All Questions
13,925 questions
2
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0
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66
views
interchange of integrals and semigroup without the semigroup being an integral operator
In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears
The formula (1.5.2) is Duhamel formula:
$$u(t) = T(t)u(...
- 677
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 578
5
votes
1
answer
483
views
Can you always extend an isometry of a subset of a Hilbert Space to the whole space?
I remember that I read somewhere that the following theorem is true:
Let $A\subseteq H$ be a subset of a real Hilbert space $H$ and let $f : A \to A$ be a distance-preserving bijection, i.e. a ...
- 63.9k
3
votes
0
answers
104
views
Comparing unitaries which are perturbatively close
Let $\mathcal{H}$ be a Hilbert space and let $H_0$ and $H_1$ be two Hermitian operators on $\mathcal{H}$. Thinking of $H_1$ as a perturbation of $H_0$, the Duhamel formula allows us to write $e^{-...
- 452
3
votes
1
answer
263
views
Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
- 1,190
2
votes
0
answers
137
views
Why a function induced by the infimum of the arclength of curves is Lipschitz?
Recently I have read a paper "Weighted Trudinger-type Inequalities" written by Stephen M. Buckley and Julann O'Shea and published by Indiana University Mathematics Journal in 1999, MR1722194,...
- 6,367
3
votes
2
answers
141
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Countable zero-sets are $C$-embedded?
I was browsing Gillman and Jerison for known relations between zero-sets, $C$-embedded sets and so on.
The spaces I'm considering are $T_{3.5}$.
There are two properties that pseudocompact spaces have
...
- 5,596
1
vote
0
answers
105
views
Can a uniform weak null set of $c_0$ be uniformly embedded into a Hilbert space?
Remark that a uniform weak null set $A$ of $c_0$ satisfied that for any $\epsilon>0$ there exists a positive number $N(\epsilon)$ such that for every $f$ in $B(\ell_1)$, the unit ball of the ...
- 16.4k
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 4,713
15
votes
1
answer
507
views
Is there an infinite subset of $\Bbb{R}$ not homeomorphic to any of its proper subsets?
Is there an infinite subset of $\Bbb{R}$ that is not homeomorphic to any of its proper subsets? Clearly, any finite subset of $\Bbb{R}$ is not homeomorphic to any of its proper subsets by mere ...
- 255
11
votes
2
answers
3k
views
“is topologically mixing” vs. “is topologically transitive” in the definition of chaos
This question is cross-posted from MSE, since it hasn't gotten an answer there for over 72 hours.
Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits"...
- 4,713
0
votes
0
answers
97
views
Heine-Borel property for (probability) measures on $\mathcal{S}'$?
For Euclidean space $\mathbb{R}^n$, it is a well-known fact that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is a Montel space. Moreover, the space of tempered distributions $\mathcal{S}'(\mathbb{R}...
- 3,477
0
votes
1
answer
150
views
When are infimal convolutions contractions?
Let $X$ be a separable Fréchet space and $\varphi,\psi:X\to \mathbb{R}$ be a lower semi-continuous and convex function with $\psi$ bounded below and coercive. Consider the infimal convolution
$$
\...
- 1,724
1
vote
0
answers
76
views
Shellable non-pseudomanifolds with dimension greater than 2
Shellability of simplicial balls and spheres (simplicial complexes whose geometric realizations are homeomorphic to balls and spheres) has been studied quite extensively. There are many explicit ...
3
votes
2
answers
228
views
Sobolev extension problems of $W^1_\infty(\Omega)$
Recently I have read the paper Whitney's problem on extendability of functions and an intrinsic metric written by Nahum Zobin and published by Advances in Mathematics in 1998. I am confused about one ...
- 3,584
0
votes
0
answers
39
views
Comonotone solution for Optimal Transport problems with supermodular surplus
In Alfred Galichon's book Optimal Transport Methods in Economics the foollowing result is stated for OT problems on the real line.
Theorem 4.3.(i) Assume that $\Phi$ is supermodular. Then the primal ...
3
votes
1
answer
102
views
Literature containing basic knowledge of homogeneous functions
Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(...
