Newest Questions
159,025 questions
3
votes
1
answer
161
views
Approximating continuous functions from $K\times L$ into $[0,1]$
Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
- 5,529
0
votes
1
answer
159
views
Can any Clifford module bundle be extended to a Dirac bundle?
I assume that the question in the title is clear, so let me talk about its relevance:
According to theorem 4.3 in Heat Kernels and Dirac Operators the index theorem
\begin{equation}\tag{1}
\mathrm{ind}...
- 339
4
votes
0
answers
313
views
What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
- 571
3
votes
1
answer
221
views
Asymptotics for number of $p$-regular partitions of $n$
The number of simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\bot}$ of the symmetric group over a field $k$ such that $\text{char}(k)=p > 0$ is the number of $p$-regular ...
- 571
4
votes
0
answers
291
views
Has anyone studied the derived category of Higgs sheaves?
Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-...
- 9,019
-1
votes
1
answer
132
views
What is an "open Baire set"?
In Measures Which Agree on Balls by Hoffmann-Jørgensen, it is stated that if $\varphi$ is a Baire function (which I presume means a pointwise limit of continuous functions), then $\{a<\varphi\}$ is ...
- 297
5
votes
2
answers
342
views
Projections in atomless von Neumann algebras
Let $\varepsilon>0$.
If we consider a sequence $\{f_n\}$ in $L_\infty(0,1)$, then there exists a very small subset $A$ of $(0,1)$ with $m(A)<\varepsilon$ such that $$\|f_n \chi_A\|_\infty =\|...
- 617
0
votes
0
answers
92
views
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
What is the highest $n\in\Bbb N$ for which a complete classification of inverse semigroups of order up to $n$ is known?
Given that there are $3{,}684{,}030{,}417$ semigroups of order $8$, I guess $n\...
- 379
10
votes
1
answer
400
views
Rigorous treatment of Ostrogradsky's instability theorem?
The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
- 1,904
3
votes
1
answer
206
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Relation between enveloping algebras and algebras of differential operators
I asked this question on math stack exchange about 3 years ago, but received no answer.
Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
- 3,318
8
votes
1
answer
390
views
Order bounded version of monotone complete $C^*$-algebras
Let $A$ be a $C^*$-algebra with self-adjoint part $A_{\operatorname{sa}}$. Then $A$ is called monotone complete if every increasing norm bounded net in $A_{\operatorname{sa}}$ has a supremum (with ...
- 12.6k
2
votes
0
answers
136
views
Absolute Galois cohomology of function fields (of high-dimensional) varieties
What is known about the absolute Galois cohomology of function fields of varieties of dimension 2 or larger? Specifically, I am interested in multiplicative coefficients $\mathbb G_m$.
I have seen ...
- 556
2
votes
0
answers
74
views
Relation of top and bottom types given multiple universes
This is something that I haven't seen mentioned in any literature.
In a type theory (extensional, intuitionistic, Martin Lof variant), given two ordered Tarski Universes such that $U_i < U_{i+1}$, ...
- 21
4
votes
1
answer
668
views
$f\in C(B_1)\cap W^{1,2}(B_1\setminus \{f=0\})$ implies $f\in W^{1,2}(B_1)$?
In a paper I am writing I need to show that a certain real-valued function $f\in C^0(B_1)$ belongs to the Sobolev space $W^{1,2}(B_1)$ ($B_1$ is the unit ball). So far I have been able to show that ...
- 1,149
8
votes
1
answer
535
views
Representation theory of $\mathrm{GL}_n(\mathbb{Z})$
I want to understand the (complex) representation theory of $\mathrm{GL}_n(\mathbb{Z})$, the general linear group of the integers. I have gone through several representation theory texts but all of ...
- 81
0
votes
0
answers
466
views
Spiegel Vermutung: no Siegel zeros iff GRH is equivalent to Goldbach's conjecture
I apologize for using German language in the title, but this question came to my mind after watching the French movie "le théorème de Marguerite" in which the protagonist gets an insight ...
- 6,984
2
votes
2
answers
89
views
Reference for article that introduces and motivates different notions of subdifferentials
I saw a tutorial/expository journal article a while ago that focused on introducing intuitively different notions of subdifferentials appropriate for general nonlinear optimization. I forgot the ...
