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4
votes
7answers
659 views

Supplementary reading for probability theory studies

Can you advise some good books covering areas which are required for serious probability theory studies (e.g. measure theory, functional analysis)? Preferably this book should have some problem sets ...
2
votes
2answers
459 views

What is the correct way to write product or sum in capital pi or capital sigma notation when you wish to exclude an index?

I have a series of terms $\{t_n : t_n = a_n x_n\}$, and I want to talk about the product of each term except $t_j$. Would any of these be an appropriate way to say that? I like this: $$\prod_{i \ne ...
13
votes
7answers
2k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : Rationals of the form $\frac{p}{q}$ where $p,q$ are primes in $[a,b]$ and some other problems in Mathematics which looks as if they are elementary but their ...
11
votes
15answers
10k views

What concepts were most difficult for you to understand in Calculus?

I'm developing some instructional material for a Calculus 1 class and I wanted to know from experience for yourself, tutoring others, and/or helping people on this site where is the most difficulty in ...
25
votes
11answers
3k views

Vivid examples of vector spaces?

When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a ...
4
votes
3answers
2k views

Group Law for an Elliptic curve

I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages. The ...
3
votes
2answers
188 views

Relationship between torsion modules and topology

I was reviewing my class notes and found the following: "The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band" In our notes we used the following definition ...
7
votes
2answers
538 views

A question on FLT and Taniyama Shimura

Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain ...
4
votes
4answers
745 views

Why does symplectic geometry have many applications in mathematics

It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you!
8
votes
8answers
7k views

Where to go after calculus?

Ok this is a bit of an unanswerable question, but hopefully someone will answer. As I have been going through college & high school there has been a kind of "path" through which you learn math. ...
51
votes
9answers
9k views

Advantages of IMO students in Mathematical Research

Everyone in this community i think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries ...
3
votes
1answer
537 views

Algebraic Topology Need!

In India, generally during the graduate years, we follow a course work pattern, unlike many places in the U.S where students are exposed to research during their undergraduate years itself. As, a ...
4
votes
3answers
715 views

What is your favorite proof that $e^{ix}$ has a period of $2\pi$?

as a function of a real variable, apparently. Part of the freedom in choosing a proof is that you get to choose what definition of $e^{ix}$ to start from -- do you use a differential equation? a power ...
40
votes
6answers
8k views

How Do You Go About Learning Mathematics?

I really like mathematics, but I am not good at learning it. I find it takes me a long time to absorb new material by reading on my own and I haven't found a formula that works for me. I am hoping a ...
47
votes
4answers
6k views

Why do we care about dual spaces?

When I first took linear algebra, we never learned about dual spaces. Today in lecture we discussed them and I understand what they are and everything, but I don't really understand why we want to ...
3
votes
4answers
947 views

Can I skip the first chapter in Rudin's Principles of Mathematical Analysis?

I am a statistician who wishes to learn real analysis in order to better understand the foundations of statistics. With that aim in mind I plan to go through Rudin's classic on "Principles of ...
5
votes
2answers
916 views

Supplement for Jacobson's Basic Algebra I

I am going to be taking a first course in abstract algebra this fall and we are using Jacobson's Basic Algebra I for the course text. It looks like a good textbook, but it is lacking a lot of ...
18
votes
7answers
6k views

Why does “convex function” mean “concave *up*”?

A function $f : \mathbb{R} \to \mathbb{R}$ is convex (or "concave up") provided that for all $x,y \in \mathbb{R}$ and $t \in [0,1]$, $$f(tx + (1-t)y) \le tf(x) + (1-t)f(y).$$ Equivalently, a line ...
15
votes
3answers
1k views

What does “only” mean?

I understand the technical and logical distinction between "if" and "only if" and "if and only if". But I have always been troubled by the phrase "only if" even though I am able to parse and ...
10
votes
5answers
3k views

Companions to Rudin?

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 ...
2
votes
2answers
286 views

Basic questions about the algebra of surfaces

When I was studying topology I remember being able to demonstrate that the set of topological surfaces with any number of punctures (including the projective plane, Klein bottle, Moebius strip, double ...
10
votes
5answers
2k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
84
votes
9answers
19k views

What is the importance of the Collatz conjecture?

I have been fascinated by this problem since I first heard about it in high school. From the Wikipedia article http://en.wikipedia.org/wiki/Collatz_problem: Take any natural number $n$. If $n$ is ...
6
votes
2answers
448 views

Elementary Number theory results that are not generalized by ring or group theory

I've taken an undergraduate course in ring and group theroy, but haven't studied number theory formally. I've noticed many important results in number theory have been generalized in group/ring ...
14
votes
6answers
4k views

Best intuitive metaphors for math concepts (of any level)

Frequently, we introduce a new concept with a formal definition, then immediately say "Intuitively, what this means is..." What are the absolute best metaphors you've seen (for concepts of any level)? ...
13
votes
1answer
876 views

Common English language mistakes in mathematical writing [closed]

Quoting from this excellent answer: If you read enough math papers you'll find that there are certain linguistic ticks that people pick up from each other So here's a question (primarily for you ...
8
votes
1answer
2k views

Elementary proof of the Prime Number Theorem - Need?

