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Algorithmics: The Spirit of Computing , 2 nd ed. Algorithms and Complexity. Combinatorics Including Graph Theory. Posamentier, Alfred S. The Fabulous Fibonacci Numbers. Numerical Analysis. Most books on numerical analysis are written to turn off the reader and to encourage him or her to go into a different, preferably unrelated, field. Secondly, almost all of the books in the area are written by academics or researchers at national labs, i. The kind of industry I use to work in was a little different than that. The problem is partly textbook evolution.

I've seen books long out of print that would work nicely in the classroom. However, textbook competition requires that newer books contain more and more material until the book can become rather unwieldy in several senses for the classroom. The truth is that the average book has far too much material for a course. Numerical analysis touches upon so many other topics this makes it a more demanding course than others. A marvelous exception to the above is the book by G. It avoids the problem just mentioned because it is based upon notes from a course.

It is concise and superbly written. It is the one I am now teaching out of. Stewart, G. Afternotes on Numerical Analysis. The first is absolutely superb. Both books are great to read, but I don't like either as a text. Acton, Forman. Numerical Methods That Work. An interesting book that seems in the spirit of the first book by Acton above is: Breuer, Shlomo, Gideon Zwas.

Numerical Mathematics: A Laboratory Approach. I would like to know however how it has done as a text. A book by a great applied mathematician that is worth having is: Hamming, R. Numerical Methods for Scientists and Engineers , 2 nd ed.. Numerical Analysis: Theory and Practice. Douglas, Richard Burden.

Numerical Methods, 2 nd ed. This is the problem with teaching the course. On the flip side of course, it covers less material e. Also, it does not give pseudo-code for algorithms. This is okay with me for the following reasons. Given a textbook with good pseudo-code, no matter how much I lecture the students on its points and various alternatives, they usually copy the pseudocode as if it the word of God rather than regarding my word as the word of God. It is useful to make them take the central idea of the algorithm and work out the details their selves.

This text also has an associated instructors guide and student guides. See the book by Cooper. Fourier Analysis. The best book on Fourier analysis is the one by Korner. However, it is roughly at a first year graduate level and is academic rather than say engineering oriented. Any graduate student in analysis should have this book. Korner, T.

The Fourier Transform and Its Applications , 2 nd ed. Another book in a similar vein has been reprinted recently I think : Papoulis, Athanasios. The Fourier Integral and Its Applications. A book with many applications to engineering is Folland, Gerald B. Fourier Analysis and its Applications. Introduction to Fourier Analysis. A fairly short book pp that is worthwhile is: Solymar, L. Lectures on Fourier Series.

Fourier Series and Integral Transforms. Another short concise work: Bhatia, Rajendra. Fourier Series. Number theory is one of the oldest and most loved mathematical disciplines and as a result there have been many great books on it. The serious student will also need to study abstract algebra and in particular group theory. Let me list four superb introductions.

These should be accessible to just about anyone. The book by Davenport appears to be out of print, but not long ago it was being published by two publishers. It might return soon. The second book by Ore gives history without it getting in the way of learning the subject. Ore, Oystein. Invitation to Number Theory. Number Theory and its History. An Adventurer's Guide to Number Theory. Here are five excellent elementary texts that last I knew are still in print.

Silverman, Joseph H. Dudley, Underwood. Elementary Number Theory , 2 nd Ed. Elementary Number Theory and its Applications , 5 th ed. Maybe the text to have. Burton, David M. Elementary Number Theory , 4 th Ed. He gets into arithmetic functions before he does Euler's generalization of Fermat's Little Theorem. However, many of the proofs are very nice. I like this one quite bit. Like Rosen, the later editions are indeed better. An Introductory Text that has a lot going for it is the one by Stillwell.

It has great material but is too fast for most beginners. Should require a course in abstract algebra. Maybe the best second book around on number theory. Elements of Number Theory. Zuckerman, Hugh L. An Introduction to the Theory of Numbers , 5 th ed. A Concise Introduction to the Theory of Numbers.

Number Theory. An Introduction to Number Theory. Lectures on Elementary Number Theory. The Theory of Numbers. Schroeder, M. Number Theory in Science and Communication , 3 rd ed. A book I like a lot is the one by Anderson and Bell. Although they give the proper definitions groups on p. It has applications and a lot of information. Well laid out. Out a very good book to have. Anderson, James A. Number Theory with Applications. The first edition was a different title and publisher but, of course, the same authors. A very short work pages at the first year graduate level covers a good variety of topics: Tenenbaum, G.

The Prime Numbers and Their Distribution. American Mathematical Society. One book that I assume must be great is the following. I base this on the references to it. Sierpinski, Waclaw. Elementary Theory of Numbers. A reissued classic that is well written requires, I think, a decent knowledge of abstract algebra. Weyl, Hermann.

