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Expectation equation, harmonic functions, do not understand why equation is true

Let $u: \mathbb{R}_+ \times \mathbb{R}^d$ be a bounded $C^2$ function whose first and second partial derivatives are uniformly bounded (or, more generally, have at most polynomial growth as $|x| \to \...
Wenliang's user avatar
1 vote
1 answer
632 views

Does sequence almost sure convergence imply almost sure convergence?

This is a cross-post of this and this questions from math.stackexchange.com since I have not received any response there. I would like to seek help here. Suppose $x(t,\omega): [0,T]\times\Omega\...
Hans's user avatar
  • 2,239
1 vote
1 answer
307 views

Convexity of discrete Fourier transform

Let $f : [0,2\pi] \to \mathbb{R}$ be a continuous convex function on $(0,2\pi)$ which is singular about $0$ and $2\pi$ but finite when evaluated at the boundaries. Assume also that $f$ is symmetric ...
spaceman's user avatar
  • 595
1 vote
1 answer
224 views

Sum of negative roots of a $5^{th}$ degree monic polynomial

Let $f(x)$ be a $4^{th}$ degree monic polynomial say $f(x) = x^4 + a_1x^3+a_2x^2+a_3x+a_4$ with the property that $a_1<0, a_4>0$ and $a_2<a_3$. They by Descartes' rule of signs we can ...
User8976's user avatar
  • 199
1 vote
1 answer
151 views

Monotone likelihood ratio of densities based on power function

Given $p,\phi,\theta \in \mathbb{R}$ such that $p>2$ and $0 \le \phi,\theta\le \pi/2$ define the density function: $$f(\phi;\theta) = \mbox{$\Large\frac{1}{p B\big(\hspace{-1pt}\frac{3}{2},\frac{p+...
japalmer's user avatar
  • 391
1 vote
1 answer
188 views

Can one show $h(x)=|2(\zeta'(x))^2-\zeta''(x)\zeta(x)|$ is a decreasing function for $x\in\mathbb{R}\cap [1,\infty)$?

This question is related to This question. When I tried to approach it I couldn't even proof that the LHS is a decreasing function on the given domain using regular methods. I have tried to write the ...
Haidara's user avatar
  • 178
1 vote
0 answers
186 views

Lipschitz continuity of an implicit function generated by a monotonic and Lipschitz multivariate function

Let $z=F(x,y)$ be a function from $\mathbb R^d\times \mathbb R$ to $\mathbb R$ satisfying the following conditions: $z=F(x,y)$ is Lipschitz continuous w.r.t. $(x,y)$; Given $x$, $F(x,y)$ is non-...
zbh2047's user avatar
  • 601
1 vote
1 answer
93 views

Is there $f$ such that $\int_0^t f(s)\,\mathrm d s<\infty$ and $|\partial_t g| (t, x) \le f(t)g(Ct, x)$ for all $t>0$ and $x \in \mathbb R^d$?

We consider the heat kernel $$ g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R,\quad (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ). $$ Then $$ \partial_t g(t, x)...
Analyst's user avatar
  • 657
1 vote
2 answers
183 views

Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$. Question 1: Can we ...
user84068's user avatar
  • 169
1 vote
1 answer
166 views

Question abouth Skorokhod representation of random variables (II)

This is a continuation of Question abouth Skorokhod representation of random variables Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ such that $$\int_{\mathbb R}|x|^pd\mu(x),~ \...
CodeGolf's user avatar
  • 1,835
1 vote
0 answers
120 views

Natural candidates for super-half-exponential which limit to half-exponential function from above

There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth. However super-half-exponentials (functions whose composition grows ...
VS.'s user avatar
  • 1,826
1 vote
2 answers
180 views

An inequality for a real function

Let $$f(z)=(1+z)^{3/4}-\left(\frac{3}{8}+\frac{\sqrt{3}}{4}\right)^{1/4}-\frac{\left(3 z+\sqrt{6} \sqrt{-1+z^2}\right)^{3/4}}{\left(2 \left(2+\sqrt{3}\right)\right)^{3/4}}.$$ Is there a simple proof ...
user67184's user avatar
1 vote
0 answers
184 views

