All Questions
15,320 questions
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(...
- 565
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
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 ...
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 ...
- 1,101
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 ...
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}...
- 6,351
0
votes
1
answer
140
views
Finding positive vectors of a special LGS
Let the following $4 \times 4$ LGS be given for which all coefficients
$a_1, a_2, a_3, a_{11}, a_{12}, ..., a_{33}$ are $>0$:
$a_1 + a_{11} \; x_1 + a_{12} \; x_2 + a_{13} \; x_3 + 0 \; x_4 = (a_{...
1
vote
0
answers
174
views
Interpolation of Sobolev spaces with constraints
Let us consider a real interval $[0, L]$, with $a\in (0, L)$, and let $I_1=(0, a)$ and $I_2=(a, L)$. We denote by $H^k(I_1)$ and $H^k(I_2)$ the usual Sobolev spaces, defined for $k\in \mathbb{N}$. Now,...
- 81
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 $...
- 407
4
votes
4
answers
2k
views
I want a smooth orthogonalization process
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
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
3
votes
2
answers
303
views
Asymptotics of A000613
The general linear group $GL_n(\mathbb{F}_2)$ acts on the powerset $2^{{\mathbb{F}_2}^n \setminus \{0\}}$ by multiplication: $A \cdot S := \{Ax \in {\mathbb{F}_2}^n : \, x \in S\}$, for an invertible ...
- 331
9
votes
3
answers
696
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
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_{...
- 407
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 ...
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 ...
- 21
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 ...
- 151
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
2
votes
1
answer
158
views
The relationship between a matrix and its coefficient matrix decomposed in Pauli matrix
For a dimension-$4$ Hermitian matrix $A$, denote pauli matrices $\{I,X,Y,Z\}$ as $\{\sigma_0,\sigma_1,\sigma_2,\sigma_3\}$ respectively. The pauli matrices form a basis of the matrix space if we take ...
- 91
0
votes
0
answers
121
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 47
3
votes
1
answer
153
views
Number of points covered by $2n$ hyperplanes in $\mathbf{F}_p^n$
For a prime $p$, fix two bases $U=\{v_1,\dots,v_n\}$ and $W=\{w_1,\dots,w_n\}$ of the vector space $V=\mathbf{F}_p^n$. We may assume $U$ is the standard basis without loss of generality.
For $s_1,\...
- 281
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\|_{\...
- 375
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 ...
- 407
8
votes
0
answers
115
views
optimal regularity for the Neumann heat equation on Lipschitz domains
$\newcommand{\R}{\mathbb R}$Let $\Omega\subset\R^d$ be a bounded Lipschitz domain, possibly non-convex (but not too nasty either, whatever that means). I am looking for well-posedness and optimal ...
- 5,380
1
vote
0
answers
72
views
How to understand "sparse graph limits"
For an $n$-vertex graph $G$, we say it is a sparse graph if $e(G)=o(n^2)$. Otherwise if $e(G)=\theta (n^2)$, we say it is a dense graph.
For a sequence of dense graphs $G_1,G_2,\dots,$ we know that it ...
- 349
0
votes
0
answers
96
views
Sufficient condition for weak convergence in Banach spaces
The question is quite elementary but nonetheless no proof or counter example comes to mind immediately.
Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ ...
1
vote
0
answers
55
views
Characterizing one-sided M-projections on real C*-algebras
Let $A$ be a real C*-algebra, and let $P: A \to A$ be a bounded linear projection. We say that $P$ is a left M-projection if the map
$$
v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
- 11
21
votes
0
answers
520
views
Is the exponent of $2$ in the Pythagorean theorem the "same $2$" as $[\mathbb{C} : \mathbb{R}]$?
I posted this question in Math StackExchange a couple years ago; due to the recent surge in interest, and following the feedback of several users, I've decided to cross-post it here. I apologize for ...
- 1,206
2
votes
1
answer
75
views
How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$
Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$.
Assume ...
