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Does $L^1$ convergence preserve the regularity of this sequence of functions?

Let $f_n$ be a sequence of $L^1(]0,1[)$ functions such that $f_n$ is non-decreasing, at least left-continuous, $f_n(0^+) <0$, $f_n(1^-) >0$, for all $n \in \mathbb N$. This sequence converges $...
W. Volante's user avatar
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89 views

Families of Riccati Equations for a closed subspace?

Let $N$ be a positive integer, fix constants $a_i,b_{i,j},c_{k,i},d_{k,i,j}\in \mathbb{R}$, and let $X$ be the subspace of all $f\in C^2(\mathbb{R};\mathbb{R}^d)$ satisfying: $f=f_0e_0+\sum_{i=1}^N ...
ABIM's user avatar
  • 5,405
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135 views

Help showing F is weakly lower semicontinuous

Given a compact subset $\Omega$ of $\mathbb{R}^N$, I wonder if $$F(u)=\int_\Omega f(u)\ dx =\int_\Omega (1-|u|^2)^2\ dx$$ is weakly lower semicontinuous (w.l.s.c) on $H^1(\Omega)$, meaning that if $\...
R. N. Marley's user avatar
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275 views

Are Bernstein polynomials bounded by their coefficients?

I am representing a function by nth order Bernstein polynomial coefficients. I have bounded the coefficients between some $f_{min}$ and $f_{max}$. From what I can see experimentally, it appears that ...
rmyas's user avatar
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0 answers
81 views

Example of a sequence of logarithmically convex functions on $\mathbb{R}$ and for all $n\in\mathbb{N}$ in the spirit of one evoked in an article

To ask this question I was inspired in some words, if I understand well, from the authors of a preprint on arXiv in section 4.1, that I believe that is [1], to ask next question. We consider the ...
user142929's user avatar
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66 views

Is there a dyadic cube decomposition where edge length is comparable to L^2 averages?

Suppose I have some measurable function $f : B_1(0) \to [0,1)$, which is pointwise very small, i.e. $\|f\|_{\infty} << 1$. I'm looking to construct some kind of dyadic cube decomposition or ...
SBK's user avatar
  • 1,179
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237 views

Semicontinuity of the Lebesgue measure of images of a family of functions

Let $\mu$ be the usual Lebesgue measure on $\mathbb{R}^m$. Suppose $f:\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^m$ is uniformly continuous, and let $U$ be a measurable subset of $\mathbb{R}^n$....
stepanp21's user avatar
  • 326
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0 answers
185 views

Express $\zeta(n)$, in particular the Apéry's constant, in terms of series involving Gregory coefficients or the Schröder's integral

The idea and motivation of this post is to know if it is possible to express integer arguments of the Riemann zeta function $\zeta(n)$, in particular $\zeta(3)$, in terms of series involving the so-...
user142929's user avatar
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55 views

Smooth compactly supported function with good scaling with respect to the fractional Laplacian

Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user avatar
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0 answers
479 views

What are the sets on which norm-closedness implies weakly closedness?

Let $X$ be a Banach space. Let $C$ be a convex, and normed-closed subset of $X$. It is well-known that $C$ becomes weakly closed subset of $X$. I want to know is there any well-know class of non ...
Red shoes's user avatar
  • 369
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0 answers
159 views

Limiting property of polylogarithm ratio

I try (without success) to figure out what could be the following limit if any... For real $s$ strictly $> 1$ and $x \rightarrow +\infty$ (x real) the limit of the polylogarithm ratio $\frac{Li_{s-...
Gianfranco OLDANI's user avatar
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0 answers
77 views

Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$ over all ...
Ben Ciotti's user avatar
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54 views

Existence of a solution for the Laplace equation with sub-linear non-linearity

At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for $$ \begin{cases} -\Delta u = u^p & \Omega \subset \...
Hheepp's user avatar
  • 371
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0 answers
221 views

Positivity of an integral

Let $f(x)=x-\theta(x)$, and for $x\ge 2$ let $J(x)=\int_x^{\infty}\frac{f(y)(1+\log y)}{y^2\log^2 y}dy$. Furthermore, $J(2)>0$. Suppose $J(x)>0$ when $f(x)=0$. Does it follow that $J(x)>0$ ...
EGME's user avatar
  • 1,018
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75 views

Does there exist $\alpha>0, \beta\in (0,1)$ such that $\dfrac{\sum_{k=1}^n a_k}{n}\le \alpha (a_1\cdots a_n)^{1/n} + \beta \max_i(a_i)$ holds?

