Reviews (1)

Reviews (3)

Like no other book on numerical methods I have read.
I sympathise with the reviewer who said this is one of the few
books on numerical methods he could stand.I will go further
and say this is a book that can be enjoyed. Example: section 2.8
"The Frequency Distribution of Mantissas" explains why the
leading digits of of decimal numbers are not uniformly
distributed, a result that is surely counterintuitive. There is
much more material of interest in this book too. It does
contain standard material too but is more readable than many
books. The author offers much practical advice and insight.
(Hamming is a famous name in applied mathematics and electical
engineering).

The Purpose of Computing is Insight, Not Numbers
Throughout the book, that motto is repeated.

By reading and absorbing the material in this book, the reader is left with the tools and the insights necessary to derive their own numerical methods.

No longer willnumerical methods be memorized as textbook formulas -- now the reader canadapt and derive a formula to solve a specific problem, instead of tryingto fit one of a small number of textbook formulas to a problem.

Thedistinction is made between numerical analysis and numerical methods, withemphasis on the latter.

The book is roughly divided into two parts. Thefirst part covers classical numerical methods, using classical erroranalysis (truncation error, roundoff error). The second part reexaminesthese methods under the frequency domain, analyzing how numerical methodsaffect various frequencies (the "transfer function"approach).

Numerical methods are derived under an information theorymodel, such as by finding a quadrature formula of the highest polynomialdegree of accuracy, given limited information about the function and itsderivatives.

Matrices and linear systems are not discussed as much as onemight expect, although one chapter convincingly leads the reader toquestion some classical methods.

The content is well-rounded, introducingmany readersto topics such as random number generators, differenceequations and summation formulas, digital filters and quantization,discrete fourier transforms and the FFT, and orthogonal polynomials. Abackground in calculus is all that is needed.

Many real-world examplesand anecdotes are cited, but without too much detail or too manyillustrations given.

This book encourages the reader to ask: "Whatinformation is available about the problem?How can it be used to solvethe problem?What are the limits of this information?"The approachis practical, not merely analytical.

This book teaches what most othernumerical books fail to teach: How to derive your own formulas, and thusyour own solutions to problems. And that is perhaps the mostimportantlesson of all.

Reviews (17)

Excellent for Self-Study
The last time I studied PDE was 7 years ago. I perceived it then as a hard subject to comprehend because of the unsightly series, symbols, subscripts ...etc.I never thought I would come across a well-organized and easy-to-follow book like Farlow's.
This book is very well-organized and class-notes oriented. It has 47 lessons. Each lesson takes about 4 pages and has a clear well-stated objective with no subtitles and misleading branching. I highly recommend this book; it is so helpful especially if you want to self-study PDE.

Good book
I used this book in an undergraduate course, and since I couldn't see the board during lectures, I relied on only the book and it was very easy to read and understand.The major drawback of this book, and I don't know if this accounts for it's abnormally low price, is that there seem to be far more errors in the solutions than most books have.About 100 pages into the book, I had encountered so many errors, that thereafter whenever my solutions were different from the solutions in the book, I wondered first if the book was wrong, not if I had done something wrong.

Features

• Unabridged

Reviews (5)

An excellent text ... in 1970, not in 2003
I used this text as a reference over 25 years ago and it was great, for its time.Today, however, there are a number of books available with a more "modern" treatment - ones more likely to provide a more realistic view of the subject matter.

Arguably, ODE is a geometry course in disguise and not a collection of "party tricks" as it is often portrayed in older texts.Analytical methods are clean and easy to convey in the classroom but, frankly, they never appear in the "real world".

If you plan to (or do) encounter ODE's in your chosen field you'd do better to spend lots of time looking at qualitative and numerical techniques, i.e., a more up to date approach.

Coddington did a great job with the subtopics he did address but in the late sixties it would have been difficult, if not impossible, to really provide the reader with a solid feel for the depth and breadth of the subject.

A great Introduction or review.
I took an undergraduate ordinary differential equations class and felt I grasped the subject quite well.I wanted an inexpensive text that I could review the subject with and I decided that I would give Coddington's book a try.I was really pleased with the order in which the text was presented which differed from the course I had taken.The author's seem to put things in a very logical order versus some texts I have seen which really confuse you by the order in which the subjects are presented.Another point that I have to make is the depth that the book has.I learned much more in reviewing this text than I ever did in any diff eq class.It shows the distinctionbetween linear and non-linear diff eq's and covered many other methods which I had not learned previously.This is a great text as a "refresher" or as a course text.I just wish I would have previously used this text to learn ordinary differential equations.