- 922
1
vote
0
answers
139
views
On counter-examples to Noether's Problem
Noether's Problem was introduced by Emmy Noether in [4]:
Let $\mathsf{k}$ be a field and $K=\mathsf{k}(x_1,\ldots,x_n)$ be a purely transcendental extension. Let $G<S_n$ be a group acting by ...
- 3,318
5
votes
1
answer
283
views
Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely ...
- 6,962
0
votes
1
answer
54
views
How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
- 825
0
votes
0
answers
108
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A surprising result with the Riccati difference equation
I was looking at the Riccati difference equation with positive and negative indices
$$
R_n=\frac{aR_{n-1}+b}{cR_{n-1}+d}\quad n\in[0,N]\\
R_n=\frac{-dR_{n+1}+b}{cR_{n+1}-a}\quad n\in[-N,0]\\
$$
along ...
- 119
5
votes
1
answer
327
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Generalizations of Chevalley–Shephard–Todd's Theorem?
Major Edit
I will reformulate my question signicantly, given Anton Geraschenko's comment. The old version of the question is bellow.
For simplicity, my base field is $\mathbb{C}$. If $G<\...
- 3,318
3
votes
0
answers
96
views
Examples of tensor-triangulated categories not satisfying the local-to-global principle
From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of ...
- 767
-1
votes
1
answer
148
views
Weighted sum of zero-mean random variables
Let us say we have two independent random variables $X$ and $Y$, with both $E[X] = E[Y] = 0$.
Is it true that for any random weight variable $0 \le W \le 1$ (e.g., $W$ dependent on $X$ and $Y$) we ...
- 367
2
votes
0
answers
181
views
How to prove this weighted sum inequality with non-increasing sequences?
Problem
I have two non-increasing sequences, $X = (x_1, x_3, x_5, \ldots, x_{n-1})$ and $Y = (y_1, y_3, y_5, \ldots, y_{n -1})$, $n$ is an even integer. I want to prove this inequality:
$$
\sum_{i=1}^{...
- 29
3
votes
1
answer
279
views
Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
- 3,318
2
votes
0
answers
250
views
Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
- 319
3
votes
1
answer
188
views
Completeness of exponentials with frequencies in a periodic set
If $f \in L^2[0,1]$ then it follows from the uniqueness-theorem for Fourier series that if
$$
\int_0^1 f(x)e^{-2\pi i x n} dx = 0, \quad n \in \mathbb Z
$$
then $f=0$ almost everywhere. Now let $F \...
- 161
1
vote
1
answer
75
views
Constructing set-truncations of types from universes
This is a follow-up question from my previous question titled Constructing coproduct types and boolean types from universes, where I showed how every basic operation in set theory/topos theory could ...
- 2,624
1
vote
0
answers
125
views
Interpolating sequences are strongly separated
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
- 151
3
votes
1
answer
213
views
What can we say about the reciprocal of a reduced regular continued fraction?
For positive integers $a>b>0$, we can represent $a/b$ uniquely as $$\frac{a}{b}=a_1-\cfrac{1}{a_2-\cfrac{1}{\cdots-\cfrac{1}{a_n}}}=:[a_1,\dots,a_n]^{-}$$ with $a_i\geq 2$, and this is called ...
- 213
6
votes
1
answer
372
views
Maximizing a sum minus its maximal summand
This is a followup to a question that appeared on m.SE:
Maximize $\displaystyle f(\pi)=\left(\sum_{i=1}^{n}{i\pi_i}\right)-\max_{1\le i\le n}{(i\pi_i)}$ over permutations $\pi\in S_n$.
The problem ...
- 1,190
2
votes
1
answer
138
views
Explicit upper and lower bounds for a certain support function
Let $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ be a sequence of nonincreasing nonnegative real numbers. Define the set, for $t > 1$,
$$
B_t = \Big\{b \in \mathbb{R}^n : b_i \geq 0, \sum_i b_i^2 \...
- 380
8
votes
2
answers
489
views
Amalgamated product acting on CAT(0) cube complex
I was reading the following result from the book Metric spaces of non-positive curvature by Bridson and Haefliger.
