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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$^4$, ...
Sebastien Palcoux's user avatar
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0 answers
78 views

Nowhere dense covering number of a connected $T_2$ space

This is a generalization of an older question. If $(X,\tau)$ is a connected $T_2$ space with more than 1 point, we define its nowhere dense covering number $\nu(X)$ by the smallest cardinality that a ...
Dominic van der Zypen's user avatar
3 votes
0 answers
187 views

Families of unbounded operators

Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\...
Joey's user avatar
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0 answers
237 views

Who defined the term "Carleman-matrix" and also their properties as they are?

Working for a couple of years with "Carleman-matrices" I became aware that it is difficult to trace down who actually introduced the name for this type of matrices. Wikipedia lets you alone with this ...
Gottfried Helms's user avatar
3 votes
0 answers
131 views

quantitative winding number?

In light of Tao's discussion of Quantitative Continuity I've decided to post some thoughts on Winding number. The Jordan curve theorem is non-trivial because closed curves can be very complicated. (...
john mangual's user avatar
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143 views

Is an Abelian topological group compact if it is complete and Bohr-compact?

A topological group $G$ will be called Bohr-compact if its Bohr topology (i.e., the largest precompact group topology) is compact and Hausdorff. A topological group $G$ is Bohr-compact if it admits ...
Taras Banakh's user avatar
3 votes
0 answers
126 views

An identity of operator norms and de Leeuw's theorem

Let $$Hf(x_1,x_2)=p.v.\int_{-\infty}^\infty f(x_1-t,x_2-S(x_1,x_1-t))\frac{dt}{t},$$ $$T_\lambda f(x)=\lim_{\epsilon\to0}\int_{|x-y|\ge\epsilon}e^{i\lambda S(x,y)}(x-y)^{-1}f(y)dy, $$ where $S(x,y)$ ...
Right's user avatar
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92 views

Arithmetic progressions inside non meager sets

If $A \subseteq \mathbb{R}$ is non-meager Borel set, then $A$ contains arithmetic progressions of every finite length. I know that this is false if we do not assume that $A$ is Borel. In particular, ...
George's user avatar
  • 31
3 votes
0 answers
156 views

Topology of the Hamel basis in a TVS

Let $V$ be a complex topological vector space, and let $I$ be a Hamel basis of it. Then as a subset $I\subset V$ acquires an induced topology, becoming a topological space. For a topological space $X$ ...
Bedovlat's user avatar
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125 views

Commutative discrete cyclic operator groups on topological vector spaces

Let $V$ be a complex Hausdorff separable topological vector space of infinite dimensions. Does there exist a commutative discrete subgroup $A\subset\mathcal{L}(V)$ of continuous operators on $V$ with ...
Bedovlat's user avatar
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52 views

Integral estimate in the Levitan's paper "On expansion in eigenfunctions of the Laplace operator"

Let $D \subset R^m$ be a domain in a $m$-dimensional Euclidean space, $P \in intD$, and $t > 0$ so small that the sphere of radius $t$ centered at the point $P$ sits in $intD$. Let $\phi : D \...
user109413's user avatar
3 votes
0 answers
89 views

Trace of a weighted composition operator on Bergman space

I am reading a series of papers by Pollicott, Jenkinson and coauthors which make use of the following type of result: Theorem: Let $\mathbb{D} \subset \mathbb{C}^d$ be a bounded, connected open set. ...
Ian Morris's user avatar
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3 votes
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208 views

Is the homeomorphism group of a Polish space a measurable group?

Let $X$ be a Polish space. Let $H(X)$ be the set of homeomorphisms $h \colon X \to X$, equipped with the "evaluation $\sigma$-algebra", namely $\sigma(h \mapsto h(x) : x \in X)$. (Note that for any ...
Julian Newman's user avatar
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0 answers
63 views

Is the collection of Schur convex functions sequentially compact?

We know in ROCKAFELLAR's convex analysis chap 10 that the collection of uniformly bounded convex functions on compact set is sequentially compact. I wonder if it is still true for the collection of ...
Sung-En Chiu's user avatar
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0 answers
465 views

Fractional sobolev spaces

On the whole space $\mathbb R^d$, the fractional Sobolev space $H_s(\mathbb R^d)$ of order $s\in \mathbb R$ can be defined as the subspace of tempered distributions $T$ such that $\mathcal F T \in L^...
Thomas's user avatar
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0 answers
190 views

Error term in the Euclidean Weyl law

Let $\Omega\subset \mathbb R^n$ be an open bounded set with smooth boundary. The Laplacian on $\Omega$ with Dirichlet boundary conditions has discrete spectrum $\lambda_1\le \lambda_2\le \ldots$ that ...
Dario's user avatar
  • 381
3 votes
0 answers
64 views

A constraint satisfaction problem on matrix sum involving symmetric group

Given $n\in\Bbb N$ what is the smallest with $m>n$ we need such that there is a non-negative $\epsilon<1$ and $\Phi_i,\Psi_j\in\Bbb C^{m\times m}$ at every $i,j\in[n]$ ($[n]=\{1,\dots,n\}$) such ...
Turbo's user avatar
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3 votes
0 answers
210 views

