The probability class I recently finished (taught at an upper-undergraduate or lower-graduate level) used the text by Grimmett and Stirzaker. I really disliked this book.   I am familiar with measure theory, so it is fine if the book is measure-theoretic. However, I want to make sure the book doesn't neglect to provide a clear explanation of a the basic concepts of probability, and provide exercises for the basic problem-solving techniques. Also, I always love texts that have good motivation and intuitive explanations for things (I guess I prefer some motivating discussion rather than a totally formal text like Rudin, or arguably Ahlfors).   I think model texts for what I'm looking for are Spivak's Calculus and Dummit and Foote's Algebra. What I love about these texts are the large amount of (enlightening) exercises.   Update: Still looking for a good probability text. Feller's text has come up as a suggestion. Does that one have good exercises?
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Question is quite straight... I'm not very good in this subject but need to understand at a good level.
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I would like to find a book that introduces me gently to the subject of stochastic processes without sacrificing mathematical rigor. It would be great if the book has lots of examples and that the book is designed for undergraduates. Just like how there are rigorous undergraduate abstract algebra books (I am thinking of GAllian's contemporary modern algebra here).   I already studied measure theory, and prob. theory.  Thank you.
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Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff on his own.  He asked me recently if there was a 'mathematical' version of .  The point being, if there was an "introduction to proofs" book which took a more hand-holding approach. He's also expressed fondness for the .  However, metamath's proofs are obviously a bit too low level to learn anything serious.  At one point in our conversation, I mentioned the dependently typed programming language , but I felt it's not quite as helpful until he at least gets some experience with  (for which the target audience of Agda is Haskell programmers, hence the similarity in syntax).  Anyone have any good suggestions on a book?
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Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would   Prove all theorems used. Use modern "mathematical notation" as opposed to "physics notation", especially with respect to linear algebra and differential geometry. Have examples that illustrate both computational and theoretical aspects. Have a range of exercises with varying degrees of difficulty, with answers.   An ideal text would read a lot more like a math book than a physics book and would demand few prerequisites in physics. Bottom line is that I would like a book that provides an axiomatic development of general relativity clearly and with mathematical precision works out the details of the theory.  Addendum (1): I did not intend to start a war over notation. As I said in one of the comments below, I think indicial notation together with the summation convention is very useful. The coordinate-free approach has its uses as well and I see no reason why the two can't peacefully coexist. What I meant by "mathematics notation" vs. "physics notation" is the following: Consider, as an example, one of the leading texts on smooth manifolds, John Lee's Introduction to Smooth Manifolds. I am very accustomed to this notation and it very similar to the notation used by Tu's Introduction to Manifolds, for instance, and other popular texts on differential geometry. On the other hand, take Frankel's Geometry of Physics. Now, this is a nice book but it is very difficult for me to follow it because 1) Lack of proofs and 2)the notation does not agree with other math texts that I'm accustomed to. Of course, there are commonalities but enough is different that I find it really annoying to try to translate between the two...  Addendum (2):  For the benefit of future readers, In addition to suggestions below, I have found another text that also closely-aligns with the criteria I stated above. It is, Spacetime: Foundations of General Relativity and Differential Geometry by Marcus Kriele. The author begins by discussing affine geometry, analysis on manifolds, multilinear algebra and other underpinnings and leads into general relativity at roughly the midpoint of the text. The notation is also fairly consistent with the books on differential geometry I mentioned above.
