All Questions
23,892 questions
3
votes
0
answers
61
views
Is this bipartite equivalent of 1-walk-regular graphs known?
A graph $G$ is 1-walk-regular if
for each vertex $v$ the number of closed walks of length $\ell$ starting (and ending) at $v$ depends only on $\ell$ but not on $v$.
for each edge $vw$ the number of ...
- 13.6k
3
votes
1
answer
488
views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=2$ ...
CommunityBot
- 1
5
votes
1
answer
379
views
Nuclear vs Banach spaces: compactness properties
A question about the meaning from following excerpt from german wikipedia adressing interesting crucial feature of nuclear spaces opposing them from Banach spaces (transl.):
While normed spaces, ...
- 6,028
1
vote
0
answers
53
views
Moduli space of stable bundles with fixed Chern class is bounded
This is an excerpt on a set of lecture notes I'm going over:
Classically, a lot of interest in stable vector bundles is due to the fact that stability allows the study of moduli of vector bundles via ...
- 73
9
votes
2
answers
493
views
Reference Request for global Hölder continuity of solutions to elliptic PDEs
This is question that was also posted on MathStackExchange Link, where it was suggested I post this question on MathOverflow. Please note that the answer given in Link does not help as I do not have ...
1
vote
0
answers
52
views
How large can the normalizer of $\mathrm{Ad}(G)$ in $\mathrm{GL}(\mathfrak{g})$ be?
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Ad{Ad}$Let $G$ be a real Lie group with Lie algebra $\mathfrak g$ (say reductive/semisimple if it makes the question easier).
I am interested in ...
- 63.9k
2
votes
2
answers
243
views
A necessary and sufficient condition for three diagonals of a hexagon to be concurrent
When talking about the condition for the three diagonals of a hexagon to be concurrent, we will think of Brianchon's theorem. Using software, I discovered a necessary and sufficient trigonometric ...
- 76
7
votes
2
answers
851
views
Proof of Littman-Stampacchia-Weinberger theorem on the fundamental solution for elliptic PDEs
Where can I find a (readable and self-contained) proof of the following result?
Let $\Omega$ be a Lipschitz domain of $\mathbb{R}^n$, with $B(0,1) \subset \Omega$. Let $u$ be the solution of $$-\...
- 6,367
0
votes
0
answers
141
views
The tensor product of two Fredholm operators
What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
- 6,367
6
votes
1
answer
2k
views
Finite element method inverse estimate
$\DeclareMathOperator\diam{diam}$Looking for a proof in the literature of the following lemma:
Let $K\subset\mathbb{R}^d$ be a bounded domain. Let $P_X$ be a finite dimensional subspace of $\mathcal{...
CommunityBot
- 1
2
votes
0
answers
47
views
Norm density of evaluation functionals in the space of weak$^*$ continuous multilinear functionals on products of dual Banach spaces
Let $K$ be a compact (metrizable) space and let $C(K)$ be the Banach space of continuous real-valued functions on $K$, equipped with the supremum norm. It is then well known that the dual space $C(K)^*...
- 6,367
0
votes
0
answers
46
views
What's the problem in using spanning Bessel sequences that are not frames to decompose vectors?
This is related to a question I recently asked on math.SE.
Consider a subset $G\equiv \{g_k\}_{k\in\mathbb{N} }\subseteq\mathcal H$ in a separable Hilbert space $\mathcal H$, and suppose $G$ spans the ...
- 342
-1
votes
1
answer
61
views
Asking for some references on correlations of joint optimization problems
Here are two problems that I am trying to understand, and it would be nice if someone could provide references on whether there is some structure theorem for these problems that have been studied in ...
- 12.7k
0
votes
0
answers
67
views
Projection of a gaussian random vector onto a convex body
Let $K \subset \mathbb{R}^n$ denote a convex body. Let $\Pi_K$ denote the projection onto $K$,
$$
\Pi_K(y) = \mathrm{arg\,min}_{x \in K} \|y - x\|,
$$
where $\|\cdot\|$ denotes the usual Euclidean ...
- 420
17
votes
3
answers
2k
views
Optimal 8-vertex isoperimetric polyhedron?
I know from Marcel Berger's
Geometry Revealed:
A Jacob's Ladder to Modern Higher Geometry
(p.531)
that it is not yet established which polyhedron in $\mathbb{R}^3$ on 8 vertices achieves the optimal ...