Algebraic Theory of Numbers. First around It requires a first course in abstract algebra it often refers to proofs in Stewart's Galois Theory which is listed in the next section Abstract Algebra. Stewart, Ian and David Tall. However, in the early 's there appeared three popular books on the Riemann Hypothesis. All three received good reviews. The first one Derbyshire does the best job in explaining the mathematics in my opinion.

Springer “Undergraduate Texts in Mathematics” (UTM) Books List

Although the subject is tough these books are essentially accessible to anyone. Derbyshire, John. Prime Obsession. Joseph Henry Press. Sabbagh, Karl. Farrar, Straus, Giroux. Abstract Algebra. Note, that at this time the only book I have listed here that could be considered really elementary is the one by Landin. Landin, Joseph. An Introduction to Algebraic Structures. A First Course in Abstract Algebra , 5 th ed. Herstein was one of the best writers on algebra.

Some would consider his book as more difficult than Fraleigh, though it doesn't go all the way through Galois Theory but gets most of the way there. He is particularly good I think on group theory. Herstein, I. Abstract Algebra , 3 rd ed. It is thorough, fairly consise and beautifully written. He is very strong on motivation and explanations. This is a four-star book out of four stars. It is one of the best books around on group theory. His treatment there I think should be read by anyone interested in group theory.

Bibliographic Information

Topics in Algebra , 2nd. It is certainly viable as a text, and I definitely recommend it for the library. Childs, Lindsay N. A Concrete Introduction to Higher Algebra , 2 nd ed. Starting with matrix theory it covers quite a bit of ground and is beautifully done. I like it a great deal. Note that some people consider this book undergraduate in level. Artin, Michael.

Margaret W. Abstract Algebra and Solution by Radicals. The following is a fairly complete text which is strong on group theory besides other topics. Hungerford, Thomas W. Abstract algebra , 2 nd ed. The Theory of Algebraic Numbers , 2 nd ed. A Modern Course on the Theory of Equations. Polygonal Press. Introductory Algebraic Number Theory. Let me mention several books on Galois Theory. As a rule even if some of these books do not presume a prior knowledge of group theory, you should learn some group theory before hand.

The first of these books has a lot of other information and is certainly one of the best: Hadlock, Charles Robert. Field Theory and Its Classical Problems. Galois Theory , 3 rd ed. May be the best introduction. My favorite is the book by Stillwell. I don't think much of it as text, but it is a great book to read. Despite the title, it is very much a book on Galois Theory. Elements of Algebra: Geometry, Numbers, Equations. Fields and Galois Theory. A Course in Galois Theory. Galois Theory. Notre Dame. Another succinct book similar to Artin's in every way is Postnikov, M.

Foundations of Galois Theory. It is not a book for a first course in abstract algebra. Rotman, Joseph. Galois Theory , 2 nd ed. The Fundamental Theorem of Algebra.

A Course In Modern Geometries

Great special study. If you are looking for applications of abstract algebra, you should look first to Childs. An elementary undergraduate small collection of applications is given in: Mackiw, George. Applications of Abstract Algebra. Hardy, Darel W. Applied Abstract Algebra. Ash, Avner, and Robert Gross. Group Theory Virtually all books on abstract algebra and some on number theory and some on geometry get into group theory.

I have indicated which of these does an exceptional job in my opinion. Here we will look at books devoted to group theory alone. One of the most elementary and nicest introductions is: Grossman, Israel and Wilhelm Magnus. Groups and Their Graphs. However, if you are comfortable with groups, but are not acquainted with graphs of groups Cayley diagrams get this book.

Graphs give a great window to the subject. The MAA published a lavish book that seems to be designed to supplant Grossman and Magnus just above this. I prefer Grossman and Magnus for their conciseness for the elementary material. Howeever, the newer book is dazzling. It spends a long time motivating the group concept emphasizing the graphical and other visual approaches. The second part goes much deeper than Grossman and Magnus and in particular gives maybe the best treatment of the Sylow theorems that I have seen.

Carter, Nathan. Visual Group Theory. It is quite good. I think it needs a second edition. The first few sections strike me as a little kludgy I know, there should be a better word-but how much am I charging you for this? Armstrong, M. Groups and Symmetry. Smith, Geoff and Olga Tabachnikova. Topics in Group theory. I think this is the best on undergraduate group theory. Would be a good text does anyone have an undergraduate course in group theory?

Humphreys, John F. A Course in Group Theory. A rather obscure book that deserves some attention; despite the title, this book is more groups than geometry there are books on groups and geometry in the geometry section. Also, it has some material on rings and the material on geometry is non-trivial. It is very good on group theory.

