5 comments about Matrix Analysis.

1. I bought this book hoping to learn about matrix analyis. I did not. This book is simply a reference manual with plenty of theorems, axioms etc. with little explanation. They give it to you rough and row. NOT A SINGLE SOLVED EXAMPLE, and not even solutions for the exercises given in the book are provided. If you intend to learn about matrix analysis, as I did, let not the 5 stars review mislead you. Don't make the same mistake, this book is not for you.



2. I agree with other commentators who remarked that the book is better suited for someone already versed in linear algebra. For the student new to all this, the text can be, shall we say, too formidable?
   
   A good usage is when you have studied the subject, perhaps several years ago, and need a concise refresher.
   
   The strong aspect of the book is the emphasis on numerical calculations. Rather than about proving theorems. Don't worry that it was printed in 1990! While computers have heavily improved, thanks to Moore's Law, the maths of course has not. All the algorithms explained here are still germane to number crunching of linear systems. As another take, look at the Amazon page for the book, for the section about other books that cite this one. Notice the preponderance of computational books.



3. THis book covers some key aspects in matrix analysis.
   
   Would certainly recommend this book.



4. Definitely, not a lot of attention has been paid to pedagogy by the authors of this book. However it forms an excellent summary of most of the theory and is very good for one who understands it and one who wants a ready reference in the subject.



5. No matter what the blurb says this is a graduate level book. You cant teach yourself linear algebra using this book. Having said that this is a fantastic book for the initiated. Concise. Consistent. Well-written. A very helpful index. I refer to it regularly and it rarely disappoints. This is a must have. Absolutely 5 stars.

3 comments about An Introduction to Manifolds (Universitext).

1. 
   i think there is a jump from ugrad analysis/alg/top etc to early grad school concepts. i didnt know category theory, i only had the flimsiest notion of a manifold, etc etc. and this book fills in that jump wonderfully. it does the right mix of analysis-differential topology-topology so that you can go read a book like bott and tu later (that's what it was designed for).
   so im having a good time with it.



2. This is an excellent book. I wish that more books on advanced mathematics were written in this style. In contrast to most books on manifolds that tend to be very difficult for beginners to follow, Prof. Tu has made every effort to make this subject understandable to the nonexpert.
   
   Greg Chirikjian
   Professor, Mechanical Engineering
   Johns Hopkins University



3. This is my favorite book on Differentiable Manifolds. After reading this book the reader will obtain a solid background on the following essential notions: Charts and atlas of a manifold; tangent vectors (as derivations); differential of a smooth function between manifolds; submanifolds and embeddings; quotient spaces; partitions of unity; vector fields; vector bundles; differential forms and de Rham cohomology. And on the road, the reader gets a gentle exposure to Lie groups, Lie algebras; and some basic notion of Category and Functors.
   
   I found the following aspects of the book especially attractive:
   > Clear style of writing: The author is the coauthor of the acclaimed "Differential Forms in Algebraic Topology". See the comments for that book. The clarity has not decreased at all.
   > Bite-sized sections: The materials contained in each section is approximately equal to that of a 50-minute lecture. This helps readers who plan self-study.
   > Right amount of topics: This is not an encyclopedia on manifolds. However, it does contain the ``absolute must'' one should know about manifolds. And it does such a good job in presenting it, the reader will be left with a solid understanding on those essential topics.
   
   I first read this book as a Physics student and had no trouble reading it. I later switched discipline to Mathematics, and I know that this book has helped me appreciate the beauty of Mathematics. I thank the author for writing such an wonderful book.

1 comments about Introductory Analysis.

1. This book will build a good base for calculus. The only problem with the book is that there are not enough examples nor enough variability in the exercises. Sometimes you would have to use another precalculus book as reference. In overall it has great subjects in logistic and mathematical theorems with their proofs,and you would learn how to build those proofs.

5 comments about Proofs from THE BOOK.

1. We used it for an undergraduate statistics seminar series; and that was largely a success. PFTB contains many of the most elegant proofs I have ever seen. Reading through it is, by itself, an enjoyable journey through some aspects of many branches of mathematics. A few of its chapters may not address the average undergraduate math population quite well, due to its depth of reasoning and assumption of prior knowledge in Abstract Algebra and Geometry; but those who reads it are likely to be more advanced anyway. In general, an excellent book.



2. One of the most important mathematicians in the last quarter of a century was, arguably, Paul Erdos. At least we could say he was the one who travelled the most, and who had more collaborations with other mathematicians in the areas of Graph Theory, Combinatorics, etc.
   
