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by **CTrombley** » Thu May 12, 2011 4:09 am UTC

I like to spend my weekends learning about the history of mathematics. I (among other things) do turbulence research, and everyone knows the giant of the Field is Kolmogorov. Are there any good biographies of the great mathematician? In fact, a general history of Soviet Mathematics would be greatly appreciated.

by **Quaternia** » Sun May 15, 2011 4:05 pm UTC

Kolmogorov wrote a three-volume book with Aleksandrov and Lavrent'ev, where they especially emphasize Soviet contributions to mathematics (Mathematics: It's Content, Methods, and Meaning),
but it has by far more math than biography.
(It was translated and re-edited by Dover as a one-volume book in English)

by **doogly** » Sun May 15, 2011 4:07 pm UTC

Which I picked up when the math department was moving buildings, and left many books out for the taking!

by **CTrombley** » Sun May 15, 2011 6:33 pm UTC

Awesome, thanks a bignum!

by **Ordinata** » Wed May 18, 2011 1:56 pm UTC

tameree wrote:I am looking to develop my mathematical mind as much as possible. The "highest" classes I have taken are Calculus I & II as well as linear algebra. I'd be looking for either a bunch of books or some sort of guide (similar to the guide "How to become a good theoretical physicist" for physics) that ranges from algebra up to topology, complex analysis and partial differential equations.

For an absolute starter I really like Biggs' "Discrete Mathematics", from Oxford Press. The first time you go beyond arithmetics and calculus, and maybe linear algebra, into a bigger field of mathematics, this is exactly the kind of stuff you will need to learn. Apart from covering some basics of logic, combinatorics and algebra (stuff that is used in practically ANY other course in ANY kind of math,) it is also an introduction into more formal theorem+theorem=proof reasoning, and more like that. It's very basic, but covers a LOT of ground. A lot of IMPORTANT ground. I'll give a little overview of the awesome stuff it contains:

chapter 1, statements and proofs. Yeah, this is basically a chapter on how to actually do formal axiom and proof based mathematics. Stuff the professors will assume you know, or at best impart to you little by little.
chapter 2, set notation. Once again stuff you'll be assumed to know in EVERY other part of mathematics.
chapter 3, the logical framework. Explains things like contrapositive statements, equivalence of statements, and logical notation.
chapter 5, functions. Formal concept, bijections, composition, etc.
The above chapters are absolutely crucial for all other mathematics, and if you're a uni student you'll notice these things are used in EVERY math course you EVER take.

It also introduces the integers and rationals, and the usual operations on them, in a formal (algebraic) way. This includes induction.
Then combinatorics, especially enumerative combinatorics, i.e. how to count the number of elements in a set (binomials and stuff like that).
And finally an introduction to algebra. Groups, rings, etc.

And this is just what I think is the most important. If i had to characterize it, I'd say it was a crash course in formal math. And it's well-written. Yeah, I'm a fanboi. I can still enjoy some of the exercises. If you already know these things don't buy it though, it will not serve as a reference book, it is very basic.

by **SkipSandwich** » Thu Jun 02, 2011 8:14 pm UTC

I'd like to start studying cryptography. I've taken up to Calc 2 and Linear Algebra. What are some good Crpyto books for someone looking to start on the subject?

by **Dopefish** » Thu Jun 02, 2011 9:35 pm UTC

Does anyone have any experiance with the book "Elementary Analysis: The theory of Calculus" by Kenneth Ross?

At a glance it seems fairly basic (which could be a good thing), but in the eyes of those of you who really know what your talking about, is it too basic, or would it be a decent first taste of analysis?

by **doogly** » Fri Jun 03, 2011 12:36 am UTC

If you've done a very proofy linear algebra class, you might find it a little too hand holdy, a brief viewing suggests. If you're getting used to the culture of rigorous proof by way of analysis, then it might be just right!

by **nehpest** » Fri Jun 03, 2011 6:33 pm UTC

I decided to teach myself about tensors this summer, and so I picked up The Absolute Differential Calculus: Calculus of Tensors by Levi-Civita through my uni library. It arrived, and I found that a) the style of exposition is pretty different than I'm used to, and b) there are no exercises of any kind in the book.

I'm an engineering student, but I'm interested in learning tensors mainly so I can understand relativity in its native tongue. Is the book I have going to equip me with that knowledge, or should I find a different book? If you suggest staying with Levi-Civita, can you suggest a resource with exercises?

