All Questions
12,776 questions
10
votes
1
answer
443
views
Analytic continuation gives a covering space (and not just a local homeomorphism)
Let $\mathcal{G}$ be the space of germs of holomorphic functions defined on open subsets of $\mathbb{C}$, topologized in the usual way. There is a natural map $p\colon \mathcal{G} \rightarrow \mathbb{...
- 111
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
$$
v_P: A \to C_2(A), \quad x \mapsto \begin{pmatrix} P(x) \\ x - P(...
- 11
2
votes
1
answer
75
views
How to show $\lVert\Delta u_n- \Delta u\rVert_{L^2(0,T; \,H^2(\Omega))} \to 0$ ? $(\Omega \subset \mathbb{R}^2)$
Let $u_n, \nabla u_n, \Delta u_n, \nabla \Delta u_n, \Delta^2 u_n$ be uniformly bounded in $L^2((0,T) \times \Omega)$ where $\Omega \subset \mathbb{R}^2, u=\Delta u =0$ on $\partial \Omega$.
Assume ...
- 101
0
votes
0
answers
100
views
Construct a bi-Lipschitz mapping that maps a cube to a ball which has the same center with the cube
A mapping $f: \mathbb{R}^n\to \mathbb{R}^n$ is said to be $K$-bi-Lipschitz, $K>1$, if
\begin{equation*}
\dfrac{1}{K}\leqslant \dfrac{|f(x)-f(y)|}{|x-y|}\leqslant K,
\end{equation*}
for any $x,y\in \...
- 69
4
votes
0
answers
322
views
Monstrous moonshine, Dedekind eta function, and the hypergeometric function
I. Monstrous Moonshine
Let $q = e^{2\pi i\tau}$ and $\tau = \sqrt{-d}$ or $\tau = \frac{1+\sqrt{-d}}2$ for positive integer $d$. Given the Dedekind eta function $\eta(\tau)$, consider the known ...
- 12.6k
0
votes
0
answers
48
views
When inclusion between two Kobayshi hyperbolic manifolds is distance decreasing?
Suppose that $X$ and $Y$ are two Kobayshi hyperbolic complex-analytic manifolds such that $X \subset Y$. It is known $d_Y(x_1, x_2) \leq d_X(x_1, x_2)$ for all $x_1, x_2 \in X$. In other words, the ...
- 41
2
votes
1
answer
121
views
Constructing a function $u$ such that $\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta<\infty$, but $u\notin H^{1/2}$
For $u\in \mathcal{S}'(\mathbb{R})$, define, if finite,
$$\Lambda(u)^4=\int_{\mathbb{R}^2}|\eta-\xi||\hat{u}(\eta)|^2|\hat{u}(\xi)|^2\,d\xi\,d\eta.$$
Using the triangle inequality $|\eta-\xi|\le |\eta|...
- 860
2
votes
0
answers
40
views
Characterization of critical point of an integral operator
I have an integral operator and I wonder how I can characterize the critical point.
I'll give a simplified example so maybe people can comment on and I can maybe generalize in another question.
...
- 115
0
votes
0
answers
42
views
Geometric alignment of adaptive models on evolving manifolds
Let $(M_t)_{t\in[0,T]}$ be a smooth family of compact $d$-dimensional Riemannian submanifolds of $\mathbb{R}^n$. Consider a function $f_t : \mathbb{R}^n \to \mathbb{R}$ evolving over time $t \in [0,T]$...
4
votes
0
answers
148
views
Some questions on Hardy's spaces
In the paper http://www.numdam.org/item/CM_1976__33_3_261_0.pdf, the authors have asked in Page 285 whether the Hardy space $H^p$ embeds isometrically into the Hardy space $H^q$ for $1\leq q<p<...
- 1,879
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$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
$$
This operator ...
- 109
3
votes
1
answer
228
views
Minimum of a subharmonic function
Let for $j=1,\dots, m$, $z_j$ be distinct points from the unit disk $|z|<1$ and let $$g(z)=-\sum_{k=1}^m \log \frac{(1-|z|^2)(1-|z_k|^2)}{|1-z\overline{z_k}|^2}.$$ It seems that $g$ has a unique ...
- 37
3
votes
1
answer
219
views
Moment problem, ergodicity and spectral gap on the space of tempered distributions
Let $\{ S_n \}_{n=0}^\infty$ be a collection of tempered distributions where $S_0:=1$ and $S_n$ is a tempered distribution on $\mathbb{R}^n$.
Just below formula [5] in p.122 of the Fröhlich paper, ...
- 3,477
2
votes
1
answer
141
views
(Sub)Optimality of random transport
Problem Setup: Consider the intervals $I_R = [a_R, b_R]$ and $I_B = [a_B,b_B]$. Let $F_R$ and $F_B$ denote the CDF of distributions with support on the intervals $I_R$ and $I_B$. I draw $k$ red and ...