   One of his dreams was to assemble a collection of all the most beautiful proofs in Mathematics, in which God would maintain the perfect proofs in mathematical theorems. This collection would be called The Book. Unfortunately, he didn't see his dream fulfilled while among us, but after his death Agner and Ziegler wrote this excellent book, "Proofs from THE BOOK", as a nice excerpt of what The Book could have been.
   
   Topics in this book range from Number Theory to Analysis, and also discussing Combinatorics and Graph Theory. Most of the proofs are accessible to everybody with some mathematical training, they are beautiful and surprisingly simple.
   
   I just love this book, in the last five years or so I have bought several copies of it, as it has been my favorite gift for people who want to discover a little more about the beauty of Mathematics.



3. NOTE: This review is JUST for the Kindle edition.
   
   The Kindle edition is completely worthless, because it is missing many symbols. It appears to have been done using OCR, and it was confused by mathematical symbols. For example, there are some places where I THINK it was supposed to be the greek letter phi, but it comes out as a left parenthesis and a right parenthesis. At least with that you can figure out what it was supposed to be. There is much worse--places where symbols are completely gone. E.g., there is a place where you just get a capital sigma with a subscript giving a summation limit, a blank space, a less than sign, and another blank space. So, the proof is saying the some of *something* is less than *something else*.
   
   This is a shame, because the book itself, from what I can see, is EXCELLENT.



4. The Kindle edition of the book is missing or misrepresenting math symbols in so many places it makes it completely unreadable.



5. This is a different kind of math book. It is not a textbook. Neither is it a monograph on a special math topic. It is a eclectic collection of proofs largely from number theory, combinatorics, graph theory and discrete mathematics in general. It is inspired by what the famous peripatetic mathematician Paul Erdos would like to see in God's BOOK. In my view, this book is best appreciated by dipping in occasionally when you are in the mood or when you have read a proof in another book and would like to see how it is done in this book. Most of the proofs are accessible to undergraduates or even college students.

5 comments about Real and Complex Analysis (McGraw-Hill International Editions: Mathematics Series).

1. While I would not recommend this text to someone wishing to teach herself real and complex analysis, having this book in your personal mathematics library is a must for anyone seeking to further her education in higher mathematics. It's one of the most commonly used undergraduate texts, referred to by some as the "Bible". If you can afford it though, I would recommend that you pick up a copy of Baby Rudin to use as a reference.
   
   The first two chapters in combination with Bartle's text on Lebesgue Integration and Measure makes for a killer introductory course in Measure Theory.
   
   Oh, and if you can solve the problems in Rudin's book, you can do pretty much anything, so it's a major confidence booster!



2. There are some excellent reviews here for this outstanding book, so I will try to avoid repetition. In preparation for my qualifying exams in graduate school, two of my colleagues and I did all of the exercises in Rudin (give or take a couple, no more). What I found striking at the time was how Rudin took three subjects -- measure theory, functional analysis, and complex analysis -- and weaved them together seamlessly. It is not that I believed them to be separate subjects, but until then I hadn't realized just how they all fit together. Really, this book is superb.
   
   A word of warning, though. Rudin's prose is concise, and his proofs leave you wondering if you'd ever be able to reproduce them on your own. It is what we in the business are used to call 'elegant'. It pays to work in groups, persevere, and go over everything twice or more. Good luck.



3. I love this book, even though I have not absorbed more than a small portion of it yet. I find this to be a much better book than the "baby Rudin", which struck me as dry, overly concise, and without motivation. This book provides ample motivation, and although it proceeds in great generality, proceeds at a reasonable pace.
   
   The best thing about this book, however, is the spirit of it--the integrated approach to analysis that Rudin takes is unique and greatly appreciated--Rudin is, like Lang, a testimony to the fact that the best mathematicians do not draw artificial lines between different areas within mathematics. Rudin presents the material in ways that connect to other areas of mathematics and will help the reader become a better mathematician, even if she never directly uses any of the material contained in this volume.
   
   I would not recommend this book as a first exposure to measure theory or complex analysis--it is advanced and requires a great deal of background to fully understand and appreciate. But I think this is a book that any serious mathematician should add to their collection and eventually work through. People wanting to learn measure theory might look to the book by Inder K. Rana, or to the classic book by Royden. For more elementary treatments of complex analysis I would recommend the classic by Ahlfors, Theodore Gamelin's book, or the book by Greene and Krantz.



4. A few words for the person thinking of buying and using this book.
   
   First, go for it. Don't be scared. But you need to be prepped a little on how it all fits together. Roughly, it breaks up into a course on real analysis (with quite a lot of supplementary material, especially on Fourier analysis), and then breaks into complex analysis in chapter 10.
   
