Reviews (1)

Reviews (1)

Great Introduction on Counting
This book is a good introduction to counting (combinatoric type counting).The book has answers to most of it's problems and recommends you try the problems before looking at the answers. Most of the problems are simple, and hit hard on the idea the section of the book is trying to get across.

Reviews (1)

What an introductory book on combinatorics should be
Little did I expect of a book that has seemingly not garnered that much attention among professors and students (at least the lack of reviews in Amazon.com might serve as a confirming instance of that speculation), but come exam time, the book proved that such prejudice is outright foolish.

Reviews (2)

worst probability book ever
I'm sorry, beginner? This book is supposed to be a beginner??? First, it's written like a novel... not going anywhere, really. They say written examples, yeah, right. You NEVER see any problems worked out, and the questions, very very AMBIGUOUS! I got about 20 different answers from 20 different ways of interpretting each problem at the back of each section. The book isn't layed out well at all. The answers in the back of the book make no sense. It gives the reader virtually no understanding of how the author got the answers. Plus, and I mean PLUS... this book is full of errors, both gramatical and in the formulas... I'm sorry, if you're going to write a book, make sure you get the formula's correct. Every prob and stat class I have, he tells us of the errors in this book.
Oh, a big no no for this book is in some of the questions at the end of each section, they assume knowledge of the next sections. Come on. This book has very few examples. For the kinds of questions it asks, it's not good enough that it have one or two examples, then ask more complex questions.
This book is all wrong. Worst book I've ever owned, probably. The only thing good about it is its size. It's easy to fit anywhere, but that's no reason to buy a book. Did the author or publisher even proof-read, I wonder. No wonder this book never had a second edition.

Reviews (3)

what!
I had this as a textbook in a junior level number theory course, and I am sorry to say that it is the worst textbook I have yet used.This book contained very little actual information and even less mathematics. I'm sad to say that it read like a book whose purpose is to sell you another book.I would have to say that the real message of the book is:

There are lots of great topics in number theory, but you won't find them in this book.

The rest of the class seems to agree with me and we recomend that you spend (less) your money on a better book (for example, Tattersall's Elemntary number theory in nine chapters).

A model of perfection for undergraduate textbooks
This is a very pleasant introduction to number theory. Each chapter is preceded by a preview and concluded by a discussion to make the main ideas clear and well-motivated and to show how things fit in the big picture by discussing the historical development. The book starts with the very basics and moves via some pearls like the four square theorem and quadratic reciprocity to a culmination with algebraic number theory. A readable and elementary introduction to algebraic number theory is especially valuable today because, as Stillwell argues in his preface, this is the proper setting in which to learn of rings and ideals. Now days, of course, the custom is to pull these concepts out of a hat in the mysterious context of "abstract algebra" where there is no apparent reason to introduce them whatsoever. Fortunately, Stillwell has provided us with an equally enjoyable book on algebra, so now we can only hope that some day the curricula will change accordingly.

Isbn: 0387953205
Sales Rank: 237928
Subjects:  1. Mathematics    2. Number Theory    3. Science/Mathematics    4. Theory Of Numbers    5. Mathematics / Number Theory   

Reviews (1)

Reviews (20)

Excellent!
This is probably the best book on linear algebra I have seen.

It approaches linear algebra from a theoretical point of view (i.e. linear maps instead of matrices), it is not "watered-down", and yet it is accessible to undergrads. The key reason for this is because Dr. Axler has kept the book well-focused, has put in only the necessary results needed, and has tied these elements together clearly. I obtained the book, because like the author, I never liked some of the more standard proofs of "classical" theorems. However, even the material at the beginning is superbly organized and well thought out, and a joy to read. His use of a single lemma in chapter 2 for example, (the so-called "linear dependence lemma") makes many of the results given later on in the chapter (and the book), trivial to prove. One of the reasons Axler can use this lemma so effectively, is because he is careful about his notation, and uses ordered tuples instead of sets of vectors. This is just one example of where his care in such matters pays off immensely for the reader.

I agree with one of the earlier reveiwer that his use of side-comments, though uncommon in texts, is very helpful and enjoyable.

