfinite math", he writes, and then blows the subject off with a jest (p 115). And there are plain errors: "when n<0, (p+q)^n becomes the Binomial Theorem" (p 117). Finally, the subject-matter itself is questionable: modern mathematicians still regard infinity as an intractable concept that leads to preposterous contradictions, as Archimedes and Galileo found and as Wallace's own examples demonstrate. "Is the area of an infinitely-long and wide sheet of paper infinity squared?" "Are some infinities bigger than others?" If questions like these have cogent answers at all, it is going to take someone more coherent than Wallace to explain them.



5. I was expecting an exciting book.
   I was disappointed.
   This book has no chapters, lots of text message abbreviations, and many phrases ending in a period.
   
   Three-quarters of this book is background information.
   
   When the payoff comes, actually talking about infinities,
   the reationship among alelf null, cardinality c, and alef 1
   is left as a "problem for the reader" for 20 pages!

5 comments about A Primer on Wavelets and Their Scientific Applications, Second Edition (Studies in Advanced Mathematics).

1. This is simply the best book I have come across on introducing wavelets.

   I am sure that within the first 20 pages, which are easy to understand and make for a very quick read, you will begin to see the beauty of this theory and will applaud the author's exposition.

   While this book does not need much more than basic linear algebra, the author does not shy away from the mathematics where necessary - he simply motivates it by providing an intuitive understanding of the equations, so it's easy to follow.

   In the very first chapter, he describes the wavelet method using examples that can be worked out by hand. (This is also mentioned in another review and contributed to me buying this book. I was doing research on another wavelet book on the site when I came across this book and it's reviews.) This helps to fix and brilliantly clarify the main ideas behind the theory. Armed with this knowledge, the reader can better appreciate the more sophisticated wavelet functions. But, the basics would be firmly planted by this book. This is rarely seen in other books dealing with this thoery.

   This book is great for someone who wants to learn about this topic. It also is an excellent book for those who have an advanced graduate degree in applied mathematics since it demonstrates how to truly understand complex concepts.

   The discussion is motivated with real world examples such as removing noise from signals, image enhancements, etc. These are useful examples that you can relate to. There is enough in this book and the downloadable software that you may want to undertake meaningful projects yourself. That is the confidence that you get from this book.

   An outstanding quality is that the book is thin. This is a strong motivating factor since it immediately sends the message that "this method can't be that hard to grasp if the book is so short." And, this subliminal message keeps your spirits up as you read this masterpiece.

   Wavelets is a mathematically intensive technique, and it seems that most authors want to show how "macho" they are by shrouding the basics under a heavy cloak of complex equations. But, true "machoism" is displayed by how deftly an author can bring a difficult topic to the lay person. James Walker does this remarkably well.




2. This lovely little book helps the novice to get an idea of the math which underlies wavelets;-- and at the same time to learn how one readily gets hold of software that is convinient,-- that will make it easy for anyone to start playing around with it. The author also explains in plain English the wavelet aspects, and some of the mathematical constructs, behind audio denoising, signal compression, image recognition, speech recognition and more.



3. This is the best introductory level book on wavelets I have read. It is written beautifully and is one of the few books that provides insight as to how and why wavelets are useful.

   Everyone should start out their investigation of wavelets with this book first. Highly recommended.




4. At times, particularly at the beginning, the author seems almost patronizing, but generally the tone and level of the book is just where it should be for what seeks to be a primer. However, it is unfortunate that one of the primary functions in the software is not more fully described in the text; it is only described as the method of preparing many of the examples.
   
   Overall, an excellent start for somebody who knows little or nothing about wavelets.



5. This book is an excellent one to learn the fundamental concepts of the wavelet transform. I recommend to study this book before starting or while studying another more mathematical text.

4 comments about Theory and Application of Infinite Series.