Thanks!

by **doogly** » Fri Jun 03, 2011 6:59 pm UTC

If you want to understand relativity, just pick up, say, Schutz's General Relativity. It introduces all of the tensors you will need.

by **mdyrud** » Fri Jun 03, 2011 8:25 pm UTC

Hey, I've been looking at grad schools recently, and many of them require that you be able to read mathematical papers in German, French, or Russian. I've taken a semester of German, so I know a little bit. Does anyone have recommendations for some lower level books in German for me to start on? My first thought was something in Discrete Mathematics or some other introductory course with a wider range of vocabulary.

by **B.Good** » Sat Jun 04, 2011 5:12 am UTC

mdyrud wrote:Hey, I've been looking at grad schools recently, and many of them require that you be able to read mathematical papers in German, French, or Russian. I've taken a semester of German, so I know a little bit. Does anyone have recommendations for some lower level books in German for me to start on? My first thought was something in Discrete Mathematics or some other introductory course with a wider range of vocabulary.

I don't have a recommendation and although I have no prior experience with the foreign language requirement, the professors and graduate students I've talked to about this subject say that the exam is incredibly easy. In most (maybe even all) cases, you even get to use a dictionary. I think if you continue to take German with any kind of regularity during your university career or even keeping your knowledge of German fresh after a certain point, you will be way more than prepared for the foreign language requirement.

by **Cornish** » Fri Jun 17, 2011 7:53 am UTC

mdyrud wrote:Hey, I've been looking at grad schools recently, and many of them require that you be able to read mathematical papers in German, French, or Russian. I've taken a semester of German, so I know a little bit. Does anyone have recommendations for some lower level books in German for me to start on? My first thought was something in Discrete Mathematics or some other introductory course with a wider range of vocabulary.

The germans have a translation fetish, so I'd recommend finding a english book you're comfortable with and would be semi-comfortable with if it were a german book and search the internet for a german version of it, quite a lot of chance that one exists.

by **Atomic Piranha** » Wed Jun 22, 2011 5:55 am UTC

I'm a college student home for the summer and I'm looking for some math to do for fun and to keep my brain working. Does anyone know of any good books with lots of interesting math problems/puzzles to work through? Wikipedia tells me that recreational mathematics is a developed field but I've never really heard of it before.

by **Inpriss Sorce** » Sat Jun 25, 2011 6:50 pm UTC

Atomic Piranha wrote:I'm a college student home for the summer and I'm looking for some math to do for fun and to keep my brain working. Does anyone know of any good books with lots of interesting math problems/puzzles to work through? Wikipedia tells me that recreational mathematics is a developed field but I've never really heard of it before.

Anything by Martin Gardner.

by **raike** » Sun Jul 03, 2011 7:45 am UTC

Atomic Piranha wrote:I'm a college student home for the summer and I'm looking for some math to do for fun and to keep my brain working. Does anyone know of any good books with lots of interesting math problems/puzzles to work through? Wikipedia tells me that recreational mathematics is a developed field but I've never really heard of it before.

There's a cute book by Alfred Posamentier called Math Charmers. It's pretty interesting, and makes for a good read and think.

Inpriss Sorce wrote:Anything by Martin Gardner.

I second this a million times.

by **pmgarvey** » Tue Jul 05, 2011 2:01 am UTC

I've finished Gilbert Strang's Introduction to Linear Algebra. What's good to follow this up with in the Linear Algebra/Numerical analysis direction? I really liked Strang's style. I was tempted to go for his other book on Linear Algebra, but was afraid it might just be a rehash.

by **Legume** » Fri Jul 08, 2011 1:59 am UTC

I would recommend Advanced Engineering Mathematics by Greenberg. It really helps connected together alot of the principles from Calc 3, Vector Calculus, Differential Equations, Fourier Transforms, Laplace Transforms, and Complex Analysis. College courses generally focus on each part discretely, and doesn't really give you the overall picture very much. For a book to review the mathematics as well as learn about some of the underlying principles like the Sturm-Liouville Theorem.

by **raike** » Sat Aug 06, 2011 8:34 pm UTC

I'm taking an introductory PDE course next semester using Partial Differential Equations: An Introduction by W A Strauss. Has anyone here used it before, and if so, is it good as a stand-alone book, or do you recommend pairing it with any other book? Also, I couldn't find anything on this, but are the first and second editions very different?

by **Giallo** » Thu Sep 08, 2011 5:29 pm UTC

I'm looking for a good introductory book to statistics, anybody has some hint?

by **Dopefish** » Thu Sep 08, 2011 6:21 pm UTC

Giallo wrote:I'm looking for a good introductory book to statistics, anybody has some hint?

