All Questions
13,927 questions
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Is there a Tychonoff space $X$ such that ....?
$X$ is not a separable submetrizable, i.e.($iw(X)>\omega$) $X$ has not a countable injective weight.
There is a Baire isomorphism 1-class between $X$ and a separable metrizable space $Y$.
- 1,101
0
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1
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104
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Poisson Equation across a Hypersurface [closed]
Let $\mathbb{B}(0,1) \subset \mathbb{R}^3$ denote the unit ball. Let $\Gamma = \{x_3=0\}$. Let us assume $f \in L^2(B)$ .Consider the problem
$ \triangle u = f $ in $\mathbb{B}$ in the weak sense such ...
- 4,115
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1
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218
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Heat semigroup dissipative
Consider the heat semigroup on $L^1(\mathbb{R}).$ I would like to know if the generator of this semigroup is dissipative in the sense of this definition.
On $L^2$ it would be completely trivial, but ...
- 167
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129
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Fundamental Surfaces in 3-manifolds
Given a 3-manifold $M$ with a triangulation $T$, will every essential surface in $M$ be a fundamental one? If not, then what are the conditions on $T$ so that these essential surfaces become ...
- 1
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235
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Criterion for weak compactness
Let $F$ be a metrizable locally convex space (you may assume it is a Banach space), and let $E$ be a complete locally convex space (you may assume it is a Frechet space). Let $T$ be a continuous ...
- 5,529
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110
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Number theory for operator bound
Let $\gamma_i$ be such that for even $i$ $\gamma_i=1$ and for odd $i$ $\gamma_i$ shall have absolute value $1$ and the product of all of the odd ones is also on the complex unit circle but not 1 or -1....
- 501
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82
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Semi-embeddings and weak compactness
Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space.
Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}...
- 5,529
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1
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112
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A classical fact about linear operators in Hilbert spaces
Studying the formulations which arise in hybridized mixed methods (say, mixed finite element method + hybridization), I got stuck with a rigorous proof of the following simple fact:
Let $\varphi$ be ...
- 256
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1
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268
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Linear operator has one-dimensional kernel
Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
- 329
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233
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Is $(\ell^1(\mathbb N_0),\sigma(\ell^1,\ell^\infty))$ not quasi-complete?
In Jarchow's Locally Convex Spaces this not being quasi-complete is asserted on page 206 referring to Corollary 11.4.4 on page 228 saying that a Banach space is reflexive if and only if its closed ...
- 3,584
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1
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151
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A Bi-Lipschitzian application
We say that $\Omega$ is a star-shaped domain (with respect to the origin) of $\mathbb R ^n$ if :
$$\Omega = \{x\in \mathbb R ^n : \left \| x \right \| < g(\frac{x}{\left \| x \right \|})\}\; \...
- 291
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votes
2
answers
99
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Characterization of separable functionals
Under what conditions is a functional $f(\vec{a},\vec{b})$ separable with rank $r$? That is, when can it be expressed as
\begin{equation}
f(\vec{a},\vec{b})=\sum_{i=1}^r A_i(\vec{a})B_i(\vec{b}),
\end{...
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130
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Bounded components of the complement [closed]
Hi everyone: Suppose $A\subset \mathbb{R}^{m}$ ($m>1$) is a closed set with empty interior. Which are the necessary and sufficient conditions that $A$ have a neighborhood $V$ such that every ...
- 411
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267
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Invariance of sets under Schrödinger equations
We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$
$$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$
$$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$
...
- 115
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1
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150
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Solutions to Schrödinger equation parameter dependence
This is somewhat unrelated to what I normally do in mathematics, which is why it may be obvious to some of you, but I was puzzled by this:
If we look for classical solutions on $[0,1]$ to
$$-y''(x) =...
- 305
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1
answer
364
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About Semi-simple Commutative Banach Algebra
Is the norm on a unital semi-simple commutative Banach algebra with $\|I\|=1$, unique? ($I$ denotes the identity element)
- 2,118
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198
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The eigenfunctions of an operator commuting with all rotations.
When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator ...
- 199
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1
answer
697
views
How much do we know about this "local" Hardy-Littlewood maximal function?
