All Questions
4,446 questions with no upvoted or accepted answers
3
votes
0
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102
views
Find a certain triangulation subordinate to a given covering of a manifold
Let $\{U_\alpha\}$ be a covering of a smooth manifold $M$. Replacing it by a refined covering if necessary, we may assume some good properties of it, like, (1) any intersection $\cap_{i=1}^k U_{\...
- 2,789
3
votes
0
answers
103
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Inequality concerning BV norm
Let $u(x) \in L^1( \mathbb{R}^n) \cap BV(\mathbb{R}^n)$ and let $\rho\ge 0$ be the standard mollifier on $\mathbb{R}^d$, supported in unit ball with $\int_{\mathbb{R}^d}\rho \,dx=1$ and define $\rho_\...
- 101
3
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0
answers
74
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A question about a theorem in 'Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators'
I have asked this question on MathSE and someone advised me to ask it here. The link is .
I'm studying the paper Quantum dynamical semigroups generated by noncommutative unbounded elliptic operators ...
- 31
3
votes
0
answers
489
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Generalization of Carleman coefficients to multivariable functions - Carleman tensor?
Recently I learned about a matrix called
Carleman matrix. It is a matrix used to represent function iteration with matrix multiplying.
Carleman linearization is a technique used to embed a finite
...
- 133
3
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0
answers
83
views
Is $\Box_{n\in\omega} \mathbb{R}$ metacompact?
Is $\Box_{n\in\omega} \mathbb{R}$ (that is $\mathbb{R}^\omega$ endowed with the box topology) metacompact?
- 48.9k
3
votes
0
answers
97
views
An example of a particular vn-algebra
Let $A$ be a vn-algebra. Let us suppose $e$ is a finite projection in $A$ and $x$ is an isometry (meaning $x^*x=1$) in $A$ such that $e$ does not commute with $x$. Then $\{q_n=x^nex^{*n}\}$ forms a ...
- 4,058
3
votes
0
answers
223
views
Sobolev space under Mellin transform
The Mellin transform is known to be an isomorphism see wikipedia
between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$
where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
user127612
3
votes
0
answers
112
views
Is a relatively weakly compact subset of $W^{1,1}(\Omega)$ metrizable?
Let $\Omega$ be a domain with smooth boundary. Let $S\subset W^{1,1}(\Omega)$ be a relatively weakly compact set.
Is it true that $(S,w)$ is metrizable?
Since $S$ is relatively weakly compact, it ...
- 1,245
3
votes
0
answers
208
views
Identification of ultrafilters with measures
We know that each ultrafilter $p$ on $\mathbb{N}$ can be identified with a finitely additive $\{0,1\}$-valued probablity measure $\mu_{p}$ on the power set of $\mathbb{N}$.
Now my question is which ...
- 883
3
votes
0
answers
183
views
Parabolic variational inequality: regularity of the time derivative in $L^2(0,T;H)$?
Let $V \subset H \subset V^*$ be a Gelfand triple of Hilbert spaces. Take $f,\psi \in L^2(0,T;H)$ and consider the VI: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V^*)$ such that
$$u(t) \leq \psi(t) ...
- 73
3
votes
0
answers
358
views
New/useful method for summation of divergent series?
Questions
$$ S(n,x) = x+e^x + e^{e^x} + e^{e^{e^x}} + \dots \text{$n$ times}$$
Also obeys (see background for argument):
$$ \frac{1}{2 \pi i} \oint e^{S(k,x)} \frac{\partial \ln(\frac{\int_0^\...
- 237
3
votes
0
answers
206
views
Do these limits exist?
Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[G]$ is ...
- 2,118
3
votes
0
answers
228
views
Categorical features of Hilbert spaces
Does the category of Hilbert spaces and bounded maps have any particular categorical feature which can be studied systematically?
I mean, I know that it's a $*$-category, but it seems to have much ...
user95283
3
votes
0
answers
116
views
Obstruction to the existence of a complex-valued determinant function
Let $\mathcal A$ be a Type $II_1$ von Neumann algebra, equipped with a finite trace $\tau: \mathcal A \to \mathbb C$. Further, denote by $\mathcal A^\times \subset \mathcal A$ the group of invertible ...
- 1,255
3
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0
answers
85
views
Existence for $-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$
Let $\Omega$ be a smooth bounded domain. Consider the equation
$$-\Delta u + (g(x)-\Delta u)^+\varphi(u) = f(x)$$
$$u|_{\partial\Omega} = 0$$
where $f,g$ are smooth functions on $\Omega$ and $\varphi$...
- 251
3
votes
0
answers
650
views
description of dual space of space of Radon measure equipped with topology of weak convergence
Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence":
$$
\mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n ...
- 325
3
votes
0
answers
88
views
Which spaces are still Lindelöf after forcing with a Suslin tree?
