1 comments about Infinite Dimensional Analysis: A Hitchhiker's Guide.

1. The monograph covers advanced mathematical methods for economists. It includes chapters on general topology, topological vector spaces, Riesz spaces and Banach lattices, measure and integration, etc. While the book does not contain (hardly) any economics, the mathematics covered is selected under the aspect of later applications to economics. The book contains for example a long chapter on correspondences, a topic which is hardly covered by any standard math book. The presentation of the mathematics is throughout clear and precise. The advantage of the book is that it covers a wide range of mathematical topics, which could not be found together in a book before. Graduate students in economic theory can use it as a text book, but it can also be used as a reference book. The only lacks of the book are that there are no exercises and that not all math areas important to economics (e.g. differential topology) are covered. Overall, this is an excellent book and should become part of the library of everybody interested in mathematical economics.

5 comments about Partial Differential Equations: An Introduction.

1. I have never commented on a book, up until now... and I do so only because I don't think that this book gets enough credit.
   
   People have complained Strauss may not have explained some proofs in as much detail as he could have, people complained that he didnt give enough examples, I think this is more of a problem with the readers than the writers. If you need someone to hold your hand through every step and detail, I think you should reconsider why you are studying what you study.-
   
   I am an undergraduate at NYU, one of the best research institutes for PDE's. I thoroughly enjoyed reading this book, it gives an amazing description of what PDE's are, how to solve them, and how they are used in science. One thing I REALLY enjoyed about this book was it did not do what many other books do: first dive into seperation of variables and focused only on that. Instead Strauss shows how to solve first and second order equations without boundary conditions, giving a very elegant prose doing so!
   
   However, I think much of the problem that people are having with this book is that it's not a "one-size fits all." (Which I don't think any book can be!) If you are a Scienctist or Engineer and just want to learn PDE's to solve problems in science.. find another book, because this book is not the book for you.
   
   That being said, if you are Mathematics student or interested in a more deep study of PDEs this is really a good book for you. You definitely should have taken Calc. 1-3, Linear Alegbra, ODE, and I recommend one semester of Analysis (for function spaces) before tackling this book, that is what I had, and I loved this course.
   
   PDE is a difficult subject/course and Strauss does an amazing job at explaining it, if someone like me can get PDEs so well from this course, than I seriously believe that complaints about this book is due to fault in the readers and not the writer.



2. I have only read bits of this book, but every time I read it I come to the same conclusion. I think the previous reviewers have highlighted the key problem with this book. A previous reviewer wrote: 'If you are a Scientist or Engineer and just want to learn PDE's to solve problems in science find another book, because this book is not the book for you.' There is some truth in this. However, in the preface Strauss wrote: 'This is an undergraduate textbook. It is designed for juniors and seniors who are science, engineering or mathematics majors.' It would appear that the book is simply not very suitable for the wider audience it was intended for - in particular it is too advanced for less experienced undergraduates. Also, despite containing numerous applied examples, these examples are dealt with so briefly that they would make little sense to a scientist or engineer who has not already studied the applied material in depth on other courses. This is somewhat inevitable as the book attempts to include a wide multitude of examples, and so its strength is also its weakness. This book is not a 'one size fits all' but was clearly advertised as one and ends up simply belittling itself in trying to be one thing whilst actually being another. It is a very good reference, however, and is very valuable to scientists and engineers who have already studied much of the material in an applied context, but the text is somewhat disjointed as it reads more like a catalogue of PDEs rather than as a 'how-to-do-it from first principles' manual. The book looks attractive, but every time in the past that I started reading it I soon put it down again! However, I have begun to read the book seriously from cover to cover. Maybe a taught course that is constructed around the book would work, indeed it is a valuable source of examples, but this is not a good book for self-taught study, except perhaps to those with more maths than a typical junior undergraduate. I would however recommend it as a must in the library of anyone who deals with PDEs on a frequent basis, or who wishes to teach the subject. In short - great for the right audience but the right audience is not exactly the one advertised in the preface! (This is a common failing of many textbooks). Since I began to seriously read this book from cover to cover and I am finding it fairly straight-forward, but then I have studied PDEs before. However, those who do not like the book should be comforted, because at first I never liked it either - but give it a chance and read it carefully and remember that many of the examples will not make complete sense until you have studied the science behind them in other courses. In this sense, the text is a valuable reference to senior science and engineering graduates. I think for someone new to the topic the text probably skips the odd crucial sentence of explanation. However, for an advanced mathematics, science or engineering undergraduate the strength of this book is that it puts everything together and so is a valuable reference and valuable to consolidate what you have studied on other courses. However, because it is so all-inclusive, it would need to be several hundred pages longer if it were to describe everything clearly from first principles (even assuming competence with ODEs). So, if you still don't like it, then come back to it in a year or so. I shall update this review if, as I continue with the text, my view changes.



