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. 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. 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.

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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. 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.

### Featured Courses

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. 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.

## From Calculus to Chaos: An Introduction to Dynamics by David Acheson

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.

## Cornell MAE Nonlinear Dynamics and Chaos

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.

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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. 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. The great thing about it is that the problems are actually interesting. I have found this to be a good source for classroom examples.

Mosteller, Frederick. Fifty Challenging Problems in Probability with Solutions. Duelling Idiots and Other Probability Puzzlers. Fuzzy Stuff logic and set theory. Some books in this area are better than others.

## From Calculus to Chaos: An Introduction to Dynamics by David Acheson (1998-01-08) by David Acheson

By in large though, it is a lot of bull about ad hoc, not particularly robust, algorithms. Claims of anything new and profound are general pompous bullstuff. Fuzzy methods are trivial if you have knowledge of probability and logic. In my view the aspiring applied mathematician can not do better than to study probability. A book of practical statistics as opposed to mathematical or theoretical statistics is the one by Snedecor and Cochran. It is rigorous but does not use calculus. It uses real life biological data for examples but is fascinating. It is a wonderfully well written and clear book.

A real masterpiece. Anyone who actually does statistics should have this book. But remember, though it does not require calculus it does require mathematical maturity. My feeling is that if you want to use this book but do not know calculus you should go back and take calculus. Snedecor, George W. Statistical Methods , 8 th ed. Iowa State.

A great book. The best books about statistics for the layman are very likely: Tanur, Judith M. Statistics: A Guide to the Unknown , 3 rd ed. This is a great book. See also Bennett. Salsburg, David. The Lady Tasting Tea. Without using a single formula it does a much better job of telling the layman what statistics is about than does the usual introductory text. It is also of interest to the professional. A classic applied book that is readable and thorough and good to own is: Neter, John, Michael K.

Kutner, Christopher J. Nachtsheim, William Wasserman. Applied Linear Statistical Models, 4th ed. My favorite text on mathematical statistics is definitely the following. It is a large text with enough material for a senior level sequence in mathematical statistics, or a more advanced graduate sequence in mathematical statistics.

It is very well done. Dudewicz, Edward J. Modern Mathematical Statistics. Introduction to the Theory of Statistics. The Cartoon Guide to Statistics. An elementary book that does a nice job on statistical tests and which might be of interest to the practitioner is: Langley, Russell. Practical Statistics Simply Explained. The book by Box, Hunter and Hunter is wonderful at exploring the concepts and underlying theory. The book by Saville and Wood is worth considering by the serious student. Although its mathematics is simple and not calculus based this is the way theory was developed and this is also touched upon in the book by Box, Hunter, and Hunter.

Hicks, Charles R. Fundamental Concepts in the Design of Experiments. Stuart Hunter, and William Gordon Hunter. Saville, David J. And Graham R. Statistical Methods: A Geometric Primer. My favorite book on regression is the one by Draper and Smith. The book by Ryan is particularly elementary and thorough. Draper, Norman R. Applied Regression Analysis. Modern Regression Methods. The book by Thompson is for the practitioner. Stuart, Alan. Ideas of Sampling , 3 rd ed. If forced to use time series analysis for purposes of forecasting I almost always will use double exponential smoothing possibly embellished with seasonal attributes and built-in parameter adjusting.

The bible of times series analysis is Box and Jenkins. The book by Kendall and Ord is fairly complete in its survey of methods. I like the book by Bloomfield. Box, George E. Jenkins, Gregory C. Times Series Analysis: Forecasting and Control. Keith Ord. Time Series , 3 rd ed. Edward Arnold. Practical Nonparametric Methods , 2 nd ed. Statistical Distributions , 2 nd ed.

The best single book on general operations research is Hillier, Frederick S. Introduction to Operations Research. Let me mention four. All these discuss the simplex method. I will soon make recommendation s on interior point algorithm books however they are covered in Rardin. A very elementary book that does a great job teaching the fundamentals with pictures is: Gass, Saul I. An illustrated Guide to Linear Programming. Linear Programming. Integer and Combinatorial Optimization.

Integer Programming. Nonlinear Programming. Republication of McGraw-Hill; Optimization in Operations Research. William H. Cunningham, William R. Pulleyblank, Alexander Schrijver. Combinatorial Optimization. It is a book that I would recommend to any student getting into optimization.

I think it is a must-have for any serious student of OR. Kaplan, Wilfred. Networks and Algorithms. However, let me mention what I like best: By far the best book for comprehensiveness is: Law, Averill M. David Kelton. Simulation Modeling and Analysis , 2 nd ed. Carson, II.

Discrete-Event System Simulation. Simulation and the Monte Carlo Method. A future edition will cover both decision theory and games of the J H. Conway variety. An early classic of extremely elementary nature is the one by Williams. It precedes the widespread use of linear programming. Williams, J.