Although i am very much new to "Analytic Number Theory", there are some non mathematical questions which puzzle me. First of all, why was G.H.Hardy so much keen to have an elementary proof of the ...
3
votes
4answers
267 views

Definition of an Algebraic Objects

How did the definition of Algebraic objects like group, ring and field come up? When groups were first introduced, were they given the 4 axioms as we give now. And what made Mathematicians to think of ...
5
votes
1answer
377 views

undergraduate courses emphasizing theory building?

I was wondering if anyone had any experience with an undergraduate course that emphasized the building of mathematical theories or if they'd ever heard of this being done? How did the class work (did ...
1
vote
6answers
633 views

How to write down proofs?

I'd like to write down proofs. I use first-order logic and natural deduction. Formulas tend to be long, too long for LaTeX. Writing formulas with the computer is also a slow process. Handwriting is a ...
26
votes
9answers
3k views

Does a Person Need a Mathematics Degree in order to Publish in a Mathematics jounal?

I am a neophyte amateur mathematician. I have been reading a lot about journals and the topic of peer-review in mathematics journals. Does one have to have professional credentials or have a Doctorate ...
5
votes
12answers
2k views

Best Quotes by a Mathematician [closed]

If i were to ask this: Which quote by a "Mathematician" do you like, then what would be your answer!
7
votes
4answers
362 views

Are the computable reals finitary?

In the comment thread of an answer, I said: The computable numbers are based on the intuitionistic continuum, and are not finitary. To which T.. replied: Computable numbers are not based on ...
5
votes
3answers
920 views

Mathematical paradoxes?

What are some interesting mathematical paradoxes? What I have in mind are things like the Banach-Tarski paradox, Paradox of Zeno of Elea, Russel's paradox, etc.. Edit: As an additional restriction, ...
15
votes
12answers
1k views

Fields that require both CS and pure math

I'm mainly a CS major, but I also want to learn more advanced mathematics. I see lot of cross over between CS and applied math, same can't be said about pure math. Definition of 'advanced': ...
41
votes
14answers
2k views

Surprising Generalizations

I just learned (thanks to Harry Gindi's answer on MO and to Qiaochu Yuan's blog post on AoPS) that the chinese remainder theorem and Lagrange interpolation are really just two instances of the same ...
0
votes
4answers
416 views

What is your favorite estimation exercise? [closed]

A fun question I ask students or interviewees (in engineering) is: This is not my question, this is an example: Using only what you know now, how many cans of soda would you estimate are ...
3
votes
2answers
588 views

What's the most effective ways of teaching kids - times tables?

I'd like to help a 6 year old who already has a pretty good grasp of 2, 5, and 10 times tables.
2
votes
1answer
751 views
5
votes
2answers
349 views

Examples of well-displayed mathematics on the internet

An off-topic question asked at Mathoverflow by Andrew Stacey; but one which fits here: I'm interested in hearing of examples of mathematical (or, at a pinch, scientific) websites with serious content ...
3
votes
1answer
160 views

Resources for getting maths on to the web

An off-topic question posed at Mathoverflow by Andrew Stacey, but one which fits here: One thing that came out of Terry Tao's recent blog posts on this matter (first post and follow up) is that it's ...
3
votes
0answers
1k views

List of interesting and well-explained textbooks [closed]

In light of the List of Interesting Math Blogs I thought it would benefit the community if we have a list of textbooks for specific topics. Trusted users and/or I will edit this post depending on ...
9
votes
3answers
2k views

Correct usage of the phrase “In the sequel”? History? Alternatives?

While I feel quite confident that I've inferred the correct meaning of "In the sequel" from context, I've never heard anyone explicitly tell me, so first off, to remove my niggling doubts: What does ...
12
votes
7answers
3k views

Choosing a text for a First Course in Topology

Which is a better textbook - Dugundji or Munkres? I'm concerned with clarity of exposition and explanation of motivation, etc.
30
votes
19answers
2k views

Which mathematicians have influenced you the most?

This question is lifted from Mathoverflow.. I feel it belongs here too. There are mathematicians whose creativity, insight and taste have the power of driving anyone into a world of beautiful ideas, ...
13
votes
5answers
2k views

Why does Benford's Law (or Zipf's Law) hold?

Both Benford's Law (if you take a list of values, the distribution of the most significant digit is rougly proportional to the logarithm of the digit) and Zipf's Law (given a corpus of natural ...
20
votes
15answers
3k views

Useful examples of pathological functions

What are some particularly well-known functions that exhibit pathological behavior at or near at least one value and are particularly useful as examples? For instance, if $f'(a) = b$, then $f(a)$ ...
19
votes
8answers
8k views

Online Math Degree Programs

Are there any real online mathematics (applied math, statistics, ...) degree programs out there? I'm full-time employed, thus not having the flexibility of attending an on campus program. I also ...
12
votes
7answers
1k views

Real world uses of homotopy theory

I covered homotopy theory in a recent maths course. However I was never presented with any reasons as to why (or even if) it is useful. Is there any good examples of its use outside academia?
53
votes
21answers
3k views

Your favourite maths puzzles

Okay, so this question was bound to come up sooner or later- the hope was to ask it well before someone asked it badly... We all love a good puzzle To a certain extent, any piece of mathematics is a ...