- 2,370
8
votes
1
answer
428
views
Wishart matrices: are eigenvalues and eigenvectors independent?
Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries.
In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ ...
- 1,295
1
vote
0
answers
52
views
High order parabolic PDEs on manifolds: Reference request
I recently became interested in parabolic PDEs of order 4 on surfaces. However, I have a very little background in parabolic PDEs. I discovered Lunardi's book (Analytic semigroups and optimal ...
- 363
1
vote
0
answers
86
views
Gamma convergence via density argument: Looking for references
I am looking for a reference or result dealing with Gamma via density argument.
Let me elaborate more my wish. I am actually trying to establish the Gamma convergence (precisely only the liminf) of a ...
- 869
17
votes
4
answers
2k
views
Eulerian number identity
The Eulerian number $A(n,m)$ is defined as the number of permutations $\sigma \in S_n$ having precisely $m$ descents, i.e. indices $i$ such that $\sigma(i)>\sigma(i+1)$.
The wikipedia entry on ...
- 10.5k
1
vote
0
answers
40
views
relatively weakly compact sets in the projective tensor product of $\ell_p $ and a Banach space $X$
We will use the notation in [1].
A sequence $(x_n)$ in $X$ is called weakly $p$-summable ($p\ge 1$) if $(x^*(x_n))\in \ell_p$ for each $x^*\in X^*$. Equivalently, a sequence $(x_n)$ in $X$ is ...
4
votes
1
answer
180
views
Analytic function with values in $L^1$
Suppose that $(\Omega, \Sigma, \mu)$ is a measure space.
Let $D$ be the unit open disk and $F : D \rightarrow L^1(\mu)$ be an analytic function. Is it true that for a.e. $w \in \Omega$ the function $F(...
- 54.2k
0
votes
1
answer
114
views
Geometric interpretation of a Grammian-like function
Let $\mathbf{v}, \mathbf{w} \in \mathbb{R}^n$ and consider the following function $f : \mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$:
$$
f(\mathbf{v},\mathbf{w}) = \|\mathbf{v}\|\|\mathbf{w}...
- 188k
0
votes
0
answers
50
views
About extreme case on complex interpolation
I'm trying to prove some equality of spaces via complex interpolation with the usual Calderon functor $[,]_\theta$. If $(E_0,E_1)$ is a compatible couple, it is known that $$[E_0,E_1]_j, j=0,1,$$ is a ...
11
votes
2
answers
1k
views
A generalization of the law of tangents
The law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides.
Let $a$, $b$, and $c$ be the lengths of the three ...
- 188k
2
votes
1
answer
79
views
What is weak convergence of random permutons?
In various papers on permutons you can find statements similar to this (see Maazoun's thesis)
For any $n$ let $\sigma_n$ be a random permutation of size $n$. TFAE:
$(\mu_{\sigma_n})_n$ converges in ...
- 677
2
votes
0
answers
83
views
3/2 Sobolev Norm on the boundary of a bounded open subset of $\Bbb R^n$
Let $\Omega\subset\mathbb{R}^{n}$ be a open bounded set and $\partial\Omega$ be the boundary of $\Omega$.
Following the reference text by Alois Kufner, Oldřich John and Svatopluk Fučík, Function ...
- 6,367
7
votes
1
answer
360
views
Strictification for closed monoidal categories
The strictification theorem for monoidal categories states that every monoidal categorically is monoidally equivalent to a strict monoidal category. Is there a strictification theorem for closed ...
- 10.7k
16
votes
2
answers
1k
views
CH in non-set theoretic foundations
I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681
I ...
- 35.5k
3
votes
1
answer
67
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb R$
This question is based on a very similar question posted by me yesterday. A very nice solution was provided by Aleksei Kulikov. Here I modify my question slightly.
Let $Q_t$ be the heat semigroup on $...
- 24.2k
12
votes
1
answer
735
views
Parametrisations for null temperature functions: nonuniqueness of solutions to the heat equation
Disclaimer. I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition....
- 6,367
0
votes
0
answers
16
views
Representing a periodic strip operator as a tensor product of operators
I hope this question is not trivial, but here goes. I want to consider a bounded operator on $\mathcal{H}=\ell^2(\mathbb{Z}\times \{0,...,N-1\})$ that is a discrete Schrodinger like operator.
...