Meromorphic continuation of resolvent of free Laplacian on homogeneous Sobolev space

Let $n \ge 2$. Set $\dot{H}^1(\mathbb{R}^n)$ to be the homogeneous Sobolev space, defined as the Hilbert completion of $C_0^\infty(\mathbb{R}^n)$ with respect to the norm $\| \varphi \|^2_{\dot{H}^1} \...
JZS's user avatar
  • 481
3 votes
0 answers
269 views

Finite dimensional representation of tensor product

Let $A$ and $B$ be $C^*$ algebras, and let $\pi:A \odot B \to B(H)$ be a $*$-representation of the algebraic tensor product on a finite dimensional Hilbert space $H$. Let $x \in A \odot B$. Since $H$ ...
burtonpeterj's user avatar
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3 votes
0 answers
169 views

A spanning set for an annihilator set on a Banach space

Let $(z_n)$ be a $H^\infty$-interpolating sequence on the open complex unit disc $\mathbb D$. If $A$ is some Banach space of analytic functions on the disc, denote by $X$ the closed subspace of all ...
user106480's user avatar
3 votes
0 answers
280 views

Helmholtz-Hodge decomposition

I have a question regarding a decomposition of a vector field. So fix $ 1<p<\infty$ and let $ \Omega$ denote a smooth bounded domain in $ R^N$. Now let $ F $ denote a smooth vector field $F:\...
Math604's user avatar
  • 1,385
3 votes
0 answers
91 views

The numerical range of a composition of two operators

For a problem I'm working on, I need the following implication. $A,B$ are two closed densely defined operators on a Hilbert space $H$. I'll be a bit vague about the setting, add assumptions at will as ...
Piero D'Ancona's user avatar
3 votes
0 answers
103 views

Extension of the Gagliardo Inequality

The Gagliardo Inequality generalizes Fubini's Theorem: let $f_j$ be $d-1$ non-negative measurable functions over ${\mathbb R}^{d-1}$. Let us form the function $$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$ ...
Denis Serre's user avatar
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3 votes
0 answers
57 views

Integration of Weyl operators multiplied by quasifree state over a symplectic space

I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
Tiju Cherian John's user avatar
3 votes
0 answers
214 views

Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?

For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
Rajesh D's user avatar
  • 698
3 votes
0 answers
94 views

Multiplicativity of $\zeta$-function regularized determinant

Let $A$ be a selfadjoint elliptic differential operator on a compact manifold. In mathematical physics and differential topology one often defines its determinant using the $\zeta$-function ...
asv's user avatar
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3 votes
0 answers
129 views

Equivariant $K$-homology with $G$-compact support

Let $G$ be a discrete countable group and let $A$ be $\sigma$-unital $G$-$C^*$-Algebra. For a proper locally compact Hausdorff $G$-space $X$ the equivariant $K$-homology with $G$ compact support and ...
Jack123's user avatar
  • 31
3 votes
0 answers
177 views

Interesting stipulation about completely monotone functions

This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
user avatar
3 votes
0 answers
225 views

Defining a trace-class operator with a Bochner integral

I had asked this question previously on Math.StacheExchange but did not get an answer there in several months. This isn't strictly speaking research level mathematics but I hope it is sufficiently ...
Adomas Baliuka's user avatar
3 votes
0 answers
148 views

Full free product of $B(\mathcal H_i)$

It struck me that I know nothing about the full (universal) free product of the $B(\mathcal H_i)$ amalgamated over $\mathbb C$ for Hilbert spaces $\mathcal H_i$ with identified unit vector $\xi_i$. So ...
Chris Ramsey's user avatar
  • 3,984
3 votes
0 answers
128 views

Stable homotopy equivalence

Let $\alpha:A \rightarrow B$ be a *-homomorphism of $C^*$-algebras. Then $\alpha$ ist a stable homotopy equivalence if there exists a $*$-homomorphism $\beta: B \otimes \mathcal{K} \rightarrow A \...
Blubb91's user avatar
  • 31
3 votes
0 answers
178 views

A point concerning absolute value of functionals

Let $M$ be a von Neumann sub-algebra in $B(H)$. Let $\phi$ be a normal functional on $M$. Assume $\psi$ is a normal functional on $B(H)$ with $\psi_{|_M}=\phi$ (note that $\phi$ and $\psi$ may have ...
ABB's user avatar
  • 4,058
3 votes
0 answers
790 views

What is the optimal constant for the injection of $H^1$ into $L^\infty$ on an interval?

Let $I\subset \mathbb R$ be an interval, $1\leq p\leq \infty$, and $W^{1,p}(I)$ the usual Sobolev space. It is known that the injection $W^{1,p}(I)\hookrightarrow L^\infty(I)$ holds, i.e. there exists ...
JKLM's user avatar
  • 31
3 votes
0 answers
235 views

Is this "differentiation map" uniquely determined by these properties?

