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159,068 questions
7
votes
1
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Why is 3 a bad constant in the Vitali covering lemma?
Hi,
recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
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3
votes
1
answer
636
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Non-existence of integral with respect to Poisson Random Measure
Let $\xi$ be a Poisson Random Measure of intensity $\mu$ (informally $\mathbb E\xi = \mu$).
(For $f \ge 0$, say) when does $\xi f = \infty?$ Kallenberg (Foundations of Modern Probabilility) claims ...
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8
votes
4
answers
5k
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Symmetric tensor products of irreducible representations
I wonder if there is a way to compute the symmetric tensor power of irreducible representations for classical Lie algebras: $\mathfrak{so}(n)$, $\mathfrak{sp}(n)$, $\mathfrak{sl}(n)$.
The question ...
- 10.7k
14
votes
2
answers
3k
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What do intermediate Jacobians do?
On a smooth complex projective variety of $\dim X=n$, we have $n$ complex tori associated to it via $J^k(X)=F^kH^{2k-1}(X,\mathbb{C})/H_k(X,\mathbb{Z})$ (assuming I've got all the indices right) ...
- 28.9k
11
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2
answers
3k
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adding an n-th root to Q_p
What can be said about extensions à la $\mathbb{Q}_p(\sqrt[n]{a})/\mathbb{Q}_p$? Ramification behaviour, valuation ring, ...?
I find it hard to say anything general - for example, as a function of ...
- 5,163
36
votes
3
answers
4k
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What is the right version of "partitions of unity implies vanishing sheaf cohomology"
There are several theorems I know of the form "Let $X$ be a locally ringed space obeying some condition like existence of partitions of unity. Let $E$ be a sheaf of $\mathcal{O}_X$ modules obeying ...
- 156k
7
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2
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Wants: Polynomial Time Algorithm for Decomposing a Multiset of Rationals into Two Additive Subsets.
First, allow me to say that this problem was posed to me by a professor in the department. It is related to his research in a way that I do not know. However, since I couldn't come up with anything ...
- 4,842
1
vote
2
answers
3k
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The sharp 3x3 lemma: a proof by universal properties?
I was reading this paper a while ago, and I couldn't figure out how to prove a lemma that was left as an exercise by only using universal properties and the definition of an abelian category.
I'll ...
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13
votes
4
answers
2k
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A comprehensive functor of points approach for manifolds
This seems unrealistic, because the topology on a manifold doesn't have anything to do with the properties its structure sheaf, but I figured I might as well ask. This wouldn't be the first time I ...
- 26.8k
8
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2
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819
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Are there interesting monoidal structures on representations of quantum affine algebras?
Is there a good monoidal structure on a category of integrable representations of a quantum affine algebra? In the ordinary affine Kac-Moody case, there is the usual tensor product (symmetric, adds ...
- 3,929
5
votes
1
answer
2k
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Finding unknown integer-valued polynomials using inequalities
I've come across this interesting inequalities problem recently, which seemed straight-forward at first glance but has proven interesting enough to ask about it here.
Suppose you are given the ...
- 61.9k
5
votes
1
answer
1k
views
Orders of field automorphisms of algebraic complex numbers
Let $Aut(\bar{Q})$ be the automorphism group on the field of algebraic complex numbers. The order of an element $f \in Aut(\bar{Q})$ is the least natural number $n$ (if there exists one) such that $(f)...
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10
votes
1
answer
635
views
Free action of SL_2(F_p) on a sphere
Let $p>2$ be prime. Then for abstract reasons the special linear group $\text{SL}_2({\mathbb F}_p)$ possesses a free action on some sphere (one has to check that any abelian subgroup of $\text{SL}...
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5
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0
answers
539
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An inverse eigenvalue problem on Jacobi matrices
I am interested in trying to design a Hermitian Jacobi (tridiagonal) matrix $H$ that has specific properties. The basic property, which is simple enough to construct, is that for an $N\times N$ matrix ...
- 25.8k
9
votes
1
answer
399
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Existence of hyperelliptic curve with specific number of points in a family
Hi,
the following question was posed to me, it apparently has applications for linear codes. Let n>1, and $K = \rm{GF}(2^n)$. Let $k$ be coprime to $2^n-1$. Does there always exist $a \neq 0$ in $K$ ...
- 1,411
13
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2
answers
648
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Functions separting points in Hausdorff spaces
A colleague in algebra asked me this, and I couldn't answer it. On the Wikipedia page for "epimorphism" it is claimed that in the category of Hausdorff spaces and continuous maps, a function is epi ...
