All Questions
1,491 questions with no upvoted or accepted answers
1
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63
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Direct proof of fact $u \in C(U)$ satisfies $|Du| \ge 1$ in sense of viscosity if and only if property holds
Is it possible anybody could sketch me a direct proof of the fact that $u \in C(U)$ satisfies $|Du| \ge 1$ in the sense of viscosity if and only if the following property holds?
If $V \subseteq U$ is ...
- 349
1
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0
answers
105
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Positivity of solution of Poisson equation
Let $B$ denote the unit ball centered at the origin in $R^N$ and take $N \ge 3$. Let $( \phi_k(\theta), \lambda_k)$ for $ k \ge 0$ denote the Eigenpairs of $ -\Delta_\theta$ on $S^{N-1}$ which are $L^...
- 1,385
1
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0
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99
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simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
- 43.2k
1
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0
answers
105
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compactness of sequence of harmonic functions
Let $ \Omega$ denote a smooth bounded domain in $ R^N$ and let $u_m \in C^\infty( \overline{\Omega})$ harmonic functions. We also suppose $ u_m$ is bounded in $L^2(\Omega)$ (uniformly in $m$).
...
- 1,385
1
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0
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194
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Cotlar-Stein's Lemma and the Dirichlet kernel
It is well-known that Cotlar-Stein's Lemma can be used to prove the $L^2$ boundedness of the Hilbert transform. See e.g. $L^2$ boundedness of the Hilbert transform via Cotlar-Stein Lemma. Then using ...
- 171
1
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0
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331
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Verifying a claim regarding $H^1$ weak convergence and $L^2$ strong convergence on a surface
I'm reading a paper whose first section discussed $H^1$ maps defined on star-shaped sets, but I got stuck in verifying a claim for quite a while. I'm now thinking the claim is wrong, but it's hard to ...
- 1,350
1
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0
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99
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Existence of a viscosity solution
Setup
I'm trying to find sufficient conditions for the existence of a viscosity solution to the following PDE,
$$
f(t,s,z) + \partial_sf(t,s,z) \\
+ \sum_{i=1}^{\infty} \left[
\partial_{z_i} f(t,s,z)...
- 5,405
1
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0
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128
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determine when $e^{ikx}$ can be boundary value of a holomorphic function
Assume that $\Gamma=\{x+if(x): x\in \mathbb{R}\}$ is a graph, separating $\mathbb{C}$ into two connected components. Let's denote the one below $\Gamma$ by $\Omega$.
My question is, for what curves $...
- 31
1
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0
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311
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Estimating an integral involving Bessel functions
I would like to preface this question by saying that I have asked a series of questions on this topic on Math Stack Exchange, but have almost never received any fruitful responses, with the exception ...
- 161
1
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0
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53
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Given a fixed convex domain $\Omega$ in 3D, for what value $c$ the function $f(c) := \int_{\partial \Omega} |x-c| d \sigma_x$ gets its minimum?
Let $\Omega$ be a bounded smooth convex domain in $\mathbb{R}^3$, then consider the following minimization problem:
$$\inf_{c \in \overline{\Omega}} f(c), \quad f(c) := \int_{\partial \Omega} |x-c| ...
- 1,350
1
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0
answers
123
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Generalization of concave envelope
Let $g:\mathbb R_+\to\mathbb R$ be a measurable function (which could be supposed to be bounded and Lipschitz if required). Let $\mathcal P$ be the collection of probability measures $\mu$ on $\mathbb ...
- 1,835
1
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0
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58
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A question on Integral inequality
Let $0 < \epsilon < 1$. Consider $\{a_n\}_{n \geq 1} \in l_2$ and $L(t) = 1+\epsilon t$. Let $x$ be fixed such that $0 < x < L(t)$. Does there exist $\tau \geq 0$ such that the following ...
- 11
1
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0
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106
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Improper integral of products and ratios of probability density functions
I am trying to find out whether the following integral is finite. The integrand consists of product of probability density functions.
$\int \frac{f(x_1,x_2, x_4^*)}{f(x_1^*,x_2, x^*_4)}\frac{f(x_1,...
- 11
1
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0
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181
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Stochastic increasing convex ordering
Consider $n \geq 2$ and the simplex
\begin{equation}
\Delta=\{(p_1,\cdots,p_n) \in \mathbb{R}^{n} \mid \forall i, p_i \geq 0 \text{ and } \sum_{i=1}^{n}{p_i}=1\}
\end{equation}
Suppose that $\Delta$ ...
- 111
1
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0
answers
86
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Least restrictive condition such that $-y''(x)+q(x)y(x)=\lambda y(x)$ has two solutions
Let $\lambda \in \mathbb{R}$ and consider some function $q$ on the interval $[0,1].$
What is known to be the least restrictive condition on $q$ such that there are two linearly independent $H^2$ ...