1. Excellent book for consulting with lots of examples and problems. Very well written but with the problem of very old notation. Everything you need to know about series is in this book. Very good to use in problems seminars



2. The last chapter on the Euler-MacLauren summation formula, and attendant interrelations among the Zeta function, Bernoulli Numbers and Bernoulli Polynomials is alone worth three times the price of this gem. Chock full of recipes and explanations of many of those little annoying points you don't understand fully. Do you REALLY understand what 'asymptotically equal to (~)' means? Heartily recommended!



3. To be honest alot of the work does not make immediate sense. Knopp leaves alot of important details out in his proofs and sometimes tends to rite to informal a proof of theorems. Theres no answers to the questions and i found that i became bored while reading the book. There are better. I have a large collection of books and the infinite series sections or chapters in them are better than alot of this book. I think that its main problem is that it cannot be easily accessible to beginners although it claims this. well apart from the begin chapters.



4. Anything to do with "infinity" is fascinating. Much of the history of mathematics has been a duel between those who see "infinity" as a delusion and impediment to progress, and those who see it as the greatest tool in the mathematician's toolbox. Infinite series, which may be loosely defined as sums of an infinite number of terms (numbers), take on some of this fascination. Although this book will appeal mainly to the professional mathematician, there is enough historical and elementary material to profit many college students- and possibly even some high school students.

   Professional mathematician will find this book useful for filling in gaps left by topics not covered in traditional courses. An example is the detailed discussion of Euler's summation formula, which goes far beyond the simplified form usually encountered in textbooks. Another fascinating topic covered is divergent series, and methods by which meaningful sums can be assigned to these. There is something counterintuitive -- and, frankly, mind-boggling -- about many of these results.
   
   Mathematicians can be put into several categories: 1) applied-mathematicians/computer-scientists/engineers concerned with solving practical problems, 2) those concerned with pedagogy and the history of mathematics, 3) epistemology and rigorous proofs, and 4) formalists. The fourth category, formalists, is difficult to define, but may be described as those that emphasize obtaining new results through formal (technical) manipulations, without undue concern regarding the meaning of the intermediate steps. The greatest exponents of this art were Euler and Ramanujan, though Fourier, Dirac and Heaviside are also solid members of this camp.

   I take this digression because I feel that this book mainly appeals to the fourth type of mathematician. Although there are some general results in the theory of infinite series, any competent mathematicians can, in a few minutes, write a dozen infinite series which defy summation. As an example, the series associated with the Riemann zeta function of EVEN arguments were first summed by Euler. The sums arising from ODD arguments have defied summation to this day. Why this should be so is intriguing, but unknown. Incidentally, Euler's method of summation will make a "rigorists" hair stand upon ends. But he got the job done!

4 comments about Infinity: Beyond the Beyond the Beyond.

1. Lillian Lieber and her husband Hugh created some of the most wonderful books in the fields of mathematics, logic, and relativity. Although some of my fondest childhood memories are the hours I spent trying to fully grasp the meaning in her books, I find these same books to be no less enjoyable today as an adult. I cannot recommend her books highly enough.



2. As an Army brat, I found this book in the school library on the Naval base in Tianan, Tiawan in 1958.

   As a 10th grader with a fondness for math, it was great. I think I'd seen a little bit about transfinite numbers in George Gamow's "1 2 3 Infinity", but this was an amazing tour of transfinite numbers, written so it could be understood by T C Mits. I learned a lot from it -- a real mind stretcher. I later recognized other books by the same author by the illustrations -- If you know her other books, nothing more need be said.

   I've not seen the book in over 40 years, but decided I needed to find a copy -- it's one of the favorite books I read before college. I was looking at my copy of "The Education of T.C.Mits" and decided to see what I could find.




3. This is a great book. I first found it in my high school library. For the uninitiated, who would have thought there were different levels of infinity? This book explains infinity in a readable and entertaining way. It is too bad this book is out of print as I suspect it would still be in high demand. It would make a great title for a book club. Somebody needs to republish it!