The "local" Hardy-Littlewood maximal function is given by $$(M_\phi f)(x)= \sup_{0<\epsilon<1}|\phi_\epsilon \ast f|(x),$$ which is similar to the classical Hardy-Littlewood maximal function : $$...
- 171
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1
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328
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Discrete Calderon-Zygmund operators
I would like to know whether there exists a Calderon-Zygmund theory discrete singular kernels. In particular I am interested when the discrete operator $T$ with kernel $K(n,m)$ given by
$$(Tf)(n)=\...
- 153
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votes
1
answer
174
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Applying min-max to find a critical point in a ball
Let $\mathbb B^n$ be an open unit ball in $\mathbb R^n$ and let $F$ be a smooth function on it. Let $\frac{1}{2}\mathbb B^n\subset \mathbb B^n$ be an open ball or radius $\frac{1}{2}$. Let $\mathbb B^...
- 14.3k
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183
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Partially ordered set of compatible topologies
Let $X\neq \emptyset$ be a set and let ${\cal J} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ be a collection of non-empty subsets of $X$. We say that a topology $\tau$ on $X$ is ${\cal J}$-compatible ...
- 48.9k
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247
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Gradient bounds on Newtonian potentials
Suppose $N \ge 3$ and let $\Phi(x):= C_N |x|^{2-N}$ is the fundamental solution. Let $\Omega$ denote a bounded domain in $ R^N$.
Consider $ -\Delta u(x) = f(x) $ in $\Omega$ with $u=0$ on $ \...
- 1,385
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1
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445
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Domain of the Stokes operator
Let
$\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$)
$\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$
$\mathcal H:=\overline{\mathfrak ...
- 167
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votes
1
answer
138
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Space time Lesbesgue spaces
I have a function which lives in $f(x,t)∈L^2(0,T;H^{1/2})∩L^\infty(0,T;L^2)$
for a certain time interval. I also know that $\partial_{t} \ f(x,t)∈L^2(0,T;H^{−1})$. Can I assure that the function lives ...
- 13
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1
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217
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Reproducing Kernel Hilbert Spaces with positive kernels
In my research I'm dealing with the following question.
Let $E$ set, $K:E \times E \to \mathbb R$ a positive type function, and $\mathcal H := \mathcal H(1+K)$ (in the sense of the Moore theorem). ...
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272
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A condition for Laplacian
Let $u\in L^{2}(\mathbb{R}^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(\mathbb{R}^{2})$ where $c>0$.
Is true $-\Delta u \in L^{2}(\mathbb{R}^{2})$?
Thank you in advance.
- 135
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2
answers
609
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Strictly convex norm on an infinite-dimensional Hilbert space
Question: Consider the Hilbert space $ H=\ell^2(\mathbb{Z})$. Let ${\rm L}(H)$ be the set of all linear
operators on $H$ onto itself. Find a norm $N$ and a domain $DN\subset {\rm L}(H)$ for $N$ ...
- 21
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348
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Request for references about computing or estimating Rademacher complexity
Is Rademacher complexity defined for any space of functions?
Or are there restrictions on the function space over which this can be defined?
For example is the Rademacher complexity defined or has ...
- 617
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2
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137
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Level sets and integral of functions of two variables
Let $f_1,f_2$ be two positive functions on $\Omega_1, \Omega_2 \subset R^2$ with $f_1|_{\partial \Omega_1}=f_2|_{\partial \Omega_2}=0$. For every $\lambda>0$, denote the the area of the domain ...
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179
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Dense subspaces of $L^p(0,T;X)$
Given a Banach space $X$ and $1\leq p<\infty$, let's define the space $L^p(0,T;X)$ as the set of all strongly measurable functions $f:(0,T)\mapsto X$ such that
$$\int_0^T\Vert f\Vert_{X}^pdt<\...
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1
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193
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$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$
Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
0
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1
answer
163
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Norm of derivative of rank one projector
I asked this question on math.stack but I got no answer, so I try here.
Let $\phi(t)$ be a solution for the nonlinear Schroedinger equation\begin{equation}
i\partial_t\phi(t)=-\Delta\phi(t)+(V*|\phi|^...
- 113
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1
answer
371
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Is there relation between vector valued RKHS and interpolation space?
Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . In a different context interpolation space is defined in the wikipedia link: https://en....
- 495
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1
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119
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Rank of a generall linear group over a finite field [closed]
What is the rank (minimal number of group generators) of the group $GL(n,F)$, when $F$ is a finite field of odd order? I found that $SL(n,F)$ is $2$, but I can't find this information.
- 19
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491
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Is this set of function belongs to $L^\infty$?
Let $\Omega\subset \mathbb R^N$ be open bounded with smooth boundary. Let $u\in SBV\cap L^\infty(\Omega)$ be given. We write
$$
Du = \nabla u\lfloor \mathcal L^N + (u^+-u^-)\otimes \nu_u\mathcal H^{N-...
- 679
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52
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Compact $R_1$-spaces
A space $(X,\tau)$ is said to be $R_1$ if for all $x,y\in X$ with $cl(\{x\}) \neq cl(\{y\})$, there are disjoint open sents separating $cl(\{x\})$ and $cl(\{y\})$.
If $X$ is compact and $R_1$, does ...
user62017
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732
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Reproducing Kernel of a RKHS of continuous functions may not be continuous in two variables together
Let $\mathcal{K}$ be a Hilbert Space of continuous functions on some topological space, where point evaluations are continuous linear functional on $\mathcal{K}$.
That is $\mathcal{K}$ is RKHS, ...
- 3
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1
answer
57
views
If $X$ has the "discrete" covering property, how about $X^2$?
We say that a space $X$ has covering property (C) if the following holds:
(C) For any open cover ${\cal U}$ of $X$ there is a closed discrete set $D\subseteq X$ and a map $\varphi: D\to {\cal U}$ ...
user70876
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1
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781
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How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?
Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary.
My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that
$\...
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154
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Sobolev type embedding
Consider a compact manifold $M$ and a point $q \in M$. Let us say that
that the following inequality holds:
$$
\Vert \varphi u\Vert_{L^p} \leq C\Vert \varphi u\Vert_{H^1},$$ where $\varphi \in C^\...
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299
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Can the image of a disk have nontrivial Hausdorff measure for $1 < d < 2$?
I am reading a blog which talks about a $C^1$ diffeomorphism $f: \mathbb{D}\{ x^2 + y^2 < 1\} \to \mathbb{R}^2$ and estimates the Hausdorff dimension of its image $\mathcal{H}_\sqrt{2}^d (f(\mathbb{...
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114
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Priestley topologizability and connected components
This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley space....
user70876
0
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1
answer
216
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Upper bound for a ratio of modified Bessel functions
I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...
- 1
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1
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517
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Injective inclusion map from RKHS function space to $L_p(\mu)$
Let $X$ be a measurable space, $\mu$ be a $\sigma$-finite measure on $X$, and $H$ be a separable reproducing kernel Hilbert space over $X$ with a measurable kernel $k$.
At a certain part in a proof I ...
- 103
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1
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146
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Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?
We formally write the solution of nonlinear Schrödinger equation (NLS) as follows:
$$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$
where $U(t)= e^{it\Delta} $(free ...
- 1,051
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votes
1
answer
197
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How to minimize this sparse quadratic function?
There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
- 1
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1
answer
568
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How to solve this differential equation with an infinite sum?
I would like to find solutions of the following differential equation:
$ \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x)$
For example in space of function from $\mathbb R^*$ to $\mathbb ...
- 1,199
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1
answer
247
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Galerkin Projection on Integral Operators
I am looking at a research paper that mentions integral operators (which in this case is brought up in reference to shading equations that are integral operators) and it says that we can create a ...
- 103
0
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1
answer
705
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Continuity of a Functional
A certain functional $T$ is defined as:
$$T(F)=\int_{(0,1)}F^{-1}(s)M(ds)$$
where $M$ is a probability measure with support $[\alpha,1-\alpha]$,for $\alpha>0$.
The result that above functional is ...
- 111
0
votes
1
answer
163
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Existence of half-planes with respect to regular open sets of the Euclidean plane
I initially asked this question at math.stackexchange.com but there was no reaction, so I thought this may be a good idea to transfer it to mathoverflow.net
Let $\langle\mathrm{r}\mathscr{O},\mathord{...
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