- 25
7
votes
0
answers
249
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
0
votes
0
answers
20
views
Decomposition of measures orthogonal to the algebra $R(K_1 \times \ldots \times K_n)$ - Can it be done via projection-preserving products of bands?
See "Measures orthogonal to tensor products of function algebras" by Marek Kosiek. Here, it is described for the two-dimensional case. It uses another, more general, approach to OB Bekken's ...
- 63
3
votes
1
answer
287
views
Expectation comparison inequality for concave function of symmetric random variables
Suppose that $X_i$, $i\in[n]$ are
independent symmetric
random variables. I think the conjectured result holds in greater generality, but we can additionally assume that each $X_i$ takes the values $\...
- 6,541
3
votes
0
answers
117
views
Which sigma-ideals in a sigma-algebra are contained in an ideal of null sets?
Let $X$ be a Polish space and $\mathcal{B}(X)$ be the $\sigma$-algebra of Borel subsets of $X$. Given a Borel probability measure $\mu$ on $X$, we write $\mathcal{N}(\mu) := \{ B \in \mathcal{B}(X) : \...
0
votes
0
answers
112
views
Vector field connecting two points
I'm now working on somehow an inverse problem of an ODE:
Suppose we have a ODE on $\mathbb{R}^{n}$: $\dot{x} = f(x)$, denote the solution to the ODE starting at $a$ as $x_{f,a}$(t).
Now there is a ...
- 1
2
votes
0
answers
80
views
Surjectivity of kernel operator with kernel in $L^1(\nu \times \mu)$
Let $ \mu $ and $ \nu $ be two finite and non discrete measures. Let's begin with a well-known fact. Let $ k \in L^2(\nu \otimes \mu) $, then we can define an operator $ \tilde{T} $ as follows:
$$
\...
2
votes
1
answer
79
views
There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?
Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
- 677
-1
votes
1
answer
86
views
how take weak derivative of norms in hilbert spaces?
Let the following be hilbert spaces with dens inclusions $V ↪H=H^* ↪V^*$. Where $H^*$ and $V^*$ are the duals. $H$ has the product $(*,*)$ and $V×V^*$ has the product $⟨*,*⟩$.
Let $u∈L^2 ([0,T];V); ...
2
votes
1
answer
128
views
Density of smooth functions in weighted Sobolev space
Let $\rho(x)=e^{-\phi(x)}$, where $\phi$ is an even polynomial with positive leading coefficient. I am interested in a proof of the fact that the space of smooth compactly supported functions $\...
- 23
2
votes
1
answer
142
views
Bounded differentiation operator on compact intervals with $L^2$ norm
It is known that the differentiation operator $D$ is not bounded on $C^1([0,1])$ with $L^2$ norm (counterexample: $f(x)=x^n$). Now I am wondering whether there is an infinitely dimensional subspace ...
- 153
3
votes
1
answer
375
views
Dimensionality reduction for total variation
Let $P_i,Q_i$, $i\in[n]$,
be distributions on a finite set $\Omega$.
We will use $P^\otimes_{i\in[n]}$ to denote $n$-fold products of distributions.
For each $i\in[n]$, define the
dimensionally-...
- 6,541
4
votes
1
answer
54
views
Krein-Rutman for integral transforms: proof of convergence to leading eigenvector
Disclaimer: This is a question in functional analysis, on which I don't have much background. It arose from me trying to prove on my own a folklore result in probability theory.
Consider an integral ...
- 312
1
vote
2
answers
310
views
Dirichlet Series that fail to be L-functions
For $\sigma \in \mathbb{R}$, let each $\mathbb{C}_\sigma = \{s \in \mathbb{C} : \Re(s) > \sigma\}$. For a sequence $a_n \in \mathbb{C}$, consider the Dirichlet series $D(s) = \sum_{n\ge 0} a_n n^{-...
- 1,124
5
votes
1
answer
261
views
Counter example for Hadamard Differentiability
I am having a hard time while trying to fully understand Hadamard differentiability.
I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
- 53
2
votes
0
answers
55
views
Distance between a Hölder function and a Sobolev ball
Let $\Omega$ denote $[0, 1]^n$ and let $\|\cdot\|_{k, p}$ and $|\cdot|_{m, \alpha}$ denote norms of Sobolev space $W^{k,p}(\Omega)$ and Holder space $C^{m, \alpha}(\Omega)$, respectively.
My question ...
- 420
0
votes
1
answer
128
views
Characterizing the integral as a function of $n$
Let $\alpha \in [0,3], \beta \geq 1, \lambda \geq 1$ and fix $n \in \mathbb{N}$. Consider the function $f(x;\alpha, \beta, \lambda) = x^{\alpha}\exp(-\lambda x^\beta)$. Let $I(n; \alpha,\beta,\lambda) ...