Let $a_1\ge a_2\ge \cdots\ge a_n\ge 0$ be given non-negative numbers. My question is the following: Is there any $\beta \in(0,1),\ \alpha>0$, such that $$\dfrac{a_1+\cdots+a_n}{n}\le \alpha (a_1\...
Samrat Mukhopadhyay's user avatar
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0 answers
63 views

Feller semigroups and fractional operators

Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user avatar
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1 answer
246 views

Cross entropy loss is not twice differentiable?

I was reading a recent theory paper in machine learning by Kenji Kawaguchi and Leslie Pack Kaelbling https://arxiv.org/pdf/1901.00279.pdf and the authors seem to suggest in section 2.2 that cross-...
ted's user avatar
  • 283
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0 answers
117 views

Harnack Inequality for uniformly elliptic PDE via constructing a singularity

I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
David's user avatar
  • 1
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0 answers
72 views

Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
Boby's user avatar
  • 671
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115 views

If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$

Let $$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$ that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$ Question: Let $\|\...
Idonknow's user avatar
  • 623
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1 answer
212 views

Exterior cone condition for $\mathrm{supp}\, u$ and Lebesgue points of $u$

Let $u:\mathbb{R}^n \to \mathbb{R}$ be an $L^1$ function with compact support. Let $\bar x \in \partial \mathrm{supp}\, u$ and assume that $\mathrm{supp} \, u$ satisfies the exterior cone condition at ...
user avatar
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0 answers
113 views

Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$

Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$. If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user avatar
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1 answer
275 views

An extension for lower semi continuous lower bounded real valued functions class

Let $(X,d)$ be a complete metric space. I need some explanations about the class of all functions like $f$ which have $f:X \to \mathbb{R}\cup\{ +\infty\}$, be a lower bounded and, for all $y \in X$ we ...
M. Reza. K's user avatar
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1 answer
51 views

Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
megggs's user avatar
  • 13
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0 answers
170 views

What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?

I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking, in that the residuals from the ...
Paul B. Slater's user avatar
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0 answers
75 views

Dense Egoroff theorem

Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given. ...
ABIM's user avatar
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0 answers
124 views

Does Hartogs's Theorem for complex-analytic functions hold for real-analytic functions? [duplicate]

Recall a very famous theorem due to Hartogs for complex analytic functions of several variables. Hartogs's Theorem Let $f$ be a holomorphic function on a set $G \setminus K$, where $G$ is an open ...
Boby's user avatar
  • 671
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0 answers
126 views

$p$-volume of $n$-dimensional hyper-ellipsoids

I read that the unit hypersphere has maximum volume for dimension five and would like to generalize this result. (If you think that integrating over an $n$-dimensional $p$-hyper-ellipsoid area ($x_1^...
Hauke Reddmann's user avatar
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0 answers
64 views

Probability of collision of sums of vectors

Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function. Does there exist a random matrix $R \...
Christopher's user avatar
0 votes
1 answer
113 views

Verifying that a map to $L^2_{\text{loc}}$ is continuous

Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
geometricK's user avatar
  • 1,903
0 votes
0 answers
112 views

On certain integrals of exponential functions with respect to Gaussian measures

I have questions about the integral $$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$ for $a,b,c>0$. What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
S.Z.'s user avatar
  • 505
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0 answers
60 views

Solution of a functional equation with cosine transform

What are the functions verifying: $$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$ With $\lambda$ a constant ? (Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
Bertrand's user avatar
  • 1,199
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0 answers
324 views

Adjoint of differential equation

Motivation: Consider the ODE $$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation $$y'(t)=A^*y(t).$$ I ...
Umberto's user avatar
  • 83
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0 answers
58 views

$N-$Green function in $\mathbb R^N$

Let $N \geq 3$. Does there exist solution of the following equation $$-\Delta_N G + G^{N-1} = \delta_0,$$ where $-\Delta_N = - \text{ div}(|\nabla \cdot |^{N-2} \nabla \cdot )$ denotes $N-$Laplace ...
nguyen0610's user avatar
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0 answers
299 views

When convolution with exponential kernel is bounded

Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying $...
Saj_Eda's user avatar
  • 395
0 votes
0 answers
93 views

Changing Couplings of Discrete Random Variables

Let $X,Y$ be two discrete random variables. Two joint mass distributions (couplings) with marginals $X$ and $Y$ and with entries $p_{i,j}=\mathbb{P}_1(X=i,Y=j)$ and $p_{i,j}'=\mathbb{P}_2({X=i,Y=j})$ ...
The Substitute's user avatar
0 votes
0 answers
124 views

Reference for the Hardy maximal function on the torus

I am searching for a reference for the (sharp) Hardy maximal function on the torus $\mathbb{T}^2:=\mathbb{R^2}/\mathbb{Z}^2$, for instance I would need result result of the following type : if $g\in H^...
Ayman Moussa's user avatar
  • 3,425
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0 answers
311 views

Approximations of Polylogarithm and Lerch transcendent?

