All Questions
13,927 questions
1
vote
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141
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Can a closed null-homotopic curve be filled in by a disc?
Let $U\subseteq\Bbb R^n$ be an open set and $\gamma\subset U$ a closed null-homotopic curve in $U$ (i.e. it can be contracted to a point). Then is there an embedded disc $D\subset U$ with boundary $\...
- 13.6k
8
votes
1
answer
264
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Does the continuous image of a disc contain an embedded disc?
Let $\phi:\Bbb D^2\to\Bbb R^n$ be a continuous mapping of the 2-disc $\Bbb D^2$ that is injective on the boundary $\partial\Bbb D^2=\Bbb S^1$. Does its image contain an embedded disc with the same ...
- 13.6k
6
votes
2
answers
348
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Does there exist a framework for determining if a power series is "differentially algebraic"
It is a well studied problem to take a function $f$ expressed (usually expressed as a solution to a differential equation w/ some initial conditions) and ask if it has an "elementary closed form&...
- 13.1k
5
votes
2
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342
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Projections in atomless von Neumann algebras
Let $\varepsilon>0$.
If we consider a sequence $\{f_n\}$ in $L_\infty(0,1)$, then there exists a very small subset $A$ of $(0,1)$ with $m(A)<\varepsilon$ such that $$\|f_n \chi_A\|_\infty =\|...
- 4,426
2
votes
0
answers
104
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Unordered configuration space with non-distinct points
Consider a topological space $X$, a natural number $n>0$ and
the quotient topological space $Q_n(X)$ of $X^n$ by the equivalence relation : $x\sim y$ if and only if
there is a permutation $\sigma$ ...
- 1,035
3
votes
1
answer
161
views
Approximating continuous functions from $K\times L$ into $[0,1]$
Let $K$ and $L$ be compact Hausdorff spaces, let $f:K\times L\to [0,1]$ be continuous and let $\varepsilon>0$. Can we find continuous $g_{1},...,g_{n}:K\to[0,+\infty)$ and $h_{1},...,h_{n}:L\to[0,+\...
- 2,830
1
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0
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210
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Is this a well known space? Perhaps homogeneous Sobolev-like space?
The homogeneous Sobolev space $\dot H^s(\mathbb{R}^n) $ is often defined as the closure of $\mathcal{S}(\mathbb{R}^n)$ under the norm
$$ || |\omega|^s \widehat{f} ||_{L^2(\mathbb{R}^d)} =\int_{\...
- 197
1
vote
0
answers
125
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Interpolating sequences are strongly separated
I am reading Agler and McCarthy's Pick Interpolation and Hilbert Function spaces. In Chapter 9, titled "Interpolating Sequences", the authors say that every interpolating sequence is ...
- 151
3
votes
2
answers
749
views
History of limit point compact -/-> compact example
A standard example in elementary topology (e.g. Munkres) of a space that is limit-point compact (every infinite subset of the space has a limit point) but not compact is the minimal uncountable well-...
- 4,786
82
votes
5
answers
6k
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How do the compact Hausdorff topologies sit in the lattice of all topologies on a set?
This question is about the space of all topologies on a
fixed set X. We may order the topologies by refinement, so
that τ ≤ σ just in case every τ open set is open in σ.
...
- 2,183
2
votes
0
answers
98
views
Geometric interpretation of uniform convexity condition
I first want to recall the moduli of uniform smoothness (US), uniform convexity (UC), asymptotic uniform smoothness (AUS), and asymptotic uniform convexity (AUC). Throughout, let $X$ be an infinite ...
- 41
1
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1
answer
126
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Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?
Does there exist an essentially nonzero function $f:[0,1] \mapsto \mathbb{R}$ so that
$$
\int_0^1 |y-x| f(x) \, dx = 0
$$
for every $y \in [0,1]$? I think I see how to show that any such $f$ can't be ...
- 6,486
11
votes
0
answers
172
views
Can the nowhere dense sets be more complicated than the meager sets?
Suppose $X$ is a completely metrizable space with no isolated points. Let $\mathcal{ND}_X$ denote the ideal of nowhere dense subsets of $X$, and let $\mathcal{M}_X$ denote the ideal of meager subsets ...
- 18.6k
4
votes
0
answers
197
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Compactness of the unit ball in the space of Radon measures w.r.t. the Kantorovich-Rubinstein norm
This question was posted previously but has not attracted any responses so I am repharising it in a slightly different language hoping to reach a wider community
Let $(X,d)$ be a pointed metric space ...
- 437
1
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0
answers
170
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Generalization of Borel functional calculus
[Repost from https://math.stackexchange.com/questions/4802593/generalization-of-borel-functional-calculus]
Let $A$ be a normal operator on a Hilbert space $V$. The continuous functional calculus gives ...
