All Questions
13,927 questions
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172
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Can we build a continuous function from "fibers"/preimages defined over a topological base?
I am looking for some proof insight or literature references for a statement which, if it's actually true, is probably a pretty trivial thing. I hope the question has not been asked here before, and ...
- 273
0
votes
1
answer
557
views
Paley-Wiener type theorem for integral functions with compact support
If $f\in L^{1}(\mathbb{R}^n)$, and $f$ has compact support, then how can I show that $\hat{f}$ cannot satisfy $\hat{f}(x)=O(e^{-\epsilon|x|})$ for any $\epsilon>0$?
This is similar in the spirit ...
- 879
0
votes
1
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272
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Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result
Let $F:\mathbb{R} \to \mathbb{R}$ be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. $F(r) = r^{\frac 1m}$.)
Let $\Omega \...
- 266
0
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1
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277
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both convex and superharmonic function on manifold concave?
M is a non-compact Rimannian manifold without boundary.
$f\in W_{loc}^{1,2}(M)$ satisfies $\Delta f \leq c$ in the weak sense, i.e.
$$
-\int_M \langle \nabla f,\nabla \psi \rangle dvol \leq c\int_M \...
- 689
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votes
1
answer
130
views
Extending affine maps defined on weakly closed sets to the whole topological space
Given $C$ a weakly closed convex subset of a (real) Banach space $B$, with $0\in C$ and $\varphi:C\longrightarrow \mathbb{R}$ weakly continuous, with $\varphi(0)=0$, can we extend $\varphi$ to a $\...
- 647
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votes
1
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241
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Nonlocal (parabolic) PDEs in the Sobolev space setting
Can someone recommend me some literature on nonlocal parabolic problems (eg. of the form
$$u_t + (-\Delta)^s u = f$$
where the nonlocal operator is the fractional Laplacian)
in the setting of Sobolev ...
- 155
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votes
2
answers
797
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If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?
Let
$$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is ...
- 193
0
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1
answer
212
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Some convergence similar to weak-$\ast$ convergence on the space of finite measures
I have a question:
Let $D$ be the space of cadlag functions defined on $[0,1]$ and $V$ be its subspace consisting of $x$ with finite variation and $x(0)=0$.
Define $TV(x)$ as the total variation ...
- 1,835
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715
views
The dual space of the Dirac measures on an Abelian group
Let $G$ be a Hausdorff locally-compact Abelian group and $L^2(G)$ the Hilbert space of two-integrable complex functions on the group.
Question. What would be natural vector space $\mathcal{R}$ of ...
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2
answers
664
views
Defining surface integral on boundary of $C^1$-domain
Let $\Omega$ be a bounded $C^1$ domain with bounded boundary $\partial\Omega$. Can someone point me to a reference where the surface integral of a measurable function $f\colon \partial\Omega \to \...
0
votes
1
answer
162
views
Extracting moments from a special Z-transform
Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that
\begin{equation}\frac{X_n-n\mu}{\sqrt{n}\sigma}...
- 41
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1
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452
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Relative interior and dense subsets
(This is a cross-post from here.) Let $A,B\subseteq \mathbb R^d$ be non-empty, such that $B\subseteq \overline A.$ For $S\subseteq\mathbb R^d$ define the relative interior of $S$ by $$\text{ri}(S)=\{s\...
- 215
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2
answers
319
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Fixed point theorem that does not require the hemi-continuity of the set valued map?
All of the fixed point theorem I have seen (like Kakutani and Brower, Browder) required the set valued map to be hemi-continuous (lower). Is any fixed point theorem that can assure the existence of ...
- 383
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2
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212
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Is there a normal space that is not uniformly normal
Let $(X,\mathcal D)$ be a uniform space and $A,B\subseteq X$. Let's say $A$ is uniformly inside $B$ and write $A\le B$ iff there's some entourage $D$ for which
$$(\forall a\in A)(D[a]\subseteq B)$$
A ...
user31967
0
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1
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252
views
Which algebra of functions can be represented as $C(X)$
I don't know if this problem is known or not, so any information would be appreciated:
Let $\cal A$ be an $\Bbb{R}$-algebra of (not necessary continuous) real valued functions defined on a ...
- 9
0
votes
1
answer
496
views
Trace, eigenvalues and functional calculus
Let $T$ be a (possibly unbounded) self-adjoint operator on a Hilbert space. Assume that we for some reason know that the point spectrum of $T$ consists of a finite number of eigenvalues $\lambda _1, \...
- 450
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votes
1
answer
168
views
local moments of measures whose Fourier transform vanish in an interval
Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form
\begin{equation*}
\int_{-\delta}^{+\delta}|h|(dt)\le C(\...