A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$

Recently I was trying to construct a counterexample to the statement "If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are ...
Alexander Kuleshov's user avatar
1 vote
0 answers
244 views

Möbius function and polynomials

Let $\mu$ be the Möbius function. It is well known that $\sum_{n|k} \mu(n) = 0$ for $k>1$. What could be said about the polynomials $R_k = \sum_{n|k} \mu(n) x^n$ for $x \in [0,1]$? There does not ...
A413's user avatar
  • 433
1 vote
1 answer
87 views

Oscillating sums

Let $\left\{a_k\right\}_{0\leqslant k\leqslant n}$ and $\left\{b_k\right\}_{0\leqslant k\leqslant n}$ be two positive sequences such that $c_1a_k\leqslant b_k\leqslant c_2 a_k$ for all $k$ for some ...
coco's user avatar
  • 539
1 vote
1 answer
119 views

A non-polynomial Young function satisfying a power-like condition

This post asked, essentially, for an example of a "non-polynomial" invertible increasing function $f\colon[0,\infty)\to[0,\infty)$ such that $f(0)=0$ and \begin{equation} f(cu)f(t)\le f(...
Iosif Pinelis's user avatar
1 vote
1 answer
918 views

Pros and cons of probability model for permutations

I am studying probability model of random permetuation Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k inversions ($inv(\pi)$). The analytic approach was considered by L....
Mikhail Gaichenkov's user avatar
1 vote
1 answer
234 views

Zeroes of elementary polynomials without involving closed-form solutions

Consider the following two polynomials, where $n$ is an integer: $$ p_n(x) = x^3-\frac1nx-\frac2n, \\ q_n(x) = x^2-\frac2n. $$ For any $n$, let $x_p=x_p(n)$ and $x_q=x_q(n)$ be the unique positive ...
chrisv's user avatar
  • 21
1 vote
1 answer
171 views

Superharmonic extension 2

This question is a simplified version of the one in the MO post Superharmonic extension. Suppose $K$ is a compact of $\mathbb{R}^m$ ($m\geq2$), and $U(x)=\log\frac{1}{|x|}$ if $m=2$, and $=|x|^{2-m}$ ...
M. Rahmat's user avatar
  • 411
1 vote
1 answer
236 views

Continuity of the solution of a Pde system

Let $\rho_1:[0,1]\to [0,1]$ and $J:\mathbb R\to \mathbb R^+$ both continuous and bounded. I have the following system of PDE's \begin{align} \begin{cases} \frac{\partial}{\partial t} u_0(t,r)=- J* ...
user268193's user avatar
1 vote
2 answers
90 views

Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?

Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
Akira's user avatar
  • 835
1 vote
1 answer
237 views

Poisson kernel, expectation, an absolute value comes in

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)}\...
Edward Hoenn's user avatar
1 vote
1 answer
387 views

$L^p$ compactness for a sequence of functions from compactness of product with cut-off

Fix $p \in [1,\infty)$. Let $f_n:[a,b] \to \mathbb R$, $n \in \mathbb N$, be a sequence of $C^1$ functions. For every fixed $m\in \mathbb N^*$, suppose that the sequence of functions $$\{f_{n}\psi_m(...
Zac's user avatar
  • 161
1 vote
1 answer
239 views

Reference request for weak solutions of an Elliptic PDE

Edit : I just learned that all weak solutions are $C^\infty$, so this question, by Willie, seems more appropriate than the current one. I want to find weak, non trivial, continuous, solutions of $$\...
Rajesh D's user avatar
  • 698
1 vote
1 answer
319 views

Is $(f \ast K)'' \in L^1(\mathbb R)$ for $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$?

