All Questions
18,180 questions
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On the operators from $l_{p}$ into Tsirelson's space $T$
Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
- 3,343
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answers
81
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Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?
According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
$\...
- 121
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votes
0
answers
54
views
Left introversion operators associated to function spaces on semigroups
I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
- 1
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1
answer
123
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The monotone operator in $BV$ space
I am considering the following minimizing problem:
$$
\min_{u\in BV(\Omega)}\{\frac12\|u-u_0\|_{L^2}^2 + |u|_{TV(\Omega)}\}
$$
where $u_0\in BV(\Omega)$, $\Omega\subset \mathbb R^2$ is open bounded, ...
- 679
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44
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Is it possible to estimate the Interaction information of three variables without knowing their joint distributions?
I want to have a measure of the "synergy" between two players in a game. Each player has its own win ratio (won/played), which I'm modeling as two binomial distributed random variables X and Y. A ...
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0
answers
260
views
Concluding that the Poisson kernel is indeed the Cauchy distribution?
See here.
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. We see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\...
- 13
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votes
1
answer
186
views
Poisson kernel, $E^{(x, y)}\text{exp}\{i\theta X_t - \theta Y_t\} = e^{i\theta x - \theta y}$
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Let $W_t = (X_t, Y_t)$. How do I see that for any $\theta \in \mathbb{R}$ and any $t \ge 0$, we have$$E^{(x, y)}\text{exp}\{...
user76167
0
votes
1
answer
81
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An asymptotic set containment problem [closed]
Given a set of $n\in\Bbb N$ integers $\mathcal S$, suppose we choose two sets:
$$\mathcal S_{\mathsf{small}}\subseteq\mathcal S$$
$$\mathcal S_{\mathsf{big}}\subseteq\mathcal S$$ with cardinalities
...
user76479
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0
answers
371
views
Harmonic function with Dirichlet boundary condition
Consider the domain $D = \{(x_1, x_2,.., x_n) \in \mathbb{R}^n : 0 \leq x_i \leq 1\}$. Let $D$ be divided into two parts $D_1$ and $D_2$ by the hyperplane $H = \{x_1 = \frac{1}{2}\}$. My question is: ...
- 1
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0
answers
355
views
Summing up costs over a Markov chain
I apologize in advance if this question is too simplistic to be appropriate for MathOverflow. I have inquired in multiple places but have found little to indicate that this is a previously studied ...
- 101
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0
answers
510
views
Composition of upper semi-continuous real valued function with upper semi-continuous matrix valued function
Say that a matrix valued function $A: \mathbb{R} \rightarrow \mathbb{R}^{n \times n}$ is upper semi-continuous at $x_0$ if
$$ \limsup_{x \rightarrow x_0} A(x) \preceq A(x_0), $$
where $\preceq$ ...
- 1,590
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answers
76
views
What is the success probability of this stochastic process?
Suppose you have $k$ black balls and $X\cdot k$ white balls.
The procedure start with you having a bag containing $y\le k$ white balls (e.g. $k+1,\ldots k+y$).
In every iteration:
A single white ...
- 1
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0
answers
2k
views
Probability two random intervals overlap
I'm working on an algorithm for orthogonal line intersection detection and am trying to analyze some things about it. For simplicity, we can consider the problem as follows:
Given N randomly ...
- 101
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answers
115
views
When do block sequences yield disjoint subspaces?
Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to $(...
- 3,115
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194
views
Derivative of a cdf with respect to a parameter
Given two independent Random Variables $X$ and $Y$ with known distributions, I would like to know if I can say that the expression
$$
\operatorname{Pr}( f (t'+Y-X)+Y-X < z)
$$
is increasing in ...
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0
answers
63
views
The union of weighted compact supported continuous function
Let $\Omega\subset \mathbb R^N$ be open. Given a weight function $v\geq 1$ such that $v\in L^1_{\text{loc}}(\Omega)$ and $l.s.c$. Also supposethere exists a Lipschitz continuous sequence $v_n$ such ...
- 679
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107
views
Under which conditions is the union of conic hulls of sets in a cartesian product equal to $\mathbb{R}^N$?
