All Questions
4,446 questions with no upvoted or accepted answers
3
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108
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On the derivatives of the solutions of the heat equations with Neumann boundary condition
Let $\Omega$ be a smooth domain with compact closure. More precisely, $\Omega$ is a $C^\infty$ domain. We write $\partial \Omega$ for the boundary of $\Omega$, and denote by $n$ the inward unit ...
- 721
3
votes
0
answers
465
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Orthonormal basis of eigenvectors of Hamiltonian - Is there any theorem justifying the physicist approach?
In his book The Principles of Quantum Mechanics, Dirac states:
"We call a real dynamical variable whose eigenstates form a complete set an observable."
To Dirac, any observable has a ...
- 1,305
3
votes
0
answers
124
views
Initial topology for a topological ring
Given a topological ring $R$ and an arbitrary (thus not necessarily surjective) epimorphism $q: R \to S$ of underlying rings is there a finest topology on $S$ such that 1) $S$ is a topological ring ...
- 103
3
votes
0
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278
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Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure
Let $(X, | \cdot |)$ be a Banach space.
I am interested in whether one can extend the definition of the Kullback-Leibler divergence
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 67
3
votes
0
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73
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Does the incompressible Navier-Stokes equation have a smooth solution if the initial vorticity is smooth and $p$ integrable for any $p$?
Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$.
If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $...
- 173
3
votes
0
answers
233
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Is it possible to reconstruct the compact space $X$ from the space of measures $M(X)$?
Let $X$ be a compact Hausdorff topological space and $C(X)$ the Banach algebra of continuous functions $u:X\to\mathbb C$ (with the usual $\sup$-norm). It is well-known that the structure of Banach ...
- 7,426
3
votes
0
answers
55
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Injectivity of a convex combination of squares $A^*A$ in $\ell^\infty$
Consider two operators $A,B: \ell^p \to \ell^p$ ([defined and]bounded for all $p \in [1,\infty]$) as well as their adjoints $A^*,B^*:\ell^p \to \ell^p$. Assume $A^*A$ and $B^*B$ have trivial kernel ...
- 4,432
3
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0
answers
77
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What is the name of the (possibly well-known) class of $\pi$-monolithic compact spaces?
A compact space $X$ is called ${\it \pi-monolithic}$ if whenever a surjective continuous mapping $f:X\rightarrow K$ where $K$ is a compact metric space there exists a compact metric space $T\subseteq ...
- 1,101
3
votes
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208
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Dominated convergence theorem for Banach limits
The notion of a Banach limit is usually defined for the space of bounded sequences, but one can define it for more general spaces (see "What is a generalized limit?" and "Do ...
- 151
3
votes
0
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144
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Action of the mapping class group on curves and triangulations
Consider an orientable surface $S$ of arbitrary genus, possibly with boundaries, and with marked points and/or punctures. I will assume that every boundary has at least one marked point so that the ...
- 1,435
3
votes
0
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77
views
Unitary with entries $(i,j)$ only on equidistant lattice points $\|i-j\|^2 = c^2 \in \mathbb{N}$
My research needs help in finding examples of unitary matrices $U$ which have entries
\begin{align}
U_{ij} = \begin{cases} \alpha_{ij}, \ \text{ if } \|i-j\|^2 = c^2 \\ 0 , \text{ otherwise} \end{...
- 41
3
votes
0
answers
68
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Relationship between Hardy-Orlicz space and the corresponding Orlicz space
For $p \in [1, \infty]$ the Hardy space $H_p$ is defined as the space of all analytic functions $f$ on the open disk satisfying
$$\|f\|_{H_p} = \sup_{0 < r < 1} \|f(r\cdot)\|_{L_p(\mathbb{T})} &...
- 565
3
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0
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138
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Continuity of parameter integral
Given two compact Hausdorff spaces $K$ and $L$, a bounded and separately continuous function $f:K\times L\to \mathbb C$, and a complex measure with finite variation $\mu$ on $L$ endowed with the Borel ...
- 16.5k
3
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84
views
A weighted $W^{2,p}$ estimates
Let $\Omega$ be a bounded smooth domain and $u\in W^{2,p}(\Omega)\cap H^1_0(\Omega)$. By the classical $L^p$ theory of second order elliptic equation, we have
$$
\|\nabla^2u\|_{L^p(\Omega)}\leq C\|\...
- 379
3
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0
answers
92
views
Asymptotic uniform convexity conditions for subsets of the $B_X$
The following question is relatively straightforward and almost looks like an exercise from a textbook but I have no idea how to handle it. The problem is related to spaces with asymptotically ...
- 1,073
3
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0
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324
views
Would you help me to find this expression?
I'm studying a paper, which I'll include a small piece here. And I'm struggling to calculate
$$C_n\|u_{m,n}\|^{\left(\frac{2*}{2}\right)^k\frac{2*-q}{(r_k)^k}}_{L^{2*}(\Omega)}$$
Where $\Omega$ is an ...