- 101
0
votes
1
answer
127
views
Under what conditions does $x^TA^{-1}y> 0$ hold? $A$ is a symmetric positive definite matrix,$A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$
This is a tricky problem I encountered in my research. $A\in \mathbb{R}^{n\times n}_+, x,y\in \mathbb{R}^{n}_+$, i.e. $\forall 1\leq i \leq n, 1 \leq j\leq n, A(i, j)>0, x(i), y(i)>0$.
As known, ...
- 27
2
votes
1
answer
172
views
Diagonalize almost symmetric tridiagonal matrix
I begin with an $n \times n$ real symmetric tridiagonal matrix. However, I replace the non-zero elements in the first and last rows with zeros, so it is no longer symmetric
$$M = \begin{bmatrix} 0 &...
- 151
0
votes
0
answers
100
views
Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube
A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if
\begin{equation*}
\dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K,
\end{equation*}
for any $x,y\in \...
- 69
2
votes
1
answer
121
views
Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$
For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite,
$$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$
Using the triangle inequality $|\eta-\xi|\le |\eta|...
- 860
0
votes
0
answers
68
views
Inequality between product of companion matrices and power of Pisot number
Let $d\geqslant 2$ be an integer and consider a convergent sequence of "companion" matrices
$$A_k := \begin{pmatrix}
a_{k,1} & a_{k,2} & \cdots & a_{k,d} \\\
& ...
- 101
2
votes
0
answers
40
views
Characterization of critical point of an integral operator
I have an integral operator and I wonder how I can characterize the critical point.
I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question.
...
- 115
0
votes
0
answers
42
views
Geometric alignment of adaptive models on evolving manifolds
Let $(M_t)_{t\in[0,T]}$ be a smooth family of compact $d$-dimensional Riemannian submanifolds of $\mathbb{R}^n$. Consider a function $f_t : \mathbb{R}^n \to \mathbb{R}$ evolving over time $t \in [0,T]$...
0
votes
0
answers
94
views
Infinite sequence of PSD non-moments in two variables
Define a 2d sequence to be a mapping $a: \mathbb{N}^2 \to \mathbb{R}$ (where $\mathbb{N} = \{0, 1, \dots\}$). Here are two definitions of types of 2d sequences:
We say that a 2d sequence $a$ is a ...
- 161
4
votes
0
answers
148
views
Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
- 1,879
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
3
votes
0
answers
108
views
A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$
Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
- 1,171
4
votes
1
answer
132
views
Direct characterization of finite-dimensional $1$-injective Banach spaces
It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk ...
- 2,793
7
votes
0
answers
269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 109
3
votes
1
answer
219
views
Moment problem, ergodicity and spectral gap on the space of tempered distributions
Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$.
Just below formula [5] in p.122 of the Fröhlich paper, ...
- 3,477
2
votes
1
answer
141
views
(Sub)Optimality of random transport
Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
- 25
7
votes
0
answers
249
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
0
votes
0
answers
20
views
Decomposition of measures orthogonal to the algebra $R(K_1 \times \ldots \times K_n)$ - Can it be done via projection-preserving products of bands?
See "Measures orthogonal to tensor products of function algebras" by Marek Kosiek. Here, it is described for the two-dimensional case. It uses another, more general, approach to OB Bekken's ...
- 63
3
votes
1
answer
287
views
Expectation comparison inequality for concave function of symmetric random variables
Suppose that $X_i$, $i\in[n]$ are
independent symmetric
random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
- 6,541
3
votes
0
answers
117
views
Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?
Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
0
votes
0
answers
112
views
Vector field connecting two points
I'm now working on somehow an inverse problem of an ODE:
Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).
Now there is a ...
- 1
4
votes
1
answer
103
views
When do the nonzero eigenvalues of a directed graph Laplacian have the same absolute value?
Question: Let $G$ be a strongly connected directed graph on $n$ vertices with Laplacian $L(G)$. Then $L(G)$ has one zero eigenvalue $\lambda_1=0$ and $n-1$ nonzero eigenvalues $\lambda_2,\ldots,\...
- 145