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Special functions, orthogonal polynomials, harmonic analysis, ordinary differential equations (ODE's), differential relations, calculus of variations, approximations, expansions, asymptotics.

-1
votes
0answers
22 views

On Compactness in $M_{b}(\Omega)$

i need some compactness results in the space of measures $M(\Omega)$ where $\Omega$ is an open subset of $R^n$. I want to search results in the various topologies, but in the specific i need results ...
5
votes
2answers
194 views

asymptotic for li(x)-Ri(x)

Is it true that $$\operatorname{li}(x)-\operatorname{Ri}(x) \sim \frac{1}{2}\operatorname{li}(x^{1/2}) \ (x \to \infty),$$ where $$\operatorname{Ri}(x) = \sum_{n = 1}^\infty \frac{\mu(n)}{n} \...
6
votes
1answer
245 views

Dominated convergence 2.0?

During my research, I came across the following question. Let $(f_n)_n$ be a sequence in $C^2([0,1])$ converging pointwise to $g \in L^1([0,1])$. Assume that: $\forall n\in\mathbb N, f_n''<h$, ...
0
votes
0answers
48 views

Derivative of the flow with respect to initial time

Let $f \colon \mathbb R^d \to \mathbb R^d$ be a smooth (say, globally bounded) vector field. For $t,s \in \mathbb R$ and $x \in \mathbb R^d$ let $X(t,s,x)$ be the unique solution of the problem $$ \...
1
vote
0answers
47 views

Linear dependence of solution?

Consider the function $f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$ In the following we want to study the ...
3
votes
2answers
238 views

Prove $\int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1$

The question is to prove: $$ \int_0^{\infty}{\frac{1}{e^{sx}\sqrt{1+s^2}}}ds < \arctan\left(\frac1x\right),\quad\forall x\ge1. $$ Numerically it seems to hold true. So I have made some attempts to ...
4
votes
1answer
66 views

Convergence of a real sequence (stochastic approximation)

Let $(\gamma_n)_{n \geq 1}$ be a sequence of positive real numbers satisfying $$\sum_{n \in \mathbb{N}}\gamma_n = + \infty \text{ and }\sum_{n \in \mathbb{N}}\gamma_n^2 < + \infty$$ I would like ...
1
vote
2answers
152 views

Fourier transform of a generalized function on the plane

Is there an explicit formula for the Fourier transform of the generalized function of 2 variables $$\frac{1}{x+y^2+i0}?$$ Remark. Equivalent question: consider the Schroedinger equation one the ...
4
votes
0answers
112 views
+100

Sobolev space under Mellin transform

The Mellin transform is known to be an isomorphism see wikipedia between $M:L^2(0, \infty) \rightarrow L^2(-\infty, \infty)$ where $$M(f):= \frac{1}{\sqrt{2\pi}}\int_0^{\infty} x^{-\frac{1}{2} + is} ...
-4
votes
1answer
95 views

Hamiltonian functions [closed]

I have a Hamiltonian function given by $$H(x,y)=(b/2) y^2+(ax-x^3)y+(a^2/ 2 b) x^2=C$$ ($C \in \mathbb R$). I would like to draw the curve of $H(x,y)-C$ by using critical curves of this curve. But ...
8
votes
2answers
266 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
3
votes
0answers
72 views

Wavefront set and Duhamel's principle

Consider the Cauchy problem: $$ \frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0, $$ where $A$ has real principal ...
7
votes
1answer
180 views

Proof of Green's formula for rectifiable Jordan curves

$\newcommand{\Ga}{\Gamma}$ I am trying to find a proof of Green's formula for rectifiable Jordan curves $\Ga$ (and the corresponding interior regions $R$). There is a proof by Ridder, followed by ...
11
votes
1answer
370 views

Nonlinear Schrödinger equation with discrete Laplacian

In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
0
votes
1answer
56 views

Harnack inequality for fractional laplacian

Let u be a positive solution of $s\in (0, 1) $ \begin{equation} \left\{\begin{aligned} (-\Delta )^{s} u &= 0 \text{ in } (-2T, 2T)\\ u &=g\quad\text{in}\quad \mathbb R\setminus(-2T, 2T). \...
1
vote
2answers
83 views

One inequality connected with the linear second order ODE

Is the following statement true? Let $ a>0, b>0, h>0 $, $x(t)$ be the solution of the differential equation $ \ddot{x}+a \dot{x}+bx=h$ with initial conditions $x(0)=u<0 , \dot{x}(0)...
4
votes
1answer
151 views