- 63
3
votes
0
answers
61
views
Dual space of Carleman functions
Let $X$ be the space of all weakly measurable functions $\gamma:\mathbb{R}^n \to L^2(\mathbb{R}^n)$ (modulo functions that are 0 almost everywhere) for which
$$\|\gamma\|_X^2 := \sup_{\|g\|_{L^2}=1} \...
- 141
3
votes
0
answers
108
views
$ f,g\in \mathrm{VMO} $ but $ f\cdot g\notin \mathrm{VMO} $
We say a function $ f\in L^1_{\mathrm{loc}}(\mathbb{R}) $ is in $\mathrm{BMO}(\mathbb{R})$ if
$$\|f\|_{\mathrm{BMO}}=\sup_{I}\frac{1}{|I|}\int\limits_I |f(y)-f_I|\, dy<\infty$$ for all intervals $I\...
- 1,219
3
votes
0
answers
159
views
$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian
Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
- 1,407
3
votes
0
answers
171
views
Covering number $C^k$-balls in $C(\mathbb{R}^n)$
Fix a positive integer $n$ and and an non-negative integer $k$. The Arzela-Ascoli theorem guarantees that for a given positive integer $k$ and a given $L>0$ the set
$$
Ball_{C^{k,1}([0,1]^n)}(0,L)
...
- 5,405
3
votes
0
answers
64
views
Algebraic characterisation of the end space of a proper geodesic space in terms of non-continuous functions
$\DeclareMathOperator\Bf{B_\mathrm{f}}\DeclareMathOperator\Bc{B_\mathrm{c}}\DeclareMathOperator\Cf{C_\mathrm{f}}\DeclareMathOperator\Cd{C_\mathrm{d}}\DeclareMathOperator\Cc{C_\mathrm{c}}$Based on a ...
3
votes
0
answers
122
views
A space with independent tightness
Recall that the tightness of a topological space $X$ is defined as the least cardinal $\kappa$ such that for every non-closed subset $A$ of $X$ and every point $x \in \overline{A} \setminus A$, there ...
- 4,413
3
votes
0
answers
148
views
Topologically characterizing metrizable spaces
There are some well-known theorems that imply that some metrizable spaces, when satisfying other topological properties, are unique up to homeomorphism. Here are a few examples, where "perfect&...
3
votes
0
answers
170
views
A version of the Nash-Moser inverse function for unbounded domains?
Do there exist versions of the Nash-Moser inverse function theorem applicable to spaces of unbounded smooth functions on unbounded domains in $\mathbb{R}^n$? Any reference would be appreciated but ...
- 557
3
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214
views
Extended adjoint of Volterra operator
Let $V$ be a Volterra operator on $L^2 [0,1]$.
Does there exist a nonzero operator $X $ satisfying the following system
$VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator?
$$ V(f) (x) =\...
3
votes
0
answers
252
views
Characterization of a Bochner/strongly measurable function solely as a random element
Be $(\Omega, \mathcal{A}, P)$ and $E$ a probability space and a Banach space respectively.
This paper of G.A. Edgar contains a proof that, for a function $X: \Omega \rightarrow E$, being weakly ...
- 31
3
votes
0
answers
131
views
Unital commutative dual Banach *-algebras whose $w^*$-closed ideals are principal
Let $A$ be a commutative Banach *-algebra. For a given ideal $I$ of $A$, we say that, it is principal if there is a projection $p$ (i.e. $p^2=p=p^*$) in $A$ with $I=Ap$.
Q. Any characterization ...
- 4,058
3
votes
0
answers
275
views
Schur-Horn theorem for principal submatrices
The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2,\...
- 6,351
3
votes
0
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120
views
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2)$ then there is a subsequence such that $u_n(t)\rightharpoonup u(t)$ almost everywhere?
If $u_n\rightharpoonup u$ in $L^2(0,T;L^2(\Omega))$. Can we find a subsequence such that $u_{n_k}(t)\rightharpoonup u(t)$ almost everywhere on $[0,T]$?
I'm not sure if this question is trivial or not,...
- 153
3
votes
0
answers
160
views
Non-emptiness of spectrum $\sigma(a)$ in non-Archimedean Banach algebras
I'm trying to determine whether or not the standard proof that the spectrum of a point in a unital Banach algebra is non-empty can be adapted to prove the same thing over certain non-Archimedean ...
3
votes
0
answers
229
views
$f(x)>0$ and $f(y)>0$ implies $f(x+y)>0$, then there must exist an linear function $g$ such that $g(x)>0$ iff $f(x)>0$?
Background: Let $x,y\in\mathbb (0,+\infty)^n$. $f$ is a continuous function on $\mathbb R^n_+=(0,+\infty)^n$. Consider the following condition (1), the sign of $f(x+y)$ is dependent on the sign of $f(...
- 263
3
votes
0
answers
180
views
Stinespring's theorem: can we choose the dilation to be an isometry?
Let $A$ be a $C^*$-algebra and $\varphi: A \to B(H)$ be a completely positive contractive map. Stinespring's theorem says that there exists a $*$-representation $\pi: A \to B(H')$ and a bounded ...
- 175
3
votes
0
answers
187
views
Analogue of Kolmogorov/Arnold superposition for general manifolds?