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I'm embarrassed to say that I have a PhD and hold an asst professorship, but get tripped up when reading statistics research. I am in a field of Business that is similar to IO Psychology or Social Psych. I spend too much time reading applied stats books, but I find even with all the reading I don't have a firm grasp of what I'm actually doing. Everything is very 'seat of the pants.' (As sad as it seems, I think this is not a unique situation among the faculty in the social sciences...) The biggest problem comes when I need to apply a rarely used stat technique. I can find an article from a mathematical stats journal with the equations that would solve my problem, but I don't have the math to convert those into code. I am forever relying on other prof's R packages, and crossing my fingers hoping it will work (I can't even check to verify if it did or not). It's been over 15 years since I took Calculus and Algebra in undergrad, and I think I want to start at the beginning and truly understand probability and statistics.   I am starting with  and  for a quick refresher of the basics -- I know it's hard to believe, but in an applied research field we rarely have use for sin or cos. I'm even trying to finally learn how to correctly do a proof, using the books from Velleman ("") and Houston ("") -- I'm serious about doing this right and understanding the subject. From there I want to move on to (correctly) learn the Calculus and Linear Algebra I need to tackle probability and statistics. I was thinking of using  and  books. But  comes highly recommended as well. After that I am completely at a loss. Further, I don't know how far to go into Calculus or Linear Algebra before I reach diminishing returns. (Apostle introduces Probability in the  -- is it vital that I work through everything preceding it before tackling Probability?)  So my question is: if you had to do it over again with the goal of truly, deeply understanding statistics, where would you start? What books are the modern path to deep understanding? I would like to follow a modern path so that I can understand current research in statistics, including Bayesian approaches. But not in a machine learning context (which seems to be the all the rage at the moment), rather a social science / design and analysis of experiments / multilevel modeling context. Perhaps my goal would be the work of Andrew Gelman;  showed me how I should be looking at modeling and statistics (simulation, uncertainty estimates everywhere, bayesian inference, and so on).  How should I go about relearning this material with that end goal in mind?    Update 1: Possible texts, starting from scratch with a focus on proofs and deep understanding. Not necessarily one after another.  Relearn the basics:    by Velleman  by Houston  and  by Gelfand (for understanding why and how instead of what)  by Simmons (for reference)  by Lockhart (for inspiration)   Calculus (which one(s), and how deep?):    by Strang  and  by Apostol   by Spivak ()   by Curant (and II/1 II/2?)   Linear Algebra (which one(s) and how deep?):    by Strang  by Searle  by Gentle   Probability (which one(s)?):    and  by Feller  (for intuitive understanding)  by Hoel, Port, Stone  by Resnick (for measure theoretic / modern approach?)  by Mosteller   Core Statistics (which one(s)?):    by DeGroot and Schervish  Casella and Berger   Other suggestions? Again with the goal of understanding and developing (or at least implementing) new methods in hierarchical modelling (generalized and linear).
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I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd however want something more comprehensive, therefore I'm asking you for tips!  Hopeful for answers ^^
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My goal is to pick up some commutative algebra, ultimately in order to be able to understand algebraic geometry texts like Hartshorne's. Three popular texts are Atiyah-Macdonald, Matsumura (Commutative Ring Theory), and Eisenbud. There are also other books by Reid, Kemper, Sharp, etc. Can someone outline the differences between these texts, their relative strengths, and their intended audiences?   I am not listing my own background and strengths, on purpose, (a) so that the answers may be helpful to others, and (b) I might be wrong about myself, and I want to hear more general opinions than what might suite my narrow profile (e.g. If I said "I only like short books", then I might preclude useful answers about Eisenbud, etc.).
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I'm currently attending a somewhat disorganized seminar on combinatorics that follows no textbook. So far we have covered the orbit-stabilizer theorem, some recursion, and we're heading into the Möbius inversion formula.   Can anyone suggest a text that approaches combinatorics at this level for a 2nd-3rd year undergrad who already knows some algebra and the more basic combinatorics like combinations, permutations, stars-and-bars, generating functions? Most introductory combinatorics books I've found are more suited to a discrete math class and cover stuff which I already know. I'm looking for something to supplement this lecture. Thank you.
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I read single variable calculus this semester, and the course is using Thomas Calculus as the textbook. But this book is just too huge, a single chapter contains 100 exercise questions! Now I'm looking for a concise and complete textbook. I'm not interested in routine, computational exercises, but rather some challenging problem sets.  I have quite a strong basic knowledge of calculus from high school, but I still have difficulties in solving a few questions from past exam papers. So I'm looking for more challenging exercises. In fact, I'm looking forward to solving Putnam level questions.  Please suggest some textbooks with these features. Thanks in advance.