4. Beware! This is not Lillian Lieber's original work. It has been abridged. Approximately one third of the original text and presumably the drawings have disappeared. In the forward, Barry Mazur, states plainly that he zapped Lillian's preface, chapter 1, one half of chapter 17, and all of chapters 18 through to 24. Gone is Lillian's introduction to SAM, Lillian's spirit creature of Science, Art, and Mathematics. Why did Mazur do this? He thought the Liebers digressed too much. He wanted them to stay on track with the main subject, transfinite mathematics. He thought that some of their worldly concerns speak less to a modern audience than they did to their readers in 1953. However we have to take Dr. Mazur's word for it, as the sections are deleted and you can no longer judge for yourself. Despite my misgivings I give a 5 star rating as what is left is still beautiful. However you may wish to try the used book market to get the original version.

5 comments about Infinity and the Mind: The Science and Philosophy of the Infinite (Princeton Science Library).

1. What is infinity? How do we train our minds to understand the idea? This one of the hardest questions to answer for non-professional mathematicians, and one that Rucker address superbly - and, believe it or not entertainingly in this excellent book. And once you think you grasped that, how about a higher level infinity? Next one? Infinite series of higher level infinities? Sound very scary, and it is. It takes an amazing capacity to explain these concepts to a (relative) layman, and Rucker has it in abundance. An exhilarating intellectual tour de force, perhaps comparable to climbing mount Everest - infinite number of times, with deep philosophical, and perhaps, religious connections, presented in a light, funny, and yet rigorous manner. The book also provides a history of the concept of the infinite, and interesting people who developed it. A must read for a curious mind.



2. Rudy Rucker, son of a cleric and mathematics whiz kid, produced this book on `Infinity and the Mind' years ago, but reading and re-reading it, I continue to get insights and the chance to wrap my mind around strange concepts.

   `This book discusses every kind of infinity: potential and actual, mathematical and physical, theological and mundane. Talking about infinity leads to many fascinating paradoxes. By closely examining these paradoxes we learn a great deal about the human mind, its powers, and its limitations.'

   This book was intended to be accessible by those without graduate-level education in mathematics (i.e., most of us) while still being of interest to those even at the highest levels of mathematical expertise.

   Even if the goal of infinity is never reached, there is value in the journey. Rucker provides a short overview of the history of 'infinity' thinking; how one thinks about divinity is closely related often, and how one thinks about mathematical and cosmological to-the-point-of-absurdities comes into play here. Quite often infinite thinking becomes circular thinking: Aquinas's Aristotelian thinking demonstrates the circularity in asking if an infinitely powerful God can make an infinitely powerful thing; can he make an unmade thing? (Of course, we must ask the grammatical and logical questions here--does this even make sense?)

   Rucker explores physical infinities, spatial infinities, numerical infinities, and more. There are infinites of the large (the universe, and beyond?), infinities of the small (what is the smallest number you can think of, then take half, then take half, then take half...), infinities that are nonetheless limited (the number of divisions of a single glass of water can be infinite, yet never exceed the volume of water in the glass), and finally the Absolute.

   `In terms of rational thoughts, the Absolute is unthinkable. There is no non-circular way to reach it from below. Any real knowledge of the Absolute must be mystical, if indeed such a thing as mystical knowledge is possible.'

   At the end of each chapter, Rucker provides puzzles and paradoxes to tantalise and confuse.

   * Consider a very durable ceiling lamp that has an on-off pull string. Say the string is to be pulled at noon every day, for the rest of time. If the lamp starts out off, will it be on or off after an infinite number of days have passed?

   Rucker explores the philosophical points of infinity with wit and care. He explores the ideas behind and implications of Gödel's Incompleteness Theorem, and leads discussion and excursion into self-referential problems and set theory problems and solutions.

   He also discusses, contrary to conventional wisdom, the non-mechanisability of mathematics. We tend to think in our day that mathematics is the one mechanical-prone discipline, unlike poetry or creative arts and more 'human' endeavours. But Rucker discusses the problems of situations which require decision-making and discernment in mathematical choices that no machine can (yet!) make.