My intro stats book was "Stats: Data and Models" by De Veaux, Velleman, and Bock. I don't know how it compares to other stats books, but compared to my other textbooks I enjoyed the way it was written (e.g. footnotes that in some sense make fun of the material), and theres plenty of carefully worked through examples. I enjoyed it sufficiently that I even read the material that wasn't covered by it in my actual course, so that's saying something.

I'm hardly a stats expert though, so those who have had more exposure to stats books (or have actually taught it) might be able to better direct you.

by **skullturf** » Thu Sep 08, 2011 9:13 pm UTC

Giallo wrote:I'm looking for a good introductory book to statistics, anybody has some hint?

I'm a big fan of Freund's Mathematical Statistics. That was my favorite statistics textbook as an undergraduate.

I have also taught introductory undergraduate statistics, though not from Freund. I taught out of Moore and McCabe, which I also think is a good textbook.

For context, I should point out that as an undergraduate, I was a mathematics major who enjoyed discrete mathematics, logic, and philosophy.

Somebody majoring in, say, biological sciences might have a different preference.

by **Giallo** » Thu Sep 08, 2011 9:59 pm UTC

I'm 1st year of maths in university :wink:

I'm probably going to have a course on statistical and quantum physics next semester, so I think it could be good to know a little bit of stats first...

by **mattbell** » Mon Sep 19, 2011 9:40 am UTC

Hi Everyone,
Is there any good ebbok for integral calculas? I need it urgently.. any help will be highly appreciated.

Thanks

by **Thesh** » Sun Nov 06, 2011 12:18 am UTC

I'm looking for a good book on number theory that is approachable to someone who took calculus and linear algebra 6 years ago, forgot most of it, and hasn't really studied anything math related since them. I have an interest in cryptography and I am writing a cipher based on what I have learned in the last year, and I also want to study cryptanalysis. I picked up a book on cryptanalysis and the second chapter, which talks about number theory, moves too fast for me since it's designed for someone with a strong math background.

So for now, I need something that will allow me to build a foundation to learn more advanced number theory. Also, any other books that will help me understand the various papers on cryptography and cryptanalysis would be appreciated, as well as books that focus more on number theory as it relates to cryptology and computer science that I can read after this book would be nice.

by **samual** » Tue Nov 08, 2011 5:27 pm UTC

Please any can help me??I have to know about district mathematics which consider the book of graph theory.Has anyone had experience with "Calculus with Analytic Geometry" by Simmons? It's what they're using for the MIT OpenCourseWare resources however I don't want to pick it up if it's a dud.

by **Swiftwind** » Wed Nov 09, 2011 1:59 am UTC

Hey y'all,

I was looking for a book on stochastic calculus and found the following: Stochastic Differential Equations: An Introduction with Applications

Has anyone used this text before for self-study? If so, do you think an undergraduate in math would be sufficient mathematical maturity for the text?

by **Quaternia** » Tue Nov 22, 2011 2:28 am UTC

Thesh wrote:I'm looking for a good book on number theory that is approachable to someone who took calculus and linear algebra 6 years ago, forgot most of it, and hasn't really studied anything math related since them

I think "A Friendly Introduction to Number Theory" by Silverman might be a good place to start. The early chapters are easy, and the latter chapters are not too hard. Overall on the easier side of things. It gives a decent first look at elliptic curves near the end, which you might find useful given your interests.
After that you can get into the harder stuff:
If you're looking for a working foundation in abstract algebra there's a bunch of good books, I think this thread has covered this before.
Steven J. Miller and Ramin Takloo-Bighash have a book called An Invitation to Modern Number Theory; it is much harder and is an analysis-heavy treatment of the subject. They propose a bunch of small research projects, most of which are computationally intensive, so maybe this would interest you.
Saban Alaca and Kenneth S. Williams have a good reference book, Introductory Algebraic Number Theory; it might be worth picking up to get through papers that assume a lot of definitions, but I would suggest taking a look at it first before buying it to see if it would help you.
There's a book called Algorithmic Number Theory by Eric Bach and Jeffrey Outlaw Shallit, it's up on google books actually. Maybe you'll like it.

