Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

Mathematics: Queen and Servant of Science (Spectrum) Review

Mathematics: Queen and Servant of Science (Spectrum)
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Mathematics: Queen and Servant of Science (Spectrum) ReviewI will keep this very brief. I am an electrical engineer and computer scientist, and, after a couple of decades in the "commercial world" , am always amazed at the "unreal effectiveness" of mathematics when applied to this and the real world. Bell's book captures this essence in a timeless tome that must be required reading and regular re-reading for all aspiring. and indeed, practising, mathematicians, engineers and scientists. It is a source of both inspiration and "bringing back to earth" to those that read it. No serious practitioners library should be without it.Mathematics: Queen and Servant of Science (Spectrum) Overview

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Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) Review

Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics)
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Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) ReviewThis book has two principal aims. In the first half of the book, the aim is the study of discrete time and continuous time Markov chains. The first part of the text is very well written and easily accessible to the advanced undergraduate engineering or mathematics student.
My only complaint in the first half of the text regards the definition of continuous time Markov chains. The definition is introduced using the technical concepts of jump chain/holding time properties. This doesn't tie out well with the treatment of the discrete time case and may seem counter-intuitive to readers initially. However, the author does establish the equivalence of the jump chain/holding time definition to the usual transition probability definition towards the end of Chapter 2.
The second half of the text deals with the relationship of Markov chains to other aspects of stochastic analysis and the application of Markov chains to applied settings.
In Chapter 4, the material takes a serious jump (explosion?) in sophistication level. In this chapter, the author introduces filtrations, martingales, optional sampling/optional stopping and Brownian motion. This is entirely too ambitious a reading list to squeeze into the 40 or so pages allocated for all of this, in the opinion of this reviewer. The author places some prerequisite material in the appendix chapter.
Chapter 5 is a much more down-to-earth treatment of genuine applications of Markov chains. Birth/Death processes in biology, queuing networks in information theory, inventory management in operations research, and Markov decision processes are introduced via a series of very nice toy examples. This chapter wraps up with a nice discussion of simulation and the method of Markov chain Monte Carlo.
If the next edition of this book removes chapter 4 and replaces it with treatment of an actual real-world problem (or two) using genuine data sets, this reviewer would be happy to rate that edition 5 stars.Markov Chains (Cambridge Series in Statistical and Probabilistic Mathematics) Overview

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Probability Through Problems Review

Probability Through Problems
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Probability Through Problems ReviewDoing problems is the best way to study mathematics. The question to most authors is that: How to select problems that are nice to the students. This book did quite well. As I see, the authors gave the exact right pace to offer the students an exicting course, and give the students a thorough understanding.
If you want to master this subject in a month, or even in a shorter time, this book may be the right choice.Probability Through Problems Overview

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Discussion of the Method: Conducting the Engineer's Approach to Problem Solving (Engineering & Technology) Review

Discussion of the Method: Conducting the Engineer's Approach to Problem Solving (Engineering and Technology)
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Discussion of the Method: Conducting the Engineer's Approach to Problem Solving (Engineering & Technology) ReviewThis book came recommended to me by a professional engineer. As I started turning the pages during a first skim read, it struck me that Koen has brought together a huge amount of experience on engineering with a deep understanding of philosophy (to his credit, both Western and Eastern) plus a range subjects from classical literature, world religion and the vagarities of world languages and forged them into a brilliant synthesis of remarkable clarity and originality.
His central thesis is "All is heuristic" (All is rule of thumb). He has surrounded this argument with a phalynx of other heuristics (59 in total)that range from the practical (e.g., at some point in the project, freeze the design) to the metaphysical (e.g., sincerity of belief and the inability to disbelieve are poor justifications for claiming that a belief is true) to the paradoxical (e.g., if a concept produces paradoxes, unexplained complexities or unexpected departures from expected results, better consider it a heuristic)
In writing this book, Koen has both mastered and melded a number of seemingly imiscable disciplines - philosophy, linguistics, theology - with his own professional field of engineering (he is professor of Mechanical Engineering at University of Texas at Austin and a fellow of the American Nuclear Society). It is reminiscent of the way that Thomas Acquinas reconciled Christianity with Philosophy.
This is no mean feat, and Koen's book, unpretentiously entitled "a discussion", is an intellectual tour of the first order. Of course, his many references mean so much more if you are familiar with them. If you haven't, be sure to look them up - your life will be immeasurably enriched. In any event, Koen illuminates a path to greater understanding. His prose is very engaging and the book is well suited for general audiences. It is one of those books that begs to be read and re-read.
One can only wonder if Koen's book had been available earlier (it was published in 2003), would we have been faced with such disasters as Challenger and Columbia, as the Ford Explorer, Chevy Corvair, Ford Pinto, Bhopal, etc. A little more humility with the inherent uncertainties of engineering life might have made a positive difference. This book strives hard - and I believe succeeds - at doing just that. Bravo, Professor Koen for shedding new light on old problems. It comes as no surprise that Koen has won high awards for teaching excellence (W. Leighton Collins and Centennial Medallion) - both from the American Society of Engineering Education.Discussion of the Method: Conducting the Engineer's Approach to Problem Solving (Engineering & Technology) Overview