- 11
1
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0
answers
447
views
Largest possible variance for log-concave distributions on a bounded interval
Let $f$ be the density of a log-concave probability distribution on the interval $[0,1]$ (with respect to Lebesgue measure). To be concrete, suppose that $f(x) = \exp( - \varphi(x))$, for some convex ...
- 251
1
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0
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106
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Identifying a notion of integration
Let $f$: $I\longrightarrow\mathbb{R}$ be a (not necessarily bounded) function on an interval $I\subseteq\mathbb{R}$.
Suppose $f$ admits a function $F$: $I\longrightarrow\mathbb{R}$ such that
(1) $F$ ...
- 277
1
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0
answers
117
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The eigenfunction of modified $1$-laplace equation?
Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation
$$
-\Delta u=0
$$
has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
- 679
1
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0
answers
161
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level sets portrait near a critical point
Let $f:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a smooth ($C^{\infty}$) function and $O$ be an
isolated critical point of $f$. I am looking at the local level sets diagram
near $O$ from topological ...
- 45
1
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0
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197
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A certain measure on Banach algebras
According to the comments of Nate Eldredge I did revise the question. In particular I change "$C^{*}$ algebras" to "Banach algebras".
Is there a reference who introduce the following measure on ...
- 356
1
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0
answers
71
views
The jump set of $SBV$ function for different value of parameter in image denoising problem
The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary,
$$
u_\...
- 679
1
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0
answers
111
views
Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?
I also put this question on MSE here
Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable).
Let $\...
- 679
1
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0
answers
104
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Show this function is strictly concave
Please help me show that $f(w)$ is strictly concave in $w\in[0,\infty)$:
$f(w)=\sum_{j=1}^N P_j (w)\cdot u_j $
where
$P_j (w)=\sqrt{w}\int _{-\infty}^{\infty}\Pi_{k\neq j}\{\Phi[\sqrt{w}(v-u_k)]\}...
1
vote
0
answers
60
views
Optimizing sum of approximate and exact functions
This is a research question that I had asked in Math.SE about a month ago, but even after putting a bounty on it, I did not get any answers.
I have two real values functions, where one ($g(w;x):\...
- 189
1
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0
answers
460
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Quotient of two smooth functions extension
Assume we are given smooth functions $f, g: U \to \mathbb{C}$, where $0 \in U \subset \mathbb{R}^n$ is open and $0 \in g^{-1}(0) \subset \{x_n = 0\}$. Furthermore, suppose that $\nabla g \neq 0$ on ...
- 501
1
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0
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84
views
extension for a complex operator
Let be $\lambda>0$. Put
$$ L_{\lambda}=\Big[-\frac{\partial^{2}}{\partial z \partial \overline{z}}+\lambda^{2}|z|^{2} +\lambda\Big(\overline{z}\frac{\partial}{ \partial \overline{z}}-z\frac{\...
- 135
1
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0
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192
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The decay rate of the spectrum of the Gaussian kernel on compact manifolds
It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
- 617
1
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0
answers
93
views
Multimodal property of polynomial logistic distribution
Let $P(x)$ be a polynomial (of an odd degree $n$) strictly increasing on $(-\infty, +\infty).$
Then $F(x)=\displaystyle \frac{1}{1+\exp\{-P(x)\}}$ is a distribution function of a polynomial logistic ...
- 783
1
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0
answers
334
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A problem on Markov chains and Dirichlet forms
Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy
$$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$
$$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$
$$c(x,x)=0\text{ for ...
- 369
1
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0
answers
120
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Interpolation functional for BV spaces?
Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
- 121
1
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0
answers
109
views
Pointwise convergence of a sequence of approximate limits of BV functions
So, let $\Omega\subset \mathbb R^2$ bounded and consider a sequence of functions $\{u_k\}_{k\in\mathbb N}\subset BV(\Omega)$ and $u\in BV(\Omega)$ such that $u_k\rightarrow u$ weakly* in $BV(\Omega)$. ...
1
vote
0
answers
143
views
stochastically decreasing sequence converges in distribution
Let $(X_i)_{i=1}^\infty$ be independent nonnegative integer valued random variables. Suppose that $X_n \succeq X_{n+1}$ (in the stochastic dominance sense). Does it follow that $X_n \overset{d}\to X$ ...
- 457
1
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0
answers
69
views
Norm-averaging reference request
(Apology in advance for the broadness of this question) I recently came across a relatively simple application where I needed to "balance" the "spreaded-out-ness" of a function with the "peaked-ness" ...
- 123
1
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0
answers
190
views
Roots of generalized homogeneous polynomials
A polynomial $P(\xi)$ of $n$ real or complex variables is homogeneous of order $m$ with respect to $\lambda \in \mathbb{Z}_{+}^n$ if $$P(\xi) = \sum\limits_{\substack{(\alpha, \lambda) = m \\ \alpha \...