Is it possible to deduce that $$(f \ast K)'' \in L^1(\mathbb R)$$ if $f \in L^1 \cap L^\infty(\mathbb R)$ and $K \in BV(\mathbb R)$? What I can prove is that $(f \ast K)' \in L^1 \cap L^\infty$. Is ...
Hiro's user avatar
  • 131
1 vote
1 answer
348 views

Baire class 1 and (uncountably many) discontinuities

Consider a function $f:[0,1]\to[0,1]$ which is continuous on a co-meager set $C\subset[0,1]$ and discontinuous on $D=[0,1]\setminus C$. Suppose that $D\cap I$ is uncountable for every open interval $I\...
Alessandro Della Corte's user avatar
1 vote
0 answers
47 views

Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
inoc's user avatar
  • 339
1 vote
1 answer
76 views

Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius

I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
Ryan Hendricks's user avatar
1 vote
0 answers
102 views

Proving that a quantity is positive (Gaussian density and Gaussian CFD)

$\def\R{\mathbb R}$$\def\aha{{1/2}}$$\def\maha{{1/2}}$ Hi everyone, I am interested in the following problem: Let consider the heat equation problem: $$\forall (t,x) \in \mathbb{R}_+\times\mathbb{R}, ~...
NancyBoy's user avatar
  • 393
1 vote
1 answer
300 views

Convergence of concave/convex function

Let assume that you have a sequence of twice differentiable functions $(f_n)_{n\in\mathbb{N}}\in\mathscr{C}^2(\mathbb{R})^{\mathbb{N}}$. Let suppose that for each $f_n$, it exists a $x_n\in\mathbb{R}$ ...
NancyBoy's user avatar
  • 393
1 vote
1 answer
114 views

question about $TGV^2$ space

Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and $$ TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(...
JumpJump's user avatar
  • 679
1 vote
1 answer
471 views

k-th largest root in common interlacing polynomials

In their proof of the celebrated Kadison-Singer conjecture, Marcus, Spielman and Srivastava exploited so-called interlacing families which are originally defined for their work on Ramanujan graphs. ...
Federico Magallanez's user avatar
1 vote
1 answer
368 views

Does the almost sure convergence of absolutely continuous r.v.'s imply the weak convergence of the pdf's in $(L^\infty)^*$?

The following question was asked in a comment at Almost sure convergence vs convergence of probability density functions : Suppose that $(X_n)$ is a sequence of random variables (r.v.'s) converging ...
Iosif Pinelis's user avatar
1 vote
1 answer
117 views

Shrinking subset with disjoint unions

Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be disjoint finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that ...
pi66's user avatar
  • 1,209
1 vote
0 answers
56 views

Extension of this maximisation problem : finite or not?

$\mathcal M$ is the space of real $d\times d$ matrices and $\mathcal S\subset \mathcal M$ is its subset consisting of positive semidefinite elements. We consider the distance the product space $\...
Fawen90's user avatar
  • 1,399
1 vote
2 answers
231 views

A real root of a cubic equation for a stationary point

Let us consider the quartic polynomial in $x$ \begin{equation} F(x) = (2 a p +2)x^4+ (6a(1-a)p^2+(6-12a)p-6)x^3 + p(2(a-2)(a-1)a p^2 + 3(5a^2-9a+2)p +12a-18)x^2 - p^2 ((a-2)(4a^2 ...
Vladimir's user avatar
  • 371
1 vote
1 answer
191 views

Good upper bound for $\Gamma(1-b,\log(a))-\Gamma(1-b,N\log(a))$, where $a,b \in (0, 1)$ and $N \ge 1$

Let $a,b \in (0, 1)$ and $N \ge 1$, and consider the incomplete gamma function $x \mapsto \Gamma(1-a,x)$. Question Is there a simple bound (involving 'simple function's) for the expression $\Gamma(1-...
dohmatob's user avatar
  • 6,853
1 vote
0 answers
92 views

Alberti rank-one theorem and reduction of the study of BV function to the two-dimensional case

By Alberti rank-one theorem, could it be possible to reduce the study of a function $u \in BV(\mathbb{R}^N, \mathbb{R}^N)$ to the study of a function $\tilde{u} \in BV(\mathbb{R}^2, \mathbb{R}^2)$? At ...
user avatar
1 vote
1 answer
136 views

