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4 votes
0 answers
116 views

Lipschitz extension of a flow can still be a flow?

Consider a map $\Phi: [0,T] \times \mathbb{R}^d \to \mathbb{R}^d$, and assume that there exists a set $U \subset \mathbb{R}^d$ such that $\Phi\rvert_{[0,T] \times U}$ is $L$-Lipschitz. It is well ...
tommy1996q's user avatar
7 votes
0 answers
150 views

The space of analytic associative operations

This question is a follow-up to this old one of mine. Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
Noah Schweber's user avatar
2 votes
0 answers
170 views

finite dimensionality of a subspace of a Banach space

Let $H$ be the space of measurable functions on $(0,1)$ such that $$ \|u\|_{H}^2 = \int_0^1 x^2\,|\partial_x u|^2\,dx + \int_{0}^1 |u|^2\,dx <\infty.$$ Let $C>0$ be a constant. Suppose that $W \...
Ali's user avatar
  • 4,135
6 votes
1 answer
331 views

A combinatorial identity involving increasing functions from $\{1, \dots, n\}$ to itself

This is related to the post An order statistics problem with some interesting geometry. The following identity arose in the context of the problem. Fix an integer $N \geq 2$. Let $\mathcal S_N^+$ ...
Nate River's user avatar
  • 6,155
23 votes
5 answers
2k views

PDEs and algebraic varieties

Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
Puzzled's user avatar
  • 8,998
5 votes
1 answer
489 views

Does coefficient-wise limit preserve real-rootedness?

Let $P_n$, $n=1,2,\ldots$ be polynomials with real roots only (and real coefficients), and $P_n$ converge to a non-zero polynomial $Q$ coefficient-wise. Does it follow that $Q$ has real roots only? ...
Fedor Petrov's user avatar
2 votes
0 answers
120 views

Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
B.Hueber's user avatar
  • 1,171
0 votes
0 answers
34 views

It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?

Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
Ilovemath's user avatar
  • 677
7 votes
0 answers
254 views

$C^0$-limit of volume-preserving maps on $\mathbb R^n$

Let $f_k:B_1\rightarrow \mathbb R^n$ be a sequence of injective differentiable volume-preserving maps (i.e. $\mu(f_k(A))=\mu(A)$ for any measurable $A\subset B_1$) that converges uniformly to $f:B_1\...
Tian LAN's user avatar
  • 435
11 votes
2 answers
1k views

Twice continuously differentiable implied by existence of limit

I have the following question. Let $f,g:\mathbb{R}\to\mathbb{R}$ be two continuous functions (vanishing at infinity) and assume that $$ \frac{f(x+t)+f(x-t)-2f(x)}{t^2}\to g(x) $$ for all $x\in X$ when ...
Sonam Idowu's user avatar
2 votes
0 answers
41 views

Blow up for certain classes of distributions

Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...
Ali's user avatar
  • 4,135
3 votes
1 answer
145 views

Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?

We endow the space $\mathcal P_2^a (\mathbb R^d)$ of absolutely continuous probability measures with finite second moment with the Wasserstein distance $W_2$. Let $\mathcal H (\mu)$ be the relative ...
Akira's user avatar
  • 835
2 votes
2 answers
615 views

In what sense does the Hermite expansion of a bounded smooth function converge?

Let $f : \mathbb{R} \to \mathbb{C}$ be a smooth and bounded function. If we denote by $\{ H_n(x) \}$ the sequence of normalized Hermite polynomials, then the Hermite expansion of $f$ is defined as \...
Isaac's user avatar
  • 3,477
22 votes
1 answer
4k views

A challenging (for me) limit calculation

How to calculate the following limit $$ \lim_{n\to\infty}\sqrt{n}\underbrace{{}\sin(\sin(\sin(\sin(\cdots\sin(\frac{1}{\sqrt{n}})\cdots))))}_{n \text{ sin's}} \text{?} $$ ${}{}$
C. WANG's user avatar
  • 549
2 votes
1 answer
159 views

A compact embedding claim

Let $U= (0,1)\times (0,1)$. Consider the weighted Sobolev spaces $H_1$ with the norms $$ \|u\|_{H_1}^2 = \int_0^1 (\int_0^1 x\,|u(x,y)|^2\,dx) \,dy$$ Let $H_2$ be the weighted Sobolev space with the ...
Ali's user avatar
  • 4,135
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
2 votes
2 answers
274 views

Is a simple closed curve always a free boundary arc?