Reviews (3)

Excellent
Silverman's book is much clearer than other books on calculus that I have seen recently.Everything is concise and there is no extra "fluff".Some may wonder about some seemengly strange orderings in the book such as introducing the differential before introducing limits but this serves to help create an intuitive understanding of limits before explaining their mathematical meaning.Otherwise, the problems range from extremely simple to challenging.All of the problems are reachable by just having read the previous section and are not out of the reach of a careful reader. Most of the problems have solutions and / or hints at the back of the book.The book's title includes the word "applications" and the book indeed does contain them.After each chapter on a new topic, the following chapter contains applications.In addition, the last two chapters on differential equations and multi variable calculus are better than in most introductory books.Essential Calculus is suited for class study or for self study and should be accessable to all with high school mathematics under their belt.

A Challenging (in a good way) Book
I am a graduate student in computer science, and I've forgotten quite a bit of math from college. I picked up this book and made a point to read it over the summer and do as many problems as I could. It is a very "tight" book, in the sense that there is not a lot of fluff (hence the "essential" in the title). The book seemed to maintain a nice balance between too hard and too easy so that I was always challenged. It is a very good book for me, and has a sufficient level of rigor so that I feel like I get some practice in that area as well. I would highly recommend it to someone wishing to learn calculus. In addition, it's always nice to find an author you like, because it usually means other books by that author will be good. As a bonus, it has solutions (or hints) to all the exercises so you can check your progress.

Reviews (15)

GREAT BOOK
This is a classic text and a great book.I found that this book helped me truly understand many concepts which other books simply made confusing.

This should be the Standard
I really believe that this book does an excellent job at teaching such a difficult topic."Advanced Calculus" is just packed with proofs and stimulating problems.This should be the text used to teach the subject.If you intend to tutor yourself on the topic or you are actually taking the class, this book is a must.I am currently using this as a secondary text to an advanced calculus class I am taking, and, as far as I'm concerned, this is the only text I need.This book does, in such a small package, more than you'll ever need.I recommend one purchases this book at the multi-varialbe calculus level and use it through your time in analysis courses.This is a must have for all math majors.

A bit of a hodge podge but still a useful reference
This text is very definitely written in the grand old non-geometrical style which some might find a bit more difficult to follow but the theory is sound with very good coverage of differentiation and integration including the essential elements of topology (it discusses compact sets without referring to them as such).My main complaint is that evidently in this era, it was stylish to omit certain details of the proofs. Other than that, it is still a serviceable text if used as a reference (especially considering the low price).

Reviews (7)

Biggest return for the biggest investment
This was the second-hardest book I ever read.Honestly, it took me years and years to get through it.I even had to buy a 2nd copy, because I kept getting frustrated and throwing the first copy across the room until it was destroyed.So yes, this book requires a substantial effort to read.

But the payback!!I've gotten more return on investment from this book than from any other book I've ever read.If you dilligently read and master this book, you will be able to analyze and solve problems your collegues just can't.

The basic idea behind Kolmogorov complexity is straighforward: a good measure of the complexity of an object is the length of the shortest computer program which will construct that object. From this basic idea an amazing variety of insights and powerful techniques have been developed, and this book is quite comprehensive in cataloging and explaining them.

For computer scientists and working programmers, probably the most useful result of Kolmogorov complexity would be the "Incompressibility Method", which is a powerful technique for the analysis of the runtime of algorithms. Typically, it is relatively easy to figure out what the best case or the worst case runtime of an algorithm is.Until now, it was hard to calculate the average runtime of an algorithm, because it usually involved a tricky counting problem, to enumerate all possible runs of the the algorithm and summing over them.The incompressibility method eliminates the need for doing these complicated enumerations, by letting you perform the analysis on a single run of the algorithm which is guarunteed to be representative of the average runtime of the algorithm.If you program for a living like I do, this will give you an edge, because if you can accurately predict that the worst-case runtimes almost never happen, you can usually simplify and streamline your programs by optimizing it for the average case.If your competitors are wasting time optimizing for a worst case which almost never happens--at the expense of _not_ optimizing for the average case, you win bigtime.

For philosophers of science and AI/knowledge representation folks, the most useful results of Kolmogorov complexity are probably the contributions of Kolmogorov complexity to Baysianism.To be a Baysian is to follow a two step process: (STEP 1) for every possible sentence, assign to it a number between 0 and 1 which represents how certain you are that that sentence is true.This initial assignment should be a probability distribution over all possible sentences.It should be a "good" probability distrubution, but of course it won't be perfect, since you don't know everything.(STEP 2) when confronted with new evidence, e.g. an observation, update your current "good" degrees of belief by using Bayes' law, to yield a new "better" set of degrees of belief.