- 25
8
votes
3
answers
616
views
Uniqueness of Neumann series
Let $f$ be an entire function. Then there exist numbers $a_0,a_1,\ldots$, independent of $z$, such that
$$f(z)=\sum_{n=0}^\infty a_n J_n(z),\quad \forall z\in\mathbb{C}$$
where $J_n$ is the Bessel ...
- 317
7
votes
1
answer
959
views
a claim for a proof of the invariant subspace problem [closed]
Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states
Does every bounded operator on a separable Hilbert space have a non-trivial ...
- 87
4
votes
0
answers
78
views
Higher-dimensional analogue of the relation between stable Higgs bundles and constant curvature metrics
In Hitchin's famous paper[1] on the self-dual Yang-Mills equations, he discussed the relation between the stable Higgs bundles and the Teichmüller space for a compact Riemann surface. Namely, through ...
- 353
3
votes
1
answer
127
views
Can doubly parabolic Blaschke product (BP) contained in another doubly parabolic BP?
Let $f:\mathbb{D}\rightarrow\mathbb{D}$ be a degree $d$ doubly parabolic Blaschke product with Denjoy-Wolff point at $z=1$. That is, $f(1) = 1$, $f'(1)=1$ and $f''(1)=0$.
Let $U \subset \mathbb{D}$ be ...
6
votes
1
answer
335
views
Existence of pairwise quasi-complementary but not complementary subspaces
Let $𝑋$ be an infinite-dimensional Banach space (complex or real). A subspace of $𝑋$ means a closed linear submanifold. Subspaces $M$ and $N$ of $X$ are quasi-complementary if $M\cap N=\{0\}$ and $M+...
- 391
2
votes
0
answers
120
views
On mollifiers acting between $L^2$ and Sobolev spaces
(I'm reposting here this question from MSE as it didn't receive any answer for two weeks.)
Consider a sequence of finite lattices in $\mathbb{R}^n$ defined by
$$L_k= [-k,k]^n \cap 2^{-k}\cdot \mathbb{...
- 505
0
votes
0
answers
78
views
What does analytic uniformly in $s$ mean?
Suppose I have a complex vector space $V$ with finite basis $\{e_{1},...,e_{s}\}$. Then, I can consider the algebra $\mathcal{U}$ of formal polynomials on the variables $e_{1},...,e_{s}$. Suppose ...
0
votes
0
answers
80
views
Relationship between two minimization problems
Let $1 \le p < n$ and $p^* = np/(n - p)$. Let $B \subset \mathbb{R}^n$ be a closed ball and let $\Omega \subset \mathbb{R}^n$ be an open set containing $B$. We denote by $W^{1, p}_{B}(\Omega)$ the ...
2
votes
1
answer
106
views
Dual space and conditions for weak convergence in Orlicz Space not having $\Delta_{2}$ property
I am interested in conditions for weak convergence on Orlicz spaces where the corresponding Young function, $\Phi:[0,\infty) \rightarrow [0,\infty)$, does not have the $\Delta_{2}$ condition, i.e. ...
- 21
2
votes
1
answer
49
views
Is any submetrizable linear topology linearly submetrizable?
Let $E$ be a vector space. A topology $\tau$ on $E$ is called (linearly) submetrizable if there is a (linear) metrizable topology $\pi$ on $E$ which is weaker than $\tau$, i.e. $\pi\subset\tau$.
Is ...
- 5,529
2
votes
1
answer
247
views
Reconstruction of Riemann surface from a germ of holomorphic function
Let $\Sigma$ be a compact Riemann surface of genus $g$, and $f: \Sigma \to \mathbb{C}$ a
meromorphic function. Take $U \subset \Sigma$ an open disk in $\Sigma$ biholomorphic to a disk
in $\mathbb{C}$, ...
- 5,230
3
votes
1
answer
187
views
Is this property preserved under weak$^*$ convergence?
Let $1 \le p < n$ and let $p^*$ be the Sobolev conjugate of $p$, i.e. $p^* = np/(n - p)$. Let $(\Omega_m)$ be an increasing sequence of bounded, convex and open sets such that $$ \lim_{m \to \infty}...
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 177
0
votes
1
answer
96
views
Existence of a complemented basic sequence
Let $X$ be an infinite-dimensional Banach space (complex or real). A subspace of $X$ means a closed linear submanifold. If $S$ is a non-empty subset of $X$, then $[S]$ denotes the closed linear span ...
- 391
3
votes
1
answer
176
views
Are measurable maps with countably separated image in a Banach space always strongly measurable?
Let $(E,\|.\|)$ be a (not necessarily separable) Banach space and $\Sigma_E$ the Borel $\sigma$-algebra (w.r.t. the norm topology). Let $(\Omega,\Sigma_\Omega)$ be a measurable space (which we can ...
- 285