   Now on the first part--You might be tempted to ask "what am I learning?" as you start on the first chapter. It seems like Rudin is taking you the longest way possible. Is he torturing me?, you wonder. Can't you make this more concrete?, you ask. Keep going, and you will begin to see what he's up to. The reason he wrote chapter 1 the way he does is because (note) it involves no structure *except* the space X and the sigma algebra M. He's showing you, in other words, what you can rely on no matter *where* you are, no matter what measure you are using. That taken care of, it's off to chapter 2, where he stirs in a topology T. Two main goals: the Riesz Representation theorem (version 1) and the construction of Lebesgue measure. Try hard to make it through this and see what Rudin is doing. Make some time to read through it, and you will really gain some insight into what Lebesgue measure on R^k does to the study of analysis, extending it beyond Riemann. Personally, I'm still not sure it's 100 percent the best way to do things, but--like I said--it's pretty cool stuff after the hard work is finally over.
   
   Next, he goes through some chapters on different sorts of function spaces that are more or less self contained (all in good time, m'boy), and then he comes back around in Complex Measures (chapter 6) to really hammer in some important things--especially the Radon-Nikodym theorem. Now you're ready to learn about differentiation in chapter 7. By now, you will have seen for yourself what I am talking about--the book really gives it to you straight, and best of all, when you're done learning out of it, it continues to be valuable as a reference because of the meticulous organization of each chapter.



5. I normally don't review books that already have this many reviews, especially when I agree so much with the reviews that already exist. But I'm teaching out of green Rudin for the first time this semester, 20 years after getting to know the book well as a student, and I find myself so enthusiastic about it again, that I just had to chime in with an "Amen" to the other positive reviews. When it comes to mathematical writing, it doesn't get any more exquisitely elegant than this.
   
   Probably all our reviews are irrelevant, however, because there are probably very few discretionary purchases of this book: There will be nearly a one-to-one correspondence between buyers of the book and students in classes for which it is required. For them, I can only recommend skipping the outrageously expensive hardback (which even at its high price is pretty cheaply constructed nowadays) and opting for the more reasonable international paperback edition.

5 comments about Problems and Solutions for Undergraduate Analysis (Undergraduate Texts in Mathematics).

1. The only way to study mathematics is by solving a large number of problems: this book permits the eager student to challenge his acquired knowledge. A must to all the students wanting to really master undergraduate analysis.



2. This book is not just a "solutions manual." There are detailed proofs for all of the problems in the book. This book is essential if you want to truly learn real analysis.



3. This collection of problems is really quite good. First, the problem is spelled out entirely, so you don't need to refer to the text. It is self-contained, for the most part. Each of the solutions is worked out with generous detail and the problems are chosen to accentuate certain basic results that are easy to take for granted unless you see the difficulties that arise in their absence. I recommend this text for self-study as a running start to Polya and Szego's more difficult problem book.



4. I agree that learning math is about working problems till you puke, then wiping off the puke and working some more (as a friend who was also a Marine Corps veteran used to say). This book is perfect for that. I wish my school would've used the textbook related to this solutions manual rather than the textbook we did use, which had no solutions manual :(



5. A wonderful volume to complement Lang's Second Edition of Undergraduate Analysis. Feel confident, AFTER carefully studying and FULLY understanding any section of the textbook (having previously read a logical and sequential order of topics that will permit you to accomplish this, besides MAYBE having the opportunity to discuss what you learned with any "down-to-earth", yet knowledgable university professor) to consult the Solutions Manual to the textbook ONLY AFTER YOU'VE COME UP WITH YOUR OWN COMPLETE/PARTIAL SOLUTION TO A PROBLEM in order to address any kind of difficulty related to your solution or to assist you in any doubts you may have.
   
   Even the smartest students out there (yes, I mean YOU... and don't tell me you don't need it because I don't believe you) will benefit tremendously from this manual, IF YOU REALLY KNOW HOW TO USE IT AND TAKE ADVANTAGE OF IT. This Solutions Manual will save you some time and enlighten your learning experience. It is NOT meant for cheating or learning any quicker than you possibly can.
   
   Good luck and thanks for taking the time to read this review!

2 comments about Schaum's Easy Outline: Calculus.

1. I'm a graduating senior and i just bought this book to get myself ready for college this year! It's an amazing book and is easy to understand.



2. This book should help you in reviewing your notes that you took in class. As the title of the book suggests, this is a crash course for general calculus.

   Figures and graphs in the book should help you understand many different problems and how their solutions are derived. You should expect to solve many of the problems in the book yourself, on paper, also. That is the only way to understand mathematics; by doing it yourself. This book should help you get started in the right direction because of the ample examples present in it.