- 859
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votes
1
answer
216
views
Find a special element in group algebra
Let $$G=\langle x, y, z\mid xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle,$$ denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the ...
- 1,528
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1
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151
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Is there any result concerning on the metric dimension of inverse limit?
To be specific, my question is as follows:
Question: Let $X$ be an inverse limit of compact metric spaces $(X_i, d_i)$, then does it hold
$\dim(X, d) \leq \sup_i \{\dim (X_i, d_i)\}$ for some ...
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1
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321
views
How to handle a scalar product in an integral?
I am having a problem with a certain inequality I try to understand. I think it's just a basic idia (/trick) I'm missing, but I can't seem to find it.
Here's a simplification of the problem:
$ \...
0
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1
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319
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continuty of volume of a convex set in Rn [closed]
Let O(X) be the metric space of all compact subsets of a compact set X in Rn and let L be an element of O(X). Let vol(L) be the volume of L. How do we prove that vol(L) is a continuous function on O(X)...
- 1
0
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1
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402
views
A question on cofinite topology.
Let $X$ be a countably infinite (or larger) set with the cofinite topology. for every $x\in X$ is there exists a family $\xi\subset\tau$ such that $\lbrace x\rbrace=\bigcap\xi
$ ? If the answer is yes,...
- 13
0
votes
1
answer
321
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Increasing regularity for $L^2$ function
Suppose that we have a function $u$ on $\mathbb{R}^2$ such that $r\frac{\partial}{\partial\theta}u \in L^2(\mathbb{R}^2)$. Suppose it is also given that $u$ lies in some fractional Sobolev space $H^s(\...
- 1
0
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1
answer
305
views
Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
- 10.5k
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1
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499
views
How identify bounded Borel measurable functions
Let $S$ be a topological semigroup, and $M(S)$ be bounded, regular complex Borel measures on $S$. How can we identify bounded Borel measurable functions with elements in $M^*(S)$?
- 109
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231
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A question on linearly lindelof space
Let $X$ is a linearly lindelof subspace of $Z$ and $b$ is not $\omega$-separated from $X$, i.e., for any closed $G_\delta$ set $P$ of $Z$ which contains $b$, $P\cap X \not=\emptyset$. If $\tau < \...
- 654
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1
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814
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a problem in functional analysis that erdos solved in 2 lines
https://math.stackexchange.com/questions/261685/paul-erdoss-two-line-functional-analysis-proof .
does anyone know about what the problem was and what was his solution.
[Edit by quid:] please follow ...
- 2,106
0
votes
1
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851
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Example of a completely regular spaces
A topological space $X$ is an $EF$-space if if for any
two collections $\mathcal{U}$ and $\mathcal{V}$ of clopen subsets
of $X$ with $\bigcup \mathcal{U}\cap \bigcup
\mathcal{V}=\emptyset$, we have $\...
- 192
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1
answer
483
views
Absolute norms and 1-unconditional sums
Absolute norm
Let $X$ and $Y$ be Banach spaces. Let $Z=X\times Y$ a norm $\|\cdot\|_N$ on $Z$ is called absolute if there is a function $N\colon R^2\rightarrow R$ such that
$$
\|(x,y)\|_N=N((\|x\|, \|...
- 3
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1
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87
views
Question regarding closure of sets defined by the vanishing of holomorphic functions
Consider the following subsets of $\mathbb{C}^n$ given by
$$ X := \{x \in \mathbb{C}^n: f(x) =0, ~~g(x) \neq 0 \} $$
$$ Y := \{ x \in \mathbb{C}^n: f(x) =0, ~~g(x) =0, ~~h(x) \neq 0 \} $$
where $f, g$...
- 3,245
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1
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549
views
One-dimensional Hausdorff measure of preimages
Let $\Omega$ be an open subset of $\mathbf{R}^n$. For a mapping $f: \Omega\to \bf{R}^n$, what kind of condition ensures that the one-dimensional Hausdorff measure of $f^{-1}(E)$ is zero whenever $E$ ...
- 1,881
0
votes
1
answer
916
views
Bounding derivative of a function
Consider $a(t)\in\mathbf{L}^{2}(\mathbb{R})$ and $a(t)>0$, is a low pass smooth function with $\hat{a}(f)=0, |f|>f_{max}$. Can we have a upper bound on the following,
$\Big|\frac{a'(t)}{a(t)}\...
- 151
0
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2
answers
765
views
About a generalization of the Radon Nikodym Theorem
Im trying to prove a generalization of the Radon Nykodym theorem, but im having troubles even for finite measures, could someone help?
