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7 votes
2 answers
608 views

What is the name for a set endowed with a Lipschitz structure?

I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
Taras Banakh's user avatar
  • 41.9k
7 votes
3 answers
2k views

A question on fractional derivatives

I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of ...
user avatar
7 votes
1 answer
461 views

Does second countable and functionally Hausdorff imply submetrizable?

A topological space $\mathbf{X}$ is functionally Hausdorff, if for any two distinct $x, y \in \mathbf{X}$ there exists a continuous function $f_{xy} : \mathbf{X} \to [0,1]$ with $f(x) = 0$ and $f(y) = ...
Arno's user avatar
  • 4,727
7 votes
1 answer
899 views

Is a closed subset of an extremally disconnected set again extremally disconnected?

Let $T$ be a compact Hausdorff extremally disconnected set (so $T$ is a compact Hausdorff space, such that the closure of each open subset is again open). Let $S \subseteq T$ be a closed subset. ...
AlexIvanov's user avatar
7 votes
4 answers
2k views

Invariant means on the integers

Let $A\subseteq\mathbb Z$, as usual we define the lower Beurling density $d^{-}(A)=\lim\inf_{n\rightarrow\infty}\frac{|A\cap[-n,n]|}{2n+1}$ and the upper Beurling density $d^+(A)=\lim\sup_{n\...
Valerio Capraro's user avatar
7 votes
2 answers
2k views

Uniform bound on the eigenfunctions of the Laplacian

Is it possibly to have $L_\infty$ bounds on the eigenfunctions of the Laplacian operator on bounded regular domains with Dirichlet condition? I found several papers by Sogge but these are pretty ...
John Zheng's user avatar
7 votes
3 answers
911 views

A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question). One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
fosco's user avatar
  • 13.6k
7 votes
1 answer
2k views

Minimize Energy for Charge Distributions

I am considering [positive] charge distributions $\rho:M\rightarrow\mathbb{R}_+$ (nonnegative reals) with unit charge $\int_M\rho=1$ for convenience. Here $M$ is a nice-enough region, say a ...
Chris Gerig's user avatar
  • 17.5k
6 votes
1 answer
298 views

What is the height (or depth) of $[\mathbb{N}]^\infty$?

(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.) Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$, partially ordered by $\...
Boaz Tsaban's user avatar
  • 3,104
7 votes
1 answer
606 views

Weak* continuity of positive parts, again

Bill Johnson pointed out to me yesterday that the map $$f \mapsto f^+ = \max(f,0)$$ is not weak* continuous on $l^\infty$. Nonetheless, I think I can prove that if $V$ is a linear subspace of $l^\...
Nik Weaver's user avatar
  • 42.8k
7 votes
1 answer
2k views

Orthonormal bases on Reproducing Kernel Hilbert Spaces

Recall that a Hilbert space $\mathcal{H}$ is a reproducing kernel Hilbert space (RKHS) if the elements of $\mathcal{H}$ are functions on a certain set $X$ and for any $a\in X$, the linear functional $...
T. Le's user avatar
  • 577
7 votes
1 answer
754 views

Closed convex hull in infinite dimensions vs. continuous convex combinations

tl;dr: When is the closed convex hull of a set $K$ equal to the set of "continuous" convex combinations of $K$? I am essentially asking for the most general, infinite-dimensional analogue of ...
user163625's user avatar
7 votes
1 answer
907 views

Lebesgue differentiation theorem holds on locally doubling space?

It's known that for a metric space with doubling measure $(X,\mu)$, the Lebesgue differentiation theorem holds , i.e. If $f:X\to \mathbb{R}$ is a locally integrable function, then $\mu$-a.e. points ...
mafan's user avatar
  • 471
7 votes
1 answer
183 views

Stability Question for Isotopies Between Compact Sets

Suppose $X, Y$ are compact sets in $\mathbb{R}^2$ and $F$ is an ambient isotopy carrying $X$ onto $Y$. Is there an ambient isotopy $F'$ agreeing with $F$ on $X$ and which is constant in a ...
John Samples's user avatar
7 votes
0 answers
626 views

Does local strict contractibility imply ANR?

Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ ...
Bruce Blackadar's user avatar
7 votes
3 answers
3k views

Is a connected separable locally euclidean Hausdorff topological space second countable?

