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1500 questions
58
votes
2 answers
For a finite set A of positive reals, prove that the set A + A - A contains at least as many positive as negative elements
I am currently working on a proof that would need to use the following theorem that I cannot prove:
"Let $A$ be a finite set of positive real numbers. Then, the set $A + A - A$ contains at least as many positive elements as negative elements. ($0$…
Timo Reichert
- 633
58
votes
7 answers
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 the solution to the heat equation, in a very nice…
Randy Qian
- 591
58
votes
82 answers
Prominent non-mathematical work of mathematicians
First of all, sorry if this post is not appropriate for this forum.
I have a habit that every time I read a beautiful article I look at the author's homepage and often find amazing things.
Recently I read a paper of Andrew Hicks and when I opened…
C.F.G
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58
votes
22 answers
Which high-degree derivatives play an essential role?
Q. Which high-degree derivatives play an essential role
in applications, or in theorems?
Of course the 1st derivative of distance w.r.t. time (velocity), the 2nd derivative (acceleration),
and the 3rd derivative (jolt or jerk) certainly play…
Joseph O'Rourke
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58
votes
5 answers
Arriving at the same result with the opposite hypotheses
I am pretty distant from anything analytic, including analytic number theory but I decided to read the Wikipedia page on the Riemann hypothesis (current revision) and there is some pretty interesting stuff there:
Some consequences of the RH are also…
user141414
58
votes
6 answers
How does one use the Poisson summation formula?
While reading the answer to another Mathoverflow question, which mentioned the Poisson summation formula, I felt a question of my own coming on. This is something I've wanted to know for a long time. In fact, I've even asked people, who have…
gowers
- 28,729
58
votes
7 answers
In what respect are univalent foundations "better" than set theory?
It was an ambitious project of Vladimir Voevodsky's to provide new foundations for mathematics with univalent foundations (UF) to eventually replace set theory (ST).
Part of what makes ST so appealing is its incredible conciseness: the only…
Dominic van der Zypen
- 45,374
58
votes
4 answers
Is it possible to have a research career while checking the proof of every theorem that you cite?
A colleague raised the above question with me; more precisely he said:
Suppose that a mathematician were resolved not to publish any theorems
unless they had checked the proof of every theorem that they cite (and
recursively the proofs of all…
John Stillwell
- 12,258
58
votes
1 answer
Square root of dirac delta function
Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
DLN
- 807
58
votes
0 answers
Grothendieck's Period Conjecture and the missing p-adic Hodge Theories
Singular cohomology and algebraic de Rham cohomology are both functors from the category of smooth projective algebraic varieties over $\mathbb Q$ to $\mathbb Q$-vectors spaces. They come with the extra structure of Weil cohomology theories -…
Will Sawin
- 135,926
58
votes
6 answers
Has decidability got something to do with primes?
Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this.
Motivation:
Recently Pace Nielsen asked the question "How do we…
abcdxyz
- 2,744
58
votes
2 answers
Can a subset of the plane have nontrivial $H_2$ or $\pi_2$?
This is a question that occurred to me years ago when I was first learning algebraic topology. I've since learned that it's a somewhat aesthetically displeasing question, but I'm still curious about the answer.
Is it possible for a subset of…
Timothy Chow
- 78,129
58
votes
9 answers
How do they verify a verifier of formalized proofs?
In an unrelated thread Sam Nead intrigued me by mentioning a formalized proof of the Jordan curve theorem. I then found that there are at least two, made on two different systems. This is quite an achievement, but is it of any use for a…
Sergei Ivanov
- 32,149
58
votes
12 answers
What areas of pure mathematics research are best for a post-PhD transition to industry?
I have a student who is looking to start a PhD in pure mathematics. She is talented and motivated, and will do quite well. She is still in a phase of her development where she is still open to the possibility of working in a broad range of research…
Jon Bannon
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58
votes
8 answers
Inverse gamma function?
This is an analysis question I remember thinking about in high school. Reading some of the other topics here reminded me of this, and I'd like to hear other people's solutions to this.
We have the gamma function, which has a fairly elementary form…
jeremy
- 2,139