3. As an intro to PDE book, it's simply put, terrible. The author must have forgotten that he was writing a book aimed at undergraduates. No simply examples, not a single one. The book jumps straight into theory right from the first chapter and beyond. The exercises involve very intricate proofs and there are no simple computational exercises, none. If you've taken math analysis or some kind of easier PDE course, then maybe you'll like the book, but if you're like me and you go from ODE's to PDE's than the transition is way too rough.



4. I've spent the past seven years or so working on analytical and numerical solutions to the various PDEs that price financial derivatives. My focus has been very much on getting and extending useable answers. When it comes to PDEs specifically, I'm mostly self-taught, but my background in real variables and functional analysis is solid.
   
   In some ares of mathematics, a single, classic text whose exposition is top-notch, or that inspires despite its exposition, manages to cover the field well. For example, I'm thinking of books like Halmos (Measure Theory), Rudin (Real and Complex Analysis), or Segal and Kunze in real variables and integration; Lax or Reed and Simon I (Functional Analysis) in functional analysis; Lang (Algebra) in algebra; and Kelley (General Topology) or Milnor (Topology from the Differentiable Viewpoint) in topology. (Full citations are in Listmania; see my Amazon profile.)
   
   I've yet to find a single reference for PDEs that addresses all of my questions, but several books taken together manage nicely. I jumped around in these books when I was learning the subject, and I'm convinced that such cherry-picking is the best approach for PDEs, since the field is so broad in theory and applications. The downsides, of course, are expense and potential confusion from conflicting notation and approach.
   
   Ignoring just for the moment the vast area of approximate solutions by discretization and perturbation techniques, here's who seems to be best for what, when the problem involves linear PDEs:
   
   :: Need quick intuition: Farlow, Myint-U and Debnath, Brown and Churchill;
   
   :: Need more theory: Stakgold (Green's Functions), Evans, Folland;
   
   :: Need help on modeling: Strang (CSE), Stakgold (BVPs), Haberman (Applied PDEs), Farlow;
   
   :: Don't understand how concepts relate: John, Levine, Garabedian, Strauss, Carrier and Pearson (PDEs);
   
   :: Can't find tough enough exercises: Carrier and Pearson (PDEs), Kevorkian;
   
   :: Need inspiration or deep intuition: Courant and Hilbert (both volumes), Zeidler (Nonlinear Functional Analysis 2A [Linear Monotone Operators], Applied Functional Analysis [especially AMS 108]).
   
   I've ranked books very subjectively within each category on a composite of things like relevance, completeness, clarity, and ease-of-use. And I should stress that I'm no doubt ignoring many fine favorites purely through unfamiliarity.
   
   WHERE DOES STRAUSS FIT? I repeat, all of these books address each of the needs in some measure, but no one is adequate for all. The terse treatment and broad coverage in Strauss are great for tying concepts together and revealing their logical relationships. This is especially evident in the superb Chaps. 1 and 2-3, as well as in Chaps. 9 and 10, which treat the Cauchy Problem and BVPs in space, respectively.
   
   Chapter 11's discussion of eigenvalue problems, and particularly their asymptotics, is remarkable at the book's level but nowhere near that in Garabedian or especially that in Courant and Hilbert, which is the original synthesis of work beginning with Weyl to which Courant and Hilbert each contributed in important ways. (The notes to Sec. XIII.15 of Reed and Simon IV [Analysis of Operators] have the history of Dirichlet-Neumann bracketing, the main methodological advance.) Both of Stakgold's works also discuss this problem but not as well as Strauss.
   
   I've done very little teaching (and I wasn't very good at it!), so my views should be taken with a grain of salt perhaps larger than usual, but if I chose Strauss as a text, I'd have to believe either that my lectures would fill in the gaps that Strauss so clearly has or that other books on my syllabus could take up the slack. If you're having trouble learning "from Strauss," the problem may lie not with the book, but with an incomplete course, since Strauss is, in many ways, only a good set of summary notes. Again, it's good as far as it goes, but it doesn't go the whole way; that's why you need the other books.
   
   CHOOSING A SINGLE REFERENCE. If I were packing for a desert island, I'd take Levine and Garabedian, since everything I need can be backed out of their presentations with some effort. In many ways these books can be thought of usefully as a set. Both are written at an intermediate level, meaning that techniques less sophisticated than those involving function spaces are fair game. Levine spends 700 pages on separation-of-variables, Fourier analysis, and transform methods, applied to parabolic and elliptic equations in general and the diffusion (heat) equation in particular. Garabedian picks up just where Levine leaves off to treat the Cauchy Problem for hyperbolic equations and the Dirichlet and Neumann Problems for elliptic equations. His book is also roughly 700 pages in length and like Levine's is a model of clarity.
   