- 63.9k
1
vote
2
answers
622
views
Kähler manifold with Ricci-flat Kähler form
hallo,
I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
- 586
1
vote
1
answer
209
views
Rate of convergence of mollified functions in $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
1
vote
0
answers
63
views
$\operatorname{ker}(q_I \otimes^{\text{min}} q_J) $ is a primal ideal of $\mathcal{A} \otimes^{\text{min}} \mathcal{B}$
In the proof of Theorem $4.1$ of the paper titled continuous bundles of $C^{\ast}$-algebras and tensor products following result is mention with a reference to Proposition $3.3$ of the paper "A. ...
- 1,115
5
votes
1
answer
104
views
When do two measured foliations on a surface define a Riemann surface structure?
Let $S$ be smooth surface of finite type, i.e. it has genus g and n punctures (assume $S$ to have negative Euler characteristic). We know by Hubbard-Masur theorem that given a measured foliation $(F,\...
- 680
1
vote
0
answers
48
views
Connected pre-images spanning $n$-cubes under dimension reducing maps
Let $I^n = [0,1]^n$ be the $n$-dimensional hypercube. For a continuous function $f: I^n \to \mathbb{R}^m$ with $m < n$, we're interested in the existence of points $p \in \mathbb{R}^m$ whose ...
- 63.9k
2
votes
1
answer
255
views
Differential equation involving square root
I am absolutely not familiar with differential equations. However, I am facing the following differential equation:
\begin{equation}
a(x)y^{\prime}(x)+b(x)y(x)=c(x)\sqrt{y^{2}(x)+d(x)}
\end{equation}
...
- 176
0
votes
1
answer
106
views
Convergence of mollified functions in weighted $L^p$ norm
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\supp}{\operatorname{supp}}
$
Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a ...
- 5,005
0
votes
0
answers
43
views
Locally uniformly convexity in kernels (generalized definition of graphon) with cut norm
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
2
votes
0
answers
104
views
When do filtered colimits commute with finite products in Top
It is well known that filtered colimits commute with finite products (more generally any finite limit). This is not the case in general in Top due to Top not being cartesian closed. My question is is ...
- 41
1
vote
0
answers
158
views
Density of Schwartz distributions in the space of distribution
Let $\mathscr S(\mathbf R^3)$ and $\mathscr D(\mathbf R^3 )$ be the space of Schwartz function and test function respectively, $\mathscr S'(\mathbf R^3)$ and $\mathscr D'(\mathbf R^3)$ be their duals....
- 3,065
13
votes
3
answers
2k
views
Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9)...
- 35.5k
3
votes
0
answers
152
views
Topological counterexample for M(K, Y1 × Z1) being a subbasis of the compact open topology of C(X,Y×Z)
We are trying to answer whether the following mapping is continuous and open
$$C(X, Y \times Z) \to C(X, Y ) \times C(X, Z)$$ (the topological spaces being provided with the compact-open topology). We ...
- 39
3
votes
1
answer
103
views
Topology on set of "real lower bounds"
Specific question: Is there a name for the "topology of real lower bounds"? This is the order topology for the ordering $\supseteq$ on the set
$$
\mathbb{LB} = \bigl\{ [t, \infty) \mid t \...
- 18.6k
3
votes
3
answers
583
views
Topology induced by context-free language
Is there any way to reasonably define a topology on a context-free-language language? In other words, given a context-free grammar (or perhaps a grammar from an interesting subclass of context-free ...
- 260
2
votes
1
answer
174
views
A topological space has the homotopy-type of a CW-complex, then is it locally contractible?
Let $X$ be a topological space which has the homotopy-type of a CW-complex. As well-known, a CW-complex is locally contractible.
Question: Is $X$ locally contractible? If not, can some one give a ...
- 5,596
5
votes
1
answer
247
views
Does a "good" homotopy equivalence between pairs imply homotopy equivalence between quotient spaces?
If $(X,A)$ and $(Y,B)$ are (good) pairs of topological spaces, and $f:X\rightarrow Y$ is a homotopy equivalence such that the restriction $f\restriction_A$ is a homotopy equivalence between $A$ and $B$...