   Do not expect to get rid your other calculus books and use this as the sole text for reviewing Calculus. This book is meant to be a companion to your other calculus textbooks, not their replacement. As good as it may be, you still need detailed explaination of rules and theories about Calculus, which are not explained in this book.

5 comments about Riemann's Zeta Function.

1. I had a hard time putting this book down. Who would have thought that for a book about mathematicians and a function? Ok, I am a mathematician and physicist, but still, this really brought to life the exploration and probing of the bizarre and fascinating universe we have created in mathematics.
   
   I highly recommend this book to anyone who likes the history of science and mathematics and is excited by discovery. Even if you only know algebra, I believe that you'll be able to follow the key mathematical concepts and implications.
   
   (Note: four stars is a really, really good rating from me. I haven't given a five star rating yet, which is reserved for literature that is truly mind-blowing.)



2. Undoubtedly the best technical exposition on the Riemann Zeta Function, complete with derivations, proofs, and examples of calculations. As a special treat, it includes a translation of Riemann's original paper on the Zeta function, a photocopy of one page of Riemann's notes, and detailed references to papers of earlier authors of papers and books on the Riemann Zeta function, including a table of values of zeta(½ + it) for t=0 to 50 in increments of 0.2 [Haselgrove, Royal Society Math Tables, Cambridge Univ. Press, 1960].



3. Comme vous auriez pu vous en douter, vu l'adresse où vous avez envoyé le livre, je suis français
   Ecrivez moi en français, et je vous répondrai



4. a wonderful exposition full of incredible formulae; a careful account of what Riemann could have thought of when writing his famous paper; the text contains calculations of zeta's non trivial zeroes in an old fashion way but still illuminating. Lacks schema of contours used in calculating
   complex integrals but this is a very minor flaw; on the other side, one can find copies of two pages handwritten by B. Riemann which may show how casually this great genius could work and may explain why his followers had such difficulties in proving and following his ideas; the study of zeta means some herculean calculations of which Edwards gives a fine taste; this book is worth the pain to work with; one can compare his first proof of zeta's functional equation with those given in Ahlfors's "Complex analysis" or Tenenbaum's book: "Introduction to analytic an probabilistic theory of numbers"; my favourite formulae are:
   1) Von Mangoldt's formula for the psi function with summation over the nontrivial zeroes of zeta.
   2) the formula for the product of zeta and gamma as an improper integral and its sequel using hankel contour.
   3) the Siegel-Riemann formula for Z(t), this Z function being derived from the values of zeta on the line re(z) = 1/2.
   This is the very book to remove oneself on a desert island with (but don't forget a laptop to verify tha accuracy of Backlund's estimations of the fifteen first non trivial zeroes of ... zeta.
   By the way, do you happen to know which are the trivial zeroes ?



5. While this is a strong mathematical treatment of Riemann's Zeta function, the steps between equations are very terse and not intuitively obvious. A little more time could have been spent filling in steps between equations. This is not a book to read but to study. If you have not had graduate level mathematics, choose another book.

2 comments about Solving Mathematical Problems: A Personal Perspective.

1. I came across this book after reading about Terence Tao, a recent Fields medalist. It's interesting to see a book like this by such an accomplished mathematician. The book gives practical approaches to solving the types of math problems encountered in math olympiad competitions. I am not, nor have I ever been, a math olympian, but I found the book to be entertaining and useful for intellectual fortification purposes.



2. This charming book explains why math olympiad puzzles are fun, and gives 15 year old insights ( in two senses --- most of the text was written by the 15 year old Terence Tao, but revised with some additional good exercises 15 years later by 30 year old Fields medalist Terence Tao.) The style is chatty, and the advice and worked examples are very good and do not require any math beyond pre-calculus. The level of difficulty is just right for would be high school math competitors, and adults with some math who enjoy a mild challenge.

3 comments about Schaum's Outline of Theory and Problems of Combinatorics including concepts of Graph Theory.

1. In its usual way schaum's series gives out another book which is both helpful yet concise. This book gives the essential grounding for combinatorics and graph theory without being overly gargantuan encyclopedia..ample problems set the tone for a future mathematician. they could've done better though..hence not the perfect 5 !