This question arose from considering for a connected smooth Hausdorff manifold the (possible) equivalence of the following properties: (1) paracompact, (2) metrizable, (3) second countable, (4) ...
TaQ's user avatar
  • 3,584
7 votes
1 answer
246 views

A notion of restricted injectivity for Banach spaces

I apologize in advance if this is well-known. Let $X$ be a Banach space. Let's call only for this post that $X$ is self-injective if for every closed subspaces \begin{equation} A\subseteq B\subseteq X ...
Onur Oktay's user avatar
  • 2,605
6 votes
1 answer
223 views

Minimal Hausdorff topologies compatible with a bunch of functions

Let $X$ be an infinite set, let ${\cal F}$ be a set of functions $f: X\to X$. We say that a topology $\tau$ is compatible with ${\cal F}$ if every $f\in {\cal F}$ is a continuous function $f:(X, \tau)\...
Dominic van der Zypen's user avatar
6 votes
1 answer
1k views

Prove that the flow of a divergence-free vector field is measure preserving

On page 3 of this preprint, after recalling the definition of flow generated by a vector field, the authors remark that "a necessary condition for a flow $\varphi_t(\cdot)$ generated by $a(t, \cdot)$ ...
Riku's user avatar
  • 839
6 votes
1 answer
1k views

Lipschitz function of independent subgaussian random variables

This question was asked here, but I have reason to believe that it's a serious research question appropriate for this forum (also, the answers given at the link aren't satisfactory). ​If $X\in\mathbb{...
Aryeh Kontorovich's user avatar
6 votes
1 answer
227 views

Quantum group representations from (convolution) matrix units?

Let $A=F(\mathbb{G})$ be the algebra of functions on a finite quantum group with a Haar state $$h=:\int_\mathbb{G}:F(\mathbb{G})\rightarrow \mathbb{C}.$$ There is a convolution product on $A=F(\...
JP McCarthy's user avatar
  • 1,037
6 votes
3 answers
655 views

When does the generalized Cantor space embed in a $\kappa$-compact space

The generalized Cantor space is the space $2^\kappa$, with basic open sets $$ [\sigma] := \{f\in 2^\kappa : \sigma\subseteq f\}, $$ for $\sigma\in 2^{<\kappa}$. A space is $\kappa$-compact if ...
Boaz Tsaban's user avatar
  • 3,104
6 votes
1 answer
261 views

Convergent filters generated by (not necessarily countable) chains

Suppose $\langle X,\mathscr{O}\rangle$ is a topological space and let $\mathscr{O}_x$ be the family of all open neighbourhoods of $x\in X$. Let $\mathscr{F}$ be the filter generated from $\mathscr{O}...
Rafał Gruszczyński's user avatar
6 votes
2 answers
1k views

Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks
Ali's user avatar
  • 4,143
6 votes
1 answer
680 views

Is there an operator algebraic reformulation of the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
Sebastien Palcoux's user avatar
6 votes
3 answers
2k views

Properties of the category of compact Hausdorff spaces

What, from a categorical rather than topological point of view, are the interesting properties of the category of compact Hausdorff spaces? In particular, is it the case that every monomorphism is ...
Theo111's user avatar
  • 71
6 votes
1 answer
765 views

An equivalence relation on the space of polynomials in one complex variable

Let $P(z)$ be a polynomial with complex variable $z$. We consider the following distribution for the roots of $P(z)=0$: the distribution is a triple $(n_{1},n_{2},n_{3})$ where these integers are ...
Ali Taghavi's user avatar
6 votes
2 answers
729 views

Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)?

The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a ...
user's user avatar
  • 83
5 votes
1 answer
493 views

Modulus of continuity of flow for non-Lipschitz vector fields satisfies Osgood condition

An Osgood modulus of continuity is an increasing function $\omega:(0,1]\to(0,1]$ such that $\int_0^1\frac{dt}{\omega(t)}=\infty$. We say a vector field $X$ satisfies Osgood condition with modulus $\...
Liding Yao's user avatar
5 votes
1 answer
419 views

When is there an unbounded tower in $[\mathbb{N}]^\infty$?

(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.) This question assumes familiarity with combinatorial cardinal ...
Boaz Tsaban's user avatar
  • 3,104
5 votes
0 answers
228 views

What is the smallest number of hyperplanes covering $\ell_2$?

For a Banach space $X\ne \{0\}$, let $\mathrm{cov}_H(X)$ be the smallest number of hyperplanes covering $X$. By a hyperplane in a Banach space I understand any closed affine subspace of codimension ...
Taras Banakh's user avatar
  • 41.9k
5 votes
1 answer
3k views

Inner product of linear bounded operators between Hilbert spaces

Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces. Can we equip $L(X,Y)$ with a natural inner product? I think it should look like $\...
shuhalo's user avatar
  • 5,327
5 votes
2 answers
454 views

Is each locally compact group topology on the permutation group discrete?