   Although both books have been available for some time, basic approaches to the three classic, second-order, linear equations and their variants--the gist of a first course--have changed so little in the past 50 years that publication date may not be as much a factor in selecting a single reference as it would be in some other areas. Indeed, it's well worth reading Fourier's original memoir on heat conduction, possibly modulated by a modern treatment like Carslaw and Jaeger (Conduction of Heat in Solids) or Crank (Mathematics of Diffusion). Levine (Chap. 13) also contains a technical precis of Fourier's original approach.
   
   If I found that I needed greater depth, meaning function spaces, I'd turn first to Courant and Hilbert or to either of Zeidler's state-of-the-art books. The treatment in C&H is profound and downright majestic. Many have spent a productive professional lifetime in these books, and C&H-2 comes close to being the sort of reference I describe in the second paragraph. Because of its age, it's possible to see in the books' discussion much of the intuition of Hilbert and Sobolev spaces that was later covered with layers of rigorous abstraction. Zeidler's discussion of that abstraction is simply the clearest that I've found anywhere. It's extraordinary that any author works as hard as Zeidler to convey mathematical ideas, and for this reason his books are among my favorites across all topics.
   
   TRANSITIONING TO DISCRETE APPROXIMATIONS. If I also took Gil Strang's new book to ease the transition to discrete approximations and eventually building and evaluating numerical code, I'd forget the rest of the list without worries. Indeed, as Courant mentions in the preface of C&H-2, there was to have been a brief third volume dealing with discrete approximations for existence and construction of solutions. Strang would stand nicely in its stead, if not for existence, then certainly for construction.
   
   Actually doing numerical work is another matter entirely, of course, and I've given some idea of the books I've found useful for these problems in my brief review of Chung's book on computational fluid dynamics.
   
   HOW ABOUT PERTURBATION SOLUTIONS? Finally, if I closed my eyes and pretended that I'd always separate variables, so I only needed to worry about perturbation solutions of hairy ODEs, I'd toss in Bender and Orszag and feel pretty good about analytical approaches. (If you're lost in B&O, and it does have its moments, try Holmes, which is a more accessible survey at less depth.)
   
   If I just couldn't bring myself to make that assumption, I'd take Kevorkian and Cole (Multiple Scale and Singular Perturbation Methods), which deals in part with perturbation solutions of PDEs directly, and Verhulst, which is a bit longer on intuition.
   
   The brave might also consider Van Dyke (Perturbation Methods in Fluid Mechanics), which deals specifically with singular perturbations of the Navier-Stokes equations, their many variants, and the other equations of fluid mechanics. To go this route you'd have to believe that you could adapt Van Dyke's results to whatever problems you ran into, which can be real work. Hinch's short book is useful as a complement.
   
   ONCE MORE...WITH FEELING! If I were learning things from scratch again, I'd sleep with Farlow under the pillow and Garabedian under the bed, regardless of what textbook my instructor had chosen. For the careful Farlow raises at least as many questions as it answers--purists have my guarantee that they'll hate it. You need to supplement Farlow with greater depth in your areas of interest, and in most instances Garabedian cleans things up nicely without doing violence to the concepts.
   
   An added plus is that since the books are reprints, published by Dover and AMS Chelsea, respectively, their cost is quite reasonable, even though Garabedian is beautifully printed and library-bound (would you believe sewn-in signatures and useable inside margins?).
   
   This review is a lot longer than I'd first intended, and its recommendations are in many ways idiosyncratic but certainly worth their cost. Partly, I think that's a function of the field itself. There seem to be as many approaches to learning PDEs as there are backgrounds and interests. The diversity of sources is likewise broad, and their quality is quite high. The beauty and power of the subject have lured many first-class mathematicians, like a striking number of the authors mentioned above, into writing basic texts. In the end someone's treatment will answer your questions, pretty much no matter what they are, if you just have the patience to look around. After enough looking, of course, you'll find you can answer many of your own questions.
   
   One of the things that makes real-world PDEs in whatever field such fun is that getting an answer is all that's important. It doesn't matter what books you use, what willing help you receive from whom, or how you reformulate a problem to make it more tractable, as long as a result that answers the real question (sometimes a rather elastic notion) eventually emerges from your efforts. There's much to be said for learning the field in the same no-holds-barred way, and I hope my remarks can get you started in that direction.