2. Combinatorics is an area of mathematics that is frequently looked on as one that is reserved for a small minority of mathematicians: die-hard individualists who shun the limelight and take on problems that most would find boring. In addition, it has been viewed as a part of mathematics that has not followed the trend toward axiomatization that has dominated mathematics in the last 150 years. It is however also a field that has taken on enormous importance in recent years do its applicability in network engineering, combinatorial optimization, coding theory, cryptography, integer programming, constraint satisfaction, and computational biology. In the study of toric varieties in algebraic geometry, combinatorics has had a tremendous influence. Indeed combinatorial constructions have helped give a wide variety of concrete examples of algebraic varieties in algebraic geometry, giving beginning students in this area much needed intuition and understanding. It is the the advent of the computer though that has had the greatest influence on combinatorics, and vice versa.The consideration of NP complete problems typically involves enumerative problems in graph theory, one example being the existance of a Hamiltonian cycle in a graph. The use of the computer as a tool for proof in combinatorics, such as the 4-color problem, is now legendary. In addition, several good software packages, such as GAP and Combinatorica, have recently appeared that are explicitly designed to do combinatorics. One fact that is most interesting to me about combinatorics is that it gave the first explicit example of a mathematical statement that is unprovable in Peano arithmetic. Before coming across this, I used to think the unprovable statements of Godel had no direct relevance for mathematics, but were only interesting from the standpoint of its foundations.

   This book is an introduction to combinatorics for the undergraduate mathematics student and for those working in applications of combinatorics. As with all the other guides in the Schaums series on mathematics, this one has a plethora of many interesting examples and serves its purpose well. Readers who need a more in-depth view can move on to more advanced works after reading this one. The author dedicates this book to the famous mathematician Paul Erdos, who is considered the father of modern combinatorics, and is considered one of most prolific of modern mathematicians, with over 1500 papers to his credit.

   The author defines combinatorics as the branch of mathematics that attempts to answer enumeration questions without considering all possible cases. The latter is possible by the use of two fundamental rules, namely the sum rule and the product rule. The practical implementation of these rules involves the determination of permutations and combinations, which are discussed in the first chapter, along with the famous pigeonhole principle. Most of this chapter can be read by someone with a background in a typical college algebra course. The author considers some interesting problems in the "Solved Problems" section, for example one- and two-dimensional binomial random walks, and problems dealing with Ramsey, Catalan, and Stirling numbers. The consideration of Ramsey numbers will lead the reader to several very difficult open problems in combinatorics involving their explicit values.

   Generalized permutations and combinations are considered in chapter two, along with selections and the inclusion-exclusion principle. The author proves the Sieve formula and the Phillip Hall Marriage Theorem. In the "Solved Problems" section, the duality principle of distribution, familiar from integer programming is proved, and the author works several problems in combinatorial number theory. A reader working in the field of dynamical systems will appreciate the discussion of the Moebius function in this section. Particularly interesting in this section is the discussion on rook and hit polynomials.

   The consideration of generating functions and recurrence relations dominates chapter 3, wherein the author considers the partition problem for positive integers. The first and second identities of Euler are proved in the "Solved Problems" section, and Bernoulli numbers, so important in physics, are discussed in terms of their exponential generating functions. The physicist reader working in statistical physics will appreciate the discussion on Vandermonde determinants. Applications to group theory appear in the discussion on the Young tableaux, preparing the reader for the next chapter.

   A more detailed discussion of group theory in combinatorics is given in chapter 4, the last chapter of the book. The author proves the Burnside-Frobenius, the Polya enumeration theorems, and Cayley's theorem in the "Solved Problems" section. Readers without a background in group theory can still read this chapter since the author reviews in detail the basic constructions in group theory, both in the main text and in the "Solved Problems" section. Combinatorial techniques had a large role to play in the problem of the classification of finite simple groups, the eventual classification proof taking over 15,000 journal pages and involving a large collaboration of mathematicians. Combinatorics also made its presence known in the work of Richard Borchers on the "monstrous moonshine" that brought together ideas from mathematical physics and the largest simple group, called the monster simple group.

   The author devotes an appendix to graph theory, which is good considering the enormous power of combinatorics to problems in graph theory and computational geometry. Even though the discussion is brief, he does a good job of summarizing the main results, including a graph-theoretic version of Dilworth's theorem. Combinatorial/graph-theoretic considerations are extremely important in network routing design and many of the techniques discussed in this appendix find their way into these kinds of applications. The author asks the reader to prove that Dilworths' theorem, the Ford-Fulkerson theorem, Hall's marriage theorem, Konig's theorem, and Menger's theorem are equivalent. A very useful glossary of the important definitions and concepts used in the book is inserted at the end of the book.




3. Not bad for those that have become accustomed to extensive math language within a text. The underlying concepts are explained well, however the density of material does take something away. Graph and group theory explanations should be more comprehensive. Considering the complexity of the various topics being presented this book is kind of good.