by **ars111** » Sun Dec 04, 2011 2:32 pm UTC

I know that book that I was most impressed by is "Linear Algebra Done Wrong" written by Sergei Treil. It supposed to be a first linear algebra course for mathematically advanced students. Even so I red a lot of books, this caught my eye because a way of understanding algebra on a different point of view.

by **doogly** » Sun Dec 04, 2011 2:49 pm UTC

Huh. I'm surprised the preface isn't specifically comparing the book to Axler's Linear Algebra Done Right. This book I am more familiar with. Have you also seen that one?

by **Darrell88** » Fri Dec 23, 2011 5:27 pm UTC

Hi everybody. I was wondering if someone had a good suggestion for a book on advanced number theory . Also I'm looking for a good book on intuitionism. I've tried Intuitionism , An Introduction by Heyting but I hate the Galileo "Discourses and Mathematical Demonstrations Relating to Two New Sciences "-like conversation format

by **jmark1107** » Mon Dec 26, 2011 5:35 pm UTC

I used Vector Calculus by Paul C. Matthews. It's nice and quick but it doesn't cover some things a standard Calc III book will cover but it also covers topics such as cartesian tensors and the divergence theorems etc. In all it's quite a good book.

by **kace1991** » Thu Jan 26, 2012 2:57 am UTC

I'm in my freshman year, and I will like to have some alternative books to the ones recommended in my classes (they tend to recommend only books made by our own professors

).

So, the key cuestion is... those of you who are studying math at more advanced levels, in retrospect, what books you wish you've had when you were starting your career?

As orientation, this year my subjects are linear algebra, real analysis (first term about sequences, second term is about functions), and other subject which is basically applied math (basic cryptography, how GPS works, google's PageRank algorithm... things like that).

If the books you're thinking in a cover different subjects, I'm also interested, as long as it's not quite advanced

by **doogly** » Thu Jan 26, 2012 3:26 am UTC

Linear Algebra Done Right is a good way to look at lin alg from a different sort of perspective.
Counterexamples in Analysis is a fantastic book that is exactly what's on the cover. Would you like a function whose derivative is finite but unbounded on a closed interval? Page thirty seven. A discontinuous linear function? Thirty three. And so forth. Pretty dope.

by **kace1991** » Thu Jan 26, 2012 5:09 pm UTC

Those two seem interesting, specially the last one, thanks!

I would also like to find a good one centered in exercises, (one with includes the solutions) for analysis... most of what I found are just lists of unsolved problems and that's not quite pedagogic

by **Talith** » Wed Feb 01, 2012 6:58 pm UTC

Could anyone suggest a text or set of lecture notes that an undergrad with an introductory knowledge of algebraic topology could use to teach himself co/bordism theory? I've been told not to get my hopes up but thought I'd post just in case someone knows something my lecturer doesn't.

by **doogly** » Wed Feb 01, 2012 9:54 pm UTC

Milnor's Topology from a Differentiable Viewpoint might be a good place to start, mostly because Milnor is awesome (it has a bit on cobordism that might get you introduced and able to go to something more specific from there.)

by **dhokarena56** » Sat Apr 07, 2012 7:19 am UTC

Does anybody have any suggestions for a good book about first-order logic? On Amazon I can find this and this- if you're too lazy to click the links it's ones by Dover Books and Leigh Cauman, respectively.

by **gorcee** » Mon Apr 09, 2012 3:11 pm UTC

Does anyone have any experience with the Stein and Shakarchi series in Analysis? I'm seriously contemplating ordering the series from Princeton University Press.

They have four books, Real Analysis, Complex Analysis, Fourier Analysis, and Functional Analysis. I'm probably at early graduate-level abilities in Real and Complex Analysis, and want some books that I can reliably use for self-study as well as supplemental reference as I proceed with my graduate program in the near future.

by **GyRo567** » Thu Apr 19, 2012 8:38 am UTC

I've heard great things about their books on complex & Fourier analysis (the one on functional analysis just recently came out, so I don't know anything about it), but the real analysis text is not particularly good. If you want an introduction to Lebesgue integration & measure theory, Royden is a much better option, and if you want a reference text, Stein & Shakarchi is too scattered & full of examples to match something like Wheeden & Zygmund.

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