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6 votes
0 answers
220 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
  • 517
2 votes
1 answer
140 views

Invertibility of message passing with invertible parametrization

Consider the message passing framework defined by, $$f(\boldsymbol{x}_i)= \boldsymbol{x}_i + \sum_{j \neq i} (\boldsymbol{x}_i -\boldsymbol{x}_j) g(\|\boldsymbol{x}_i -\boldsymbol{x}_j\|^2),$$ for $i=...
PonderingPolynomial's user avatar
3 votes
1 answer
198 views

Can gradient zero implies that a function is constant with Hörmander vector fields

Let $X=(X_1,\cdots,X_m)$ be a system of Hörmander vector fields defined on $\mathbb{R}^n$. The Sobolev space $W_{X}^{1,p}(\Omega)$ is defined by $$W_{X}^{1,p}(\Omega):=\{u\in L^p(\Omega)|X_iu\in L^p(\...
Houa's user avatar
  • 561
3 votes
0 answers
90 views

Upcrossing lemma and subharmonic functions

I have been studying the upcrossing lemma for submartingales, which asserts that if $X_n$ is a non negative submartingale, and $ \lambda>0$ then if we denote by $U_n$ the number of $[0,\lambda]$-...
an_ordinary_mathematician's user avatar
0 votes
0 answers
103 views

Who first gave a result stronger-or-equal to this one on ODEs

After some thinking I've come to the following conclusion. Consider the initial value problem $$\text{(P)}\begin{cases}x'(t)=f(t,x(t)),\quad t\geq t_0\\x(t_0)=x_0 \end{cases}$$ where $f:D\subset\...
aleph2's user avatar
  • 637
1 vote
0 answers
86 views

The intersection of $ n $ cylinders in $ 3D$ space

I posted the question on here, but received no answer I recently found out about the Steinmetz Solids, obtained as the intersection of two or three cylinders of equal radius at right angles. If we set ...
user967210's user avatar
3 votes
1 answer
458 views

Limit of an infinite series with quadratic arguments

I have encountered a limiting process on some infinite series. So, I would like to ask: QUESTION. Assume $n$ is an even positive integer. Is this true? $$\lim_{r\rightarrow1^{-}}\sum_{j=1}^{\infty}\...
T. Amdeberhan's user avatar
1 vote
1 answer
143 views

Projection of an element of the $n$-simplex onto subset

Let $\mathbb{S}^{n}$ denote the $n$-dimensional probability simplex and let $\{e_1,...,e_{n+1}\}$ be the canonical basis of $\mathbb{R}^{n+1}$. Consider the subset $\mathbb{S}^{n}(K) \subset \mathbb{S}...
aureliano_buendia's user avatar
6 votes
0 answers
108 views

Archimedean ordered field in which every function is smooth

In constructive mathematics, it is consistent that every function $\mathbb{R} \to \mathbb{R}$ on the Dedekind real numbers is continuous. However, it is not consistent that every function $\mathbb{R} \...
Madeleine Birchfield's user avatar
0 votes
0 answers
22 views

An auxiliary problem while constructing the system of Jordan sets on a plane

Let $\mathfrak{S}$ be a system of rectangles in $R^2$ of the form $[a,b]\times [c,d]$ where $a,b,c, d \in R$, $a<b$, $c<d$. Let $\mathfrak{A}$ be a system of simple sets based on $\mathfrak{S}$. ...
Alexander's user avatar
0 votes
0 answers
616 views

The set of continuous bounded functions $f:X\to Y$ is dense in $L^p(X,Y)$ where $X,Y$ are Polish

It is well known that the set of real-valued continuous functions with compact support is dense in $L^p(\mu)$ where $\mu$ is a Radon measure (see e.g. [Folland, Proposition 7.9]) Clearly, the set of ...
Kaira's user avatar
  • 305
4 votes
1 answer
250 views

Does a generic linear map admit a vector whose iterates span $V$?

We say a linear map $T$ on a finite dimensional vector space $V$ admits spanning vectors if there exists some vector $v \in V$ whose iterates $v, Tv, T^2 v, \dots$ under $T$ span $V$. Question: ...
Nate River's user avatar
  • 6,155
6 votes
1 answer
289 views

Archimedean ordered fields without maxima and minima in constructive mathematics

In constructive mathematics, let us define an ordered (Heyting) field $F$ to be a commutative ring with a binary relation $<$ which is irreflexive, where for all $x$, $\neg (x < x)$ asymmetric, ...
Madeleine Birchfield's user avatar
0 votes
1 answer
192 views

A continuous injection from the Hilbert cube to the real line?

Continuing an earlier "too good to be true" question that I posted recently, the same holds for the present question: Is there a continuous injection from the Hilbert cube $[0,1]^{\Bbb N}$ ...
Boaz Tsaban's user avatar
  • 3,104
1 vote
1 answer
162 views

Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?