- 633
1
vote
1
answer
128
views
Infinite direct sum decomposition of the heat semigroup on $\mathbb{R}^n$
Consider the heat semigroup $Q_t$ on $L^2(\mathbb{R}^n)$ generated by the Laplace operator $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x^2_i}$. Does there exist a direct sum decomposition
$$\oplus_{...
- 6,476
1
vote
1
answer
298
views
maximal sets of vertices that avoids a clique
I am looking for some known algorithm that finds, for a given graph, all the maximal sets of vertices that avoid a clique of some given size $k$. I'd prefer one written in MATLAB, but other languages ...
CommunityBot
- 1
3
votes
1
answer
159
views
Reference request: generalized Jacobian variety for higher dimensional variety
Let $X\subset \mathbf{P}^n$ be a hypersurface such that the singular locus of $X$ consists of a single ordinary double point. I'm trying to find a reference to the "generalized" intermediate ...
- 1,404
2
votes
0
answers
331
views
What is the spectrum of this differential operator?
My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
- 206
3
votes
3
answers
550
views
Looking for a very particular kind of non-convex functions
I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously,
It should be at least twice differentiable.
It should have a ...
CommunityBot
- 1
0
votes
1
answer
117
views
Validity of approximation method for von Mangoldt function
I'm working on a problem involving the pointwise almost everywhere convergence of multilinear ergodic averages with the von Mangoldt function inspired by this paper. Specifically, I'm looking at ...
CommunityBot
- 1
1
vote
1
answer
215
views
Compactness with respect to topology induced by total-variation distance
I've been working on a problem and at some point in the proof I need to show that the following set $$\left\{\mu \in \mathcal{P}_{ac}(\mathbb R^d): \int \varphi(x)\mu(\mathrm{d}x)\leq C\right\}$$
is ...
- 192
3
votes
1
answer
6k
views
About eigen-functions of the Gaussian kernel
If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
- 188k
10
votes
3
answers
739
views
Is there a version of Fischer-Riesz theorem for Banach space?
$( \Omega,F, P )$: a measurable space equipped with a finite measure
$(B , \Vert \cdot \Vert) $ : a Banach space with $\mathcal{B}$ as its borelian $\sigma$-algebra
$p$ : a constant bigger than $1$
...
- 41.1k
3
votes
1
answer
136
views
Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
- 75
2
votes
1
answer
196
views
Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
CommunityBot
- 1
3
votes
1
answer
79
views
Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$
Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
- 407
0
votes
0
answers
54
views
Number of indecomposable modules over representation-finite hereditary algebras
Let $A$ be a finite dimensional $K$-algebra over a field $K$ that is hereditary and of finite representation type.
It is well known that they are classified by Dynkin diagrams.
For algebraically ...
- 26.5k
1
vote
0
answers
104
views
Reference about the semiabelian variety associated to a stable curve
If a stable curve $C$ of genus $g$ is such that its dual graph has Betti number $1$, then the Jacobian of $C$ is a semiabelian variety of torus rank $1$. It is described by Mumford that the data of a ...
8
votes
1
answer
849
views
Representations of groups with the same derived group, how much control do we have over the central character?
Let $G_1 \subset G$ be the rational points of $p$-adic reductive groups sharing the same derived group. There are some well known results relating representations of $G_1$ to representations of $G$, ...
CommunityBot
- 1
1
vote
1
answer
122
views
distance in the matrix algebra w.r.t. the nuclear norm
Let $\varphi\in\mathcal{M}_n(\mathbb{C})$ and let $Z:=\mathbb{C}\cdot I=\{zI\colon\,z\in\mathbb{C}\}$ be the one-dimensional subspace spanned by the identity matrix $I$. Let moreover $\|\cdot\|_{\...
- 1,517
11
votes
4
answers
950
views
Is there a name for finite unions of intervals?
Finite unions of intervals are simple sets that are used quite often, e.g. in measure theory. (The construction of the Cantor set is a noble example). I realised that I do not have a name for them. Is ...
- 6,367
11
votes
1
answer
340
views
Number of odd-dimensional irreducible representations of $S_n$
In this paper the structure of odd-dimensional irreducible representations of the symmetric group is described, but what is the asymptotic behaviour of the number of such representations? (Or, if it ...
- 12.9k
2
votes
1
answer
204
views
A continuous analogue of the notion of Hilbert basis
Let $X$ be a locally compact space, let $H$ be a Hilbert space and let $\beta:X\to H$ be a continuous function such that the linear subspace of $H$ spanned by $\beta(X)$ is dense in $H$. I would like ...
- 12.6k