Dover See Thie. A fine elementary book is: Straffin, Philip D. Game Theory and Strategy. Game Theory , 3 rd ed. Lectures on Game Theory. A well written text at the senior level emphasizing economics is: Romp, Graham. Game Theory: Introduction and Applications. Stochastic Markov Decision Processes will be covered in a future edition.

Stochastic Processes and Queueing. See the first books in probability. A classic that seems out of print is: Parzen, Emanuel. Stochastic Processes. An inexpensive paperback republication of merit is: Ross, Sheldon. Applied Probability Models with Optimization Applications. Dover, Fundamentals of Queueing Theory , 3 rd ed. Queueing Theory: For Services and Manufacturing. Inventory Theory and Scheduling. I am not to smitten with the books in this area. For the second edition I will try to do better. Until then, there is one excellent book in print. There is almost certainly an excellent book to appear.

The book by French is excellent and is out of print and shouldn't be. The books by Conway et al and Hadley et al were published in the sixties and are out of print and despite that are first rate if you can get your hands on them. The book to have these days: Silver, Edward A. Pyke, and Rein Peterson.

Inventory Management and Production Planning and Scheduling , 3 rd ed. So I would bet this will be a must have book for its area: Lawler, E. Lenstra, and A. Rinooy Kan. Theory of Sequences and Scheduling. Scheduled for A book that never should have gone out of print: French, Simon. Ellis Horwood. Maxwell, and Louis Miller. Theory of Scheduling. Analysis of Inventory Systems. Introduction to Sequencing and Scheduling. This is a new area for me. There are a lot of books giving contradictory advice or useless advice.

Investment theory is inherently mathematical, but there is a mathematical offshoot known as "technical analysis. Some of it is as bad as astrology. The better technical analysis stuff is basically a dead end, or perhaps I should say deadly end. The book by Malkiel aresses it well.

General Physics. I really haven't gotten around to this area yet. Secondly, I prefer to learn most physics from specialized sources for example to study mechanics, how about using a book just on mechanics? One series you are sure to hear about is the great series by Feynman. Be aware, that it is probably more useful to people who already have a knowledge of the subjects. Also, it is a great reference. It deserves its reputation as a work of genius, but in gneral I would not recommend it to someone just beginning to learn physics.

There is a great classic, very readable, by a major thinker, full of history, that goes back to Mach, Ernst. The Science of Mechanics , 6 th English ed. Open Court. Classical Mechanics. University Science Books. Newtonian Mechanics. A couple of concise well written first books for the student who has been through the calculus sequence: Smith, P, and R. Mechanics , 2 nd ed. A First Course in Mechanics. Classical Mechanics , 4 th ed. Very nice!! Elements of Newtonian Mechanics. Classical Mechanics: A Modern Introduction , 2 nd ed.

Woodhouse, N. Introduction to Analytical Dynamics. A couple of thorough books: Greenwood, Donald T. Principles of Dynamics , 2 nd ed. Textbook of Dynamics , 2 nd ed. Wiley actually it is not clear who published this. Three undergraduate books in order of increasing difficulty: Chorin, Alexandre J. A Mathematical Introduction to Fluid Mechanics , 3 rd ed.

Topics in Fluid Mechanics. Fluid Mechanics. Thermodynamics and Statistical Mechanics. There are several books for laymen on the second law of thermodynamics. The first by Atkins is well illustrated--basically it is a coffee table book. Atkins is one of the best science writers alive. The book by the Goldsteins does a thorough job of discussing the history and concepts of thermodynamics. It is also very good. Atkins, P. The Second Law. Four Laws that Drive the Universe. Goldstein, Martin, Inge F. It is probably of less interest to nerds.

An unusual book in format that is aimed at the serious student, but is definitely worth having: Perrot, Pierre. A to Z of Thermodynamics. The Physics of Chance.

Basic Engineering Thermodynamics. Understanding Thermodynamics. The book by Lawden is fairly brief. Lawden, D. Principles of Thermodynamics. Statistical Physics: A probabilistic Approach. Classical and Statistical Thermodynamics. Electricity and Electromagnetism. An elementary "coffee table" book would be: Fowler, Richard J. Electricity: Principles and Applications , 4 th ed. Also, another fine book with Schey in the section on Vector Calculus is the book by Marsden and Tromba.

Unless my memory is suffering the ravages of alcohol, the 4 th edition has a much more thorough treatment of Maxwell's equations of electromagnetism than did the 2 nd edition. A book for people interested in electrical engineering and who want a single book to get them into it is: Rutledge, David. T he Electronics of Radio. A truly excellent short book; a must have for students of EE. Highest recommendation: Fleisch, Daniel.

A Students Guide to Maxwell's Equations. Lancaster, Gordon. Introduction to Fields and Circuits. The book by Skilling is a reprint of an ancient work and is highly recommended. Skilling, Hugh H. Fundamentals of Electric Waves. Dugdale, David. Essentials of Electromagnetism. American Institute of Physics. Schwarz, Steven E. Electromagnetism for Engineers. Cottingham, W. Electricity and Magnetism. Westgard, James Blake.

