All Questions
734 questions
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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 ...
1
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1
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438
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Some fun with special infinite nested radicals
Let us define the following functions:
$$f_n(x)=\sqrt{x^{n}-\sqrt{x^{n+1}- \sqrt{x^{n+2}-\cdots}}} $$
$$g_n(x)=\sqrt{x^{n}+\sqrt{x^{n+1}+ \sqrt{x^{n+2}+\cdots}}} $$
with $f(x)=f_1(x)$ and $g(x)=g_1(x)$...
- 3,098
1
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0
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186
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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-...
- 601
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2
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346
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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 ...
user365311
1
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1
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2k
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About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$
The following real-valued functions are closely related to the zeros of $\zeta(s)$ in the critical strip $\frac{1}{2}<\Re(s) < 1$.
$$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\...
- 3,098
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1
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171
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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}$ ...
- 411
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0
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244
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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 ...
- 433
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47
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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 ...
- 339
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1
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117
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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 ...
- 1,209
1
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1
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307
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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 ...
- 595
1
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1
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387
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$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(...
- 161
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1
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319
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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 ...
- 131
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368
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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 ...
- 128k
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0
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67
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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\...
- 79
1
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2
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140
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Extending a discrete singular kernel
Let $\{\phi(n)\}_{n\in\mathbb Z}$ be a sequence of complex numbers with the following properties:
$\phi(0)=0$ and $|\phi(n)|\leq \frac{C_1}{|n|}$ for all $n\neq 0$ and $C_1>0$ is independent of $n....
- 1,879
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1
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201
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Does weak continuity of Jacobians hold for non nondegenerate maps?
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, compact, connected, oriented Riemannian manifolds. (with or without boundaries).
Let $f_n \...
- 6,741
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1
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224
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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 ...
- 199
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1
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136
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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 $f:U_\epsilon(0)\to\mathbb{R}$ ...
- 1,133
1
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1
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176
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A condition on the inequality $f'(x)/(1-f(x)^2)-1/(1-x^2)\ge 0$
Assume that $f:[0,1]\to [0,1]$ is an diffeomorphism so that $(f''(x)/f'(x))'<0$ and that $f''(0)=0$. It seems to me that $$\frac{1-f(x)^2}{1-x^2}\le f'(x)$$ on $[0,1]$. But no proof so far.
The ...
- 333
1
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1
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94
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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(...
- 1,389
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1
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117
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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,\...
- 128k
1
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1
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186
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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 ...
- 3,625
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1
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236
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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* ...
- 139
1
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1
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620
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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 ...
- 959
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1
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518
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Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
- 305
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0
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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
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1
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162
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Does there exist a class of real-valued upper semicontinuos functions on $X$ such that $\mathcal{F}$ is countable?
Ian Morris quoted the following:
For any upper semi-continuous function $f \colon X \to [-\infty,+\infty)$ defined on a nonempty topological space $X$ there exists a nonempty set $\mathcal{F}\...
- 623
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1
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236
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What does Landau symbol mean in an inequality?
I'm reading about subdifferentiable function at page 232 of Villani's Optimal Transport: Old and New.
Definition 10.5 (Subdifferentiability, superdifferentiability). Let $U$ be an open set of $\...
- 835
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1
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82
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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 ...
- 809
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0
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922
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A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it
Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
- 698
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1
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193
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Quantitative finite speed of propagation property for ODE (cone of dependence)
Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...
- 839
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71
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Continuous injection of metric ball into Euclidean ball
This is a follow-up to this post.
Suppose that $(X,d_X)$ is a compact metric space with (finite) metric-capacity, defined by
$$
\kappa_X(\epsilon)\triangleq\sup\left\{
k : \exists x_0,\dots,x_k \...
- 5,405
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56
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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 $\...
- 1,389
1
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1
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212
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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{...
- 449
1
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1
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632
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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\...
- 2,239
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2
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231
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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 ...
- 371
1
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1
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239
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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 $$\...
- 698
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1
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142
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Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? [closed]
Why are the homeomorphisms from the unit circle to the unit circle preserving measure affine? The affine is composition of rotation and continue automorphism.
- 83
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0
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63
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A maximisation problem : finite or not?
Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
- 1,389
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1
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139
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Singular integral bounded by Dirichlet form?
We define for some fixed $L$
$$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$
in particular $x_1,x_2 \in \mathbb R^2.$
Let $f \in C_c^{\infty}(\Omega)$, then I am ...
0
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1
answer
127
views
Continuous extensions of tangent vector fields
Let $\Omega$ be an open subset of $S^2$ with $\bar{\Omega}\neq S^2$. Suppose a continuous tangent vector field $G$ is given on $\partial \Omega$ with $|G(y)|=1$ for all $y\in \partial \Omega$. Does ...
- 434
0
votes
1
answer
53
views
Rate of convergence of the minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
0
votes
1
answer
77
views
Decay rate of minimum point over a product space
Let $f(\theta, \epsilon)$ be smooth on $[0,2\pi] \times [0,\infty)$ such that
$f(\theta, \epsilon)$ converges to $f(\theta, 0)$ uniformly as $\epsilon \rightarrow 0$.
$f(\theta, \epsilon) > 0$ for ...
- 434
0
votes
0
answers
80
views
Alternative to the Sampling Theorem / Invertible transform with sampling criteria
I seek a transform $T$ that operates on real-valued $x(t)$, that
Is perfectly invertible
Has discrete counterpart with continuous reconstructor
Provides conditional reconstruction guarantees
...
0
votes
1
answer
375
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Bringing a Heun equation into canonical form
It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably ...
- 213
0
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1
answer
137
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Zeros of entire functions with parameter
Let $f_w:\mathbb C \to \mathbb C$ be an entire function with $f_w(0)=1$ and at least one root for any choice of $w \in (0,1)$. Assume further that for a dense set of $w$ the function $f_w$ has ...
- 192
0
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1
answer
78
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The conditions used to prove upper semicontinuous of generalized directional derivative (in Clarke sense)
Let $X$ be a reflexive Banach space, $z, x, v \in X$. $\{z_i\}, \{x_i\}$ and $\{v_i\}$ are arbitrary sequences converging to $z, x$ and $v$, respectively.
I would like to know under which conditions ...
- 15
0
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2
answers
387
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Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
0
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1
answer
347
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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 $$\text{fix}(\pi) = \{x\in \{1,\ldots, n\}: \...
- 48.9k
0
votes
1
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116
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Integrable function [closed]
Suppose that $a, b, c_1$ and $c_2$ are real constant.
Is there the necessary and sufficient conditions of $a ,b, c_1,c_2 $ for the following integration is integrable? i.e.
$$\int_1^{\infty}\int_1^{\...
- 119