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Isabelle/HOL: A Proof Assistant for Higher-Order Logic (Lecture Notes in Computer Science) Review

Isabelle/HOL: A Proof Assistant for Higher-Order Logic (Lecture Notes in Computer Science)
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Isabelle/HOL: A Proof Assistant for Higher-Order Logic (Lecture Notes in Computer Science) ReviewIsabelle is fantastic, and this is an excellent tutorial.
With Isabelle, all the mystery of math and proof goes away
and everything becomes concrete... just like programming.
The only improvement that I'd like to see is that the
tutorial be rewritten using the Isar proof language.Isabelle/HOL: A Proof Assistant for Higher-Order Logic (Lecture Notes in Computer Science) Overview

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Proofiness: The Dark Arts of Mathematical Deception Review

Proofiness: The Dark Arts of Mathematical Deception
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Proofiness: The Dark Arts of Mathematical Deception ReviewOne of the benefits of retiring from my career as a statistician is that I no longer feel it's my personal responsibility to alert friends and colleagues to the myriad ways they are being misled or deceived by the kind of abominably poor summarization of data that's pretty much the norm these days. It's just as well - who wants to be *that guy*, the crank at the table who people start to inch away from surreptitiously, avoiding eye contact all the while?
Not that I endorse misleading or deceptive data presentation - far from it. Now more than ever, as we all struggle to make sense of the avalanche of information that constantly assails us, the capacity for critical, intelligent interpretation is vital. So it's important to be able to see through the most prevalent fallacies in data interpretation, not to mention data presentation strategies deliberately intended to mislead. This latest book by Charles Seife has the laudable goal of educating the reader about some of the most common types of statistical malpractice out there, continuing a tradition established by such authors as Darrell Huff ("How to Lie With Statistics"), John Paulos ("Innumeracy"), Edward Tufte, or the authors of last year's highly successful "The Numbers Game" (Michael Blastland and Andrew Dilnot).
Unfortunately, though "Proofiness" is a well-intentioned book, it suffers from a fundamental crisis of identity. There is a major gap between what "Proofiness" promises and what Seife actually delivers. The first hundred pages cover roughly what one might expect: graphical deception by use of misleading labels or scales, comparison of apples and oranges (e.g. dollar amounts unadjusted for inflation, absence of an appropriate control group, regression to the mean), cherry-picking of data, the tendency to interpret mere random variation as systematic, nonsensical conclusions obtained by extrapolating beyond the range of observed data, overstatement of the precision of measurements, the way in which humans are hard-wired to misinterpret risk and deal poorly with calculations involving risk. Seife's exposition of these topics is lively and clear (with the major caveat discussed below). About halfway through the chapter on risk, however, he makes a major detour. His discussion of the malfeasance of those involved in the Enron debacle, the Bernie Madoff pyramid scheme, the failures at AIG, Citigroup and other institutions, and the subsequent bailout efforts has almost nothing to do with statistical trickery, focusing instead on the public policy and regulatory issues raised by the financial meltdown.
The next chapter, "Poll Cats" does return to the issues involved in conducting accurate sample surveys and presenting the data appropriately, with a reasonably clear discussion of systematic error versus random error. However, the following two chapters, "Electile Dysfunction" and "An Unfair Vote", taking up some 80 pages, really have little to do with data-related issues. Instead they provide a review of events surrounding the Florida vote count in the 2000 presidential election, the six-month circus that took place before Al Franken was eventually declared winner in the 2008 Minnesota Senate race, and a review of historical and present-day gerrymandering efforts whenever congressional redistricting comes up for discussion. Not that Seifen's review of the relevant events, and the issues they raise, is not interesting - but it is largely editorial comment on political events and, as such, it seems to belong in a different book, as does the appendix in which he discusses electronic voting. In making this criticism, I take the view that fraud, malfeasance and corruption stemming from poor public policy, faulty regulatory mechanisms, or inadequate enforcement of existing protections, really are subjects for a different kind of book than that initially described by Seifen. Though the author does return to his initial remit in the final two chapters (discussing abuse of probability and statistical arguments within the judicial system, and for propaganda purposes), overall the book does not make a coherent whole.
Then there's the caveat mentioned above, regarding Seife's exposition methods, which turns out to be a serious one, enough to prevent me from giving this book my endorsement, despite its good intentions. It's the author's predilection for coining cutesy neologisms that not only add nothing to the discussion, but actually end up seriously muddying the exposition. It's evident right there in the book's faux-cute title, "Proofiness". I wish I could say that the author offers a rigorous definition of exactly what he means by this invented term, but he doesn't. It remains unhelpfully vague throughout the book. Sadly, it's not the only example of authorial neologism run amok. "Disestimation", "Potemkin numbers", "randumbness", "regression to the moon", and the horrendous coinage "causuistry"; each of these is a neologism that adds nothing to the discussion. Many of them lack a clear definition, or when a definition is offered, the term just seems to muddy the waters. For instance, Seife uses "disestimation" to mean "overstatement of the precision of a number or measurement", indicating an error based in randomness. But the 'dis'-prefix clearly suggests a systematic error, as does the parallelism with "misestimation", a term which statisticians routinely use to indicate a systematic error. And while one applauds the author's efforts to educate his readership about the error of mistaking correlation for causation, the term "causuistry" is simply an abomination. I'm not sure where this recent trend for authors to invent their own faux-cutesy terminology, where none is needed, originates (possibly Malcolm Gladwell bears some of the responsibility), but it needs to stop.
Though I am sympathetic to the author's stated aims, his execution was such that I cannot endorse this book. Anyone interested in this important topic would be far better served by reading The Numbers Game: The Commonsense Guide to Understanding Numbers in the News, in Politics, and inLife by Michael Blastland and Andrew Dilnot.Proofiness: The Dark Arts of Mathematical Deception Overview