Question: Under which conditions on $A, B\in\mathbb{R}^{N\times N}$ is the function $f: \mathbb{R}^N\mapsto \mathbb{R}^N$,
$$f(v) = A[v]_+ + B[-v]_+$$
surjective? Here $[.]_+$ is an elementwise ...
- 123
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answers
188
views
One parameter family differentiable dependence for linear parabolic pde's
Consider for example, the Black Schole's equation
$$
\partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0
$$
on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The ...
- 383
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0
answers
173
views
Is this has anything to do with Riesz representation?
The Riesz representation is very useful in study BV space. There is a lot of version of it and one of the good one can be found in this book, page 49.
Here I come up with a question which has similar ...
- 679
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answers
54
views
Differentiation of $(u(t),v(t))_{L^2(\Omega)}$ when $u, v \in H^1(I\times \Omega)$
Let $I=(0,\infty)$. Consider $u, v \in L^2(I;H^1(\Omega))$ with $u_t, v_t \in L^2(I;L^2(\Omega))$ where $\Omega$ is a bounded doamin.
Is it true that
$$\frac{d}{dt}(u(t),v(t))_{L^2(\Omega)} = (u'(t), ...
- 1
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votes
0
answers
144
views
Unitarizability of group representations
Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...
- 21.8k
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answers
407
views
What does the Plancherel theorem say about positive-definite distributions?
I'm trying to understand the answer to this MO question: Bochner's theorem for measures of positive type, which suggests a relationship between Bochner's theorem and the Plancherel theorem.
The ...
- 3,799
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answers
117
views
Ergodicity property for continuous-time Harris positive Markov process
I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If $\Phi$ ...
- 109
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0
answers
109
views
solutions of elliptic linear pde depending analytically on a parameter
Fix $ \Omega$ a bounded smooth domain in $ R^N$ and suppose $0<w(x)$ is a smooth solution of $ -\Delta w(x)=w(x)^2$ in $ \Omega$ with $ w=0$ on $ \partial \Omega$ (were are assuming $2< \frac{N+...
- 1,385
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104
views
Why is this distribution exponential?
Take the interval $[0, 1]$.
Now sample 10000 points in this interval randomly according to the uniform distribution.
The fact is that the distribution of the distances between adjacent points on ...
- 193
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votes
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answers
165
views
Expected length of minimum spanning trees
For a simple, finite, connected and complete graph $K_n = (V(K_n), E(K_n))$ with vertex set $V(K_n)$ and edge set $E(K_n)$, we assign a non-negative independent and identical distributed random weight ...
- 1
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votes
0
answers
74
views
conditionning by a Gaussian field
I know that if $(X,Y)$ is a Gaussian vector, then $(X|Y=y)$ is a Gaussian vector which covariance matrix is explicit in function of the covariance matrix of $(X,Y)$, and does not depend on $y$.
What ...
- 1,352
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320
views
Gromov-Hausdorff distance measure between minimum spanning trees
I am trying to compare minimum spanning trees through time. I have two questions:
1-Is it possible to measure the similarity between two minimum spanning trees with Gromov-Hausdorff distance measure ...
- 1
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0
answers
266
views
Maximizing Expected Utility
I am currently trying to solve a maximization problem given by
$\max_{f(x)} \int_0^1 \int_\mathbb{R} (c-y\cdot f(x)-d\cdot (x+f(x)-b)^2) \ h(x) \ dx \ dy$.
Or in other words, I have a utility ...
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0
answers
137
views
Heat asymptotics
Consider a compact manifold $M$ with smooth boundary, with either the Dirichlet or the Neumann boundary conditions. Consider a (time-dependent) open ball $B_t \subset M$. Given a fixed $u \in L^1(M)$, ...
- 11
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358
views
Boundedness of heat semigroup on $L^1(\Omega)$
On a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary, consider the Laplacian $-\Delta$ with either the Dirichlet or Neumann boundary conditions. More generally, one can also consider ...
- 1
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0
answers
187
views
About equivalence of two fractional Sobolev/Hilbert spaces
Let $(\varphi_k, \lambda_k)$ be the eigenelements of the Neumann Laplacian. It's possible to define a space
$$H(\Omega) = \{ u \in L^2(\Omega) \mid \sum_{k \geq 1}\lambda_k^{\frac 12}|(u,\varphi_k)_{L^...