For the Gamma function $\Gamma(x+1)$, we have beautiful approximations of the function in terms of elementary function, such as the Stirling approximation and its refinements, that give sharp ...
MCH's user avatar
  • 1,324
0 votes
1 answer
347 views

Asymptotic behaviour of fixed points in permutations

For any $n\in\mathbb{N}$ let $S_n$ denote the set of all permutations (bijective maps) $\pi:\{1,\ldots, n\} \to \{1,\ldots,n\}$. For $\pi \in S_n$ we set $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
Dominic van der Zypen's user avatar
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0 answers
71 views

Existence of local minimizer

For a $f\in C^3$ function, if there is a sufficiently small $\epsilon$ $$\| \nabla F(x) \| < \epsilon$$ and a sufficiently large $\alpha$ where $$\lambda_{\min}[\nabla^2 F(x)] \ge \alpha$$ Can ...
Nikolayevich's user avatar
0 votes
0 answers
81 views

Differential operator and equivalence

Here is the problem: I have a certain PDE and there is the nonlinear terme $h$, I have as data: $f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$ Now on consider the fnction $$h(...
Gustave's user avatar
  • 617
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0 answers
140 views

Lipschitz extensions preserving the convex hull of the range

We assume that $X$ is a metric space and that $A \subseteq X$ is a subset. Let $f : A \rightarrow \mathbb R$ be a Lipschitz-continuous function with Lipschitz constant $L$. By the Kirszbraun theorem, ...
shuhalo's user avatar
  • 5,327
0 votes
0 answers
308 views

Invertible operator

We consider the operator $$T=I + {{{\partial ^2}} \over {\partial {x^2}}}:{H^2}(0,L) \cap H_0^1(0,L) \to {L^2}(0,L)$$ We hope to prove that $T$ is invertible if and only if $L = n\pi $. and for this ...
Gustave's user avatar
  • 617
0 votes
0 answers
58 views

in search of convergent daughter sequences

Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$. Question. Is there a subsequence $\{f_{...
T. Amdeberhan's user avatar
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0 answers
69 views

Quasi-stationary measure on a finite graph equals stationary measure?

Assume the simple random walk $X$ on the graph $G(V,E)$, s.t. $G$ is simple, undirected, finite, connected and let $B \subset V$, s.t. $V\setminus B$ is connected. Let $\sigma_B$ be the quasi-...
jondal's user avatar
  • 71
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0 answers
93 views

What is the class of real sequences satisfying these conditions?

I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions: $\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
zeraoulia rafik's user avatar
0 votes
0 answers
80 views

Comparison of two functions

Given a function $f$ from $R^2$ to $R$ satisfying tha following: $1)$ $f$ is a convex function which vanishes on $(0,1)$ and on $(1,0).$ $2)$ $f$ is a decreasing function on $x$ and on $y$ and $f$...
Khadija Mbarki's user avatar
0 votes
2 answers
144 views

Optimization function of two variables

Let $A, B, C, D \in \mathbb{R^*_+}$. Is it possible to solve $$ \max_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)} $$ The KKT conditions give for an extrema $(x^*,y^*)$ ...
user avatar
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0 answers
42 views

What (analytical or numerical) method can I use to solve scalar optimal problem?

I got the following optimization problem in mind and I am looking for some (analytic or numerical) methods to solve it. Can anyone have any ideas? Here is problem \begin{aligned} & {\text{...
Thomas Edison's user avatar
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0 answers
59 views

Restriction to Basis of Cadlag function

If $f \in L^2([0,T])$ then it can be written as $$ f(t) \triangleq \sum_{i \in \mathbb{N}} c_i e_i(t), $$ for some sequence $\{c_i\}$ of real numbers and a Schauder basis $\{e_i(t)\}$ of $L^2([0,T])$ ...
ABIM's user avatar
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