- 95
0
votes
0
answers
73
views
Computationally efficient solution for the measure of central tendency minimizing Lp loss for p > 1
We know that the measure of central tendency that minimizes the Lp loss is $\min_c \sum_{i=1}^n |x_i - c|^p$
For $p=1$ (L1 loss), this is the median. For $p=2$ (L2 loss), this is the mean. Both of ...
- 109
3
votes
2
answers
182
views
Maximal contractible-ish Hausdorff surfaces
For the duration of this question, let a "surface" be any connected Hausdorff topological space that is locally homeomorphic to R2. Note that we make no assumption about a countable base to ...
- 2,858
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votes
1
answer
161
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Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...
- 128k
0
votes
1
answer
509
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 684
9
votes
1
answer
1k
views
Noncompactness of the Sobolev embedding in the critical exponent case
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a Lipschitz boundary and $n > p \ge 1$.
It is well known that up to the critical exponent $p^* = pn/(n − p)$, i.e. $q < p^*$, the ...
CommunityBot
- 1
19
votes
1
answer
465
views
Large Borel antichains in the Cantor cube?
Let $2^\omega$ be the Cantor cube $\{0,1\}^\omega$, endowed with the standard compact metrizable topology and the standard product measure, called the Haar measure. The Cantor cube is considered as a ...
- 655
1
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0
answers
146
views
Intuition behind right-inverse of map from Johnson-Lindenstrauss Lemma
The Johnson–Lindenstrauss lemma states that for every $n$-point subset $X$ of $\mathbb{R}^d$ and each $0<\varepsilon\le 1$, there is a linear map $f:\mathbb{R}^d\to\mathbb{R}^{O(\log(n)/\varepsilon^...
6
votes
2
answers
4k
views
Question about Schauder bases in C([0,1]).
I would like to check a statement about Schauder bases in $C([0,1])$ to be sure that I don't lie to my students on Monday. The statement(s) that I would like to check are:
The family of monomials $\{...
- 60.6k
6
votes
1
answer
796
views
A Poincaré-like inequality
Is it true that for some real $K>0$ and all real $u\in C_0^\infty((0,1))$ we have
$$\int_0^1 (u'(x)^2+u(x)^2)\,dx\,\int_0^1 u(x)^2\,dx
\le K\Big(\int_0^1 x\,u'(x)^2\,dx\Big)^2\text{ ?}$$
- 141
11
votes
1
answer
1k
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Research topics in microlocal analysis
Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
- 63.9k
6
votes
1
answer
346
views
Best approximation of L1 function by Lipschitz function
Fix constant $L,C>0$ and $k\geq 1$ and let $f\in W^{1,k}(\mathbb{R}^d,\mathbb{R}^n)$ with $\|f\|_{W^{1,k}}\leq C$.
Is there a known estimate on the distance
$$
\|f - \operatorname{Lip}_L(\mathbb{R}^...
- 5,405
8
votes
0
answers
226
views
A variation of necklace splitting
Our problem is the following:
Let $n$ and $k$ be integers.
We are given two (unclasped) necklaces, each with $n$ colored stones: a top necklace which has $k$ colors and a bottom necklace which has 2 ...
- 63.9k
1
vote
0
answers
232
views
Show that $\mathbb{K}\cong M_{n}(\mathbb{K})$ [closed]
I would like to show the following isomorphy but not sure how to go about this:
$\mathbb{K}\cong M_{n}(\mathbb{K})$
Also in Blackadar (Operator Algebras, page 171) he states that this isomorphism ...
- 6,367
3
votes
1
answer
428
views
Any formula or estimates the Green function for the Laplacian in $3D$ periodic box?
Let $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ be the three-dimensional torus with sides identified. That is, I am considering the unit box $[0,1]^3$ with periodic boundary conditions.
In this case, I ...
- 646
3
votes
1
answer
422
views
How to find partial derivatives of the Beta Function?
I was reading the book (Almost) Impossible Integrals, Sums and Series. The author used a method involving taking partial derivatives of the Beta Function to solve some integrals.
$$B(x,y)=\int_0^1u^{x-...
- 188k
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}...
- 1,112
2
votes
0
answers
62
views
Localized estimate for divergence free vector field
Suppose $\Omega \subset \mathbb{R}^3$ is a simply connected Lipchitz domain. For a divergence free field $w\in [L^2(\Omega)]^d$, it is well known that there exists a vector field $v\in [W^{1,2}(\Omega)...
- 31
9
votes
1
answer
916
views
Inverse Fourier transform of an $L^2$ function as limit on balls
$B_m :=\{x \in \Bbb R^n : ||x|| \le m\}$ and $\mathscr{F}f$
denotes the $L^2$ fourier transform of an $f \in L^2(\Bbb R^n)$.