Previously asked and bountied at MSE with slightly different language:
Given a topological space $\mathcal{X}$, let $$\mathsf{Cl_C}(\mathcal{X})=\bigcup_{n\in\mathbb{N}}C(\mathcal{X}^n,\mathcal{X})$$ ...
- 20.5k
3
votes
0
answers
282
views
Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$
Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...
- 2,605
3
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0
answers
116
views
Multiplier on a Sobolev space
Let $b$ be a function and $W^{1,2}$ the first-order Sobolev space on some Euclidean space (edit: so the whole space, but in arbitrary dimension, although $d=1$ would be interesting for me for the ...
3
votes
0
answers
177
views
When do Polish spaces admit complete metric making them $\mathrm{CAT}(\kappa)$?
Question
$\DeclareMathOperator\CAT{CAT}$Let $X$ be a Polish space. When are there known conditions under which $X$'s topology can be metrized by a metric $d$ such that $(X,d)$ is a:
$\CAT(\kappa)$ ...
- 195
3
votes
0
answers
80
views
Every Borel linearly independent set has Borel linear hull (reference?)
I am looking for a reference to the following fact, which probably is known and could be proved somewhere by someone.
Theorem. The linear hull of any linearly independent Borel set in a Polish ...
- 42k
3
votes
0
answers
120
views
Approximation of a linear functional by linear continuous functionals
Let $X$ be a locally convex space, $T$ a balanced convex compact set in $X$, and $f:X\to\mathbb{C}$ a linear functional which is (not necessarily continuous on $X$, but) continuous on $T$. It is not ...
- 7,426
3
votes
0
answers
320
views
Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
3
votes
0
answers
154
views
What do people call functionals on holomorphic functions and on polynomials?
There are four most important functional spaces in analysis:
the space $\mathcal{C}(M)$ of continuous functions on a topological space,
the space $\mathcal{E}(M)$ of smooth functions on a smooth ...
- 7,426
3
votes
0
answers
480
views
de Rham currents/distributions on manifolds with boundaries
My main source for currents and distribution theory on manifolds in general is de Rham's Differentiable Manifolds. To recap, let $M$ be a smooth, $m$ dimensional real manifold without boundary. De ...
- 3,307
3
votes
0
answers
292
views
Are smooth functions with compact support a core for the Laplacian on compact manifolds with boundary?
If $M$ is a complete Riemannian manifold and $L$ is the Friedrichs extension of the Laplacian $-\Delta$, then it is known (first proven by Gaffney in the '50) that $C_0 ^\infty (M)$ is a core for $L$. ...
- 5,407
3
votes
0
answers
83
views
Does there exist a regular $P$-space which is strongly star-Lindelof but not star-Menger?
A space $X$ is said to be star-Menger if for every sequence $(\mathcal{U}_n)$ of open covers of $X$ there exists a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a finite subset ...
- 505
3
votes
0
answers
144
views
Is the singular value decomposition a measurable function?
$\DeclareMathOperator\svd{svd}$Consider the SVD of rectangular matrices as operators
$$\svd:\mathbb C^{n\times m}\to \mathbb U_n\times \mathbb D_{n, m}\times\mathbb U_m$$
where $\mathbb U_n$ is the ...
- 201
3
votes
0
answers
284
views
Injectivity of the convolution operator $f \mapsto (k*f(s))_{s \in S}$ via sampling at $S=\alpha \mathbb Z$
Let $S \subset \mathbb R$ be a set of sampling points, say $S = \alpha \mathbb Z, \alpha >0$.
Let $k$ be some convolution kernel and $A$ the operator which maps some $f$ to the sequence
$$
Af = (k*...
- 437
3
votes
0
answers
198
views
Geometric characterisation of polynomials between normed spaces
Let $(X, \| \cdot \|_X)$ $(Y, \| \cdot \|_Y)$ be normed space. A function $f \colon X \to Y$ shall be called an $n$-th degree (single variable) polynomial ($n \in \mathbb{N}\cup \{ 0\})$ if there ...
- 820
3
votes
0
answers
141
views
How can one construct this dendrite?
In the early 1970s Pelczynski noticed that the only surjective isometries on $C(K)$ for the following compact Hausdorff space $K$ are $\pm Id$. I believe this was the first such example.
Quoting from ...
- 1,073
3
votes
0
answers
285
views
Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
- 255
3
votes
1
answer
216
views
Existence of a quasi-open (a.k.a semi-open) map into a Cantor cube
Recall that a topological space is extremally disconnected if the closure of any open set is open.
A continuous map is quasi-open if it maps nonempty open sets onto sets with nonempty interior. For ...
- 5,529
3
votes
0
answers
96
views
A sequence of functions solving $-\Delta u_n + V u_n = u_{n-1}|_{\partial M}$
Let $M = \mathbb R^3 \setminus B_1$ where $B_1$ is the unit ball.
Let $ h \in C^{\infty}(\partial M)$ and let $u_0$ be the unique function that vanishes at infinity and solves
$$\begin{cases} -\Delta ...
- 969