All Questions
4,446 questions with no upvoted or accepted answers
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118
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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$^4$, ...
3
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78
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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 ...
3
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187
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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\...
3
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237
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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 ...
3
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131
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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. (...
3
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143
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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 ...
3
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126
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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 $S(x,y)$ ...
3
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92
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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, ...
3
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156
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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 $X$ ...
3
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125
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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 ...
3
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52
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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 \...
3
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89
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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. ...
3
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208
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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 ...
3
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63
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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 ...
3
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465
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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^...
3
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190
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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 ...
3
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64
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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 ...
3
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210
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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} \...
3
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269
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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 $H$ ...
3
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169
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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 ...
3
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280
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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:\...
3
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91
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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 ...
3
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103
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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
$$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$
...
3
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57
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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 ...
3
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214
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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\...
3
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94
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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 ...
3
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129
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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 ...
3
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177
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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 ...
3
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225
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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 ...
3
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148
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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 ...
3
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128
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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 \...
3
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178
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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 ...
3
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790
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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 ...
3
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235
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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 ...
3
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317
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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 ...
3
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589
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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 ...
3
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119
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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 ...
3
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232
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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 ...
3
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78
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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 ...
3
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285
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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$....
3
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97
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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 ...
3
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261
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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 ...
3
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245
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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, ...
3
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359
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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 ...
3
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83
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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 $...
3
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112
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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 ...
3
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answers
104
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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-...
3
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106
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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 \...
3
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0
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609
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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 ...
3
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0
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132
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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 ...