   * Consider the sentence S: This sentence can never be proved. Show that if S is meaningful, then S is not provable, and that therefore you can see that S must be true. But this constitutes a proof of S. How can the paradox be resolved?

   This is a beautifully complex and intriguing book on the edges of mathematics and philosophical thinking, which is nonetheless accessible and intellectually inviting. You'll wonder why math class was never this fun!




3. This book is spooky! I love it! The infinite is such a fascinating topic. I contemplate it on a daily basis and wrestle with some of its concepts almost constantly. This book is a must read for anyone who has ever wanted to know more on this subject. Robot conciousness is an incredible topic and really makes one think about what conciousness really means. Some of the concepts are difficult to relate to, but give it time and open your mind. If I could, I would give this book infinity - 1 stars!



4. Rudy Rucker deals with the concept of Infinity in regard to our mental conceptions and the structure of reality. The question is whether or not the concept of Infinity makes sense, and then the relation of finite thought and human consciousness relates to the possibility of infinites in the structure of reality.
   
   Rucker is a professor of Calculus and centres this discussion in the History of Mathematical stemming form the ancient Greeks. For the Greeks there was no distinction between mathematics and philosophy. He takes a mathematical approach, but converses fluently in the disciplines of Quantum Physics and Philosophy.
   
   I classified this book as Epistemology (the Philosophy of Knowledge) because the central concept is the meaning and definition of Human Consciousness. In this regard Rucker probes the meaning of consciousness and the relationship of the individual mind to the concept of Universal Mind.
   
   The title includes Infinity, because the investigation considers all aspects of the ultimate or Absolute. So at the root of this is the question of whether it makes sense for anything to be Infinite. Is there such a thing as Infinity? Are there multiple infinities? Involved is the question of whether the human mind can conceptualize an infinite thought, or is every human thought a finite thought?
   
   The reason this is a question of Epistemology is that one must consider how we know, and what a finite mind can know. Thus Rucker looks at the question in terms of many disciplines of knowledge. Basically, we are asking whether it is possible for something in the universe (one mind, and its thoughts) to know the Absolute or Ultimate reality, of which it is a part!
   
   Another term for the discipline commonly dealing with this problem is Theory of Mind. Rucker I have not looked at the concepts of Theory of Mind and the Philosophical question of the Absolute and the One-Many debate in a mathematical perspective before. This latter entails the concept of whether there is some ultimate unity to the universe, including the recent question of multiple universes, and whether the Absolute is sentient, as an active God or relatable entity.
   
   Rucker points out that any ultimate question, posed in terms mathematical, theological or otherwise, is a mystical question. He references concepts of Zen Buddhism as well as classical Western Philosophy and Christian theology. He lays a firm foundation for the problem in a historical format by reviewing the ancient Greek concepts.
   
   I had never looked at these questions from a mathematical approach. His discussion of set theory helps to see the issues involved in considering whether humans, as finite entities, can conceptualize the ultimate. He deals with the relationship between thoughts and concepts and the external objective world. Set theory and its refinements, which Rucker discusses in terms of the history of their development, provide a way of objectively evaluating whether there can be infinite.
   
   Rucker lays out the formulas in geometry and calculus, but discusses the implications from practical and theoretical perspectives in science and theology. I did not campout in the mathematical formulas, but could generally follow the arguments. But the philosophical implications and the factors discussed in the practical and theoretical scientific disciplines was very helpful. Rucker uses very practical life-situations and analogies to provide a reality for these concepts, which can seem ethereal and abstract.
   
   One of the practical aspects is a whole chapter critically evaluating ideas of Artificial Intelligence, "Robots and Souls." He asks whether an artificial intelligence can become self-developing to the stage comparable to human consciousness. He ruminates on the relationship of artificial intelligences to human consciousness.
   
   Rucker reviews the creative and ground-breaking theories and writings of Kurt Gödel, a mathematical philosopher in the 20th century. Gödel conclusively established the concept of Infinity. Rucker reports on personal discussions he had with Gödel, who was a mystic and philosopher. They discussed the concept of Universal Mind and the existence of mind beyond body.
   