1
vote
0
answers
200
views
Differentiability criterion in the Zygmund class
Let $ f: \mathbf{R}^{m} \rightarrow \mathbf{R} $ be a continuous
function, $ \omega $ be a modulus of continuity and assume
$$ | f(x+h) +f(x-h) -2f(x) | \leq \omega(|h|)|h| $$
whenever $ x,h \in \...
- 541
1
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0
answers
79
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An inequality for integral on spheres
I have a question concerning to the integral on sphere. It's maybe true and simple but I don't know how to prove it. Could anyone have some suggestions? Thanks.
Denote $S^{n-1}$ the unit sphere in $R^...
- 379
1
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0
answers
92
views
Perturbation in Besov space
$\|f\|_{B^{0}_{p,p}}=(\sum_{j\geq -1} \|\Delta_j f\|_p^p)^{1/p}$ is the Besov norm of $f$.
Here the Fourier transform of $\Delta_jf~(j\geq 0)$ is $\psi(2^{-j}\xi)\hat{f}(\xi)$ and $\psi$ is a smooth ...
- 515
1
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0
answers
90
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Expansion of a power series as integral of cosine functions
Suppose $$f(x)=\sum_{k=0}^\infty (-1)^k x^{2k} \int_0^1 p(\xi_{2k})\int_0^{\xi_{2k}}q(\xi_{2k-1})\cdots\int_0^{\xi_3}p(\xi_2)\int_0^{\xi_2}q(\xi_1) d\xi_1 \;d\xi_2\cdots d\xi_{2k-1}\;d\xi_{2k},$$
...
- 587
1
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0
answers
308
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Inverse Laplace transform of a non-negative function
Consider an entire function $f$, which is real for real arguments and satisfies $f(s)\geq 0$ for all $s\in\mathbb{R}$. Furthermore, assume this function is a Laplace transform,
$$
f(s)=\int_0^\infty e^...
- 1,488
1
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0
answers
182
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The real method of interpolation and operator ideals
Let $\overline{A} \mbox{ and } \overline{B}$ be n+1-tuples of Banach spaces and $T:\overline{A}\rightarrow \overline{B}$ be an interpolation operator; let $J(\overline{A})$ and (the corresponding, ...
1
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0
answers
102
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monotonicity of a function
I want to know if the function below is monotonically decreasing for all $a,b >0, a\neq b $
\begin{equation}
x\rightarrow \frac{\sinh^2((a-b)x)}{\sinh(2ax)\sinh(2bx)} \text{, $x >0. $}
\end{...
- 21
1
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0
answers
130
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An analytic family of in fact non-existent improper Riemann integrals
Question:
Are there any useful interpretations or "applications" of the formula
$$\intop_0^\infty e^{-ax}\frac{\sin x}{x}\,dx=\frac{\pi}{2}-\arctan(a)\qquad \forall\;a\in \mathbb{R},
$$
in which the ...
- 1,942
1
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0
answers
158
views
On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $
This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $...
- 1,396
1
vote
0
answers
157
views
Laplace method with "bad" zero set
It is well-known that if $\phi:\mathbb{R}^n \longrightarrow \mathbb{R}$ is a function with $\phi(0) = 0$, $\phi(x)>0$ if $x \neq 0$ and $D^2\phi(0) \geq 0$, then the integral
$$\int_{\mathbb{R}^n} ...
- 9,853
1
vote
0
answers
70
views
The jump set of $SBV$ function over a hyper surface
Assume $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. Also assume $S\subset \Omega$ is a smooth hyper surface such that $0<\mathcal H^{N-1}(S)<+\infty$.
Now, given a positive ...
- 679
1
vote
0
answers
75
views
Characterization of certain families of functions
For $R$ equal $\mathbb{R}$ or $\mathbb{Z}$, let $D^+_R:=\{(x,y)\in R^2\colon x<y\}$. For each natural $n$, let $F_{n,R}$ denote the set of all Borel-measurable functions $f\colon D^+_R\to\mathbb{R}$...
- 128k
1
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0
answers
205
views
A Question about compactness of an embedding into $L^p$ spaces
Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} \frac{u^...
- 371
1
vote
0
answers
488
views
concavity of a vector function
I'm given a function $g:\mathbb{R}^n \mapsto \mathbb{R}$, $g(y) = \prod_{i\in[n]} (1+y_i\cdot c_i)$, where $c_i>0$.
Let $e_a,e_b$ be two arbitrary standard vectors. It is easy to show that for any ...
- 341
1
vote
0
answers
178
views
Density of subspace with nonlocal/Wentzell boundary condition
Given the space $F$ defined by:
$$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$
I want to prove that the subspace $E$ of $F$ defined by $E=\...
- 111
1
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0
answers
42
views
Error bounds for approximation with dyadic sums of polynomials
Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?
In the 2-dimensional case the ...
- 13.2k