Extension by harmonics

Let $d \geq 2$ and $K$ some positive integer. Consider distinct points $\theta_1, \ldots, \theta_K\in \mathbb{T}^d$ and (not necessarily distinct) $z_1, \ldots, z_K \in \mathbb{C}$, does there exist ...
11
votes
1answer
462 views

A recursive formula

$$a_0 = 1, \ \ a_{n+1} = \ 1+\frac{n *a_n}{n+a_n} , \ \ n=0,1,2,3,4,...$$ I have built the above recursive formula. Some terms of sequence are: 1, 1, 3/2, 13/7, 73/34, 501/209, 4051/1546, 37633/...
-1
votes
1answer
66 views

A smooth curve and mean value theorem

Assume that $f$ is smooth function defined in the unit disk $D: x^2+y^2\le 1$, and consider the integral $$I=\int_D f dxdy=\int_0^1r \int_0^{2\pi} f(re^{it})dt.$$ Then it is clear that for $r\in[0,1]$...
2
votes
0answers
41 views

Solvability of Neumann boundary problems with singular boundary data $g \in (H^{1})^{*}$

I have a question on the solvability of Neumann boundary problems with singular data. To state my question, let $\Omega$ be a bounded Lipschitz domain (open and connected) in $\mathbb{R}^n$. In the ...
16
votes
1answer
2k views

Differential equation changing sign almost everywhere

Conjecture: Let $f:\mathbb{R}→\mathbb{R}$ be an everywhere differentiable function and assume that $f(x)+f′(x)∈ \{-1,1\}$ almost everywhere and $f'(0)=0$. Then is $f$ necessarily a constant function? ...
0
votes
0answers
103 views

Estimation of the integral $\int_a^b e^{2\pi i f(x)} dx $

Let $f$ be a $C^2$ real-valued function on the interval $[a,b]$. Suppose that $f'(x)$ is monotone on $[a,b]$ and there is $\lambda>0$ such that $$ \min_{x\in [a,b]} |f'(x)|>\lambda $$ It is ...
16
votes
2answers
418 views

Existence of an antiderivative function on an arbitrary subset of $\mathbb{R}$

Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $x$ for every $x\in I$ where $I\subset \mathbb R$ could be arbitrary. Does there always exist a function $F:\mathbb{R}\to \mathbb{R}$ differentiable ...
1
vote
1answer
173 views

An equation in Gamma function has at most (n-1) positive solutions

I have to prove some result. And for that, I need to prove this new problem. To prove, $c_{1}\Gamma(z+b_{1})+c_{2}\Gamma(z+b_{2})+\ldots+c_{n}\Gamma(z+b_{n})=0$ has at most $(n-1)$ real positive ...
9
votes
2answers
351 views

Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular

Let A is a $n\times n$ matrix given by \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \ldots < \...
2
votes
2answers
170 views

Projection of a ball in the ambient space to a manifold

Let $B_h (x)$ be the ball of radius $0<h \ll 1$ centered at $x\in \mathbb{R}^d$. Let $I=[0,1]^{d-1}$ be the unit cube in $\mathbb{R}^{d-1}$, and let $f:I \to \mathbb{R}$ be a $C^2$ function. Then $...
6
votes
1answer
481 views

(In)stability of a two-dimensional dynamical system

Consider the following system of coupled differential equations $$ \dot{x}_1(t) = -x_1(t) - \cos(\omega t)x_1(t) + \cos(\omega t)x_2(t), \ x_1(0)\in\mathbb{R},\\ \dot{x}_2(t) = -\gamma x_2(t) - \cos(\...
2
votes
0answers
148 views

A question about whether an operator can be lipschitz or not

Let assume $ X= \Omega \times (0,T) $ where $\Omega \subset \mathbb{R} $ is a bounded open set and $T>0$. Now define the operator $ \mathcal{A} : C^{‎\sigma‎, \sigma‎/2‎}(‎X‎) \to C^{‎\sigma‎, \...
3
votes
0answers
70 views

Fenchel conjugate on a Hadamard manifold

Let $M$ be a Hadamard manifold and let $F:M\to\mathbb{R}$ be a real-valued convex function on $M$. What would be the Fenchel-Young conjugate of $F$? In general for a real locally convex vector space $...
1
vote
1answer
80 views

Behaviour of solutions to $(A-r)f=0$ in the limit $r \to \infty$

Define the second order linear differential operator associated with $X$ (Here $X$ is the unique strong solution to appropriate Ito SDE) by $$A = \frac{1}{2} \sigma^2(x) \frac{d^2}{dx^2} + \mu(x) \...
10
votes
2answers
347 views

$x f'$ bounded by $x^2f $ and $f''$?

Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$ I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
12
votes
0answers
161 views

Profiles of very high dimensional functions

This question comes from trying to understand the recent success of deep neural nets. Neural networks just (crudely speaking) create a very complicated function of very many variables, and then ...
8
votes
1answer
185 views

Bi-Lipschitz extension

Given a bi-Lipschitz homeomorphism $\Phi:\mathbb{B}^n(0,1)\to\mathbb{R}^n$, (that is a bi-Lipschitz map onto the image), can one find a bi-Lipschitz homeomorphism $\Psi:\mathbb{R}^n\to\mathbb{R}^n$ ...
3
votes
0answers
207 views

Can continuity always be shown by using ε-δ? [closed]

When we learn calculus we usually: 1. Prove that polynomials, the exponential functions, the logarithmic functions, the trigonometric functions, the inverse trigonometric functions are continuous on ...
25
votes
1answer
2k views

Lindelöf hypothesis claim

I was randomly browsing, when I found this puff piece claiming a proof of the Lindelöf hypothesis by Fokas. Note that the Wikipedia article says that he claimed, then withdrew his claim in 2017, but ...
4
votes
1answer
231 views

$\sum_{k=1}^n\frac{\sin kx}{k^\alpha} >0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi, \text{and}\ \alpha \ge 1$

The Fejer-Jackson inequality as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ I conjecture that the inequality as follows holds: $...
0
votes
0answers
25 views

Classical solution of one dimensional Parabolic equation and a priori estimates

I am researching a system of pdes and it leads me to study the classical solution of one dimensional linear parabolic equation: $u_t+Lu=f,\, t\in[0,T],x\in \Omega=[0,1]$, where $L$ is non-divergence ...
4
votes
1answer
182 views

Symplectic forms and sign of eigenvalues

This question has come out while reading J. Moser "New Aspects in the Theory of Stability of Hamiltonian Systems". I'm particularly interested to the Appendix, where one investigates the stability of ...
10
votes
1answer
285 views

Asymptotic behavior of an integral depending on an integer

A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where $$ f(n) := \...
1
vote
0answers
38 views

Conditions for Poisson summation (for discontinuous functions)

Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
0
votes
0answers
60 views

the enumeration of 2 dimensional lattice walks with fixed number steps and largest distance from the end point ti the origin

There is actually an one dimensional version of this problem. For each step of the lattice walk, we can move either east for one unit or west for one unit. The problem is that given a fixed $n$ steps ...
13
votes
1answer
399 views

A question on the sine function

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$ Here I ask the ...
5
votes
0answers
65 views

Computing (formally or numerically) Green's function for the wave equation on a sphere

Consider Green's function for the wave equation on a sphere, namely, for $t>0$ and fixed $0<\theta<\pi$, $$G(\theta,t) = \sum_{\ell=0}^{+\infty} (2\ell+1)\,P_\ell(\cos\theta)\,\cos\big(\sqrt{\...
0
votes
1answer
132 views

Corner integrals of $\exp$

I feel that there is a good chance to apply certain integrals of $\ \exp(-(\sum_{k=1}^n x_k))\ $ over corners (see below) to the analytic number theory. I have obtained two formulas to start with ...
4
votes
2answers
172 views

Invariance of an integral over SO(3) under permutation of parameters

The integral $$Z_3(\lambda_1,\lambda_2,\lambda_3)=\frac{1}{2}\int_{-1}^1 I_0\left[\tfrac{1}{2}(\lambda_1-\lambda_2) (1-x)\right] I_0\left[\tfrac{1}{2} (\lambda_1+\lambda_2)(1+x)\right]\,e^{\lambda_3 ...
1
vote
0answers
58 views

density of fractal measures

Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
2
votes
1answer
71 views

Lipschitz bound on semigroups

Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator. Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$ Now if you think about ...
0
votes
2answers
112 views

Generalization of a proposition on vector analysis [closed]

Here is my question: Suppose there is a map from $\mathbb R$ to $\mathbb R^4$, say $f$. And it satisfies: for any $t$, $f(t)$ and $f'(t)$ are linearly independent; but $f''(t)$ is linearly dependent ...
1
vote
2answers
58 views

Confluent Heun Equation

Does anyone know any source in which I could find a recurrence relation for the coefficients of the series solution of the Confluent Heun Equation $$y''+\left( {\gamma\over z}+{\delta\over z-1}+\...
2
votes
0answers
114 views

Average of irrational flow on the torus

Let $$F(x,y) = \frac{1}{\sqrt{2-\sin(2\pi x) - \sin(2\pi y)}}$$ defined on $\mathbb{T}^2$. Here $\mathbb{T}^2 = \mathbb{R}^2/ \mathbb{Z}^2$ is the 2-torus. How can I show that $$ \lim_{T\...