Is it possible to extract a neighborhood around any point on a simple closed curve such that the boundary of this neighborhood intersects the curve at only two points? For a simple closed curve $\...
S.Zhang's user avatar
  • 23
1 vote
3 answers
180 views

Evaluating a sinusoidal series

Define the sequence of functions $$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$ Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...
K. Grammatikos's user avatar
1 vote
1 answer
90 views

The number of roots of pseudo-exponential polynomials

Assume that $J$ is the interval $(-\pi,\pi]$. For $k=1,\ldots,2n$, suppose that $\lambda_k$s are real functions on $J$ with $|\lambda_k|=1$, meaning that $\lambda_k(t)$ is either $-1$ or $1$ where $t\...
ABB's user avatar
  • 4,058
3 votes
1 answer
128 views

Weaker version of the lemma of K.L. Chung

Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds: $$...
giorgi nguyen's user avatar
2 votes
1 answer
264 views

Is a continuous functional on continuous functions the restriction of a continuous functional on the space of all functions?

As sets, we can consider the space $C(\mathbf{R}^n;\mathbf{R}^k)$ - of all continuous functions from $\mathbf{R}^n$ to $\mathbf{R}^k$ - to be a subset of the product space $(\mathbf{R}^k)^{\mathbf{R}^...
SBK's user avatar
  • 1,179
3 votes
2 answers
294 views

Domain of spectral fractional Laplacian

Let $(M,g)$ be a complete Riemannian manifold with Laplacian $\Delta:C^{\infty}_{c}(M)\to C^{\infty}_{c}(M)$ (think of $\mathbb{R}^{d}$ if you wish). This operator is essentially self-adjoint in $L^{2}...
B.Hueber's user avatar
  • 1,171
2 votes
0 answers
70 views

A lemma in the application of Lions's concentration compactness pricnciple in Hardy-Littlewood-Sobolev inequality

I'm encountering some problems when reading Lions' paper "the concentration-compactness principle in the calculus of variations. The limit case, Part 2". The Hardy-Littlewood-Sobolev (HLS) ...
IMOS's user avatar
  • 121
5 votes
1 answer
272 views

Is the local maximal function bounded from $W^{1, 1}$ to $L^1$?

Let $f \in W^{1, 1} (\mathbb R^d)$. For every $\varepsilon > 0$, we consider the local maximal function $M_\varepsilon f: \mathbb R^d \to \mathbb R$, defined by $$M f_{\varepsilon} (x) = \sup_{r \...
Nate River's user avatar
  • 6,155
2 votes
0 answers
216 views

When do these ODE have positive solutions?

Consider the ODE \begin{equation} x'' + q(t) x = 0 \end{equation} in the unit interval $(0,1)$, with a potential function $q(t) = 4\pi^2 - \frac{Ct}{(1 - t)^2}$ depending on a positive constant $C >...
Leo Moos's user avatar
  • 5,038
3 votes
2 answers
976 views

Are $L^p$ norms absolutely continuous?