Question. Is each locally compact group topology on the permutation group $S_\omega$ discrete? Here $S_\omega$ is the group of all bijections of the countable ordinal $\omega$. A group topology on a ...
Taras Banakh's user avatar
  • 41.9k
5 votes
1 answer
312 views

"Weird-open" maps in topology

Given topological spaces $X$ and $Y$, we define an open map from $X$ to $Y$ to be a map of sets $f\colon X\to Y$ satisfying the following condition: For each $U\in\mathcal{P}(X)$, if $U$ is open in $...
Emily's user avatar
  • 11.8k
5 votes
1 answer
294 views

Regularity of the Radon transform with respect to the original function

Consider a function $f: \mathbb{R}^{d} \rightarrow \mathbb{R}$ (whose properties are to be specified). I note $\mathbb{S}^{d-1}$ the hypersphere and the Radon transform of $f$ defined for $(t,\theta) \...
Titouan Vayer's user avatar
5 votes
2 answers
459 views

Backward heat equation and forward perturbed heat equation well posed?

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
Sascha's user avatar
  • 536
5 votes
0 answers
263 views

Are continuous self-maps of the Golomb space $\mathbb G$ dense in the space of all self-maps of $\mathbb G$?

The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ ...
Taras Banakh's user avatar
  • 41.9k
5 votes
1 answer
356 views

Are the polyhedral cones the only examples of cones that remains closed when they are added to vector subspaces?

Let $C \subset \mathbb{R}^{n}$ be a closed convex cone. If one wants to know whether the linear map $T:\mathbb{R}^{n} \to\mathbb{R}^m$ sends the closed set $C$ to another closed one, $T(C)$, it is ...
R. W. Prado's user avatar
5 votes
2 answers
231 views

Squaring a space with the fixed-point property

We say that a space $(X,\tau)$ has the fixed point property (FPP) if for every continuous map $f:X\to X$ there is $x\in X$ with $f(x) = x$. What is an example of a space $X$ with FPP such that $X^2$ (...
Dominic van der Zypen's user avatar
5 votes
1 answer
600 views

When is the generalized Cantor space $\kappa$-compact?

My M.Sc. student has the following question, that I assume has an answer in the literature, and we are looking for references. The generalized Cantor space is the space $2^\kappa$, with basic open ...
Boaz Tsaban's user avatar
  • 3,104
5 votes
3 answers
730 views

Is it possible to connect every compact set?

Let $X$ be a "nice" space: metrizable, connected, locally path connected perhaps. Let $K\subset X$ be a compact set. Is there a always a compact connected $L\subset X$ such that $K\subset L$? This ...
erz's user avatar
  • 5,529
5 votes
1 answer
287 views

Is each compactification of $\mathbb N$ soft?

Definition. A compactification $c\mathbb N$ of the countable discrete space $\mathbb N$ is defined to be soft if for any disjoint sets $A,B\subset\mathbb N\subset c\mathbb N$ with $\bar A\cap\bar B\ne\...
Taras Banakh's user avatar
  • 41.9k
5 votes
1 answer
269 views

A question on semi-stratifiable space

This question is also posted here. A space $X$ is callled semi-stratifiable space if it has a $g$-function such that: for any point $x$ of $X$ and a sequence $\{x_n\}$ of $X$ if $x \in g(n,x_n)$, ...
Paul's user avatar
  • 654
5 votes
1 answer
1k views

Is every distribution a linear combination of Dirac deltas?

My question is whether Dirac-type distributions over an Abelian group define a basis of the Schwartz-Bruhat space $\mathcal{S}(G)^\times$ of tempered distributions on $G$, so that any distribution $f\...
Juan Bermejo Vega's user avatar
4 votes
0 answers
114 views

Find at least one square-boxed subcontinuum

Recall that a plane continuum is a closed, bounded, connected subset of the plane. It is non-degenerate if it contains at least two points. (We may sometimes just say "continuum" even if we ...
Mirko's user avatar
  • 1,375
4 votes
6 answers
2k views

Real functions with finitely many zeroes

I am looking for as general a class as possible of real functions defined on $\mathbb{R}^+$ that are guaranteed to have a finite number of zeroes - no, polynomials are not enough :). Specifically, ...
Yair Carmon's user avatar
4 votes
0 answers
111 views

Does an interior point necessarily pass through the boundary under a homotopy?

It's a straightforward exercise to show that if a point moves continuously from the inside of a set to the outside, it necessarily passes through the topological boundary of the set. This question is ...
ldrinehart's user avatar
4 votes
1 answer
1k views

Quotients of standard Borel spaces

Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
SBF's user avatar
  • 1,655
4 votes
0 answers
2k views

metric entropy for Lipschitz functions

Suppose $(X,d)$ is a metric space of unit diameer and let $F$ be the collection of all $1$-Lipschitz functions mapping $X$ to $[-1,1]$, equipped with the sup-norm $||\cdot||_\infty$. I am interested ...
Aryeh Kontorovich's user avatar
4 votes
0 answers
220 views

improved regularization for $\lambda$-convex gradient flows

It is well-known that gradient-flows of convex functionals are "parabolic" in some vague sense, and accordingly solutions tend to regularize instataneously. In the abstract context of gradient flows ...
leo monsaingeon's user avatar

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