The Baysians always had a good story for Step 2--just use Bayes law.But until now, they were mostly hand-waving on Step 1--what would constitude a "good" initial probability distribution?There were many proposals (e.g. maximum entropy) but all proposals had benefits and drawbacks.What Kolmogorov complexity provides is the so-called "universal" distribution, which is guarunteed to be a "good" initial distirbution.This book devotes much time to explaining and exploring this, and shows how previous techniques, like maximum entropy, minimum description length, etc all can be seen as computable approximations to the (unfortunately uncomputable) universal distribution.This really gives a nice framework for evalutating and formulating good prior distributions.

After remarking on how hard this book was to read, I should emphasize that this is not due to bad writing on the part of the authors! Indeed, after throwing the book across the room, I was always drawn back by Li & Vitanyi's most engaging writing style to pick the book back up, dust it off, and have another go at it. If it were not for their wonderul ability to expain a very complicated subject matter, I never would have gotten through it.

An unsung hero of this book is Peter Gacs, who wrote a set of lecture notes which really could be considered to be an Urtext for this book.If you tackle this book, I highly recommend that you also get ahold of these notes, because it is sometimes very useful, when trying to puzzle out a difficult argument, to get another description/explaination of it from a different point of view. These notes are available on the web, just google for "Lecture note on descriptional complexity and randomness" by Peter Gacs.

If you're up to the challange, then buy this book, dilligently read it, swear at it--then swear by it.

A must
The book provides all the tools needed for a productive use of the theory. Written by leading experts in the field, the book is both a fascinating introduction as well as a comprehensive reference for experts.

The authors are careful to place the development of the theory in its historical context, give a face to the main players in the field and explore frictions with other lines of thought. But the main storyline is the mathematical world of Kolmogorov complexity. Neccessary background knowledge is provided, most proofs are given and the open problems are presented. Most chapters are more or less self sufficient, making it possible to skip those that are of less relevance to you. In the later chapters much thought is given to the different fields of application.

A third edition is in the making which will include recent advances. But since the authors make new discoveries available on the web, the present edition will continue for a long time to hold a prominent place in the book shelves of many computer scientist.

Excellent if you have the math...
to understand it.This book is intended for serious students of computer science or those who have some similar training - it is definitely set up as a textbook.However, that being said, if you have the background the authors' delivery is fist-class and very clear.

The reviews below give more than enough information so I won't belabour the Kolmogorov complexity here.Suffice it to say you won't find the subject detailed more fully in any other reference work in existence today.

However, this book does need to be revised and updated.There has been a lot of development in the field and the sections overviewing Solomonoff's work, in particular, could be expanded.Also, I found it hard to believe that nothing about the 'philosophical' importance of the whole induction question - this is at the core of many very important questions and should not be treated trivially.

There should also be some overview of two other areas that, in combination with the theory outlined in this text, are starting to form the nexus of a "new kind of science" (definitely not Wolfram's pathetic attempt).I refer to some information regarding non-classical logical systems as well as anticipatory computing systems.Both will, I predict, become core areas in addition to extensions to Kolmogorov/Chaitin complexity in the future.

Reviews (3)

Riddled with errors, but ---
I have never seen so many typos, omissions, and errors in a published book.Many of the examples are poorly introduced, theorems are mentioned that don't exist in the book, etc.Other than Rubik's cube, most of the other puzzles are presented in a completely incomprehensible manner.It's very annoying, in a book that's otherwise just what I want.It does give a good quick and dirty intro to the group theory needed, however.

In depth group theory via games and puzzles
I am old enough to remember the original appearance of the Rubik's cube puzzle. I examined it a few times while in a store, but never put any effort into it. Later, I looked at some of the literature that explained how "easy" it was to solve the puzzle. The solution involves the use of some advanced topics in group theory, so it is a puzzle with a mathematical twist. However, that is not the only application of group theory, there are many ways in which it can be used. Joyner shows us many of them, and provides the foundation before he tackles the problems.
This is an excellent book that can be used to either refresh your understanding of group theory or teach it to advanced undergraduates. The objects being manipulated are easy to understand, sometimes easy to build or acquire and the explanations are easy to follow. They are also different from those found in the standard group theory text. Puzzles are an area that fascinates many people, so it is often an advantage to present mathematical instruction in the form of a puzzle rather than in the standard sequence of background notation, theorem and then proof.
Finally, the author is to be commended for donating all of the profits from the book to the Earth Island Institute. It is a non-profit organization dedicated to environmental projects throughout the world. Therefore, not only can a purchase of this book do your mathematical skills some good, it can also improve the quality of life for everyone on the planet.