- 251
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votes
0
answers
377
views
Densely-defined operator with closed range: conditions for operator closed
Suppose we have Banach spaces (or Hilbert spaces) $X$ and $Y$,
and a densely-defined linear operator $A : \operatorname{dom}(A) \subset X \rightarrow Y$ that is densely-defined and with closed range. ...
- 5,327
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votes
0
answers
150
views
smoothness of green's function in wave equation
I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...
0
votes
0
answers
81
views
Approximation property of Fréchet if range is restricted to an embedded Hilbert space
Let $W$ be a separable Fréchet space, and $H\subset W$ be a separable Hilbert space that is continuously embedded (equivalently, the topology of $H$ is stronger than the subspace topology generated by ...
- 320
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answers
119
views
Estimating the number of colors in a bucket
This question was previously posted to Math Stack Exchange here.
Suppose we have a bucket containing a large (but known) number of balls. Each of the balls has a color. We don't know how many colors ...
- 1,302
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201
views
Range of a trace preserving completely positive projection
I'm working in $M_n(\mathbb{C})$, the algebra of complex $n\times n$ matrices. I managed to build a completely positive, trace preserving, star preserving, projection $P$. That is
$$\text{Tr}(P(A)) = ...
- 515
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answers
444
views
How to decide a value of learning rate for Stochastic Gradient Descent?
I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla E_n(...
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0
answers
583
views
When an integral with respect to a Poisson point process is finite?
Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ \...
- 724
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256
views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is $$\sum_{\alpha}\frac{z^{\...
- 325
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answers
114
views
Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map $...
- 5,529
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104
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Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?
The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...
- 1,771
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0
answers
127
views
Approximation Property: Characterization
Problem
Given a Banach space $E$. Denote compact sets by $\mathcal{C}$, compact operators by $\mathcal{C}(X,Y)$, and finite rank operators by $\mathcal{F}(X,Y)$.
Suppose it has the approximation ...
- 598
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156
views
Compact Approximation
This thread originated from MSE: Compact Approximation
This is meant as lemma for: Approximation Property
Given a Banach space $E$.
Denote compact domains by $\mathcal{C}$.
Denote compact ...
- 598
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votes
0
answers
351
views
Existence of a complementary closed subspace extending a given subspace
Let $H$ be an infinite dimensional (separable) Hilbert space (or any infinite dimensional Banach space in which every closed subspace has a complementary subspace). Suppose that $X$ and $Y$ are closed ...
- 3,115
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votes
0
answers
145
views
Discrete measures and discrete kernels
This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
- 215
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1
answer
136
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$\epsilon$-nearly isoclinic
Question: Two $k$-dimensional subspaces $W_1,W_2$ with associated orthogonal projections $P_1, P_2$ are isoclinic with parameter $\lambda \ge 0$ if $P_1P_2P_1=\lambda P_1$ and $P_2P_1P_2=\lambda P_2$. ...
- 23
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0
answers
117
views
The functional $L(\varphi)=\int_0^{2\pi}\frac{\sqrt{1-\varphi^2-(\varphi')^2}}{1-\varphi^2}d\theta$
Consider the $2\pi$-periodic inner product space $L^2[0,2\pi]$.
Let $F\triangleq\{f\in L^2[0,2\pi]|f(\theta)>0,(f(\theta),\cos\theta)=(f(\theta),\sin\theta)=0\}$.
Let $G\triangleq\{\varphi\in L^2[...
- 139
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0
answers
302
views
Banach space of discontinuous functions on a product space
Edit: According to comments of Eric Wofsy and Yemon Choi I edit the question.
For a (compact) topological space $X$, we put $A=\{f:X\to \mathbb{C}\mid f\text{ is bounded}\}$. We define a semi-...
- 356
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votes
0
answers
179
views
Non-reflexive linear subspace
We know that if X infinite dimensional normed space, then weak topology smaller than normed topology.
This is my problem(from russian textbook of Bogachev-Smolyanov, Functional Analysis) :
Let X be ...
