Most active questions
690 questions from the last 30 days
36
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6
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Gromov's thesis: Any property holding for all finitely generated groups must hold for trivial reasons
There is a famous aphorism attributed to Gromov (sometimes referred to as Gromov's thesis) stating roughly that any property P that holds for all finitely generated groups must hold for trivial ...
31
votes
4
answers
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Why is it so difficult to define constructive cardinality?
Consider Frege's cardinality and HoTT set-truncation cardinality, both of which can be well-defined in constructive theory (as SetoidTT and CubicalTT, respectively). Why don’t we regard them as well ...
- 1,217
41
votes
5
answers
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How does the mathematical community handle minor, non-critical errors in published papers?
I’ve noticed that even peer-reviewed mathematical articles sometimes have minor errors, like small typos or slight logical gaps, which don’t affect the main results. I’m curious about how these kinds ...
28
votes
6
answers
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Have you seen my power series?
I have a formal power series $\Phi(q,t)$ that satisfies the following functional equation:
$$\Phi(q,t)\cdot \Phi(q^{-1},-t) = 1.$$
Is there a nice known family of functions that satisfies identities ...
- 2,827
15
votes
3
answers
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What is the group of homotopy classes from the 4-torus to the 3-sphere?
Since the 3-sphere S3 is a Lie group, the homotopy classes [X, S3] of a path-connected space X into S3 naturally form a group.
What is this group [T4, S3] if X is the 4-torus T4 ?
(Since T4 has a ...
- 3,390
12
votes
6
answers
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Sum and sum of cubes are both perfect cubes
I am interested in integer tuples $(a_1, a_2, \dots, a_n)$ with $n > 3$, satisfying the following conditions:
Each $a_i \in \mathbb{Z} \setminus \{0\}$;
The $a_i$ are pairwise distinct and no two ...
- 133
18
votes
2
answers
1k
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What is this modified arithmetico-geometric mean function?
I acquired a vintage programmable calculator and thought I'd give it a spin by computing some interesting transcendental function. I wanted to compute the arithmetico-geometric mean but the calculator ...
- 8,603
14
votes
3
answers
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Where did this theorem appear?
In a 1934 paper of Erdős and Turán , whose title is On a problem in the elementary theory of numbers, they said,
… Their proof depends on a theorem of Mr. Pólya asserting that if we denote by $q_1<...
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19
votes
1
answer
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Reference request of quote about Gromov
Recently I saw a post on LinkedIn in occasion of Gromov's birthday, where a quote probably due to Milnor was mentioned: "Half of Riemannian Geometry is known to mankind, the other half is only ...
- 760
31
votes
1
answer
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Is there an “opposite” hypothesis to the (Generalized) Continuum Hypothesis?
There are many questions on this site about the (Generalized) Continuum Hypothesis, its philosophical or epistemological justifications, and various attempts at “solving” it. Because one such ...
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9
votes
2
answers
447
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When does the vanishing of $\pi_2$ imply asphericity of complex manifolds?
Let $X$ be a closed (compact without boundary) complex manifold. Assume that $\pi_2(X)=0$. Can we conclude that $X$ is aspherical? Does the vanishing of the second homotopy group (or the second ...
- 163
8
votes
3
answers
606
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Asymptotics of alternating sum of squared binomials by contour integration
I'm interested in finding the asymptotics of the following alternating sum as $n \to \infty$:
$$
\sum_{k=0}^{n} (-1)^k \binom{n}{k}^2.
$$
Of course, one can easily evaluate the sum exactly by ...
- 2,583
14
votes
1
answer
1k
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Diagonalizing Pascal's triangle
Let $D_n$ be the $n \times n$ diagonal matrix with entries $1, 2, \dots, n$.
Let $P_n$ be the $n \times n$ upper triangular matrix whose entry $a_{i,i+j}$ is given by $\binom{i+j}{i-1}$. For instance, ...
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5
votes
3
answers
615
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Why isn’t the weak-* topology normed?
I know this is a basic question, but I have been struggling to find an answer!
The weak-* topology (say on $M(X)$, where $X$ is a compact separable metric space) is induced by the metric $$d(\mu,\nu)=\...
- 560
10
votes
2
answers
646
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Is there a simple convex 3D polytope with an odd number of facets, all of which have an even number of sides?
In a simple convex 3D-polytope, every vertex is incident to exactly 3 facets. Suppose it has $F=n+2$ many facets. Then it is easy to see that it has $V=2n$ vertices and $E=3n$ edges, and fulfills the ...