- 27.7k
8
votes
4
answers
660
views
multi-index Dirichlet series
Hi,
I have recently got interested in multi-index (multi-dimensional) Dirichlet series, i.e. series of the form $F(s_1,...,s_k)=\sum_{(n_1,...,n_k)\in\mathbb{N}^k}\frac{a_{n_1,...,n_k}}{n_1^{s_1}......
- 7,442
0
votes
1
answer
980
views
Name of upper triangular/lower triangular Lie Algebra decomposition
What is the name of the Lie algebra decomposition where the positive root vectors are upper triangular and the negative root vectors are lower triangular?
- 3,929
18
votes
0
answers
517
views
Cohomological characterization of CM curves
In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...
- 797
1
vote
1
answer
569
views
Transition Functions and Complex Structure
For a complex manifold $M$, the transition functions of the tangent bundle $T(M)$ come from the Jacobian of the change-of-coordinate maps. Does there exist a related description of the transition ...
- 14.7k
5
votes
1
answer
2k
views
Polynomial with two repeated roots
I have a polynomial of degree 4 $f(t) \in \mathbb{C}[t]$, and I'd like to know when it has two repeated roots, in terms of its coefficients.
Phrased otherwise I'd like to find the equations of the ...
- 156k
6
votes
2
answers
385
views
Eigenvalues of an element in a Weyl algebra
I have an operator acting on the polynomial algebra $\mathbb{C}[x,y,z]$ that I would like to find the eigenvalues/eigenvectors of. More specifically, let $P(x_1, \ldots, x_6)$ be a homogeneous ...
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2
votes
4
answers
9k
views
$SU(2)$ and the three sphere [closed]
Can anyone give me an explicit isomorphism between $SU(2)$ and the three sphere?
What about for higher spheres? This question link text seems to indicate that there exists a homeomorphism from $SU(n)/...
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28
votes
5
answers
6k
views
Stokes theorem for manifolds with corners?
Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The Stokes theorem tells us that, for a $n+1$-dimensional manifold $M$ with boundary $\partial M$ and any ...
- 2,935
3
votes
4
answers
4k
views
How can I tell if y is a function of x in a random sample?
I have some data and believe that a given metric is a function of another metric. I have the values of both metrics and many different sets of these values. Can I tell if one is a function of the ...
- 24.1k
22
votes
5
answers
2k
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Homological algebra and calculus (as in Newton)
This question reminded me of a possibly stupid idea that I had a while back.
On page 2 of this paper, while discussing Euclid's axioms of plane geometry and spatial geometry, Manin makes an extremely ...
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1
vote
1
answer
1k
views
How can I calculate the characteristic function of these distributions? [previously: difficult integral]
How to compute this integral in general case?
$$t(x)=\int_{-\infty}^{\infty}\frac{\exp(ixy)}{1+y^{2q}}dy$$
Mathematica can compute it when q is known. For example,for q=1 this integral is
$$\exp(-{\...
- 267
8
votes
0
answers
348
views
A formula for moments of the limit distribution of singular values in the proof of the circular law
One of the steps in the proof of the circular law in random matrix theory is obtaining the limiting spectral distribution for the matrix
$(\frac{1}{\sqrt{n}} X_n - zI)(\frac{1}{\sqrt{n}} X_n - zI)^\...
- 25.8k
4
votes
3
answers
322
views
Candidate definitions for "1-braided 2-category"?
Recall that a braided monoidal category is a category $\mathcal C$, a functor $\otimes: \mathcal C \times \mathcal C \to \mathcal C$ satisfying some properties, and a natural isomorphism $b_{V,W}: V\...
- 28.1k
3
votes
1
answer
639
views
L^{p} multiplier sets
Let S be a set of integers and denote the characteristic function of S as $\chi_{S}(n)$. Define an operator on the space of trig functions by the relation $\hat{Tf}(n) = \chi_{S}(n) \hat{f}(n)$. Here $...
- 25.8k
4
votes
1
answer
173
views
factorization of the product of a matrix element and its cofactor
Hi,
this is kind of continuation of this thread to concentrate on a specific problem from linear algebra and analysis that, I think, is rather interesting for itself.
Here we go:
1) Main problem: ...
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2
votes
3
answers
885
views
Is the mapping from a scheme to its global sections a closed map?
This is a question posed to me in private communication by this user.
Given a scheme $T$, let $\Gamma (T) = Mor (T, \mathbb{A}^1)$ be the ring
of global sections. Note that there is a canonical map
$\...
CommunityBot
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1
vote
0
answers
742
views
Tensor products as isomorphic functors in category theory
An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my ...