Introduction to Geometry

Excellent at the undergraduate level for someone who has already had exposure to groups. Sullivan, John B. Groups and Geometry. William C. Permutation Groups. See combinatorics. Rotman, Joseph J. An Introduction to the Theory of Groups. Another book that goes into graduate level that is worth a look and quite inexpensive is Rose, John S. A Course on Group Theory. Note both books by Herstein do a good job, but the second is the one to have.

I have yet to meet a book that is on just point set topology that I adore. The following book which is not just on point set topology is very good: Simmons, George F. Introduction to Topology and Modern Analysis. Essential Topology. Introduction to Topology , 3 rd ed. Arthur Seebach, Jr.

Counterexamples in Topology. Lecture Notes on Elementary Topology and Geometry. Classical Topology and Combinatorial Group Theory , 2 nd ed. Basic Concepts of Algebraic Topology. A Geometric Introduction to Topology. By set theory, I do not mean the set theory that is the first chapter of so many texts, but rather the specialty related to logic. See the section on Foundations as there are books there with a significant amount of set theory. A particularly fine first book, if still in print, is Henle, James M. An Outline of Set Theory.

Notes on Set Theory. Cohen, Paul J.

A Course in Modern Geometries by Judith N. Cederberg

Set Theory and the Continuum Hypothesis. Logic and Abstract Automata and computability and languages. Matiyasevich, Yuri V. Hilbert's Tenth Problem. Hehner, Eric C. The following is a good introduction to Godel's incompleteness theorem as well as providing a very useful discussion of its abuses:. Franzen, Torkel. By foundations I do not mean fundamentals. Of the books listed here the only one of serious interest to the specialist in logic is the one by Wilder. One of the most underrated books I know is this book by Eves.

It does a very credible job of covering foundations, fundamentals and history. It is quite a little gem pp. Eves, Howard. Foundations and Fundamental concepts of Mathematics , 3 rd ed. Et al. A book I like a lot senior level in my view is Potter, Michael. Set Theory and its Philosophy. I strongly recommend it.

A slightly more elementary text is: Tiles, Mary. The four volumes of D. They are comprehensive, authoritative, brilliant. They are mathematically sophisticated and are considered by most people to be references more than texts. See General Computer Science. For graph algorithms specifically see the books by Gibbons and Even. For algorithms on optimization and linear programming and integer programming go to the appropriate sections.

The best single book on the subject is the one by Cormen, Leiseron, and Rivest. It covers a great deal of ground; it is well organized; it is well written; it reviews mathematical topics well; it has good references; the algorithms are stated unusually clearly. Cormen, Thomas H. Leiserson, and Ronald L. Introduction to Algorithms. The second one is slightly more elementary and is better written. If I were to choose one I would choose this one Aho, Alfred V.

Hopcroft, and Jeffrey D. The Design and Analysis of Computer Algorithms. Data Structures and Algorithms. The Design and Analysis of Algorithms. Practical Genetic Algorithms. Coding and Information Theory. Note that coding theory is different from cryptography. That is a different type of coding.

There is one fairly informal non-technical beautifully written book on information theory by a great engineer and it is cheap! There are two books that are quite good by Steven Roman. I suggest that one read the first. If you want to continue deeper into the subject, by all means obtain the second: Roman, Steven. Introduction to Coding and Information Theory.

Error-Coding Codes and Finite Fields. Introduction to the Theory of Error-Correcting Codes. It covers information theory and more. The author is one of the best writers on applied mathematics.

Fairly large book. Luenberger, David G. Information Science. The second edition will include recommendations on books on Digital Filters and Signal Analysis. The books listed here are all calculus based except for the book by Bennett.. An absolutely superb book for the layman, and of interest to the professional accomplishes what many other books have merely attempted.

Bennett, Deborah J. See also Tanur. An interesting book, quite philosophical, on randomness is the one by Taleb. One of the best books written for the undergraduate to learn probability is the book by Gordon. Despite the restriction to discrete probability this book is a superb general introduction for the math undergraduate and is very well organized.

A Course in Modern Geometries

Great text!! Gordon, Hugh. Discrete Mathematics. As a rule I think that the best books to learn probability from are those on modeling. For example, perhaps the best writer on probability is Sheldon Ross. Introduction to Probability Models , 6 th ed. Karlin, Samuel. An Introduction to Stochastic Modeling , rev. It is indeed a wonderful book: Hamming, R. The Art of Probability for Scientists and Engineers. Elementary Probability, 2 nd ed. Problems and Snapshots from the World of Probability. The first volume is inspiring.

The larger second volume is even more technical than the first, for example there is a chapter review of measure theory. Feller, William. Introduction to Probability Theory. Vol 2, 2 nd ed. The following is an inexpensive little reference. It requires only a basic knowledge of probability, say through Bayes' Theorem.

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