5. you should first read about the Green functions in some other books to appreciate Strauss's way of explaining things. He is clear, and easy to follow. The way he presented the material is not so advanced, Nevertheless the concepts are beyond undergraduate. This book is one of the best books in PDE's I've seen, specially chapters 7-11 and most specifically he did an awesome job for the Green functions.

3 comments about Functioning in the Real World: A Precalculus Experience (2nd Edition).

1. This textbook, although presenting a basic approach to the fundamentals of Precalculus, provides a cluttered collection of confusing explanations and poorly edited questions. The cover proves to be the best designed part of the bookl; inside there is little illustration and lots of wordy text. The book contains no full set of answers which could often leave a teacher stranded in some of its poorly phrased, misguided questions. Overall, a poorly conceived, designed, edited, and published textbook.



2. The text is very wordy and problems could be answered in the book or in a solutions manual. Very few illustrations are a problem. Our Prof. had to explain several mistakes in the text that were completely incorrect.



3. I read the preceding reviews and don't find what was said untrue. The simplicity and algorithmically solved problems, however, belie the sophisticated look at functions. My students found the work with difference equations difficult because of the symbolic manipulation, but their teacher (me), appreciated how the text took the idea of derivatives, reduced this concept into a discrete function, and concluded with the antiderivative. Very clever.

3 comments about Putnam and Beyond.

1. Another panorama of amazing math problems written by two famous math problemists: Titu Andreescu and Razvan Gelca
   Many many congratulations to them for this invaluable treasure of math problems.
   I am not absolutely able to describe this excellent book; the best way is purchasing this book. I highly recommend it to all math lovers; in particular to whom are preparing themselves to mathemaical competitions of all kinds.
   In fact I do warmly recommand all of the books by Titu Andreescu and his colleagues without exception!!!



2. The book's usefulness depends highly on the level of the reader. It is true that the book provides a succinct overview of each topic covered but the problem is that if you are not already familiar with that topic you will not be able to understand it only with the information provided.



3. This book consists of a very useful collection of Putnam-like math problems. Putnam and Beyond is organized for self-study by undergraduate and graduate students who wish to try a lot of competitive math problems.It is also useful for teachers who are preparing their bright students for IMO type (or higher) math competitions. However the book assumes a level of mathematical maturity and prior mathematical knowledge that not many college students possess. Another very useful book for math competitions is The IMO Compendium.

2 comments about A Geometric Approach to Differential Forms.

1. I highly recommend this text for anyone looking for a "gentle" introduction to forms and manifolds.
   
   When learning a topic, I believe that it is important to develop both computational proficiency and a deep conceptual understanding. I have come to understand that manipulating symbols is not sufficient and that whenever possible, understanding the underlying geometry is critical.
   
   For whatever reason, I struggled to understand forms from other sources. (Maybe I was too focused on the algebra of the wedge product.) However, Bachmann's exposition was easy to follow and very insightful. It was a revelation that all integrands are not differential forms. Also, I had read elsewhere that forms are a basis for the tangent space of a manifold. I could say the words but they contained little meaning for me. Within the first couple of days with Bachmannn's book, this as well as some other basic ideas became crystal clear. I particularly liked that he at times presents more than one geometric interpretation of an concept.
   
   Anyone who has already seen some vector calculus and now wants a very quick introduction to forms with a minimum time investment can benefit greatly from this text. In total, I spent about a month reading, A Geometric Approach to Differential Forms and I am now confident that I am ready to tackle more advanced texts on the topic.
   
   A word of caution, in the book's Preface, it is suggested that there are three possible tracks one can take with this text. In addition to an upper division track that focuses on forms and manifolds, one is a vector calculus track and another is a multi-variable calculus track. In either of the latter two cases, if that is your main interest, I would recommend a text like Marsden's Vector Calculus. It encompasses a broader base of material and it is also very well written.



2. Easy to read, but not too deep in theory or algebraic properties of differntial forms. Interesting for many exercices to solve ( with solutions !) Useful to grasp an intuitive approach to the concept, but if you are seeking a thoroughtly book on the subject this is not the book.

5 comments about Numerical Analysis: Mathematics of Scientific Computing.

1. The book was a major disappointment. I am glad that I did not purchase it for my class, but instead borrowed it. The ordering of topics and emphasis choices never seemed to make sense to me. The layout throughout most of the text is like one long, run-on sentence. The underlying structure of numerical analysis never developed and I was left swimming in meaningless details while the basics were short-changed by an over abundance of specialized algorithms. Perhaps the text's curriculum could be saved by a capable professor, but alas my professor was just as scatter-brained as the text. More pictures would also have been helpful. A replacement text I recommend, which covers the first, matrix theory portion of this book, is David S. Watkins' Fundamentals of Matrix Computations.



