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1500 questions
49
votes
9 answers
What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?
I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe that such a fight is winnable?
Question. What…
goblin GONE
- 3,693
49
votes
2 answers
How to unify various reconstruction theorems (Gabriel-Rosenberg, Tannaka,Balmers)
What I am talking about are reconstruction theorems for commutative scheme and group from category. Let me elaborate a bit. (I am not an expert, if I made mistake, feel free to correct me)
Reconstruction of commutative schemes
Given a quasi compact…
Shizhuo Zhang
- 5,365
49
votes
2 answers
Is this Riemann zeta function product equal to the Fourier transform of the von Mangoldt function?
Mathematica knows that:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)\;\;\;\;\;\;\;\;\;\;\;\; (1)$$
The von Mangoldt function should then be:
$$\Lambda(n)=\lim\limits_{s \rightarrow 1}…
Mats Granvik
- 1,133
49
votes
2 answers
Rachinsky quintets
This 1895 painting of Nikolai Bogdanov-Belsky shows mental calculations in the public school of Sergei Rachinsky. Boys in a Russian village school try to calculate $(10^2+11^2+12^2+13^2+14^2)/365$ in their heads. One of the methods of solution is…
Zurab Silagadze
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49
votes
2 answers
Is a function with nowhere vanishing derivatives analytic?
My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic?
EDIT. According to a theorem of Serge N. Bernstein (Sur les fonctions…
smyrlis
- 2,873
49
votes
4 answers
Is there an "elementary" proof of the infinitude of completely split primes?
Let $K$ be a Galois extension of the rationals with degree $n$. The Chebotarev Density Theorem guarantees that the rational primes that split completely in $K$ have density $1/n$ and thus there are infinitely many such primes. As Kevin Buzzard…
François G. Dorais
- 43,723
49
votes
4 answers
What fraction of the integer lattice can be seen from the origin?
Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$.
Say that a point $(x,y)$ of $Q$ is visible from the origin if the
segment from $(0,0)$ to $(x,y) \in Q$ passes through no other point of $Q$.
So points block…
Joseph O'Rourke
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49
votes
5 answers
are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?
I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary cotangent bundle of configuration space -- because…
symplectomorphic
- 1,189
49
votes
4 answers
Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to any integer multilpe times, it will eventually…
Daniel Soltész
- 3,004
49
votes
5 answers
If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Update: The answer to the title question is no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous theorem with amazingly tricky proof says that if we dissect a square into an odd number of triangles,…
domotorp
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49
votes
6 answers
What is Yoneda's Lemma a generalization of?
What is Yoneda's Lemma a generalization of?
I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory.
Comments are welcome why the following candidates…
Hans-Peter Stricker
- 9,628
49
votes
1 answer
Order of an automorphism of a finite group
Let G be a finite group of order n. Must every automorphism of G have order less than n?
(David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil don't!)
Reid Barton
- 24,898
48
votes
11 answers
In "splendid isolation"
While browsing the Net for some articles related to the history of the Whittaker-Shannon sampling theorem, so important to our digital world today, I came across this passage by H. D. Luke in The Origins of the Sampling Theorem:
However, this…
Tom Copeland
- 9,937
48
votes
7 answers
Why are there so many smooth functions?
I do understand that my question might seem a little bit ignorant, but I thought about it a lot and still can't wrap my head around it.
Analycity imposes very strong conditions on a map, from elementary ones like "locally zero implies globally…
Piotr Pstrągowski
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48
votes
3 answers
Classical geometric interpretation of spinors
A lot of notions in differential geometry have direct meaning in Physics. For example:
A Riemannian metric is a way to encode distances on a manifold and in Physics it is the gravitational field. The curvature of the Levi-Civita connection gives…
Benjamin
- 977