It seems too good to be possible, but: Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space? Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
Boaz Tsaban's user avatar
  • 3,104
0 votes
1 answer
148 views

Equi-coercivity of functionals on a metric space

Definition: A family of functionals $\{F_n: X\to\bar{\mathbb R}\}$ on a metric space $X$ is said to be equi-coercive if, for every $\alpha \in \mathbb{R}$, there is a compact set $K_\alpha$ of $X$ ...
Guy Fsone's user avatar
  • 1,101
7 votes
2 answers
626 views

Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
B.Hueber's user avatar
  • 1,171
1 vote
0 answers
48 views

How to derive a lower bound of a MinMax inequality?

Let $x_5,\cdots,x_n\in[0,\alpha]\cup[-\pi,\alpha-\pi]$ where $\alpha$ is a fixed angle $\in(0,\pi/2)$. The goal For a fixed $(A_{ij})_{1\leq i\leq 4,5\leq j\leq n}\in\{-1,+1\}$, verify whether it ...
tony's user avatar
  • 405
0 votes
1 answer
150 views

Property of $p$-norm in the $n$-simplex

Let $\mathbb{S}^{n}$ be the canonical simplex of $\mathbb{R}^{n}$ and let $u = (1/n,\dotsc,1/n)$. Is it true that $$\lVert x - u \rVert_p \leq \lVert y - u \rVert_p$$ implies that $$\lVert x\rVert_p \...
aureliano_buendia's user avatar
1 vote
0 answers
148 views

Rational solutions to $\cos(\lambda \pi) = \cos^2(a\pi) - \cos(b\pi) \sin^2(a\pi) $, with $a,b \in \mathbb{Q}$

In a similar vein to this question, I am trying to understand the occurrence of rational solutions $\lambda$ to the following equation $$\cos(\lambda \pi) = \cos^2 (a\pi) - \cos ( b\pi ) \sin^2 \left(...
Mary_Smith's user avatar
0 votes
1 answer
251 views

Reference request: log Sobolev inequality for uniform measure (uniform distribution over discrete set)

Suppose that $N \in \mathbb N_+$ is fixed and denote by $\mu = (\mu_0,\ldots,\mu_N)$ the uniform distribution on the set $\{0,1,\ldots,N\}$ (i.e., $\mu_n = \frac{1}{N+1}$ for each $0\leq n\leq N$). I ...
Fei Cao's user avatar
  • 730
0 votes
0 answers
126 views

A question about associated operator on continuous functions space equiped with L2 norm

For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
WaoaoaoTTTT's user avatar
3 votes
0 answers
179 views

Maximum of an integral

Assume that $a>0$ and $r\in[0,1)$. How to prove that the function $$f(p)=\int_{-\pi}^\pi \left (1+r^2+2 r \cos x\right)^{a/2} |(2+a) \cos(x+p)-a r \cos(p)| \, dx$$ attains its maximum for $p=\pi/2$...
AlphaHarmonic's user avatar
0 votes
1 answer
414 views

Necessary conditions for convergence of convolution

In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
S.H.W's user avatar
  • 61
2 votes
0 answers
95 views

Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?

Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
Analyst's user avatar
  • 657
3 votes
1 answer
146 views

Behaviour of the solution of a second order ODE

I am currently studying the following second order ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\ y(0) = 0\\ \dot y(T) = c \end{...
Falcon's user avatar
  • 452
1 vote
1 answer
66 views

Upper bound $I_R := \int_{B_R^c} |x| (P_t \ell_\nu) (x) \, \mathrm d x$ in terms of $R, \nu, t$?

Let $(p_t)_{t >0}$ be the Gaussian heat kernel on $\mathbb R^d$ and $(P_t)_{t >0}$ its induced semi-group, i.e., $$ \begin{align} p_t (x) &:= (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \...
Akira's user avatar
  • 835
7 votes
2 answers
567 views

Intuition for Agmon-Douglis-Nirenberg ellipticity

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently. I am trying to understand the definition of ellipticity of systems due to ...
G. Blaickner's user avatar
  • 1,429
3 votes
1 answer
493 views

A strange condition of convexity?

During my research, I come across this question. Let $f \in C^2(\mathbb R, \mathbb R_+^*)$ with $\forall x \in\mathbb R, f'(x) \geq |f''(x)+f(x)|$. Is it true that $\forall x \in \mathbb R, f''(x) \...
Dattier's user avatar
  • 4,074
1 vote
0 answers
162 views

Triangular and pentagonal numbers in $q$-series

Consider the following two infinite series $$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\, \sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
T. Amdeberhan's user avatar
1 vote
0 answers
96 views

Regularity of Feynman-Kac formula for a simple diffusion

Let consider the diffusion process given by: $$dX_t = \alpha(X_t) dW_t$$ where $\alpha(x) = \alpha_1\mathbf{1}_{x\geq 0} + \alpha_2\mathbf{1}_{x< 0}$ ($\alpha_1,\alpha_2>0$) and $W$ a Wiener ...
NancyBoy's user avatar
  • 393
9 votes
0 answers
1k views

How complicated can an elementary antiderivative get?