Let $\mu$ and $\nu$ two $\sigma$-finite measures in $\(X,\...
- 11
0
votes
1
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341
views
Length of intersection of intervals
Can anyone prove this statement? It seems true, but I'm finding it tricky to give a concise proof.
Fix $\alpha\in[0,1]$. Let $\mu$ be Lebesgue measure. Define $B(c,r)\equiv[c-r,c+r]$, where $[\cdot, ...
- 500
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votes
1
answer
1k
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Showing a coercivity condition for this bilinear form
Suppose $\Omega \subset \mathbb{R}^n$ is a compact domain. Let $f$ and $J$ (and also $\frac 1J$) be $C^1$ functions on $\Omega$. Consider the bilinear form $a:H^1(\Omega) \times H^1(\Omega) \to \...
- 107
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1
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229
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Complemented subspaces of $\ell_p(I)$ for uncountable $I$
I was looking for an article mimicing result of Pelczynski for $\ell_p$. I have found this one
Rodriguez-Salinas, B. (1994). On the Complemented Subspaces of $c_0(I)$ and $\ell_p(I)$ for $1 < p &...
- 1,697
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1
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488
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Discrete Sobolev space of $R^n$ valued maps
Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \...
- 17
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1
answer
156
views
Does homeomorphism preserves the family of cones?
Let me state my problem. Suppose we have a ball $B$ in standard $\mathbb{R}^3$, that is a $\varepsilon$-neighbourhood of $0$ point. Suppose we have a family of cones $X_C = \lbrace C > 0 \vert x^2 +...
- 165
0
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1
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238
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A property of a quasiperiodic function
Let F be a continuous periodic function on R^N. Let a,b be vectors in R^N. Also assume a is not parallel to b.
Does the limit of
$\varepsilon \int_0^{1/\varepsilon} F(as+b/\varepsilon) ds$
Exist ...
- 213
0
votes
2
answers
818
views
Application of inverse function theorem to get short time existence
I am reading a book on curve shortening flow. Optionally, please see this image for the page that is confusing me (I am not allowed to include it in this post since I'm new): https://i.sstatic.net/...
- 45
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3
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404
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Some Questions about zero-dimensional subsets of the unit interval related to cantor set
Let $\mathbb{P}$ denote the set of all irrational numbers in the open segment$(0 , 1)$. let $K$ be the intersection of $\mathbb{P}$ and the standard cantor set and $H=\mathbb{P}-K$. as you know these ...
- 1,788
0
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2
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146
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representation of compact supported distribution
Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
- 9
0
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1
answer
338
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The part of an operator as an analytic generator
Let the operator $A$ be the generator of an analytic semigroup on a Banach space $X$.
Let $Y$ be another Banach space embedded in $X$. We consider$A_Y$, the part of $A$ in $Y$, defined as the ...
- 271
0
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1
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864
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Sequence of smooth functions converging to sgn(x)
I'm looking for a sequence of smooth functions $f_i(x)$ converging to Sign$(x)$, each of which additionally have the following property:
\begin{equation}
f_i(x_1+x_2) = g_i(x_1, f_i(x_2))
\end{...
- 561
0
votes
1
answer
666
views
A Cauchy–Schwarz Type Inequality Involving Scaled Distributions
I have stumbled upon a rather intriguing inequality involving the product of the scaled distribution and the scaled density of a random variable. The inequality has a very attractive form, and it ...
- 197
0
votes
1
answer
438
views
Möbius Transform of a Continuous Possibility Function
In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.
There is ...
- 103
0
votes
1
answer
494
views
Sheaf of sections and local triviality
This is probably not a research level question, I'm sorry if it is inappropriate. I'm reasking here this question on math.se.
Suppose that $\xi: E \to B$ is a bundle (by which I mean simply a ...
0
votes
1
answer
498
views
Quotient of \ell_1 by space of finite sequences
The following question came up during a reading of Rudin's functional analysis. I have not been able to find any information through searching online, but I apologise if the answer is obvious, or the ...
- 11
0
votes
1
answer
2k
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What does it mean to have Zero Density (mathimatically) [closed]
I read a question that asked "prove that the set of all positive integers expressible as the sum of two integers square has zero density." Now I was under the impression that something was dense iff ...
- 93
0
votes
2
answers
415
views
Commutative *-subrings of the noncommutative C*-algebra $B(l^2)$
A $\star$-ring is a ring with an involutive anti-automorphism. The simplest example of a noncommutative $\star$-ring is perhaps $B(l^2)$, the ring of bounded linear functions on the sequence space $l^...
user2529