   It was also interesting to see this perspective on the Theory of Mind, various concepts of the Absolute, and critical analysis of the possibilities and limitations of human conception, as written almost 25 years ago, and see that most of what is known and considered now was active knowledge back then. The critical analysis Rucker provides was helpful for a fresh perspective on the methods mathematics brings to metaphysics involved now in Particle Physics and the Cosmogony now entailed by Theoretical physics on the astronomical level.



5. Rucker had finished writing this by June 19, 1981, as his preface says. Yet, he has the naivete (or perhaps the gall) to say something inane like "Set theory is, indeed, the science of the Mindscape. A set is the form of a possible thought." on p. 41. Since Zadeh published his landmark "Fuzzy Sets" paper in 1965, and Black and others had written similar ideas years earlier, along with multivalued logicians like Lukasiewicz developing possibly infinite-valued logics as far back as the 1920s, one would think that Rucker would be informed or wise enough than such statements. It appears otherwise. I find it curious that Rucker also knew Godel who did work in multivalued logics, but basically Rucker doesn't acknolwedge multivalued logics as even possible forms of thought.
   
   As for the comments about Mr. Rucker qualifying as an intellectual descendent of Hegel, they simply don't hold water. Rucker denies the property of contradiction (it is not the case that A and not A hold). Hegel accepted it and sought some other way to do logic than Aristotle's logic.

5 comments about The Infinite Book: A Short Guide to the Boundless, Timeless and Endless.

1. If you are interested in infinity and you are not familar with Cantor or Borges' "The Library of Babel", then you may be amazed by this book. Otherwise, you can find it too light. Probably good as a light summer reading.
   
   Infinity is a fascinating subject, and I thought that this book would contain a lot of interesting information in its 300 pages. I have found many quotations, a lot of superficial theology and ethics, and little information on the concept itself. I missed more depth in handling the mathematical concepts.
   
   Anyway, there is a very good part of the book (from my point of view) devoted to eternal inflation and simulated universes, especially for how the theories are introduced and chained. Even if it is not strictly related to infinity, it is the best part of the book. The chapter that describes Cantor's works is worth reading too.



2. I have not been disappointed by any of John Barrow's book so far. He has a unique gift of writing with exceptional clarity about difficult topics. This is not a typical cosmology book, but large portion is devoted to beginning, shape and future of The Universe.
   Like in his previous "Book of Nothing", author mixes philosophical and scientific musings about infinities (big and small) affecting theology, mathematics, cosmology, physics (TOE) and our existence.
   I found Georg Cantor's life and his quest for understanding "absolute infinity" (God?) quite interesting and emotional. And check how Blaise Pascal argued about believing (or not) in God, because of infinite gain (or loss!!).
   One truth emanates from "The Infinite Book": we are far, infinitely far from knowing the truth about everything (Immanuel Kant's rings the bell!). The more we learn the bigger infinite number of questions surface in front of us. Are we nearing the limits of knowledge? Professor John Barrow does not suggest it has come to this, but read about them and enjoy stretching your mind.