Published in the recreational mathematics newsletter, reprinted with permission.

For the love of Puzzles...
I just got this book yesterday and I have not read it fully, but I had to write a quick review to say how excited I am about this book. The Rubik's Cube craze hit when I was young. I loved solving the cube and have loved puzzles ever since. I did start trying to describe a solution mathematically when I was at college, but got side tracked and bogged down in some of the math. So this book was a great find for me. I am going to enjoy reading this book and following the mathematical proof. Even though there does seem to be a lot of equations and for the casual reader this might put them off, but from my first browse of the book the math isn't too complex and should be something that anyone who has taken some introductory math courses at the college level should be able to follow.

Reviews (5)

Great info about the Riemann Zeta function
This is a great resource about the Riemann Zeta function.A good chunk of the mathematics in this book is beyond me, but the value nevertheless was immense.

A great resource and important book.

Good complement to Ivic and Titchmarsh
This is by far the book of mathematics that I like most. It's not the most complete source of information about the zeta function, Titchmarshand Ivic are the authorities. However when you read this book, you have a feeling that you are following Riemann's, de la Vallée Poussin's, Hadamard's, Littlewood's, etc... steps and you understand how these mathematicians must have felt while they studied the zeta function.

It includes a translation of Riemann's original paper (On the Number of Primes...) which is very nice and most authors now seem to forget to mention (mainly because of the obscure way in which it was written).

The first chapter is devoted to the study of the paper, then it is followed another chapter proving the product formula (which was not quite proven by Riemann), then a third chapter of von Mangoldt's proof of Riemann's Prime Formula.

The fourth chapter has the famous prime number theorem and it's original proof by Hadamard and Poussin. The fifth one includes an error estimation due to Poussin for the prime number theorem, and the equivalent of the Riemann Hypothesis in terms of prime distributions.

The Euler-Maclaurin formula is introduced in the sixth chapter to calculate zeros in the critical line.
The Riemann-Siegel formula is introduced in the seventh, and then later chapters include large scale computations, Fourier analysis, growth andlocation of zeros.

Finally we have my favourite chapter, counting zeros: Hardy's theorem, which says that there are infinitely many zeros in the critical line, which was improved by Littlewood, then later by Selberg, and then by Levinson.

The last chapter is dedicated to some theorems, including an elementary proof of the prime number theorem.

Most important idea: the introduction! It will give you an idea of how these amazing people studied and did math.

Reviews (5)

garbage
i don't understand why everyone is giving this book such great reviews.as a linear programming student, i find this book extremely difficult to understand, very poorly orgranized, extremely lacking in practical examples to demonstrate the concepts that the author is attempting to describe, and basically a piece of garbage.it was obviously written by a mathematician, not someone capable of teaching.i would not recommend it to anyone.

It's a Keeper
I cut my teeth on this text in George Nemhauser's class. The book is clear and concise and does an excellent job explaining this topic to beginners. I've not come across a better introductory text yet. I still have this book in my reference library.

If you want an introduction to LP, this is the text for you.

Editorial Review

Reviews (4)

Excellent Introduction to Combinatorics
A COURSE IN COMBINATORICS covers a great breadth of topics under the label of "combinatorics," including graph theory, enumeration, and some algebra. The book is comprehensive; the instructor can pick-and-choose appropriate material from the huge array provided without detriment to understanding.

Each chapter is written in a friendly, accessible manner: plenty of interesting and instructive examples follow the clear definitions and preliminaries. To give the reader an idea of the topics presented in the book, a list of chapters follows:

1. Graphs
2. Trees
3. Colorings of graphs and Ramsey's theorem
4. Turan's theorem and extremal graphs
5. Systems of distinct representatives
6. Dilworth's theorem and extremal set theory
7. Flows in networks
8. De Bruijn sequences
9. Two (0,1,*) problems: addressing for graphs and a hash-coding scheme
10. The principle of inclusion-exclusion; inversion formulae
11. Permanents
12. The Van der Waerden conjecture
13. Elementary counting; Stirling numbers
14. Recursions and generating functions
15. Partitions
16. (0,1)-Matrices
17. Latin Squares
18. Hadamard matrices, Reed-Muller codes
19. Designs
20. Codes and designs
21. Strongly regular graphs and partial geometries
22. Orthogonal Latin squares
23. Projetive and combinatorial geometries
24. Gaussian numbers and q-analogues
25. Lattices and Mobius inversion
26. Combinatorial designs and projective geometries
27. Difference sets and automorphisms
28. Difference sets and the group ring
29. Codes and symmetric designs
30. Association schemes
31. (More) algebraic techniques in graph theory
32. Graph connectivity
33. Planarity and coloring
34. Whitney duality
35. Embeddings of graphs on surfaces
36. Electrical networks and squared squares
37. Polya theory of counting
38. Baranyai's theorem

The problems in the book are generally very rich and well-written, with helpful hints from the appendix that provide motivation but do not spoil. However, the relative difficulty of the problems is not readily made appparent, so over- or underthinking of problems often occurs with misjudgments.

For the interested high-school student to the beginning graduate, this book is ideal for the study of combinatorics. Truly a nice read that connects many areas of mathematics and combines them into a thing of true beauty.

A nice tour of combinatorics
The first word that comes to my mind when I think of this text is "encyclopedic". It contains around 40 chapters, hitting most of the high points of combinatorics that a graduate student should see. The exposition is generally good with nice examples. The one thing that I fault it for is the number of statements that the authors claim are "obvious". In a way, this is good, because it makes you pay attention and understand the material, but sometimes the statement isn't obvious until you've thought about it for an hour and written out a lengthy proof. At that point, it does become completely obvious and you can't believe that you ever thought it wasn't, so I can understand why van Lint and Wilson fell into the trap so often. (In fact, I've heard that Wilson even stumbles over some of those points in lectures.) This is a great book to have on your shelf if you need somewhere to look up combinatorial ideas.

A gentle introduction to combinatorics
This book was the text for a graduate-level course I took.The presentation is very laid-back, much like the lecturing style of one of the authors (Wilson), and so it was quite readable (unlike many other mathbooks which you have to stop every few pages and pick apart everythingbefore it sinks in).

Combinatorics is a relatively recent development inmathematics, one which is generally easy to explain, but with manydifficult open questions.Van Lint and Wilson do an excellent jobexplaining, but there are a few places where the reader needs to know somebackground to place the particular problem in the appropriate mathematicalcontext.Understandably, if the authors were to include all themathematical machinery needed, the book would be huge!Instead, they havechosen to describe as many facets of the field as possible, and thereforehave written a broad, well-balanced book which approaches the topic in anon-threatening way.

My one criticism, then, is that there is a lack ofdepth in several areas of the book, with further discussion of advancedtopics or open problems.But even so, I can appreciate the omission forthe sake of accessibility.

Reviews (11)

My God? What a book!
Godfred Harold Hardy is one of the greatest mathematicians of the 20th century. A Course of Pure Mathematics is an introduction to Analysis, but he only starts Limits on page 110. Until there he talks about real numbers, Dedekind cuts and complex numbers as vectors. I have no words to describe his style: it seems that he is talking with you. If you are looking about a book of Calculus, stop here in buy it.

Best introduction to mathematical analysis
This book is simply beyond any rating whatsoever. Giving 5 stars is to undermine the value of this classic.
The first time I got this book, I was neither aware of it not of its author. I just picked it up randomly from school library. From the contents I figured it was a book on calculus. I immediately searched for the proof that "every continuous function is integrable." This was the first book I encountered which had a rigorous proof of this.
Then I began reading the chapters sequentially thinking that this seems to be a good book on calculus. The book went much beyond my expectation and it satisfied all my mathematical curiousities. All the mysteries of calculus were revelaed. Hardy demystified calculus in the first chapter itself by creating reals out ot rationals.
The Dedekind's construction of reals as presented in this book is the best I have seen. The properties of reals were not stated as axioms (common approach in books on analysis) but rather deduced from those of rationals.
The concepts of functions, limits, continuity, derivative etc. were explained in a prosaic style which has no parallel. This was also my first book on maths which had far more english words than mathematical symbols.
After finishing the entire book I was wondering who was this guy G. H. Hardy who has written such a masterpiece.
Only a few months later I came to know that he was one of the greatest British mathematicians of the century and was responsible for making our Indian Ramanujan famous. After that I read most of his books including "A Mathematician's Apology" and "An Introduction to Theory of Numbers"
Any persons who thinks maths is dull should just read few pages from this book and I bet his old beliefs would be shattered.