Let $A$ be the set of all real-valued functions having their domain a subset of $\Bbb R$ which are at least differentiable on an open set, and for $f \in A$, let $U_f$ be the largest open set on which ...
user avatar
3 votes
0 answers
317 views

Best constant for maximal function for locally compact groups

Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much ...
BigM's user avatar
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3 votes
0 answers
589 views

Norm in a product vector space induced by a norm in $\mathbb{R}^d$

I posted this question originally here (nobody answered there): https://math.stackexchange.com/questions/2066318/is-the-following-function-a-norm Let $\| \|$ be any norm in $\mathbb{R}^d$. Consider ...
William M.'s user avatar
3 votes
0 answers
119 views

Increasing sequence of closed subspaces of $L^2$ and error estimate of a product of orthogonal projections

We define an increasing sequence of closed subspaces \begin{align*} V_{0} \subset V_{1} \subset V_{\ell} \subset \dots \end{align*} of $L^2(I)$ where $I=(0,x_{max})$, and each $V_{\ell}$ is equipped ...
user144209's user avatar
3 votes
0 answers
232 views

I've found a representation of the Itō-Stratonovich correction term and don't understand the used notion of a "trace"

Consider a Stratonovich SPDE $$X_t=X_0+\int_0^tb(s,X_s)\:{\rm d}s+\int_0^t\sigma(s,X_s)\circ{\rm d}W_s\tag 1$$ in a separable $\mathbb R$-Hilbert space $H$ with $W$ being a $Q$-Wiener process on a ...
0xbadf00d's user avatar
  • 167
3 votes
0 answers
78 views

Perscribed/Inverting Conditional Expectation

I'm having difficulty finding papers which deal with the following inversion problem. Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to ...
ABIM's user avatar
  • 5,405
3 votes
0 answers
285 views

Error estimate on convolution of mollifiers

Given $u\in W^{1}_{p}(\omega)$ with $1\leq p\leq \infty$, and the mollifier $\rho\in C_0^{\infty}(R^d)$ with support $B_1$ is a unit ball centered at the origin, $\rho\geq 0$ and $\int_{B_1} \rho = 1$....
user177196's user avatar
3 votes
0 answers
97 views

Boudedness of linear operator between generalized Orlicz spaces

I am using the notations, definitions, and results of the Section X of [1] on generalized Orlicz spaces. We say that $\varphi : \mathbb{R} \rightarrow \mathbb{R}^+$ is a $\varphi$-function if it is ...
Goulifet's user avatar
  • 2,306
3 votes
0 answers
261 views

Generalization of trace and associated determinant

The standard relation between the trace and the determinant of matrices is presented in the MO-Q "Cycling through the zeta garden" where the log and exp functions allow one to jump between additive ...
Tom Copeland's user avatar
  • 10.5k
3 votes
0 answers
245 views

Lawvere's 'Categories of space and of quantity" - the universal coefficient theorem

This is a continuation of this question about the paper Categories of Space and of Quantity by W. Lawvere. As intuitively clear by the very broad (and tentative) definitions suggested by W. Lawvere, ...
Arrow's user avatar
  • 10.5k
3 votes
0 answers
359 views

Cubical approximation theorem for cubical complexes

A version of the simplicial approximation theorem states that a continuous map between finite simplicial complexes is homotopic to a simplicial map after subdividing the domain. I have found a claim ...
Ben Knudsen's user avatar
3 votes
0 answers
83 views

Is the increasing union of disk bundles a disk bundle?

Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $...
Matthew Kvalheim's user avatar
3 votes
0 answers
112 views

Orlicz spaces and $\phi$-functions

A $\phi$-function $f$ is usually defined as a continuous function $f=\mathbb R_+ \to \mathbb R_+$ such that: (1) $f$ is nondecreasing. (2) $f(0)=0$ and $f(x)>0$ for all $x>0$. (3) $\lim_{x\to ...
Thomas's user avatar
  • 630
3 votes
0 answers
104 views

A link of four 2-tori $T^2$ in $S^2 \times S^2$

Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-...
wonderich's user avatar
  • 10.5k
3 votes
0 answers
106 views

A link of four 2-tori $T^2$ in $S^3 \times S^1 \# S^2 \times S^2 \# S^2 \times S^2$

Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary to obtain a new 4-manifold: $$(S^4 \smallsetminus D^2\times T^2) \cup (S^4 \...
wonderich's user avatar
  • 10.5k
3 votes
0 answers
609 views

Homeomorphism Between $C([0,1])$ and $C([0,1]^2)$

It is well known that there cannot be a homeomorphism between $\mathbb{R}$ and $\mathbb{R^2}$ because if we remove the origin from $\mathbb{R}$ the space becomes disconnected, but if we remove the ...
Asterix's user avatar
  • 371
3 votes
0 answers
132 views

Duality for continuous lattices based on [0, 1]

A continuous lattice may be defined as a complete lattice in which arbitrary meets distribute over directed joins. A continuous lattice is naturally regarded as an algebraic structure where the ...
Ronnie's user avatar
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