2. I think this book is lucently written and explains various aspects of numerical analysis in great detail. The proofs are stated in an understandable way and algorithms are presented clearly and in such a way that it is easy to implement them in the programming language of one's choice.



3. I bought this book for my college and it was excellent



4. Honestly, this has turned out to be a horrible book. Particularly in the disconnect that exists between the text and the problem sets. Very few of the examples are useful to read through because they are trivially simple, while the problem sets seem to take particular delight in finding the hardest tricks to be solve-able.
   
   I would highly suggest that anyone that purchases this book, already know what they are trying to learn, or have an excellent teacher that can fill in the gaps.



5. I teach a course in numerical methods every other year and use Maple as the platform for the computer solution of problems. At this time, I am using "Numerical Methods: Third edition" by Faires and Burden and am quite happy with it. However, I am always trolling for better textbooks so that it is premise with which I examined this book.
   The level of rigor is on the higher end of the scale, there are many more formal theorems and proofs than is found in most of the other numerical methods textbooks. Rather than using code in Maple or Mathematica, the algorithms are expressed in a lower level pseudocode. It has the appearance of a programming language; however some of the operations are expressed in mathematical form for brevity. This use of the mathematical syntax generally makes them easier to understand than if they were expanded out using a programming language. There are many exercises at the end of the sections, including a group meant for computer solution.
   I will not be adopting this book for my class, the level of rigor is a bit higher than my students can easily digest and quite frankly, I am hooked on using Maple as the computer platform. Nevertheless, it serves as a valuable secondary reference, where when necessary, I can look through a detailed proof of a topic that I am about to present.

4 comments about Introduction to Numerical Analysis: Second Edition (Dover Books on Advanced Mathematics).

1. This book is really one of the best books on numerical analysis I've ever seen. It is simple but complete.



2. This is a reprint of the 1974 2nd edition. So what is Numerical Analysis? It's the down-and-dirty methods of approximation and interpolation of equations that don't have closed-formed mathematical solutions. Just about every real-world problem in material engineering from pipe flow, wing design, convection currents to multivariate econometrics have to resort to numerical approximations. You'll find all the familiar names from your undergrad Math and Physics courses here (Newton, Gauss, Largrange, etc.); however, advanced methods in Numerical Analysis has changed tremendously since this book was published. Since NA is dependent on present computing power, what was once too expensive or unthinkable in the 70s can be done today. However, it's still a great introduction and a great bargain from Dover. The writing style is informal and conversationally peer-to-peer, rather than teacher-to-student. There is no historic consciousness placing methods and men in context. You won't find programming algorithms here (not even Fortran or Pascal). There are probably better books out there for what ever your specific speciality is, but at five times the price of this Dover reprint. You'll will find the old favorites here. The book covers the various finite difference approximations (forward, backward and central differences). It uses the operational approach for these. The later chapters cover splines, continued fraction and iterative methods. More importantly it covers the difference between round-off error v. truncation of divergent series in approximations -- something that still confuses practicing professionals. Be warned there have been many improvements in theory and methods in finite element methods of Fluid Dynamics and other 3D modeling (bezier and NURBS); And, the whole world of Complexity and Chaos theory happened well AFTER this book was published. Calculus and Differential EQs are prerequisite, there's no attempt at introduction in the text.



3. Although old, this is still an outstanding introduction to a wide range of topics in numerical analysis. I get impatient with the amount of detail Hildebrand devotes to some topics, but that's because those are topics where I already know the techniques and pitfalls.
   
   However, I have one serious criticism of this book. Hildebrand in very many places drags in the question of inherent errors in input data, but fails to distinguish the different views one must take depending on how one got involved with some topic. In 50+ years of doing numerical analysis and numerical software from time to time, I have come to realize that three quite different issues of inherent error occur.
   
   First, one may be working with scientists or engineers to derive results for a specific problem or set of problems. In this case, one must ask two pertinent questions, and keep asking until one gets clear answers: "How are you getting the input data?" and, "What are you going to do with the results?" Given answers to these two questions, one can do analysis and computation knowing from the start how accurate the input data is likely to be, and how much that matters to the results. Hildebrand pays little attention to the quite complicated problem of how one should do the analysis and programming in those situations.
   
   Second, it may happen that there is no input data from the real world, and hence no inherent error; the input data is conjured up out of whole cloth, as happens in many calculations in "computational physics". In those cases, one wants to produce results that accurately reflect the hypotheses provided by the people with the problem to be solved. Usually, one finds in such cases that the more accurately one can do the computation or analysis, the better one can serve one's users.
   