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I'm starting to read Baby Rudin (Principles of mathematical analysis) now and I wonder whether you know of any companions to it. Another supplementary book would do too. I tried Silvia's notes, but I found them a bit too "logical" so to say. Are they good? What else do you recommend?
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Edit:  Ok, user  made comment that truly helps narrow the focus of my question. Basically, I'm asking for a self-study course of Mathematical Methods.  Thanks to his recommendation I identified a book that is very close to the answer:  Mathematical Methods in the Physical Sciences by Mary L. Boas  Supposedly, the book covers most of the Math a person needs to learn in order to understand the simulations I presented.  Does this book suit my needs?    Hello,  I have always been interested on the idea to study Math at an Undergraduate level to be able to implement simulations like the followings:  Finite Difference Simulation of the Wave Equation    Heat Equation in Two Dimensions    Besides my precollege experience in Math, I have only been able to do very simple simulations due to my lack of knowledge of Advanced Math. Every time I ask about the inner workings of the simulations I get redirected to papers that look like foreign language to me and, unfortunately, I depend on easy-to-read tutorials to understand them, and while I have found   I'm more interested in reaching the solution myself.  I checked some of the questions related to book recommendations for self-study and while they are very helpful I'm interested in learning by example (There is a book called: "Accelerated C++ Practical Programming by Example" that fits very well my study habits)  Thanks in advance.
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I'm working my way through Calculus, 9th ed. by Larson and Edwards in independent study. The problem is that many of the exercises and examples use a lot of algebra tricks that are glossed over and assumed knowledge on the part of the student.  Now, when I took precalc in high school I was a total layabout and didn't really absorb many of the finer points as far as some specific topics go (nested absolute value inequalities, for instance) and I'm having a hard time working through many of the exercises. I have no trouble understanding the calculus aspect of the problems, but fail miserably on some of the harder algebraic aspects.   I know I'm not the only one struggling with this, though -- an overarching criticism I've found of this book is that it assumes too much algebra knowledge of undergrad math students and doesn't fully explain any steps.   What I'm really looking for is a precalc-level algebra reference with good examples that I can turn to for specific topics in algebra when I encounter a question I have trouble with.   Can anyone recommend such a thing?    Edit: I'd also like to point out that I have a kindle, so cheaper suggestions for good kindle books are welcome. That's assuming, of course, that kindle is a satisfactory medium for such a thing. I'm not sure.
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What would be your recommendation for the text which could be useful for someone starting with lattice theory? Suggestion for both books and online materials/lecture notes are welcome.    This was asked as a part of  and I did not like that it was combination of two different questions. (One of them on geometric modular lattices and the other one about text recommendation.)  I don't expect the user to post a follow-up question, since he was not seen at MSE for some time. I think that this could be a useful for some users here, so I've decided to post it myself.
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I've taken up self-study of math. (How smart can that be?) I've just about finished a course in real analysis which spent a lot of time on metric spaces and some time revisiting calculus.  I was thinking of trying abstract algebra. I would appreciate any book recommendations.  Thanks in advance.  Andrew
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I am in search of a good book for learning Stochastic Calculus from a purely mathematical/statistical point of view. Almost all the books I see are based on Finance. Also, please specify the pre-requisites for this course.  Actually, I am quite knowledgable in Measure Theory, Markov Chains and Martingale Theory. I believe this knowledge should suffice.  I am planning to self-study this subject. Please suggest me good books.  Thank you.
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I'm an undergraduate statistics student looking for a good treatment of clinical trials analysis. The text should cover the fundamentals of experimental design, blocking, power analysis, latin squares design, and cluster randomization designs, among other topics.   I have an undergraduate knowledge of mathematical statistics and real analysis, but if there's a fantastic text that requires a bit higher level of statistics or analysis, I can work up to it.
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I am looking for a good source on group theory aimed at physicists. I'd prefer one with a good general introduction to group theory, not just focusing on Lie groups or crystal groups but one that covers "all" the basics, and then, in addition, talks about the specific subjects of group theory relevant to physicists, i.e. also some stuff on representations etc.  Is Wigner's text a good way to start? I guess it's a "classic", but I fear that its notation might be a bit outdated?