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79
votes
90answers
10k views

What would you want to see at the Museum of Mathematics?

EDIT (30 Nov 2012): MoMath is opening in a couple of weeks, so this seems like it might be a good time for any last-minute additions to this question before I vote to close my own question as "no ...
7
votes
1answer
1k views

Technical trends quietly aimed at big open problems? [closed]

When I was an undergraduate 35 years ago, I made the mistake of asking some of my mathematics professors what well-known open problems they liked to think about. I got the message that this was ...
12
votes
17answers
2k views

Individual mathematical objects whose study amounts to a (sub)discipline? [closed]

Certain mathematical objects have a theory so rich that their study alone arguably constitutes a distinct (sub)discipline. My own list would begin with 1) the absolute Galois group of the rationals; ...
1
vote
0answers
163 views

Regarding a Feature of Multivariate Real Function

Any real function can be expressed as a function of the sum of two monotonic real functions? More precisely, for real function p(x, y), there exist continuous real functions P(x), h(x,y), g(x) such ...
1
vote
0answers
526 views

What is the name for(a^2 + b^2 + c^2 +…)/(a + b + c +…)? [closed]

That is, the sum of squares of some numbers divided by the sum of the numbers. The term "anti-harmonic mean" has been coined for this quantity. I'm hoping there is a better name.
45
votes
6answers
5k views

Still Difficult After All These Years

I think we all secretly hope that in the long run mathematics becomes easier, in that with advances of perspective, today's difficult results will seem easier to future mathematicians. If I were ...
44
votes
25answers
4k views

Proof synopsis collection

I hate to keep going with the big lists, but the question about one-sentence summaries of topics/areas spurred this question...and I just can't help myself! Definition (Fraleigh): A proof synopsis ...
14
votes
35answers
2k views

Basic results with three or more hypotheses

Consider the following statement of the Arzela-Ascoli theorem. Theorem. Let K be a compact topological space and let S be a subset of C(K). Then S is relatively compact if and only if S is uniformly ...
13
votes
2answers
821 views

Is there a name for sets for which it is easier to test membership than to find members---and vice versa?

This is a question my son Bob asked me. For some sets it is relatively easy to test for membership but a lot more difficult to find members, and for others the reverse is true. Here is an elementary ...
4
votes
4answers
3k views

Theorems true but wrong.

Many theorems have the form : Premise(es) implies Conclusion(s) Example A of wrongness: There are many examples in which a theorem is stated without mentioning that part of the premise is not ...
29
votes
5answers
4k views

Does a referee have to check carefully the proof ?

I have always checked very carefully the papers I was refereeing when I wanted to suggest "accept". Actually I spend almost as much time checking the maths of a paper I referee than checking the maths ...
29
votes
6answers
4k views

Why is the Gaussian so pervasive in mathematics?

This is a heuristic question that I think was once asked by Serge Lang. The gaussian: $e^{-x^2}$ appears as the fixed point to the Fourier transform, in the punchline to the central limit theorem, as ...
1
vote
2answers
791 views

Example of a function that behaves like another function

I need a function $f(x)$ with the following properties - It should be monotonically non-decreasing. For $x \geq 1$, $x + \frac{1}{x} - f(x) < \epsilon$ where $\epsilon$ is an extremely small ...
7
votes
3answers
566 views

What is the strategy for “all words valid” scrabble?

The rules for "all words valid" scrabble are exactly the same as ordinary scrabble, except that every single combination of letters is in the dictionary. To make the game deterministic, we will also ...
2
votes
3answers
679 views

Is there a simple criterion to determine if two parallelograms intersect?

see title. assume we are given two parallelograms in the plane. how can I check if the intersection is nonempty? note that I do not need to actually find the intersection.
110
votes
36answers
20k views

Demonstrating that rigour is important

Any pure mathematician will from time to time discuss, or think about, the question of why we care about proofs, or to put the question in a more precise form, why we seem to be so much happier with ...
4
votes
1answer
480 views

Elementary problem about triangles inside a convex polygon

Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
6
votes
1answer
573 views

Sum of subset of geometric series: a^2^n

The formula for 1 + a + a^2 + .... where 0 < a < 1 is $\frac{1}{1-a}$, but what if you wanted to sum only those where the exponent is a power of 2? That is, $S = a + a^2 + a^4 + a^8 + \cdots$ ...