3. This book discusses infinity. This concept has a precise definition in mathematics and since the times of Cantor we know that there are various degrees of infinity, one of the most interesting problems being whether there in an infinite between the cardinal of the natural numbers and that of the real numbers, the so called continuum hypothesis, which was proven to be undecidable in the usual Zermelo-Frankel-Choice axioms of set theory.
   In recent times, cosmologists, whether those adopting the inflationary scenario or those favouring the cyclic universe, are pondering whether the universe is infinite in space and possibly eternal in time (although some believe it had a beginning about 14 billion years ago, but may never end).
   So the topic of the book is pertinent to our age.
   Naturally, the idea of infinite is also related to the idea of God, although this is not a scientific subject, but possibly a philosophical one.
   The first part of the book is a hystorical review of the concept of infinity, from Zeno and Aristotle to Kant and Cantor, via St. Augustine. A very entertaining chapter is the one about the Hotel Infinity and all the challenges that the manager meets, quite successfully and that would be impossible in a hotel with only a finite number of rooms. The second part of the book deals more with physics and cosmology, things like the singularities at the center of black holes. It is interesting to learn that an English astronomer of the 16th century already proposed that the universe is infinite. The question of the possible topologies of the universe is discussed, although we do not know yet the answer. The important distinction between the observable universe and the universe as such is made in page 139 where the radius of the visible universe is stated to be 42 billion light years (which seems to be the correct figure if we take into account the expansion of the universe since the light emitted 14 billion years ago has reached us). Unhappily , the drawing in the next page will confound the lay reader because the radius is pictured at 14 billion light years. (There are also some other minor mistakes in the book, which would have been avoided by a careful reviewer before publishing. Another example is the graph in page 190 which suggests that expansion of the universe is decelerating, contrary to recent data of supernovas). Naturally, the limit on how fast information can spread will probably preclude us from knowing whether the universe is infinite unless we can get some degree of confidence on some basic theory that predicts this infinity.
   The book also discusses interesting problems regarding the impact on ethics of inmortality and the possibility of clones in an infinite universe (Vilenkin has explored also this idea in one of his books). Physicists have changed their views on the universe in the last 30 years when it was hoped that The Theory of Everything would be mathematically unique and would determine one universe. Instead, superstring theory has landed with a whole landscape of possible universes. So the question remains, how we happen to live in such universe that has made it possible for life to appear (at least in the Earth, possibly in many other planets) and to develop a self-conscious and inquisitive species by means of which the universe interrogates itself? The diverse answers are tabulated in page 186.
   It also has another chapter on virtual reality "à la Matrix" (simulated universes) and it also discusses the possibility that advanced civilizations are capable of cultivating universes, the way we grow cornfields or build cities.
   Another of the subjects discussed by the author is that of machines capable of supertasks . I found very interesting the 4-body configuration discovered by Xia in 1971 that , according to Newton's theory , sends the 4 bodies at infinite distance in finite time. Einstein's general relativity doesn't allow this, so that infinities did appear not only in quantum mechanics, but also in newtonian mechanics.
   One of the important conclusions of the book is that the human race is not necessarily equipped to know all things that are true about the universe. "We have no special right to expect that all truths about the Universe can be tested by observations that are within our reach: that really would be an anti-Copernican outlook" (page 198).
   The book is an eye opener for those readers not familiar with the role of infinity in the mathematical and physical sciences, but if you look for definite answers about these difficult problems you will not find them here (not in other books, of course).



4. This is a marvelous book. Infinity is a tough concept to wrap your mind around, but Prof. Barrow makes it as understandable as anyone possibly can. He goes deep into the subject -- this is by no means a superficial treatment -- yet never loses an attentive reader. His brilliant final chapter, dealing with the theory of time travel, will knock your socks off. Barrow has a nice wit, and as a special bonus he has seeded the text with some intriguing Briticisms that were new to me, but happily understandable from the context.



5. Barrow is one of my favorite authors, and his lucid explanation of fascinating topics is always a pleasure.

4 comments about Continued Fractions.

1. This is Khinchin's classic work, translated from Russian in the 1930's. Although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times, the book moves at a truly alarming pace, and the book is not suitable to be used ALONE as an introduction to continued fractions. To supplement this book if this is a first exposure to continued fractions, I would recommend C.D. Old's book, which has many more examples which can be worked through until the reader is comfortable with the topic.

   The book is brilliant and necessary for understanding continued fractions, but can't stand alone without supplemental material unless one is a professional mathematician. Khinchin frequently employs contrapositive proof formats, and there are occasional translation errors from Russian. The errors range from minor (awkward usage) to major (in one place, the translation is "negative" when it should be "non-negative", which confused me for half a day).