Also, even though the book is not application oriented, Axler does give a lot of examples of abstract defintions, which for someone learning linear algebra, is essential to have. He ties new abstract notions, like say linear maps, to things that an undergrad with modest math background would understand (like derivatives, etc). These examples I think are also crucial to a good abstract math book. Too often, an abstract math book will go from theorem to theorem. In this book, I felt like I was pacing myself. There were a lot of theorems that followed sequentially, but there were also "breather sections", where Axler will stop and take a look at what he is doing. This, I think, gives the student time to stop, reflect on what he is doing, and get a better, deeper, fuller understanding of the material.

If you can purchase one book on linear algebra, this is the one book I would suggest!

Elegant theoretical presentation of linear algebra
This is a short, elegant presentation of linear algebra appropriate for upper level undergraduate math majors with a theoretical bent.The student has perhaps taken a linear algebra course designed for engineers and scientists.Such a student is comfortable reading mathematics and writing proofs.It is meant to be read and re-read until the ideas are absorbed.The exercises are relatively easy and no answers are provided.With exercises of this sort you generally know if you are on the right track and they require you to understand the presentation in the text and process the ideas in a straight forward way.

Of course, there is nothing in this book about applications or the computational aspects of applying linear algebra.

The price is right.This could be a very useful purchase even if it's not assigned as a text.

Reviews (8)

beware, no solutions
Good book, but no solutions to any exercises. Usually Springer books do have some solutions, so it was a bit disappointing.

the missing star
Stephen Abbott is with no doubt a very talented writer in mathematics. The book is a fun to read because
of its style : each chapter starts describing a basic mathematical question that challenged the human mind
in history. This always makes you curious to read further to discover the great constructs made by the creative thinkers who solved these problems. Also, each chapter ends with adescription of related topics and some historical notes . I really like this style ....

However I did not give this book a five star rating for the following reasons :

-Some proofs contain gaps that are left as an exercise to the reader. Not all of these exercises are
staightforward however. It sometimes took me several hours to find a solution for these exercises ...
This is OK for real exercises, though it is no fun to have to spend this time filling up some basic proofs..
Sometimes I also had the impression that the hints were misleading. For example, I completed the proof on the double summation bit did not at all understood why we needed the hint prooved in exercise 2.8.4. Also when I tried to complete the
proof of the sequential criterium for nonuniform continuity (theorem 4.4.6), I did not see why we would need the hint
to take values 1/n for epsilon...
-Some explanations are missing, (maybe this will be solved in second edition). For instance :
a)please give a clear definition of what an interval is before using the name interval throughout the book.
b)In baires theorem, the author claims that every open set is either a finite or countable union of open intervals .... Please explain why ...
-Especially the 'more advanced' topics like baire 's theorem, fourier analysis, metric spaces, ... are rather presented as one big exercise. If you want to learn these topics, there are better books, providing you with much more information....
-This book only covers a limited range of topics. All the analysis is done for real variables in one dimension.

I think we need a broader scope, even for an introductionary course. My opinion is that modern analysis should start from the beginnigwith n-dimensional metric spaces, conveying your mind to the beautifull theories of normed linear spaces and banach spaces.
-Since the book is targeted to the beginning student of abstract math, it would be good idea to include some pages
(appendix) on logic reasoning like second order predicate calculus, and some basic set theory ....

So, no five stars for this edition (maybe for a next edition ??)...
Giving this beautifull book less then four stars however would be unfair, since it definitely has it strengths : the things that are explained are explained very clear and the narrative style of the author always keeps the reader interested!!! Nobody could have done that better !!

Isbn: 0387954848
Sales Rank: 1080275
Subjects:  1. Calculus    2. Mathematical Analysis    3. Mathematics    4. Science/Mathematics   

Reviews (5)

Would be better if solutions are provided
Great textbook, great afterchapter exercises! However, since the exercises are a bit challenging, it would have been better if solutions or hints for solutions are provided. By the way, I would be grateful if anyone could tell me where I can find solutions for the exercises.

Very good exposition, great problems
Real analysis is a genre with an established classic (Rudin) and a plethora of available books and resources. Unfortunately, most analysis books cost a great deal of money so the average reader will only purchase one or two texts. In evaluating which book(s) to purchase two questions should be asked:

1.) Why purchase this book rather than the classic of the genre?

2.) Is this book appropriate for me?