Result:
Let $F_0,F_1$ and $H$ be groups acting properly
by isometries on complete $...
- 349
6
votes
1
answer
320
views
Convergence of derived series
There are quite a few simple results about convergent/divergent series derived from similar ones. So, here is something that I came across recently:
Let $\sum_{n=1}^\infty a_n$ consist of positive ...
- 689
2
votes
0
answers
135
views
Minimum cost k-edge connected subgraph
The problem of finding a k-edge connected spanning subgraph with the minimum number of edges is $ \mathcal{NP} $-hard in general. Is it the case for positive weighted graphs with "fractional ...
- 21
5
votes
2
answers
430
views
Cohomology of the adjoint representation of $\mathrm{SL}_2(k)$
Let $k$ be a finite field. Do we always have $H^1(\operatorname{PSL}_2(k), k^3) = 0$, where $\operatorname{PSL}_2(k)$ acts on $k^3$ via the adjoint representation (= conjugation action on trace zero ...
- 37k
6
votes
0
answers
137
views
Is the minus class group isomorphic to the relative class group?
I think this is something I should have known, but if I ever did I forgot about it. Consider the field $L$ of $p$-th roots of unity ($p$ prime) and its maximal real subfield $L^+$. The transfer of ...
- 32.6k
2
votes
0
answers
113
views
What is known about warped product metrics satisfying conditions more general than conformal flatness?
In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...
- 659
2
votes
1
answer
131
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Estimating $\max_{\|u\|=1} \frac{ E\left[\langle u\cdot x\rangle ^4\right]}{E\left[\langle u\cdot x\rangle ^2\right]}$
Suppose we have access to samples of $x$ distributed according to unknown multivariate Gaussian in $\mathbb{R}^d$. Estimate the following quantity:
$$\max_{\|u\|=1} \frac{ E\left[\langle u\cdot x\...
- 2,252
0
votes
1
answer
270
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Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
user516435
2
votes
1
answer
297
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Projective objects in chain complexes of an abelian category: Further question
Yes, I see there are other Q&A's on this, for instance here: Projective objects in the category of chain complexes
I am wondering why a level-wise projective chain complex $P$ which is split ...
- 511
2
votes
0
answers
90
views
Subrepresentation of a representation of $\text{Sp}(2n,\mathbb{R})$ spanned by specific elements
$\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Sym{Sym}$Let $V = \mathbb{R}^{2n}$ be the standard representation of the symplectic group $\Sp(2n,\mathbb{R})$, and let $\{a_1,b_1,\dotsc,a_n,b_n\}$ be ...
- 21
1
vote
0
answers
210
views
Is this a well known space? Perhaps homogeneous Sobolev-like space?
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
- 197
0
votes
0
answers
98
views
Tangent spaces of Lipschitz sub manifolds
Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
- 403
3
votes
0
answers
172
views
Abelian characters and odd perfect numbers?
This question is about applications of abelian characters to odd perfect numbers:
Context and Definitions:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring ...
- 3,074
0
votes
0
answers
78
views
Let $A, B$ be matrices with elements in $\mathbb{Z}_n$, does $\ker A = \ker B$ imply that they are row equivalent?
Let $A, B$ be matrices with elements in $\mathbb{Z}_n$. If $A x = 0$ and $B x = 0$ have the same set of solutions, where the vectors also have elements in $\mathbb{Z}_n$, does this mean that there is ...
- 219
2
votes
2
answers
414
views
Upper bound on number of integral solutions of elliptic curves
I was studying M. Bhargava Et al's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
And came across a very fascinating ...
- 61
1
vote
1
answer
126
views
Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?
Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that
$$
\int_0^1 |y-x| f(x) \, dx = 0
$$
for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
- 136
2
votes
1
answer
107
views
Finite dimensional irreducible representations of $\frak{sl}_m$ with non-trivial zero weight spaces
For the special linear algebra $\frak{sl}_{m}$ which finite dimensional irreducible representations $V_{\mu}$ have non-trivial zero weight spaces?
For $\frak{sl}_2$ this is clear: $V_{2k\pi}$ for $\pi$...
- 157