Let $T$ be a Suslin tree and $f:T\to Y$ be continuous. ($T$ is endowed with the order topology.) Assume that the image of $T$ is contained in a Lindelöf subset of $Y$. Then, force with $T$. Which ...
- 1,855
3
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0
answers
175
views
Araki's proof of simple connectedness of the restricted orthogonal group
I am trying to understand Araki's proof of the statement that the restricted orthogonal group of a Hilbert space with a unitary structure is simply connected. This proof starts on page 114 of these ...
- 556
3
votes
0
answers
45
views
Certain limits of normal completely positive idempotents
Suppose $L_n$ is a sequence of normal completely positive mappings of $B(H)$ into itself of norm strictly less than one with the property that $L_n(L_m(A)) \to L_m(A)$ in the strong operator topology ...
- 365
3
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0
answers
80
views
On the compactification of partial semigroups
We begin by introducing some relevant definitions.
Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...
- 745
3
votes
0
answers
109
views
Is the Banach space $C(K)$ a $1$-Lipschitz comp-extensor?
Given a real number $c\ge 1$ let us say that a metric space $X$ is a $c$-Lipschitz comp-extensor if each Lipschitz self-map $f:K\to K$ of a compact subset $K\subset X$ extends to a Lipschitz self-map $...
- 42k
3
votes
0
answers
180
views
When is a minimal immersion holomorphic?
Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:
$\phi\colon X\to Y$
be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
- 2,818
3
votes
0
answers
147
views
One strong fixed-point property
Each topological space $A$ with fixed-point property is connected (all clopen subsets are trivial). This is an analog of Rice theorem (all decidable subsets are trivial). Suppose, we have a space $A$ ...
3
votes
0
answers
108
views
Radial Poincare inequality for Gaussian measures
Let $\mu$ be a zero mean Gaussian probability measure on $\mathbb{R}^n$ whose covariance is less than the identity. If $f$ is a $1$-Lipschitz real function on $\mathbb{R}^n$ such that there exists a ...
- 2,772
3
votes
0
answers
97
views
Why c.p.c order zero maps induce morphism between cuntz semigroups
Maybe a naive question:
Right now I am reading the paper ``Completely positive maps of order zero'' written by Wilhelm Winter and Joachim Zacharias. I do not quite understand the last corollary when ...
- 181
3
votes
0
answers
149
views
Classical subspaces of non-atomic Banach lattices
Tsirelson's space was the first example of a Banach space which does not have a subspace isomoprhic to any of the classical spaces $\ell_p$, $1\leqslant p<\infty$, or $c_0$. As this space has a $...
user114263
3
votes
0
answers
182
views
Prove a certain function maps to upper half plane
Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in ...
- 31
3
votes
0
answers
422
views
Isometries between subspaces of finite-dimensional vector spaces
I would like to characterise the subspaces of $\ell_p^n(\mathbb{R})$ that are isometric (for $p$ an even integer). In the literature, I have found few results related to this.
Taking $n \le m$, one ...
- 31
3
votes
0
answers
89
views
Discrete Lions Peetre interpolation
In S. Heinrich's "Closed operator ideals and interpolation," Heinrich describes two equivalent descriptions of Lions-Peetre interpolation space $(X,Y)_{\theta, p}$ for $0<\theta<1$ and $1\...
user114263
3
votes
0
answers
141
views
Is there a normal space with a $G_\delta$ diagonal which is not submetrizable?
A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. ...
- 4,413
3
votes
0
answers
154
views
$G_\delta$-diagonal and productivity of the CCC
Is there a known example of a completely regular c.c.c. space with $G_\delta$-diagonal which is not productively c.c.c.?
The non-existence of such a space is consistent (for example, under $MA$ no ...
- 1,615
3
votes
0
answers
78
views
Is every weakly Lindelof Banach space a $D$-space?
An open neighbourhood assignment for a topological space $(X, \tau)$ is a map $U: X \to \tau$ such that $x \in U(x)$, for every $x \in X$. A space $X$ is called a $D$-space if for every open ...
- 4,413
3
votes
0
answers
48
views
Questions on "The condition number of a randomly perturbed matrix"
This question is about the two vectors $w'$ and $y$ that are necessary for the argument in section $7$ (page 6) of this paper by Terence Tao and Van Vu,
https://arxiv.org/abs/math/0703307 (that ...
- 2,246
3
votes
0
answers
120
views
Schur property for a sum of Banach spaces
Suppose we have two Banach spaces X and Y each of them having the Schur property (weakly convergent sequences are norm convergent). Does it follows that X+Y has the Schur property? Note that this is ...
- 31
3
votes
0
answers
127
views
Max-Plus algebra and hyperplane arrangements
Given an expression in the Max-Plus algebra is it possible to recognize if it represents a continuous piecewise linear (CPWL) function whose polyhedral complex is a hyperplane arrangement?