On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$. In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...
ar.grig's user avatar
  • 1,133
1 vote
1 answer
82 views

What is $\left\| u \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$

What is $\left\| f \right\|_{ H_{0}^{k}, H_{0}^{k}}$ norm when $H_{0}^{k}=\left\{u \in H^{k, 2}(M) \mid \int_{M} u \operatorname{vol}_{g}=0\right\}$. I'm reading a paper Chern-Yamabe flow which said ...
Elio Li's user avatar
  • 809
1 vote
1 answer
212 views

Lipschitz aspect of a projection on the boundary of a convex

Let $C$ be a closed convex set of $\mathbb{R}^n$ $(n\geq 1)$, with asymptotic cone $C^{as}$ having for interior $\text{Int}\big(C^{as}\big)$. Let $u\in\mathbb{R}^n\setminus\{0\}$ such that \begin{...
G. Panel's user avatar
  • 449
1 vote
0 answers
67 views

Solution to recurrence relation from integro-differential dynamical system?

Consider the integro-differential equation \begin{equation} \kappa\ddot x+\dot x=2\int_0^t J_1(x_t-x_s)e^{-\epsilon(t-s)}ds.\tag{1} \end{equation} such that $\kappa,\epsilon\in\mathbb{R}$, $t\in\...
UNOwen's user avatar
  • 79
1 vote
0 answers
123 views

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 ...
CodeGolf's user avatar
  • 1,835
1 vote
2 answers
346 views

Is this relationship, $\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$, true?

According to numerical simulation, the relationship $$\sum^{\infty}_{N=1}\frac{N^{-\alpha} \, x^{N-1}}{\Gamma(N)}\sim e^{x}x^{-\alpha}$$ where $\Gamma$ is the Gamma function seems to be true. Do you ...
user avatar
1 vote
1 answer
94 views

Differentiability of some function defined as the maximum

Let $d,n\ge 1$ be fixed integers. Given some compact subset $E\subset \mathbb R^d$, consider the function $f: E^n\ni (x_1,\ldots, x_n) \longrightarrow f(x_1,\ldots, x_n)\in \mathbb R$ defined by $$f(...
Fawen90's user avatar
  • 1,399
1 vote
1 answer
117 views

On summation methods of divergent series

$\newcommand{\R}{\mathbb R}\newcommand{\N}{\mathbb N}\newcommand{\si}{\sigma}\newcommand{\CC}{\mathcal C}$This previous question introduced the following notion of a summability space. Let $\N:=\{1,2,\...
Iosif Pinelis's user avatar
1 vote
1 answer
186 views

Existence of a smooth function that approximates a characteristic function of an interval with certain property

Let $N$ be a large integer and $I = [aN, bN]$ for some $0 < a < b < 1$. Denote by $\chi_I(x) = 1$ if $x \in I$, $0$ otherwise. I was wondering if there exists a smooth function $w$ with the ...
Johnny T.'s user avatar
  • 3,625
1 vote
0 answers
103 views

Real root isolation for exponential polynomials

Suppose we are given an exponential polynomial $f:\mathbb{R}\mapsto\mathbb{R}$ $$ f(t)=\sum_{i=1}^n p_i(t)e^{\lambda_i t} $$ where $p_i(t)$s are polynomials with algebraic coefficients and $\lambda_i$...
gondolf's user avatar
  • 1,503
1 vote
1 answer
620 views

Smallest Lipschitz constant on non-convex domains

It is well known that if a function $f:U\to \mathbb C^n$, $U\subset \mathbb C^m$ satisfies $\sup_{x\in U}\|Df(x)\|_{\infty} = C < \infty$ uniformly on $U$ and $U$ is compact and convex, then $f$ is ...
dima's user avatar
  • 959
1 vote
1 answer
190 views

Proof of extended version of non-random "almost supermartingale"

In this question, a non-random version of "almost supermartingale" theorem is proved. Here, I would like to extend/apply the non-random version to the slightly different situation. I wonder ...
user550103's user avatar