I am trying to show that
If $f \in L^2(\Bbb R^n)$ then $f(y)=\lim\limits_{...
CommunityBot
- 1
4
votes
0
answers
129
views
Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
CommunityBot
- 1
19
votes
0
answers
553
views
Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
- 484
4
votes
4
answers
1k
views
An example of a non-paracompact tvs (over the reals, say)
What is an example of a non-paracompact topological vector space?
I'm aware of this question, but I don't care if my tvs is locally convex. In fact the wilder the better. The only criterion is that ...
- 35.5k
1
vote
1
answer
128
views
Compare the weight of $p\vee q$ and that of $p+q$
Let $M$ be a von Neumann algebra.
If it has a semifinite faithful normal trace $\tau$, then we have $\tau(p\vee q)\le \tau(p)+\tau(q)$.
However, for the weight (even a faithful normal state) $\omega$ ...
- 42.8k
1
vote
1
answer
118
views
A Gaussian measure $\mu$ on $\mathcal{E}'(S^1)$ by Minlos theorem and its value for Sobolev spaces $H^{\alpha}(S^1)$
I posted this question on ME as "A Gaussian measure on $\mathcal{E}'(S^1)$ by Minlos Theorem and its value for $L^2(S^1)$",
but it seems much more nontrivial than I expected... so, I post an ...
- 6,367
1
vote
1
answer
88
views
Bounded $C_0$-semigroups on barrelled spaces are equicontinuous
I have the following question: Let $X$ be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let $(T(t))_{t\geq0}$ be a $C_0$-semigroup, ...
- 63.9k
1
vote
0
answers
123
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Dependence of Sobolev embedding theorem constant on smoothness
Assume that $\Omega \subset \mathbb{R}^d$ is "nice" enough and $k$ is a positive real number. Using the Sobolev embedding theorem, we can get that
$$
\|f\|_{W^{0,2d/(d-2k)}\ \ \ \ \ (\Omega)}...
- 63.9k
2
votes
1
answer
189
views
Equivalent characterization of weak derivative in Bochner space
Let $H$ be a hilbert space. A function $v\in L_\text{loc}^1(0,T;H)$ is called the weak derivative of $u \in L_\text{loc}^1(0,T;H)$ iff
$$ \int_0^T u(t) \varphi'(t) \, dt = -\int_0^T v(t) \varphi(t) \, ...
- 2,670
3
votes
2
answers
1k
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Local coordinates on (infinite dimensional) Lie groups, factorization of Riemann zeta functions
Given a (finite dimensional) Lie group $G$ (real $k=\mathbb{R}$ or complex $k=\mathbb{C}$) and its Lie algebra $\mathfrak{g}$, one can prove (a basis $B=(b_i)_{1\leq i\leq n}$ of $\mathfrak{g}$ being ...
- 35.5k
13
votes
2
answers
767
views
Smooth Urysohn's lemma on Fréchet spaces
Let $V$ be a Fréchet topological vector space.
Let $K_0$ and $K_1$ be two closed subsets which are disjoint.
I wish to show the existence of a Fréchet-smooth function $f:V\to [0,1]$
whose restriction ...
- 2,388
23
votes
5
answers
2k
views
PDEs and algebraic varieties
Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
- 2,689
11
votes
1
answer
341
views
Density of linear subspaces in $C(K)$
Let $K$ be a compact Hausdorff space and denote by $C(K)$ the space of all real valued and continuous functions on $K$. We endow $C(K)$ with the supremum norm topology, making it a Banach space.
...
- 60.6k
4
votes
0
answers
164
views
When $X$ is homeomorphic to $\mathscr{F}[X]$?
While I was talking to some colleagues, one of them said that there exists a topological space $X$ such that $X$ is uncountable, non-discrete and homeomorphic to $\mathscr{F}[X]$ (the Pixley-Roy ...
- 63.9k
8
votes
1
answer
351
views
"Compactness length" of Baire space
Intuitively, my question is: how many times do we have to mod out by an closed equivalence relation with all classes compact in order to collapse Baire space $\omega^\omega$ to a singleton?
In more ...
- 73.2k
3
votes
1
answer
220
views
Conditional expectation as square-loss minimizer over continuous functions
It is well-known that the conditional expectation of a square-integrable random variable $Y$ given another (real) random variable $X$ can be obtained by minimizing the mean square loss between $Y$ and ...
- 128k
1
vote
0
answers
103
views
Cyclic representation isomorphic to L2 space
This question is also posted on Math Stack Exchange.
I need some help understanding a proof of the following claim: every cyclic representation is isomorphic to some $L^2$ space.
First, formal ...
- 6,367
7
votes
0
answers
150
views
The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.5k