Let $1 < K \leq \infty$, and suppose $f \in L^p (X)$ for all $1 \leq p \leq K$, for $X$ some $\sigma$-finite measure space with no atoms. Question: Is the function $p \to \|f\|_{L^p}$ absolutely ...
Nate River's user avatar
  • 6,155
-5 votes
1 answer
270 views

Calculus based on pdf [closed]

Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
Marius S.L.'s user avatar
9 votes
3 answers
553 views

Bounding the $n$-th derivatives of $\frac{1-\cos(x)}{x^2}$

Define the smooth map $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(x) := \frac{1-\cos(x)}{x^2} = -\sum\limits_{k=1}^\infty \frac{(-1)^k}{(2k)!} x^{2k-2}$. I am looking for a nice bound on $|f^{(n)}(x)...
Ben Deitmar's user avatar
  • 1,295
2 votes
1 answer
206 views

Bound for zero-crossings of heat equation

I am considering the following problem. Let $\mathcal{P}$ the classical heat-diffusion problem: $$\mathcal{P} : \left(\partial_t u (t,x)=\frac{1}{2}\partial_{xx}^2u(t,x)\text{ with }u(0,\cdot) = f(x)\...
NancyBoy's user avatar
  • 393
4 votes
2 answers
413 views

A measure assigning values in $\{0,1\}$ must be a Dirac measure?

Let $\mu$ be a measure on some measurable space $(\Omega, \mathcal F)$ such that $$\mu(B)\in \{0,1\},\quad \forall B\in \mathcal F.$$ Can we show that $\mu$ must be a Dirac measure (under suitable ...
Fawen90's user avatar
  • 1,389
1 vote
1 answer
217 views

Perturbation of matrices

Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$. Question. Does there exist a Lebesgue measurable ...
Ali's user avatar
  • 4,135
5 votes
1 answer
510 views

Norm inequality for the inclusion $L^2(\partial \Omega)\hookrightarrow H^{-1/2}(\partial \Omega)$

Let $\Omega \subset \mathbb{R}^3$ be a lipschitz domain. We then have the trace operator $\tau : H^1(\Omega) \to L^2(\partial \Omega)$ and can define the space $H^{1/2}(\partial \Omega) := \tau(H^1(\...
Mandelbrot's user avatar
0 votes
0 answers
119 views

About definition of stable solution. $Q_u(\phi) \ge 0$ for all $\phi \in C_c^1(\Omega)$ replaced by "for all $\phi \in W_0^{1,2}(\Omega)$"

I want to ask about a remark about the stable solution of elliptic PDE Remark 1.1.1. We say $u$ is stable solution of $-\Delta u=f(u) \ \text { in } \Omega$ and $u=0$ on $\partial \Omega$ if it ...
Elio Li's user avatar
  • 809
1 vote
0 answers
59 views

Study of the properties of a non-local ODE

I am studying the following non-local ODE $$\dot p(x) \nu_{\varepsilon, \alpha}(x) + \int_{x}^{2x_0}\frac{\dot p(s)}{s + \varepsilon} ds = c \quad \text{for } x \in [0,2x_0].$$ The number $x_0$ can ...
Falcon's user avatar
  • 452
1 vote
0 answers
76 views

Is this extension of n-th derivatives to ordinal-indexed derivatives trivial? [duplicate]

Let $f$ be a function defined everywhere on the real line, which is infinitely differentiable everywhere, in other words, $f$ is everywhere smooth. I define the $\omega$-th derivative, where $\omega$ ...
user107952's user avatar
  • 2,013
1 vote
1 answer
187 views

Bound the distance between two vectors on the probability simplex

Let $a,b$ be two vectors with strictly positive elements and $\delta = 1 - \frac{\langle a,b \rangle}{\|a\|\|b\|}$. Bound the following optimization problem as a function of $\delta$ $$\sup_{x>0} \...
good bandit's user avatar
1 vote
0 answers
82 views

How to calculate the Integral with confluent hypergeometric function

How to prove this.Thank you in advance Let $\delta,\beta>0$ How to prove this \begin{align} & \int^1_0 \frac{w^{1-\beta}}{(1-w)^{1+\delta}} (-t.s w)^{\frac{-\delta}{2}} e^{-\frac{w}{1-w}(s+t)}...
zoran  Vicovic's user avatar
1 vote
1 answer
120 views

Sobolev-type estimate for irrational winding on a torus

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
user197284's user avatar
1 vote
0 answers
79 views