   Third, and most difficult, is the situation where one is writing a utility routine for use by large numbers of people, most of whom one will never encounter. Everyone who has done much numerical programming faces this issue from time to time. Here the problem is that the users are likely to place absolute faith in the results, even in cases where you, as the implementer of the software or originator of the analysis, may know all too well that the results are unstable with respect to very minor variations in input data. This occurs with monotonous regularity, for example, in routines that manipulate matrices to derive such quantities as eigenvalues and eigenvectors. In my own experience, a high proportion of the actual matrices that users present to "utility packages" are ill-conditioned, and there's a reason for this. If the problem were well-conditioned, it wouldn't be a problem for the scientists or engineers or financial types who need a solution; they would know a priori from experience what the answers are. I have no good answer for how one should think about such "utility software" and neither does Hildebrand. The way I deal with it myself is to ensure that mathematically accurate results are provided even for ill-conditioned problems, and to provide documentation for users that includes the equivalent of: "If you ask this software a stupid question, it will give you a stupid (but correct) answer, so if you are unsure about the stability of your data, please call or visit or email me to discuss your specific problem."
   
   In short, despite the virtues of this book, it doesn't come to grips with the issue of numerical analysis and mathematical computation that I have found causes me more headaches than any other.



4. Certainly one of the best books on Numerical Analysis ever written. Since this subject matter is vast, it has not been covered in its entirety, but what has been covered is simply the best. Moreover, it has been written by one of the best mathematicians. A MUST READ for everyone using numerical analysis.

5 comments about Differential Equations and Their Applications : An Introduction to Applied Mathematics (Texts in Applied Mathematics, Vol. 11).

1. As an academic I rarely if ever use textbooks because I find their writing style bland and disengaging--especially in my own area (economics). Braun's text is the exception to the rule: this is a textbook that is so well written it could qualify as entertainment. At the same time the major issues in the field are covered well, complete with an overview of the linear algebra required for doing ODEs and a good introduction to nonlinearity and chaos. It may not, as one reviewer notes, provide a complete and rigorous coverage of this highly technical field, but it does what a text should do but so few achieve: it excites its readers and inspires them to delve further into this fascinating discipline.



2. I quite liked this book. It taught the material in a different order than I am used to, which gave me a more complete understanding. I would recommend it, especially to people who already know a bit about differential equations, as it is quite wordy and assumes some prior knowledge.



3. This book is extraordinarily clear as well as being concise (but never too much so) in the mathematical parts. Discussion of applications is verbose, but is kept in separate sections; this material can be omitted entirely or read later without any detrimental effect to the flow of the book. However, the discussion of the applications is interesting and deep, and would be useful (and fun) for motivated students to read.
   
   The book begins with a no-nonsense discussion of how to solve differential equations analytically. Unlike many books, it gives clear instructions to the reader as to how to know which techniques are applicable. Also, it does not introduce qualitative or numerical methods until it has already developed a number of analytic techniques, and in my opinion, this results in greater clarity than the path most books take of integrating (or should I say jumbling?) the material together. The book gradually and logically covers the ground between analytic and numerical, moving towards actually writing algorithms, which are included in the text. The emphasis is always on understanding. Exercises are straightforward and useful.
   
   My only complaint is that, in this modern age, the C programs should be included in the text and the Pascal and FORTRAN ones relegated to the index. (It is the other way around, alas.)
   
   This book is simply wonderful for anyone studying differential equations for the first time. I do not understand why undergraduate institutions use the more commercialized texts instead of ones like this. This is a great book; it would be excellent for a textbook or for self-study.



4. From a pedagogy point of view this is a bad book becuase of the way its chapters are organized and presented. There seems to be a lack of a natural order of topic in the book, specifically that theory and applications are intertwined/overlapped rather than placed in seperate chapters. Also, few exercises are presented at the end of a section.
   Putting aside the pedagogy philosophy, this is a great book for a one semester course in Differential Equations but I would rather choose "Differential Equations, The Classic 5th edition" by Dennis G. Zill.