So why buy this book rather than Rudin? It has great exposition (as does Rudin), very well chosen problems (as does Rudin), but Pugh manages to improve on the standard by supplementing his written explanations with diagrams and pictures that Rudin mostly lacks. Additonally, the price stands at something less than half the cost of Rudin's book.

Who is this book appropriate for? This text delves into the topological underpinnings of analysis. It is not an "analysis-lite" textbook a la Ken Ross's Elementary Analysis. It is a rigorous treatment of the subject, and it has a comprehensive feel to it, covering topics like Lebesgue measure and integration, and multivariable analysis in addition to the normal topics one would expect. In short, it is appropriate for somebody who is seeking the challenges and rewards of a full treatment of what for many is a difficult subject.

It is a very good book that does not shy away from difficult material that no amount of explanation or good writing will make easy to learn, but of all the analysis books I've seen, this comes the closest.

Reviews (4)

I'm Sorry
I'm sorry I had to read this book, I'm sorry anyone else had to read this book, and I hope that no one else has to suffer through this book.Pedagogically speaking, it is un-sound to state, in the first chapter and in the first several pages, a theorem and, rather than set a tone and rhytmn for the book, good 'ole Saxes leaves these theorems to the reader as exercises.Now, Saxe is not Lang and the Springer book series, namely, undergraduate texts, is just that, for undergraduates, for young students, mathematically speaking, and hence such an adverse and lazy tone should not be set in such a book.Furthermore, her direction, her flow, her in-ability to illuminate and bring forth the beauty of Functional Analysis is depressing (see Ascoli's Theorem and, while your seeing things, maybe Saxes should have seen an editor).For those of you who have had some Analysis, are familiar with basic point-set topology, and have a general idea of what it means for a mapping between to spaces to be linear, then the book you should read is that of Erwin Kreyszig.This book provides you with all the necessary tools as well as motivated and complementary exercises (some solutions are provided).Steer far away from Saxe's book.As an aside, it is obvious that no book, and hence no author, is perfect or capable of producing a 'perfect' book.The problem that arises in the case of the current book being reviewed is that considerable knowledge and or passion for a given subject does not necessarily imply that you are capable of re-casting this knowledge in the form of a book.Maybe, just maybe, if there is a future reprint, hopefully, Saxe will be able to re-work the book and create something hjigher in quality.Until that time, go else-where if you are interested inl earning Functional Analyiss.

Quite a bad book.
A lot of bla bla in this book. This book is a collection of noteswithout a guideline in mind. The more interesting subjects of functional analysis are only superficially treated. Proofs are often quite shortly illustrated and in a proof I found a gigantic error. Not worth to buy this book.

Lucid
This book is fantastic! It is an extremely readable account of the basics of the subject. I thought the Chapter on Measure Theory and Lebesgue integration were particularly well organized. Every definition was well motivated and the theorems were arranged in a very natural progression.

One thing I especially enjoyed about this book is that most of the proofs are done only for special cases of theorems, without loss of generality. For example, the Arzela-Ascoli theorem is proved for the function space C([a,b],R) (R = real numbers), but then Saxe points out what makes the proof 'tick' so that the reader may easily modify it to a more general setting (she always states the more precise versions of such theorems as well). This is great because it helps one's intuition without getting short-changed.

Finally, the book has a great wealth of historical notes and biographies which are rich in mathematical content (e.g., Saxe explains that Frechet was the first person to define a metric space even though he called it 'une class E'; Hausdorff gave it its modern name in 1914). The reader can in this way appreciate how the subject slowly developed into its present form.

Reviews (1)

Reviews (3)

DO NOT BUY THIS BOOK
As a student that has taken his clas let me tell you that Exner is possibly the worst teacher ever.DO NOT waste your money on this book unless you are studying extremely high level calculus.For those of you who only deal with practical and applicable calculus, this book is not for you.Life is not about theory and proofs, it is about being able to use what you know.If you share my theory then please don't support this man by buying his book.

A GRADUAL GUIDE TO WRITING EPSILON DELTA PROOFS
Like Exner's AN ACCOMPANIMENT TO HIGHER MATHEMATICS, this is a book intended for students taking that "before real analysis" course. This book sets its sights specifically on really understanding limits, and producing epsilon-delta proofs (which probably rate highest in the amount of fear produced in students doing proofs for the first time.) Exner talks through the scratchwork that goes into a proof (which is often ignored) and along the way gives a lot of hints on producing proofs in other situations. I strongly recommend this for any student taking a "real analysis" or "advanced calculus" course who is presently failing it ( as I once was a long time ago.....)