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The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser Review

The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser
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The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser ReviewIf you are old enough, you remember the sensation that the Rubik's Cube caused all the world over in 1980. No one is still alive that remembers the 1880 fad for the analogous two-dimensional "Fifteen Puzzle", which had fifteen numbered blocks within a four by four container and you were supposed to arrange them numerically. Mechanical puzzles can make storms like these, maybe because you can solve them over and over again, but it isn't often that word puzzles produce such fads. True, the Zebra Puzzle, a reasoning exercise consisting of fifteen seemingly unconnected statements that if regarded together the right way make a logical whole, was popular in 1962. Once you solved it, however, that was that. The Monty Hall Problem entered the public consciousness in 1990 and has been completely solved, but because the solution is so counterintuitive, it is still on the minds of many. One of those minds is that of Jason Rosenhouse, an associate professor of mathematics who has written _The Monty Hall Problem: The Remarkable Story of Math's Most Contentious Brain Teaser_ (Oxford University Press). "My original idea for this book," he writes, was that an entire first course in probability could be based on nothing more than variations of the Monty Hall problem." Indeed, some of the chapters here are full-power mathematics, with unknowns x, y, and z, summation or conditional probability symbols, and complicated equations choked with parentheses within brackets, and more. Math phobics won't get far with such stuff, but there is enough other material here, along with different explanations of the basic puzzle, that will be of interest to anyone who likes recreational mathematics in even the slightest degree.
People feel strongly that the answer the mathematicians have worked out is wrong and cannot be made right. Here is the problem: You are Monty's contestant, and he presents you with three identical doors. One hides a car, which you want, but the other two doors hide goats, neither of which you want. You pick a door, but instead of opening it, Monty opens one of the other two doors. Monty knows, of course, where the car is and where the goats are, and he only opens a door that shows you a goat; in the case where you happened to pick the door hiding the car, he chooses one of the two remaining doors randomly. So then you have one door open with a goat, and two doors unopened, including the one you picked. Monty now says he will give you a choice: you can stick to the unopened door you originally picked, or you can switch to the other unopened door. So, do you stick or switch? It is obviously a fifty-fifty chance, and like so many obvious things, it is also wrong. Rosenhouse goes on to show several ways of calculating the problem, and he is good at explaining why you are twice as likely to win if you switch. Essentially, Monty is giving you extra information when he opens that door with a goat behind it. You had a one third chance of picking the door with the car to begin with, and if you have picked that door and switch, you lose. But you also had a two thirds chance of picking a goat to begin with, and (under the conditions of the problem), if you picked a goat and switch, you can only switch to the door hiding the car.