5. This book discusses several excellent applications. I shall summarise a few of the simplest and most beautiful ones. Warfare models. Consider first a battle between two conventional armies, A and B. Each army has a constant efficiency coefficient (determined by weaponry, training, etc.): an A soldier takes out a enemies per unit time while a B soldier takes out b enemies per unit time. The battle is then described by the differential equations dA/dt=-bB and dB/dt=-aA. Dividing the first by the second gives aAdA = bBdB, which we integrate to get aA^2-bB^2=constant. The sign of this constant determines the outcome of the battle, since if, for example, there are side A troops still standing when B reaches zero then the constant must be positive. Thus the strength of an army is proportional to the square of its size, and this has an important strategical implication: never divide your forces. Now consider a battle between a conventional army A and a guerilla army G. The conventional army suffers casualties as before, dA/dt=-gG, while their offensive strategy consists in firing into the jungle more or less at random, making guerilla casualties proportional not only to the conventional army's efficiency a and size A but also the number of guerilla troops G, i.e. dG/dt=-aAG. Dividing the first equation by the second gives aAda=gdG and integrating gives (aA^2)/2-gG=constant. Thus the guerilla can divide its forces without loss, while the conventional army still does not want to divide its forces. Predator-pray systems. Consider the system of food fish and sharks. With no sharks around, the food fish would grow exponentially, x'=ax. The sharks alone, having nothing to eat, would die off exponentially, y'=-cy. In the combined system the food fish are eaten at a rate proportional to the number of encounters with sharks so x'=ax-bxy, and more sharks live as a result of this so y'=-cy+dxy. An equilibrium solution is x=c/d and y=a/b. We cannot find other solutions explicitly but we can prove that they are periodic (also very plausible from the direction field, should we choose to draw one; there are none anywhere in the book) and prove the following qualitative theorem: for any solution, the average number of food fish is c/d and the average number of sharks is a/b. Proof: Let x, y be solutions with period T. Take x'=ax-bxy and divide it by x to get x'/x=a-by. The integral of the left hand side from 0 to T is log(x(T))-log(x(0))=0, so the integral of the right hand side is also 0, so y-average=(1/T)(integral of y from 0 to T)=a/b. Similarly, taking y'=-cy+dxy, dividing by Ty and integrating from 0 to T gives x-average=c/d. Volterra used this result to explain why Italian fishers caught a larger percentage of sharks during world war I when overall fishing was reduced. If we assume that fishing by net simply catches a random handful of fish in proportion to their number then the system above becomes x'=ax-bxy-ex and y'=-cy+dxy-ey, i.e. x'=(a-e)x-bxy and y'=-(c+e)y+dxy, which is just the same system with different coefficients, making the new averages x=(c+e)/d and y=(a-e)/b. In other words: an increase in fishing benefits the food fish and a decrease benefits the sharks. Population growth. The standard model for population growth is the logistic equation p'=kp(1-p/s), where s is the maximum sustainable population. The observed periodicity of many populations is to be explained by a large population's susceptibility to epidemics, as is confirmed when we study an epidemiological model in detail later. But right after population growth we turn instead to the spread of technological innovations, which is not terribly exciting, but it can easily be translated into a simplistic model for the spread of a disease. The disease spreads in proportion to the size of the infected population p and, because of limited encounters as more people are infected, in proportion to the uninfected population (n-p), so p'=kp(n-p). But by factoring out the total population n we see that this is simply an instance of the logistic equation, where the total population corresponds to the sustainable population and the infected population corresponds to the living population. Thus mathematics tells us that the growth of a population is the spread of the decease of life.

5 comments about Schaum's Outline of Graph Theory: Including Hundreds of Solved Problems.

1. In this book one can find a practical survey of both principles and practice of graph theory, with great coverage of the subject. The outhor provides a lots of solved problems, with losts of theory proofs and all with great clarity and common reasoning. The outhor gets you enter the subject step by step from the easy problems to the hardest with great skill. Also the algorithms on graphs presented in this book, and in general the algorithmic approach of this book are presented most clearly. You wouldn't leave this book until you'l finish read it and understand graph theory. Finally you would fill that at least on one branch of mathematics you are well sitted.



2. In general, the book not requires study in advance, but it is better for reference. I'm a software engineer and the book's treatment of "Shortest Path" and "Connectivity" problems is very usefull. Good for fast remember of the subject.



3. This book was an absolute hell to contend with. I've taken two courses in Graph Theory, using Robin J. Wilson's Introduction to Graph Theory and this cheap broadsheet, respectively. Wilson's book is the one to use! It's extremely well-written, even fun to read--the reviews on Amazon will bear that out.

   In the second graph theory course that I took (to refresh and refine my understanding), the professor chose the Schaum text solely for its low cost--he thought he was doing the students a service. Hardly.

   No thought whatsoever has been put into the readability of this book. The tiny dark-grey font on light-grey paper is a simple enough design flub that makes reading past even two or three pages at a time almost unbearable. Defining terms is seen as a chore to be compacted--a single page at the beginning of each chapter might try to define 10-15 terms, just to get them out of the way. It becomes a mess of bold print that the reader is forced to continually return to because the definitions come with no context nor examples by which to remember them. In the end, the reader realizes that 2/3 of the book is just list after list of badly-worded questions following under-scripted lessons.