Reviews (5)

perhaps the best introduction to complex analysis
This is the book that really made me understand basic complex analysis.It doesn't try to give the most sophisticated or slickest presentation for experts.Instead, it gives a beautiful, concrete, down to earth explanations.The best feature is the applications.D. J. Newman is one of the world's great problem solvers, and this book includes numerous examples of how to use complex analysis to solve problems in surprising ways.Even in the more standard applications, such as summing series, the book gives many unusual examples.It concludes with Newman's proof of the prime number theorem, which is substantially shorter and clearer than many other proofs.

Not enough for getting a complete perspective.
My comment refers to the third edition of this book, but I don't think the fourth could be much better.

First of all, this title shouldn't be included in the "Graduate Texts in Mathematics" series becausethe material it covers is covered in introductory undergraduate courses.Second, eventhough the author made a great effort to include as much topicsas he could, the treatment of most of them is highly old-fashioned. I mean,he pays no attention to the most recent and elegant refinements of thebasic theory, so the student is not immediately able to understand the realimportant ideas behind the subject. For example, nowadays the proof of theCauchy integral formula is presented as a more ar less easy corollary ofthe general Stokes theorem. The Cauchy integral theorem is also obtainedeasily following the same fashion. Incredibly, the author explores thisline in one appendix, but not well done, and apparently he doesn't realizethat there is the key idea.

Also, keeping in mind that holomorphicfunctions are harmonic, most of the important results for holomorphicfunctions should follow at once from the corresponding ones for harmonicfunctions, but this old-fashioned texts don't take this remarkableimportant feature of complex analysis into account, making the treatmentinnecessarily complicated and leading the student to misunderstand bothcomplex and harmonic analysis. Eventhough the book includes a whole chapteron harmonic functions, the author doesn't use their power as heshould.

I'm afraid there are few famous introductory texts that I wouldsuggest for first-timers. The best of them is Markushevitch, unfortunatelyout of print.

There is also another serious drawback: The author pays noattention at all to boundary value problems and therefore to theCauchy-type integral, maybe the most important tool of complex analysis.The Hilbert transform is also not present.

If you have the opportunitytake a look at Muskhelishvili's "Singular Integral Equations" andGakhov's "Boundary Value Problems" and then you will understandmy point.

Lang's book could be used as a companion text and as areference for introductory courses. It's got some interestigexcercises.

Its contents are: Complex Nubers and Functions; Power Series;Cauchy's Theorem, First Part; Winding Numbers and Cauchy's Theorem;Applications of Cauchy's Integral Formula; Calculus of Residues; ConformalMappings; Harmonic Functions; Schwartz Reflection; The Riemann MappingTheorem; Analytic Continuation Along Curves; Applications of the MaximumPrinciple and jensen's Formula; Entire and Meromorphic Functions; EllipticFuctions; The Gamma and Zeta Functions; The Prime number Theorem;Appendices.

Please take a look to the rest of my reviews (just click onmy name above).

Reviews (3)

probably the best undergraduate level topology book
It starts the discussion with the familiar metric spaces and the real line and then
shows how convergence can be generalized. From this point, he then shows the
connection to topology. He also provides motivation for the results, exercises,
and historical sketches at the end of every chapter. The writing is unusually
clear.

I wish all math books were like this...
I'm a graduate student in mathematics and this is one of the few math books I can highly recommend. Topological concepts are made intuitive, so when you get to the formal definitions, they are no surprise, you evenexpect them. Sometimes it is explained why a 'seemingly obvious solution'doesn't work after all. This really helps you to see WHY things are donethe way they are. After reading this book you'll have a good basicunderstanding of what topology is all about. With this background it shouldbe no problem to tackle more advanced texts on topology.

By the way, Ilove the 'Extras' in the book. They are little pieces of history orexcursions into other parts of mathematics that can be found at the end ofeach chapter.

Reviews (8)

An Introduction to Difference Equation
El texto An Introduction to Difference Equations del profesor Dr. Saber N. Elaydi ha sido utilizado en reiteradas ocasiones, en el curso Ecuaciones en Diferencias Finitas, dictado por el profesor Dr. Julio López, de la UniversidadLa Frontera de Chile, curso del cual, este semestre soy alumno.