Don't worry if the summary in this review isn't convincing. Many who first saw the problem in a _Parade_ magazine article by Marilyn vos Savant in 1990 weren't convinced, either. Rosenhouse prints some of the responses to her article, many of them from mathematicians and many of them withering in their disapproval of her correct analysis that switching is the best policy by a factor of two. He is embarrassed by the vituperative nature of some of the professional voices in opposition. If this puzzle were not puzzling enough, Rosenhouse goes through many variables of the problem and its effects in different schools of thought (including quantum dynamics), because there is a huge amount that has been written about it. Rosenhouse says he could write a second book with material he has reluctantly left out of this one, and this one covers: what if there are four doors, what if there are n doors, what if there is another player playing against you, or what if Monty opens any of the three doors randomly. It covers the history of the problem and the similar problems that went before it, and it covers the psychological causes of sticking or switching, and studies that show how people tend to stick in all the cultures so far tested. Best of all, for this reader anyway, it made the previously counterintuitive strategy of switching feel a little more sensible.
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Differential Forms: Integration on Manifolds and Stokes's Theorem Review

Differential Forms: Integration on Manifolds and Stokes's Theorem
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Differential Forms: Integration on Manifolds and Stokes's Theorem ReviewFortunately there are several books, at an introductory level suitable for undergraduate students, on how differential forms constitute a "new" powerful mathematical technique that surpasses the outdated vector calculus. This book by Steven H. Weintraub is a very good example among others -- such as: (i) "Advanced Calculus: A Differential Forms Approach" by Harold M. Edwards (Birkhäuser, Boston, 1994); (ii) "Vector Calculus, Linear Algebra, and Differential Forms" by John H. Hubbard and Barbara Burke Hubbard (Prentice Hall, NJ, 2nd ed., 2002).
As far as I know, it was in "Gravitation" -- by Charles W. Misner, Kip S. Thorne and John Archibald Wheeler (Freeman, San Francisco, 1973) -- that a pictorial representation of forms was clearly presented to physicists for the first time. These authors went even further, explaining how "forms illuminate electromagnetism, and electromagnetism illuminates forms" (p. 105).
However, until now, it seems that in engineering forms have been disregarded -- despite early attempts by George A. Deschamps (see, e.g., his paper "Electromagnetism and differential forms", Proc. IEEE, Vol. 69, pp. 676-679, 1981), not to mention Harley Flanders's book ("Differential Forms with Applications to the Physical Sciences", Dover, NY, 1989). Perhaps the book by Ismo V. Lindell ("Differential Forms in Electromagnetics", IEEE Press/Wiley, NJ, 2004) will be able to change this sad scenario.
It seems that the difficulty lies mainly in the fact that a proper understanding of k-forms, as antisymmetric (0,k) tensors in differentiable manifolds, requires the study of technical demanding subjects such as de Rham cohomology. However, this book shows that it is possible to make an introduction to forms without mastering such concepts in topological and smooth manifolds -- although there is an extensive bibliography on this subject out there (the books by John M. Lee on manifolds are my favorite).
For more advanced readers, the book by Friedrich H. Hehl and Yuri N. Obukhov on the "Foundations of Classical Electrodynamics" (Birkhäuser, Boston, 2003) is, in my opinion, the most elegant exposition on the relation between electromagnetism and forms.Differential Forms: Integration on Manifolds and Stokes's Theorem Overview