- 11.1k
0
votes
0
answers
107
views
Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
- 41
0
votes
0
answers
45
views
Mean value property for fractional laplacian
I just started reading about fractional Laplacian. I am curious on the following questions
Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
- 41
4
votes
0
answers
107
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Reference request for a theorem of Jaworowski
Jan Jaworowski, in 2000, proved the following theorem (I came to know about it from here)
Jaworowski (2000) : Let $Y$ be a finite simplicial complex of dimension $k$ and let $n\ge 2k$. If $f:S^n\to Y$...
- 141
10
votes
0
answers
159
views
Closed sets versus closed sublocales in general topology in constructive math
This question is set in constructive mathematics (without Choice), such as in the internal logic of a topos with natural numbers object, or in IZF.
Short version of the question: if $X$ is a sober ...
- 32.5k
2
votes
2
answers
151
views
Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
- 825
2
votes
1
answer
123
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Signed measures on algebras (fields) and their boundedness properties
I asked this question here on math.StackEchange, but it might be too technical so I re-post it here.
Let $X$ be a compact Hausdorff second countable topological space. Let $\mathcal{B}$ a countable ...
- 2,830
3
votes
0
answers
81
views
Mixing flow has aperiodic orbit?
Let $X$ be a compact connected metric space with more than one point.
Suppose that $H:X\times [0,\infty)\to X$ is continuous such that $h_0=H\restriction X\times \{0\}$ is the identity on $X$, and $h_{...
- 3,317
0
votes
0
answers
64
views
Can an upper hemicontinuous correspondence be discountinuous everywhere?
Let $\phi: X \rightrightarrows Y$ be an upper hemicontinuous correspondence. If $K \subset X$ is a compact and convex set, $K$ contains an open set $U$, and $\phi(x)$ is nonempty, compact, and convex ...
- 101
10
votes
6
answers
879
views
Countable chain condition in topology
A topological space $X$ is said to have the countable chain condition (ccc) if every collection of open and disjoint subsets of $X$ is at most countable. This definition can be found in L. Steen, J. ...
- 467
5
votes
1
answer
375
views
Looking for a counterexample: Conditioning increases regularity?
Let $p(x,y,z)$ be a joint density (over $\mathbb{R}^3$) under no smoothness or regularity assumptions, besides its existence. I am looking for a (counter)example where $p(y|x)$ is less regular than $p(...
- 6,213
4
votes
0
answers
330
views
Book recommendation in functional analysis and probability
I am interested by functional analysis and probability. I would like to know if you have any books that deal with these two subjects (at a graduate level) to recommend?
I'm looking for a book that has ...
- 4,713
0
votes
1
answer
151
views
Super-reflexivity is separately determined
I've found this result that states that super-reflexivity is separably determined, i.e., if every separable subspace $Y\subset X$ of a Banach space is superreflexive then $X$ itself is superreflexive. ...
- 2,830
1
vote
1
answer
121
views
Question on complemented subspaces of a product space
Assume that we have closed subspaces $Y_1$ and $Y_2$ of Banach spaces $X_1$ and $X_2$, respectively. If the product $Y_1\times Y_2$ is complemented in $X_1\times X_2$, does it follow that $Y_i$ is ...
- 1,848
0
votes
0
answers
72
views
Domain of a Jacobi operator with unbounded coefficients
Is it possible to describe the domain of a Jacobi operator explicitly?
Let $J$ be the linear operator acting on a real sequence $(u_{n})_{n\in\mathbb{N}}$ by
$$
J(u_{n}) = a_{n+1} u_{n+1} + a_{n} u_{n-...
- 23
2
votes
0
answers
111
views
Everywhere-defined unbounded operators between Banach spaces
In this post, it is said that there are no constructive examples of everywhere-defined unbounded operators between Banach spaces; every example furnished must use the axiom of choice. This seems like ...
- 151
8
votes
1
answer
4k
views
Covering number of Lipschitz functions
What do we know about the covering number of $L$-Lipschitz functions mapping say, $\mathbb{R}^n \rightarrow \mathbb{R}$ for some $L >0$?
Only 2 results I have found so far are,
That the $\infty$-...
- 1
3
votes
0
answers
318
views
The curse of dimensionality of the Kolmogorov–Arnold neural network
The Kolmogorov–Arnold neural networks (KAN), Ziming Liu et al., KAN: Kolmogorov–Arnold Networks is inspired by the Kolmogorov–Arnold representation theorem (KA theorem). Though it is not proved in the ...
- 2,239