   Look, it's not even worth writing any more about, the text frustrates me so much. There's only two other reviews on this page, and I'd place money on them being written by the author himself. Save yourself the $$$ and the hassle, and just go buy Wilson's book. Trust me.




4. I have bought and used many Schaum's outlines on various subjects in math and science, and I would say that this outline on graph theory is one of the worst. Most Schaum's outlines give you the theory in small doses, with plenty of diagrams to explain the concepts. This outline reads more like one of the textbooks on the subject, however. Theorems and their illustrations are poorly presented, and the author could not have made the subject matter drier and more unappealing if he had tried. You might be able to get something out of it if you are a student of pure mathematics, but you will definitely be disappointed in this book if you are a computer science student. If you are already using a bad textbook for a class in graph theory, this book will only add to your collection of bad unreadable texts on the subject. For computer science students, I suggest that you check out the chapters on graph algorithms in Introduction to Algorithms by Cormen et al. That book has pseudocode, explanations, and diagrams to help you work out implementations of various graphing algorithms.



5. This book is wonderful in my eyes. However, I do not recommend most Schaum's Outlines as textbooks, but as supplements to texts. They just contain too much. This book is good reference to have if you're doing a course in graph theory or if your work involves graph theory. I highly recommend it for reference use.

5 comments about Modern Graph Theory.

1. My profile is the following: I am a phD student in theoretical computer science and I needed a good introduction book to graph theory.

   This book is just what I needed...




2. Bela Bollobas has the rare gift of having both deep mathematical insights, and the ability to eloquently communicate them in a way that is accessible to the average graduate student. In his book "Modern Graph Theory", Bollobas covers just about every exciting area of the subject, and does so in an up-to-date fashion that gives the reader a big picture of each sub-area of the field. The ability to do this not only seems difficult, but also essential, since he himself has written entire books on two of the chapters (extremal graph theory, and random graphs). Just about every major important theorem (including max-flow/min-cut Theorem, and theorems by Menger, Szemeredi, Kuratowski, Erdos/Stone, and Tutte) can be found here, and thus makes this book indispensable for anyone who does research in graph theory, combinatorics, and/or complexity theory. In my opinion the true highlights of this book are indeed those areas he knows best: extremal graph theory, random graphs, and random walks on graphs, the latter of which may be the best introduction to that subject that one will find in a textbook.

   My only complaint, at the cost of perhaps half a star, is that his discussions and proofs often seem difficult to follow, as he will state something that to him seems quite obvious, yet to this reader often seemed a bit subtle, and would hence slow down the reading. Indeed, if these off-handed remarks were included as exercises at the end of each chapter, then the number of excercises would have swelled from the current 600 to well over one thousand ! Speaking of which, these 600+ exercises, although also representing another blessing of this book in that they add another degree of depth, tend to lack "starter" exercises, and go straight to the theory. But this is to be expected from a graduate text.

   Finally, for the reader whose research significantly intersects with graph theory, but may not be ready or willing to be initiated by Bollabas into the world of graph theory, I would recommend Dietsel's graduate text on the subject. His book covers similar topics, but may be more clearly and transparently, but with less depth and insight.




3. I am, what Prof. Bollobas would call a hobby mathematician. Some popular science book arouse my interest in graph theory, and the author of that popular science book recommended this book. I feel it was a vey good introduction to the subject, even though the proofs become challenging at times. His motivation for the subject is always concise but precise, one cannot but notice, that a master of the subject is writing about it.
   
   The only distraction are the enormous number of typographical errors: I counted over 60, and this in a third corrected printing!?!



4. This book is absolutely precious! It is a little bit weird, but you can get used to it. This book's strongest points are that it is easy to jump around in it, and it contains a wealth of material. It also has incredible numbers of exercises, of greatly varying difficulty levels.
   
   The author's clarity of writing comes out particularly well in the later chapters. In particular, my favourite parts are the discussion of algebraic graph theory, and the discussion of the Tutte polynomial and connections with knot theory. There is also some beautiful use of linear algebra in various parts of the book; some rather strange and difficult results are presented very clearly.
   
   I think this book would be a great purchase for anyone wanting to engage in some self-study in graph theory, or anyone wanting a good reference on graph theory, or anyone wanting to work some hard problems (or easy problems) in graph theory, or someone choosing a textbook for a graph theory course...or...in short, anyone who wants anything to do with graph theory at all.



5. This is a very well structured book. However, this book is not amenable to easy reading. The theorem proofs are short and concise with no overt explanations. Bottom line is that reading this book is a an exercise for the brain.
   
   Being an engineer my only grouse about this book is that this book is written for mathematicians and as the author himself claims there are very few practical applications accompanying the theory. But this being a graduate text in mathematics it really cannot be expected to fulfill this need.