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Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10 Review

Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10
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Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10 ReviewThis is a wonderful resource for teachers and future teachers. It shows examples of how to begin to use reform mathematics in your classroom and more conceptually based problems vs. the traditional plug and chug methods that we had been taught years ago.Fostering Algebraic Thinking: A Guide for Teachers, Grades 6-10 Overview

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Lebesgue's Theory of Integration: Its Origins and Development (AMS Chelsea Publishing Series) Review

Lebesgue's Theory of Integration: Its Origins and Development (AMS Chelsea Publishing Series)
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Lebesgue's Theory of Integration: Its Origins and Development (AMS Chelsea Publishing Series) ReviewThis book is not intended as textbook with which to learn the subject itself - it is intended as a partly historical treatise on how the subject developed.Lebesgue's Theory of Integration: Its Origins and Development (AMS Chelsea Publishing Series) Overview

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Tools of the Trade Review

Tools of the Trade
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Tools of the Trade ReviewMr. Sally's book starts with basic, almost obvious elements of the set theory that provide the logical foundation of essentially all modern mathematics. In this way, he mines the best of the 20th century's progress in solidifying those foundations. But fear not, he does not become mired in Russelian logic.
Instead, he goes on to derive and define the properties of myriad basic mathematics, up through some of the basics of topology, complex analysis and so on. A variety of statements are left for the reader to prove, though all important theorems at this level are at least stated explicitly.
The reader will go through essentially the same mental development undergone by students fortunate enough to take the famous U of Chicago Math 207 course. Budding mathematicians have been inspired by that course for generations. The proofs, where provided, are pithy and crystal clear while the exercises have a fine distribution of difficulty from trivial to tricky.
This is not a reference text by any means, but rather, as the title implies, a means to give you the skill set necessary to understand and invent real mathematics.Tools of the Trade Overview

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Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics) Review

Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics)
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Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics) ReviewWow! What a great Table of Contents. It has all the stuff I've been wanting to learn about. So I bought the book in spite of seeing only one review of it. After one day, I'm now only at page 26, but I already have read enough to make some comments about it.
The main point about this book is that it is, as the author specifically states, LECTURE NOTES, not, I repeat, not a textbook. What are the implications of this (outside of a somewhat more chatty style than a textbook)? ["chatty" isn't quite what I mean; "smooth" might be a better word'] There are two which are noticable to me. 1) A lot of math knowledge is taken for granted. 2) It has a somewhat sloppy style to it.
Regarding point one, make sure you have a lot of math under your belt before picking up this book. By page 18 the author uses these terms without defining them: Differentiable Manifold, semigroup, Riemannian Metric, Topological Space, Hilbert Space, the "" notation, vector space, and Boolean Algebra. Fortunately for me, I have a fairly extensive math education, and self-studied Functional Analysis, so I wasn't thrown for a loop; but for many others -- brace yourselves!
Regarding point two, Here are two examples:
1) Here is a quote: "The collection of all open sets in any metric space is called the topology associated with the space." Sounds like a definition to me! Fortunately the author gives a (sloppy) definition a few lines later. By the way, the only thing the reader learns about what an 'open set' is, is that it contains none of its boundary points. All the topology books I have read define open sets to be those in the topology. This is another point of confusion for the reader. In fact, points of confusion abound in that portion of the book.
2) On page, 17, trying somewhat haphazardly to explain the concept of a neighborhood, the author defines N as "N := {N(x) | x is an element of X}" This is already a little disconcerting: x is already understood to be an element of X. So he is saying that N is defined as N(x) (which he defines to be a collection of subsets of X). This is all he has to say on the matter until, on page 26, he writes "each N, an element of N(x)". Now N isn't both N(x) and an element of N(x). This is a point which the author does not clear up. He then starts using N all over the place, yet the reader isn't sure of what he's refering to.
A couple of other things:
-When he defines terms, they is not highlighted, and are embedded in a sentence, making it difficult to find them later.
- The index is pitifully small. Typical for English texts, I know; but this *is* the 3rd millinium!
On the other hand, I have good things to say about the book, too.
I like his style of writing. If it were just more precise, it would be fine for me. I like it better than the normal higher math texts, which tend to be too laconic for me. Notice that I make a distinction between the somewhat chatty style, which I like, and the sloppiness, which is confusing. One can be chatty, yet clear. So far, the undefined math terms which I listed above were not central to the text; and one would not miss much by just reading past them. The author includes many 'comments' sections throughout the book. These are wonderful so far. They are full of comments and examples which really clear up a lot of points. His examples are very good, too, although he is very terse in stating them. The paperback is nice looking. The paper, font, etc. make for easy reading (except for the sub/super-script font, which is too small for me).
To wrap this review up, I had already pretty much learned the stuff covered in the book so far, but judging from what I have read, I will be able to learn a lot from the rest of it; and, unlike some other math books I have studied, the experience won't be too painful.
p.s. See other reviews of it on the UK Amazon site.Modern Differential Geometry for Physicists (2nd Edition) (World Scientific Lecture Notes in Physics) Overview