Or ...
- 2,246
3
votes
0
answers
168
views
Zak transform and VMO
The Zak transform of a function $f\in L^1(\mathbb R)\cap L^2(\mathbb R)$ is defined as follows:
$$
Zf(x,\omega) := \sum_{k\in\mathbb Z}f(x+k)e^{-2\pi i k\omega},\quad (x,\omega)\in Q_0 :=(0,1)^2.
$$
...
3
votes
0
answers
56
views
Name for a special kind of neighborhood assignment or for the existence thereof
Lets say temporarily that a topological space $(X,\tau)$ is weird if there is a function $\varphi:X \to \tau$ such that for all $x \in X$:
$x\in\varphi(x)$,
$\{y\in X: x \in \varphi(y)\}$ is finite.
...
- 11.5k
3
votes
0
answers
82
views
Proving the existence of a continuous function that satisfy a certain property from a finite version of this property
Let $M \subseteq [0,1] \times \mathbb{R}^n$ be a compact semialgebraic set.
In particular, $M$ can be described by a finite set of polynomial equalities and inequalities.
Let $\delta_0 > 0$ be a ...
- 745
3
votes
0
answers
1k
views
Inner Product on tensor product of Hilbert spaces is unique?
Given two Hilbert Spaces $H$ and $K$, a natural inner product on $H\otimes K$(= vector space tensor product of $H$ and $K$) is given by
$\hspace{.5in}\langle h_1\otimes k_1,h_2\otimes k_2\rangle=\...
- 165
3
votes
0
answers
198
views
Asymptotic stability of eigenvalues by compact perturbations
I need some references concerning the asymptotic stability of eigenvalues by compact perturbations. In [T. Kato, Perturbation theory for linear operators] there are some results concerning stability ...
- 1,329
3
votes
0
answers
161
views
A characterization of Cauchy filters on countable metric spaces?
Given a filter $\mathcal F$ on a countable set $X$, consider the family
$$\mathcal F^+:=\{A\subset X:\forall F\in\mathcal F\;(A\cap F\ne\emptyset)\}.$$
The following characterization is well-known.
...
- 42k
3
votes
0
answers
367
views
Reference on semigroup theory and fractional heat equation
Consider the Dirichlet problem associated to the classical heat equation $\partial_t u - \Delta u = 0$ and to the fractional heat equation $\partial_t u + (- \Delta)^s u = 0$.
Where can I find a ...
user103450
3
votes
0
answers
275
views
Seminorms on tensor products of affinoid algebras
Let $A \leftarrow C \rightarrow B$ be affinoid $K$-algebras, where $K$ is a non-archimedean field with non-trivial absolute value. Equipping $A$, $B$, $C$ with the supremum seminorms, there is a ...
- 31
3
votes
0
answers
85
views
essential norm versus invertibility
Let $X$ be an infinite dimensional Banach space and $T:X\rightarrow X$ be a bounded linear operator. If $T$ is invertible and $\lVert T\rVert_e=\lVert T\rVert$, is it true that (or when is it true ...
- 83
3
votes
0
answers
110
views
Functorial description of irreducibility of topological space?
This is a crosspost of this MSE question.
A topological space is connected if it's not the coproduct of two non-trivial spaces. Equivalently, it is connected if the copresheaf it represents preserves ...
- 10.5k
3
votes
0
answers
153
views
Identifying a Banach space-valued functions' integral notion
In Teschl's book on Mathematical Methods in Quantum Mechanics (https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf) in section 4.1 a notion of an integral for Banach space-valued functions ...
- 203
3
votes
0
answers
278
views
Interchanging direct sum and direct integral
I am interested in the concept of direct integrals. The definition I am concerned with can be found in Hall, "Quantum Theory for Mathematicians" (p. 146). Short version: We have a family of separable ...
- 31
3
votes
0
answers
241
views
Is the shift operator continuous on the real Hardy spaces?
Is the shift operator (or the translation operator) continuous on Hardy spaces $H^p(\mathbb{R}^n)$ (with $0<p\leq 1$)? i.e. given $f\in H^p(\mathbb{R}^n)$, is the following map
\begin{align}
\tau:&...
- 31
3
votes
0
answers
125
views
Partial regularity for transmission problem in corner domains
Let $n=2$ or $3$ and $\Omega \subset \mathbb{R}^n$ be an open bounded domain. Let suppose that $\Omega$ is divided in two subdomains $\Omega_1$ and $\Omega_2$ and we define $\Gamma = \partial \Omega_1 ...
- 143
3
votes
0
answers
118
views
Weak contractibility of some infinite dimensional metric spaces
Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that:
$X_{n}$ is a regular$^1$ CW-complex of constant local dimension$^3$ $n$, it is of finite type$^4$, ...