Electrodynamics: A Concise Introduction. Electricity and Magnetism , 2 nd ed. Electromagnetism: Principles and Applications , 2 nd ed. Engineering Electromagnetics. A book which I think is particularly well written and clear: Dugdale, David. Engineering Field Theory with Applications. Quantum Mechanics. There are books that try to explain quantum physics to the layman, i. For the most part it is like trying to explain Rembrandt to a person who has never possessed sight. To start off with I'll mention one of the non-mathematical coffee-table works: Hey, Tony and Patrick Walters.

The Quantum Universe. Ponomarev, L. The Quantum Dice. Institute of Physics. Quantum Mechanics and Experience. Quanta: A Handbook of Concepts , 2 nd ed. Primer of Quantum Mechanics. Introduction to Quantum Physics. Wolf, H. An Introduction to Quantum Physics. The Meaning of Quantum Theory.

Quantics: Rudiments of Quantum Physics. North Holland. A much more comprehensive treatment that can be a little hairy but nonetheless is as readable as this stuff gets: Zee, A. Quantum Field Theory in a Nutshell. Bell, J. Speakable and Unspeakable in Quantum Mechanics.

It is however rather subtle and deserves a lot of attention. A literature professor would explain that the special relativity is a nuanced paradigm reflecting in essence Einstein's misogyny. As to general relativity it can not be understood with little more than algebra. Rather, it can be described technically as a real mother-lover. On the subject of general relativity and covering special relativity as well, there is a magnum opus, perhaps even a 44 magnum opus.

This book is the book for any serious student. I would imagine that graduate students in physics all get it. It is pages long and it takes great pains to be pedagogically sweet. Tensors and everything are explained ex vacua that is supposed to be Latin for out of nothing it probably means death to the left-handed. I have trouble seeing this all covered in two semesters at the graduate level. It is formidable but it is also magnificent. Misner, Charles W. The only caveat here is that there are many fine books on special relativity and some of them are less technical.

Nonetheless the book avoids calculus. Taylor, Edwin F. Spacetime Physics: Introduction to Special Relativity , 2nd ed. Although it can be read independently, I strongly recommend reading Spacetime Physics first. Epstein, Lewis Carroll. Relativity Visualized. Insight Press. Special Relativity. Einstein's Theory of Relativity. Born was a Nobel laureate. Rindler, Wolfgang. Introduction to Special Relativity , 2 nd ed. The last two Harpaz and Hakim are very mathematical and in my judgement Harpaz is the more elementary of the two.

The book by Bergman is wonderfully concise and clear. Gibilisco, Stan. The Rile of Gravitation. General Relativity from A to B. University of Chicago. Relativity Theory: Concepts and Basic Principles. An Introduction to Relataivistic Gravitation. Seuss with tensors. Lieber, Lillian. Paul Dry Books. The best introduction I think is: Pierce, J. Almost All About Waves. Peder rated it really liked it Nov 02, Abhishek Chaudhary rated it it was amazing Jun 12, Granaal rated it it was ok Feb 19, Eduardo rated it liked it Feb 15, Christian rated it really liked it May 14, Daniel rated it really liked it Jan 26, Matthew Warren rated it it was amazing Aug 10, Tim Cooper rated it it was amazing Jul 06, Philip Cramer rated it liked it Oct 16, John rated it really liked it Nov 16, Henry Cooksley rated it liked it Mar 27, Scott Campbell rated it liked it Sep 02, Sabrina Qian rated it it was amazing Jul 18, Sophie marked it as to-read Apr 25, Martin Cohen is currently reading it May 29, Jonathan Harwell added it Jan 28, Knud van Eeden marked it as to-read Apr 27, Persephone marked it as to-read Jul 18, Ana marked it as to-read Dec 15, M marked it as to-read Jan 02, Zheng Ma added it Jan 08, Yadi marked it as to-read May 12, Mike marked it as to-read Aug 01, Sebastian Los marked it as to-read Aug 18, Greet added it Jan 27, Lutger marked it as to-read Feb 16, Not marked it as to-read May 20, Rlibrary added it May 21, Joe marked it as to-read Jul 20, YY added it Jan 17, Kenny Olsen marked it as to-read Apr 24, Tiffany marked it as to-read Jun 13, Tabitha Mburu marked it as to-read Jun 20, Clifton Callender added it Jul 09, Bob added it Jul 24, Leonardo Masolini marked it as to-read Sep 09, Krishnaaditya marked it as to-read May 21, Gianpietro added it Jun 25, Chunge marked it as to-read Sep 25, Pablo Aguirre marked it as to-read Jan 26, Emma M.

There are no discussion topics on this book yet. About David Acheson. David Acheson. Books by David Acheson. Trivia About From Calculus to