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Topology of Surfaces (Undergraduate Texts in Mathematics) Review

Topology of Surfaces (Undergraduate Texts in Mathematics)
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Topology of Surfaces (Undergraduate Texts in Mathematics) ReviewConsidering several other undergrad topology texts, e.g. Munkres, Armstrong, etc. this is the easiest to work with. Certainly the best text for self-study. The problems are not too difficult yet they help you grasp concepts as well. They are also laid out as you go; so every so often while you read the text you encounter a problem and you do it as you go. It is much better than putting them in the back of the chapters, as most text do. It is better to lay the problems in the text so you are encouraged to do them as you learn the material. The material in the text is very well explained and contrary to the previous review, is very well-suited, and with sufficient rigor, for mathematics students. The fact that this book "can be grasped at the sophomore level" as the previous revewer claims (and I agree with) lends credence to the simplicity of presentation of the material. Some reviewers I suppose aren't satisfied unless they see a hyperdense conglomeration of gobbledygook which characterizes so many mathematics texts. I don't fall into that camp and if you don't either and at the same want to begin study in topology then I highly recommend this book.Topology of Surfaces (Undergraduate Texts in Mathematics) Overview

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Foundations of Analysis (Graduate Studies in Mathematic) Review

Foundations of Analysis (Graduate Studies in Mathematic)
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Foundations of Analysis (Graduate Studies in Mathematic) ReviewLandau's most known book is this little masterpiece. If you want to see everything about numbers proved, from the beggining, assuming just logical and set-theoretical principles and the five Peano axioms, you will find it here. You will see the proof of why 1+1=2, for instance, or why a+b=b+a. Usually people learn analysis with a lot of pictures and assumptions, and every once in a while one asks himself: how does it all begin? Because sometimes you see something which ought to be evident proved, and something which ought to be proved assumed. I recall that when I first met this book I became amazed and read it through with a lot of willing. It is difficult reading, so be prepared. That's because Landau wanted to follow the axiomatic Euclidean style in its most pure way. So the book is in the non-merciful telegram style of presenting everything in terms of "Axioms", "Definitions", "Propositions". Few books before and after strove to reach such pure and clear presentation of arithmetic. Thank God some one had once the patience to write such careful and complete text! In this book the words of Edgar Allan Poe are more than anywhere true: "What I here propound is true:-therefore it cannot die:-or if by any means it be now trodden down so that it die, it will 'rise again to the Life Everlasting'".Foundations of Analysis (Graduate Studies in Mathematic) Overview

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500 Advanced Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards Review

500 Advanced Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards
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500 Advanced Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards ReviewThese are a great match to the Essential Words cards, and they even start the numbering on these at 500 so you don't mix the two sets up if you don't want to. Here is my review of the essential words:
I bought the Kaplan AND these Manhattan GRE flashcards. The Manhattan are better, and here is why:
Basically they are both the same in content. The Kapplan have a nice pronunciation guide, which is the only benefit they have over these.
The Kaplan cards (NOT THESE MANHATTAN CARDS) are cheap. They are printed on thin paper, and they are physically pretty small. Also I wish the printing on the back were upside-down from how it is now (both brands are like this). These Manhattan cards are bigger, printed on card stock but still reasonably thin, and come with a nice key-ring that fits through holes. With these Manhattan cards you load up the key ring, and blast through about 50, maybe stash them in a large pocket and get to them later, or throw them in the backpack. Those Kaplan cards you are forced to try and keep the tiny cards together in stacks. The Manhattan you will use more because they are just easier to work with.500 Advanced Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards Overview