I asked this question on MSE here. I recently learned that there are many very large numbers that have been defined, such as $\operatorname{TREE}(3)$ and many others that are too big to be written ...
pie's user avatar
  • 541
6 votes
2 answers
333 views

Attainment of maximum

A basic result in real analysis is that a continuous function $f:[0,1]\rightarrow \mathbb{R}$ attains its maximum on $[0,1]$, i.e. there is $x\in [0,1]$ such that $f(x)=\sup_{y\in [0,1]} f(y)$. A ...
Sam Sanders's user avatar
  • 4,359
-3 votes
2 answers
317 views

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]

When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
MrPie 's user avatar
  • 317
1 vote
1 answer
160 views

Differentiability of an integral of geodesic distance

Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$. Q1: Define $$ g(t)=\...
Hengchao Chen's user avatar
3 votes
1 answer
161 views

Equivalent definition for Skorokhod metric

I have a question about the Skorokod distance on the space $\mathcal{D}([0,1],\mathbb{R})$: $$ d(X,Y):= \inf_{\lambda \in \Lambda}\left( \sup_{t\in [0,1]}|t-\lambda(t)|\vee \sup_{t\in [0,1]}|X(t)-Y(\...
user1598's user avatar
  • 177
1 vote
1 answer
113 views

An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$

Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e., $$ g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d. $$ Let $f : \mathbb R^d \to \...
Akira's user avatar
  • 835
2 votes
2 answers
307 views

Preimage of null sets under a monotone increasing function

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
Julian's user avatar
  • 113
3 votes
1 answer
132 views

Existence of a density

Let $W: \mathbb R^d \to \mathbb R^{d \times d}$ be a matrix-valued function with domain $\mathbb R^d$ and taking values in the set of $d \times d$ real matrices, such that $W(x)$ is positive-definite ...
Aurelien's user avatar
  • 301
1 vote
1 answer
106 views

Upper bound $I (t) := \sup_{x \in \mathbb R^d} \int_{\mathbb R^d} \frac{|x-y|^\alpha}{t^{d/2}} \exp ( - \frac{|\psi(x) - y|^2}{t} ) \, \mathrm d y$

Let $\alpha \in (0, 1)$ and $\psi : \mathbb R^d \to \mathbb R^d$ be a $C^\infty$-diffeomorphism such that $\|\nabla \psi\|_\infty + \|\nabla \psi^{-1}\|_\infty < + \infty$. Let $$ I (t) := \sup_{x \...
Akira's user avatar
  • 835
1 vote
1 answer
170 views

fourth-order multivariate Gaussian integral

I am struggling with an integral of form $$ \int_{\mathbb R^n} y\otimes y~ \langle Ay,y\rangle \, \mathrm d N(0,\Sigma)(y). $$ I assume that it will involve the trace of some product of $R$ and $\...
Philipp Wacker's user avatar
0 votes
0 answers
122 views

Convergence of a series related to counting distinct prime factors

I am here to ask whether the following series is convergent for all real $z$. I am also asking whether this is everywhere real analytic. I conjecture that it is convergent for all real input, or at ...
Zachary Hoelscher's user avatar
0 votes
0 answers
53 views

Non-linearity of viscosity solutions

I am interested in the following problem. Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem: $$ \begin{cases} u_t = F(u_{xx}),\\ u(0,x) =...
NancyBoy's user avatar
  • 393
0 votes
0 answers
36 views

Convergence of numerical scheme for HJB equation

Convergence of numerical scheme for HJB equation has been widely studied, the key paper is the Barles's one. Essentially, the convergence is guarenteed if the scheme is: Consistent Stable Monotony ...
NancyBoy's user avatar
  • 393
5 votes
5 answers
1k views

What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$

FYI: I asked this question here couple of days ago but got no answer yet. $n$ is an integer We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
RajaKrishnappa's user avatar
7 votes
2 answers
592 views

Prove that the following function is positive

Consider the following function: $$K(x, y; t) = \sum_{n \geq 0} \frac{e^{-(2n+1)t}}{\sqrt{\pi} 2^n n!} H_n(x) H_n(y) \exp\left(-\frac{(x^2 + y^2)}{2}\right) $$ This is Mehler's kernel, and can be ...
matilda's user avatar
  • 90
10 votes
2 answers
1k views

Does a conditionally convergent sum with random signs converge almost surely?

Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
Nate River's user avatar
  • 6,155
8 votes
1 answer
1k views

When can a sum be re-signed to converge to any limit?

Let $a_n$ be a sequence of positive real numbers with $\sum a_n < \infty$. What are the necessary and sufficient conditions for the following to hold? For any $S \in \mathbb R$ with $-\sum a_n \...
Nate River's user avatar
  • 6,155
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
136 views

Inequality with convolution

I have some troubles with the following problem: A definition Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian ...
NancyBoy's user avatar
  • 393

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