2. A Y Khinchin was one of the greatest mathematicians of the first half of the twentieth century. His name is is already well-known to students of probability theory along with A N Kolmogorov and others from the host of important theorems, inequalites, constants named after them. He was also famous as a teacher and communicator. Several of the books he wrote are still in print in English translations, published by Dover. Like William Feller and Richard Feynman he combines a complete mastery of his subject with an ability to explain clearly without sacrificing mathematical rigour.

   In this short book the first two chapters contain a very clear development of the theory of simple continued fractions, culminating in a proof of Lagrange's theorem on the periodicity of the continued fraction representation of quadratic surds. Chapter three presents Khinchins beautiful and original work on the measure theory of continued fractions. The proofs of the theorems in this chapter are also entirely elementary.




3. You won't find many books on such an out-of-fashion theme as continued fractions, will you? Even less on the arithmetic side of the theory. Yes, it's true, many texts on elementary number theory provide a chapter or so about the subject, but if you want to gain a reasonably thorough picture of the field, without dwelling so much on details, you've got to resort to Kinchin's "Continued fractions": readable (no more mathematic needed than basics of analysis), complete (all fundamental conceptual aspects dealt with, included measure theory and implications on irrational numbers), brief (less than a hundred pages with virtually no applications - not even to Pell's equation!) and LIVELY in style.
   All in all a very good start for understanding this profound mathematical tool.



4. A wonderfully written, clear exposition of advanced material which, however, begins simply enough to lure one in.

4 comments about Schaum's Outline of Fourier Analysis with Applications to Boundary Value Problems.

1. Very helpful to me in my medical imaging and signal processing assignments! A good buy for any student of engineering or science, particuarly useful to the study of signal analysis.



2. This text is a good supplement to understanding the use of Fourier analysis and how it is used in real-world applications. The explanations are to the point and the solved problems are all fairly easy to follow.

   At the end of the chapter, there are exercises to test your knowledge, and most of the answers are in the back of the book. Modeling the exercises on the problems, you can usually work out what you should do for the exercise.

   This is a good study guide.




3. This Schaum's outline is unique in that you not only get a thorough coverage of Fourier analysis, but of other orthogonal functions such as Bessel, Legendre, Hermite, and Laguerre. The first chapter would be interesting to students of partial differential equations because of its excellent treatment of boundary value problems and of the different types of partial differential equations. It makes an interesting first taste of PDE solution methods. Chapter two is a traditional treatment of Fourier series and its applications. As in the first chapter, it is the applications that make the chapter unique as the Fourier series is used to solve problems in heat flow, Laplace's equations, and vibrating systems. Chapter three has a good discussion of why you would actually care if a function is orthogonal. Chapter four discusses special functions and how they are evaluated. Chapters five and six are all about applications of the Fourier integral and of the Bessel function respectively. Chapter seven uses the Legendre functions to solve problems such as finding the potential interior and exterior to a sphere given a specific charge distribution. Chapter eight finishes the guide with a discussion of Hermite and Laguerre polynomials, more from a properties standpoint than from an applications standpoint. The reader of this guide should already be knowledgeable of Calculus and differential equations, and should probably have some kind of background in physics or engineering to get the most from the book. It would be a good supplement for the student of partial differential equations or signal processing as well as the student of Fourier analysis. I think what really sets this book apart is its ability to act as a stand-alone guide to the student with the required prerequisites and to actually to inspire you to study applied mathematics more. As for my own story, I picked up a previous edition of this book 17 years ago for a coworker that thought it might be helpful in a class he was taking. He decided he didn't want it and so I began thumbing through it. I found the applications sections to be so interesting and inspiring that I wound up going back to graduate school and eventually picked up three master's degrees! If you like this Schaum's outline, you might also want to pick up "Schaum's Outline of Advanced Mathematics for Engineers and Scientists" by the same author. It is also full of mathematics inspired by real world problems in need of solution.



4. This book have a several exercises.
   IF you prepare a test, i recommend you purchase this book