Asymptotics of ${2n \choose n+k} {2n \choose n}^{-1}$ when $k$ grows with $n$

The quotient $Q(n,k) := \frac{{2n \choose n+k}}{{2n \choose n}}$ clearly converges to one for $k \in \mathbb{N}$ fixed and $n \rightarrow \infty$. Simultaneously it converges to zero, if $k$ grows ...
Ben Deitmar's user avatar
  • 1,295
2 votes
0 answers
188 views

Self-adjointness of fractional laplacian

Lets consider the following functional analytic definition of the fractional Laplacian: Consider a (complete, connected, oriented) Riemannian manifold $(M,g)$ with corresponding Laplacian $\Delta_{g}$....
B.Hueber's user avatar
  • 1,171
3 votes
2 answers
293 views

On convergence of convex-concave functions

Let $(f_n)$ be a sequence of twice differentiable functions on $\mathbb R$ such that for each $n$ there exists some $x_n\in\mathbb{R}$ such that: $f_n$ is strictly convex on $(-\infty,x_n)$, $f_n$ is ...
Iosif Pinelis's user avatar
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
8 votes
1 answer
258 views

Sequential colimit of iterated quotients of Cauchy sequences

We work in constructive mathematics. The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
Madeleine Birchfield's user avatar
5 votes
1 answer
542 views

If $f$ is bounded, decays fast enough at infinity and $\int f=0$, does this imply that $f$ is in the Hardy space $\mathcal H^1(\mathbb R^n)$?

Let $\mathcal H^1(\mathbb R^n)$ be the real Hardy space (as in Stein's "Harmonic Analysis", Chapter 3). It is well known that $\mathcal H^1(\mathbb R^n)\subset L^1(\mathbb R^n)$ and its ...
Lorenzo Pompili's user avatar
4 votes
1 answer
334 views

Is this approximation for $\pi$ enough to make this value converge? And how to find an upper bound for it

Update: \begin{align*} |I_n-J_n| = (\pi-S_n)\sum_{k=0}^n |\frac{a_kp_k(\ln\pi)}{\ln^{k+1}\pi}| \end{align*} and \begin{align*} |I_n| = \sum_{k=0}^n | \frac{a_k\pi p_k(\ln\pi)}{\ln^{k+1}\pi} -\sum_{k=...
Pinteco's user avatar
  • 521
3 votes
1 answer
211 views

Blowup of Sobolev norms in approximating a non-absolutely continuous function

Let $f: [0, 1] \to \mathbb R$ be a continuous function, and $1 <p \leq \infty$. Suppose $u_n \in W^{1, p}$ are such that $u_n \to f$ uniformly. Is it true that if $f$ fails to be absolutely ...
Nate River's user avatar
  • 6,155
5 votes
1 answer
326 views

Does Cesaro convergence along all arithmetic progressions imply convergence on a full density subsequence?

Suppose $\{x_n\}_{n \geq 1}$ is a real valued sequence such that for every $a, r \in \mathbb Z_+$, we have that $$\lim_{N \to \infty} \frac{1}{N} \sum_{i = 0}^{N-1} x_{a + ir}$$ exists and equals $L$ ...
Nate River's user avatar
  • 6,155
0 votes
0 answers
120 views

Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$

I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$: $$\frac{1}...
Josh's user avatar
  • 1
1 vote
1 answer
191 views

Concentration inequality for square roots

Given a sequence of (not-necessarily-iid) real-valued random variables $X_n$ that converge to $a\in\mathbb{R}$ in probability, suppose we have an exponential concentration inequality of the form $$ P(|...
tim523's user avatar
  • 13
3 votes
2 answers
429 views

Functional equations based on composition

I have asked this question here (*), but there are no answer. Let $n \in \mathbb N^*$, $\{a_0,\ldots,a_n\} \subset \left] 0,+\infty\right]$. We suppose $Eq : \sum\limits_{k=0}^n a_k f^k(x)=0$ have no ...
Dattier's user avatar
  • 4,074

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