El texto del profesor Elaydi, es un libro que explica de forma amena, clara y precisa, distintos tópicos de Ecuaciones en Diferencias. Comenzando el estudio de los primeros conceptos a tópicos avanzados del tema, destaco dentro de sus muchas cualidades el gran número de ejemplos y aplicaciones expuestas en distintas áreas de las ciencias, además de gran cantidad de ejercicios propuestos, lo que permite al alumno practicar e internalizar las definiciones, proposiciones y teoremas estudiados.Me parece, sin lugar a duda, el texto más claro e interesante que he leído sobre el tema. Cabe destacar que este texto contiene tópicos, que en otros no son tratados. Otra importante cualidad del texto es, que aunque se tiene sólo ediciones en ingles, el profesor Elaydi, ha escrito su libro de tal manera, que no es un impedimento para los hispano parlantes, tenerlo como libro guía en nuestros cursos.

Alex R. Sepúlveda C.
Alumno de Ingeniería Matemática
Universidad de La Frontera
Chile.

One of the Very Best Introductions on Difference Equations.
Among the few introductory texts to difference equations this book is oneof the very best ones. It has many features that the other texts don'thave, e.g., stability theory, the Z-transform method (including a study ofVolterra systems), and asymptotic behavior of solutions of differenceequations (including Levinson's lemma) are studied extensively. It alsocontains very nice examples that primarily arise in applications in avariety of disciplines, including neural networks, feedback control,biology, Markov chains, economics, and heat transfer. Unfortunately, amongdifferential and difference equations, students usually only learn aboutdifferential equations, although many problems they encounter later can besolved using difference equations. Hence it is very useful for students tostudy difference equations (or even, dynamic equations, which canaccomodate differential and difference equations at the same time), and forsuch an undergraduate or beginning graduate course, Saber's book can beused as a textbook. It is particularly useful for such a purpose since itcontains many exercises and also many programs for the calculator TI-85.Although the first edition did contain several misprints, the secondedition has eliminated most of them. Saber's superb writing style makesreading this book very enjoyable.

Reviews (4)

Excellent introduction to group theory
Please note that the other reviews here are obviously for some other book.This is not an advanced text on bifurcations and stability.It is an introductory book on group theory.I have been using this book for self study.It is well suited to this purpose.The book uses symmetry to unify and motivate the study of groups. The discussion of the symmetry groups of Platonic solids is both enjoyable in itself and useful for visualizing groups. The chapters are very short.The exercises are well suited to gaining insight into the material.

A Good Presentation For the More Advanced Student
I have used Iooss and Joseph for over 20 years now, starting with the first edition back in the early 80s.

The book is for the more advanced student, one who has a bacic working knowledge of real and functional analysis. Unfortunately, these days, few engineering and physical science students have such a background.Hence, the book would be better if it contained some supporting basic material on mathematical analysis.

The first edition contained numerous "typos".While much improved, the second edition still contains too many "typo" errors.

Overall, the book is a good source of information that should be consulted by anyone interested in bifurcation theory.The book contains material (like the bifurcation of forced T-periodic solutions) not normally included in an elementary treatment of bifurcations.

John Stensby, Professor
Electrical and Computer Engineering
University of Alabama in Huntsville
Huntsville, AL

Reviews (1)

Another fine text by Klaus Janich
Janich's previous texts on topology and linear algebra are very valuable additions to the library of many existing texts on these subjects. His new book on vector analysis is similarly valuable. It deals with manifolds, differential forms, and the generalized Stokes's theorem. This is the mathematical machinery necessary, for example, for mathematical physics and differential geometry. Janich's chapter 10 discusses classical vector analysis, relating Stokes's theorem in its modern form to the classical integrals theorems of vector analysis. He closes the book with a discussion of De Rham cohomology and differential forms on Riemannianmanifolds.

Janich's exposition and mathematical taste are, as always, impeccable. Particularly valuable is a discussion in the second chapter which relates three different ways of defining the tangent space to a differentiable manifold. Each chapter also has a multiple choice test with answers in the back of the book which the reader pursuing self study can use to test his/her knowledge of the subject. Exercises close out each chapter as well.