CommunityBot
- 1
4
votes
1
answer
313
views
Origin of Fujimura set
If we have 10 coins arranged in an equilateral triangle and we want to know the minimum number of coins we can remove so that none of the remaining coins form an equilateral triangle the remaining ...
- 13.6k
3
votes
1
answer
410
views
direct limit of free complemented subgroups
Consider the following property of an abelian group $G$:
S: $G$ is torsionfree and a directed limit of finitely generated (hence free) subgroups $\{F_i\}_i$, such that for all $i \leq j$, $F_i$ is a ...
- 2,799
15
votes
2
answers
1k
views
a weird sequence with a non-integral term
Define a sequence $(a_n)_{n \geq 1}$ by $$na_n = 2 + \sum_{i = 1}^{n - 1} a_i^2.$$
(In particular, $a_1 = 2$.)
How can you show - preferably without using a pc! - that not all terms of the sequence ...
- 748
4
votes
1
answer
796
views
5
votes
1
answer
376
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Translation of "le nilradicalisé de g"
I apologize for asking something that might well be found in a mathematical dictionary, but the similarity of the French word to an English one is frustrating my attempts to Google the answer (and the ...
- 44.7k
4
votes
1
answer
342
views
When does a certain natural construction on monoidal categories yield a Hopf algebra?
Let $\mathcal C = (\mathcal C_0,\mathcal C_1)$ be a (small) strict monoidal category. Pick a field $\mathbb K$, and let $\mathbb K[\mathcal C_1]$ be the vector space with basis the morphism of $\...
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8
votes
1
answer
578
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Transmutation versus operads
A while ago, I was reading Majid's book Foundations of quantum group theory, and Section 9.4 has a rather fascinating description of a Tannaka-Krein reconstruction result for quantum groups. In ...
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11
votes
1
answer
946
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Is ΩΣ in {simplicial commutative monoids} group completion?
Let C be the model category of simplicial commutative monoids (with underlying weak equivalences and fibrations), or equivalently the (∞,1)-category PΣ(Top), where T is the Lawvere theory ...
CommunityBot
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5
votes
2
answers
725
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Binary codes with large distance
Suppose $C \subset \lbrace 0,1\rbrace^{n}$ is a binary code with distance $\delta * n$, for $1/2 < \delta < 1$. Can $|C|$ be arbitrarily large (if I allow n to be arbitrarily large)? Note that ...
0
votes
1
answer
2k
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Equalizer objects in Set.
An equalizer in a category $\mathcal{C}$ is a couple $(E,eq)$ consisting in an object $E$ and a morphism $eq:E\longrightarrow X$ so that $f\circ eq=g\circ eq$ for every pair of parallel morphisms $f,g:...
- 18.7k
3
votes
0
answers
189
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Which local homeos to numerical space are bijective?
I am reading T. Szamuely's book on Galois groups and fundamental groups.
As preparation to the algebraic case, he recalls the topological case.
So I am wondering if a surjective local homeomorphism $f$...
- 65.4k
8
votes
1
answer
1k
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Question concerning the arithmetic average of the Euler phi function:
Let $\varphi(n)$ denote Euler's phi-function. If we let
$$ \sum_{n\leq x} \varphi(n) = \frac{3}{\pi^2}x^2+R(x),$$
then it is not hard to show that $R(x)=O(x\log x)$. What is the best known bound for $...
- 4,485
2
votes
1
answer
316
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how does the basis of an inner product space change when the domain is deformed
Assume we have a complete orthogonal system on a domain $D$, given by the eigenfunctions of the Laplacian on $D$. For example, the set $\{e^{int}\}$ on $[-\pi, \pi]$, or the spherical harmonics on ...
- 25.8k
11
votes
3
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461
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citation for first statement of the Re(s) = 6 conjecture on zeros of Ramanujan L function
Hi, for the bibliography of a paper I'm writing I seek a citation for the first statement of the conjecture that the nontrivial zeros of $F(s) = \sum_n\tau(n)n^{-s}$ all lie on the line Re(s) = 6. (...
- 18.7k
3
votes
4
answers
1k
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Ternary relations that are not binary functions
By far the most prominent elementary relations that are not functions are binary and the most prominent elementary ternary relations are in fact binary functions.
"Elementary" shall mean "part of the ...
- 6,556
3
votes
3
answers
2k
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External tensor product of two (perverse) sheaves
Motivation: I was reading through Frenkel's article on geometric Langlands program, and the external tensor product of two perverse sheaves occurred in the definition of the geometric Langlands ...
- 2,216
-3
votes
1
answer
691
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Why is a partition function of a Topological Conformal Field Theory related to Deligne-Mumford space
I find when I read a paper, Costello" The Gromov-Witten potential associated to TCFT"
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