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1500 questions
50
votes
5 answers
Integrability of derivatives
Is there a (preferably simple) example of a function $f:(a,b)\to \mathbb{R}$ which is everywhere differentiable, such that $f'$ is not Riemann integrable?
I ask for pedagogical reasons. Results in basic real analysis relating a function and its…
Mark Meckes
- 11,286
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votes
7 answers
Determinant of sum of positive definite matrices
Say $A$ and $B$ are symmetric, positive definite matrices. I've proved that
$$\det(A+B) \ge \det(A) + \det(B)$$
in the case that $A$ and $B$ are two dimensional. Is this true in general for $n$-dimensional matrices? Is the following even…
user15221
- 511
50
votes
10 answers
Definition of "simplicial complex"
When I think of a "simplicial complex", I think of the geometric realization of a simplicial set (a simplicial object in the category of sets). I'll refer to this as "the first definition".
However, there is another definition of "simplicial…
Kevin H. Lin
- 20,738
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15 answers
What math institutes offer research in pairs/research in teams?
Some math institutes offer programs in which a small number of researchers are enabled to meet at the institute for a week or more. A list seemed as if it could be useful.
Hugh Thomas
- 6,075
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votes
12 answers
Combinatorial results without known combinatorial proofs
Stanley likes to keep a list of combinatorial results for which there is no known combinatorial proof. For example, until recently I believe the explicit enumeration of the de Brujin sequences fell into this category (but now see…
Qiaochu Yuan
- 114,941
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votes
4 answers
Difficult examples for Frankl's union-closed conjecture
Frankl's well-known union-closed conjecture states that if F is a finite family of sets that is closed under taking unions (that is, if A and B belong to the family then so does $A\cup B$), then there must be an element that belongs to at least half…
gowers
- 28,729
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votes
11 answers
Publishing papers that became classics before they were submitted
Sometimes the following happens: a result is proven, but the author never submits a paper for publication. In some cases, a preprint appears. In some cases, the proof is so short that it can be presented at a conference or lecture series, and the…
Kostya_I
- 8,662
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votes
4 answers
When is $L^2(X)$ separable?
I have never studied any measure theory, so apologise in advance, if my question is easy:
Let $X$ be a measure space. How can I decide whether $L^2(X)$ is separable?
In reality, I am interested in Borel sets on a locally compact space $X$. I can…
Bugs Bunny
- 12,113
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votes
13 answers
Is amateur research in mathematics viable?
After a long reflection, I've decided I won't go to graduate school and do a thesis, among other things. I personally can't cope with the pressure and uncertainty of an academic job.
I will therefore move towards a master's degree in engineering and…
Nilav
- 61
50
votes
15 answers
Explicit computations using the Haar measure
This question is somewhat related to my previous one on Grassmanians. The few times I've encountered the Haar measure in the course of my mathematical education, it's always been used in a very theoretical setting: in the right setting, it exists,…
Thierry Zell
- 4,536
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votes
5 answers
Usefulness of Nash embedding theorem
Nash embedding theorem states that every smooth Riemannian manifold can be smoothly isometrically embedded into some Euclidean space $E^N$. This result is of fundamental importance, for it unifies the intrinsic and extrinsic points of view of…
Ettore Minguzzi
- 1,341
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votes
5 answers
On Euler's polynomial $x^2+x+41$
This is an elementary question about something way outside my area of expertise. A well-known observation due to Euler is that the polynomial $P(x)=x^2+x+41$ takes on only prime values for the first 40 integer values of $x$ starting with $x=0$,…
Allen Hatcher
- 19,387
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votes
7 answers
Dividing a cake between $n-1$, $n$, or $n+1$ guests
A housewife is waiting for guests and has prepared a cake. She doesn't know how many guests will come, but it will be $n-1$, $n$, or $n+1$.
What is the minimal number $f(n)$ of pieces the cake should be cut to make it possible to divide between…
Lviv Scottish Book
- 4,435
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votes
6 answers
Intuition for the last step in Serre's proof of the three-squares theorem
Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer $a$ is the sum of three squares if and only if it is not of the form $4^m (8n + 7)$ : first one shows that the condition…
Qiaochu Yuan
- 114,941
50
votes
7 answers
Is all non-convex optimization heuristic?
Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic programming is almost as easy, and there's a good deal of…
DoubleJay
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