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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Review

How to Think Like a Mathematician: A Companion to Undergraduate Mathematics
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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics ReviewI am reading a great book that is in author's words intended for undergraduate students of mathematics, but that in my opinion offers much more to a motivated reader.
"How to Think Like Mathematician" by Dr. Kevin Houston is a very engaging, readable and pragmatic text on mathematical "technique". It is a non-pompous, well written, valuable, easy to follow and understand valuable set of lessons and tips on understanding and adopting mathematical method, language, theorems, proofs and techniques.
It is an introductory text, so for more in depth treatment of the subjects such as proofs and number theory you may need to look further into books such as Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) orThe Princeton Companion to Mathematics but if you are looking to understand the mathematical method and how to be able to read and write "serious" math this is an ideal book.
I specially have to point out Dr. Houston's writing style - it is engaging, humorous, but substantial, pedagogical and never trivial.How to Think Like a Mathematician: A Companion to Undergraduate Mathematics Overview

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500 Essential Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards Review

500 Essential Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards
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500 Essential Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards ReviewI bought the Kaplan AND these Manhattan GRE flashcards. The Manhattan are better, and here is why:
Basically they are both the same in content. The Kapplan have a nice pronunciation guide, which is the only benefit they have over these.
The Kaplan cards (NOT THESE MANHATTAN CARDS) are cheap. They are printed on thin paper, and they are physically pretty small. Also I wish the printing on the back were upside-down from how it is now (both brands are like this). These Manhattan cards are bigger, printed on card stock but still reasonably thin, and come with a nice key-ring that fits through holes. With these Manhattan cards you load up the key ring, and blast through about 50, maybe stash them in a large pocket and get to them later, or throw them in the backpack. Those Kaplan cards you are forced to try and keep the tiny cards together in stacks. The Manhattan you will use more because they are just easier to work with.500 Essential Words, 1st Edition: Manhattan GRE Vocabulary Flash Cards Overview

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Engineering mathematics handbook: Definitions, theorems, formulas, tables Review

Engineering mathematics handbook: Definitions, theorems, formulas, tables
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Engineering mathematics handbook: Definitions, theorems, formulas, tables ReviewFirst of all my review is on the 1979 version of this book. It has been in my possession that long. Once again today, it helped me drag up a mathematical fact I had long forgotten.
In 1976 I graduated from college with an associates degree in engineering. I had high school algebra, geometry, trigonometry and calculus behind me. Also with my two year degree, I had calculus, analytical geometry and differential equations as well. I needed a single reference that would cover everything. Then I happened upon this book in a bookstore. What a prize this was!
Since that time, I have completed both my B.S.M.E and a Masters in Mechanical Engineering and I have this dog eared book beside me at all time. From looking up linear algebraic equations, differential calculus, vector analysis, complex variables, all the way to the formula for the eccentricity of an ellipse for my son this evening, this book has baled me out of referring to ten other books. It is well worth the investment.
Ironically, years later when discussing a geometric equation with a co-worker he commented that he relied on a great reference he found in book store many years earlier while attending college. You got it, it was the same book.
What I can't speak to is the last two chapters that are new to this edition. Mine only goes to Chapter 20.
What I also should note, as a Mechanical engineer using this, I found it incredibly useful. I have no idea if a person that does not regularly use geometry, trigonometry, algebra, calculus, linear algebra, etc., would get very much use from it or not. For students in this field I found it to be very useful. Basically, read the table of contents Amazon provides and make your own judgement.Engineering mathematics handbook: Definitions, theorems, formulas, tables Overview

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Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover Civil and Mechanical Engineering) Review

Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover Civil and Mechanical Engineering)
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Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover Civil and Mechanical Engineering) Reviewover a period of thirty-plus years, i've benefited countless times from this dizzyingly broad and accurate handbook. it must be seen in that light: handbook. but, as such, there is simply no competition, at least for the engineer doing mathematical evaluations or predicting performance. i myself even find it enjoyable to just read in it. also to remember: it is not a compendium of computer-based techniques; indeed, it is a handbook of actual mathematics. i appreciate this small forum as an opportunity to say: Thanks, K&K.Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review (Dover Civil and Mechanical Engineering) Overview

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