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

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3
votes
2answers
353 views

Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!
1
vote
2answers
247 views

What is known about $\displaystyle \sum_k{a^{b^k}}$?

What is known about $\displaystyle \sum_k{a^{b^k}}$? I am very interested in the possible applications of this series. I have asked about this on Mathematics Stack Exchange here. I'm wondering if ...
0
votes
0answers
59 views

Trying to solve for total derivatives at a stationary point (maybe using the implicit function theorem)

Suppose we have a function $F(q) \in \mathbb{R}$, where $q=(q_1, \dots, q_n) \in [0,1]^n$, at least thrice differentiable in $(0,1)^n$. We fix the value of one variable $q_i \in (0,1)$, then maximize ...
1
vote
1answer
399 views

Monge–Ampère type equation

Let $B(x,y) \geq 0$ be a function defined for $x, y \geq 0$ such that $B(x,0)=B(0,y)=0$ and $B''_{xx}\leq 0, B''_{yy}\leq 0$ (i.e. it is bicocncave function). I am looking for the solutions among of ...
3
votes
1answer
86 views

Taylor Series Expansion [closed]

I am reading an article and came across this expression and would appreciate some explanation. "We have a function $$u(F)=2\Big[\frac{F-L}{L}-ln\frac{F}{L}\Big]$$ Since $u''(F)>0$ a Taylor ...
3
votes
1answer
209 views

Can I approximate Schwartz functions which integrate to zero by $C_0^\infty$ functions which integrate to zero?

Let $X$ be the closed subspace of Schwartz space $\mathcal{S}(\mathbb{R}^N)$ defined by \begin{equation*} X=\left\{f\in\mathcal{S}(\mathbb{R}^N):\quad \int f\; dx=0\right\}. \end{equation*} My ...
3
votes
0answers
72 views

Monotonicity of solution of very simple integral equation [closed]

Given p>1 denote by x=x(p) the solution of $$ \int_0^x\frac{dt}{1+t^p}=1. $$ Prove that $p\to x(p)$ is a decreasing function of $p$ on $(1,\infty)$
10
votes
1answer
252 views

smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...
3
votes
1answer
145 views

Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...
1
vote
0answers
77 views

Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...
2
votes
0answers
200 views

More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...
-2
votes
1answer
118 views

a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...
2
votes
0answers
82 views

Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diffusing Particles. The picture below shows the idea how permutations and inversion numbers reflect ...
4
votes
2answers
318 views

Abstract ODE; PDE; uniqueness of solution

I have a somewhat vague question regarding an abstract ODE in a Banach space. Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...
17
votes
2answers
653 views

What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular, What properties are common to ...
0
votes
1answer
207 views

Theorem with an example [closed]

i have this theorem in the paper they gives an example: but here $H_1$ is not satisfied ! How to correct it please?
1
vote
0answers
216 views

Error term in formula for products of necklaces

Let us consider the $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$, where the number of fixed necklaces of length n composed of a types of beads $N(n,a)$ can be calculated via totient function: ...
1
vote
1answer
117 views

On the geometry of roots of a sum of complex linear fractions

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex ...
9
votes
2answers
320 views

On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation ...
4
votes
1answer
165 views

Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral $$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$ which frequently occurs in ...
2
votes
0answers
83 views

Quick estimate of attractor of non-linear dynamical system [closed]

Say I have a system of form $$ \frac{dy}{dt} = f(y), $$ and it is know this system has an attractor. Can I quickly for given $\varepsilon$ guess some point, such in its $\varepsilon$- neighbourhood ...
4
votes
1answer
290 views

solution of functional equation $f^{\circ k}(x) = x$

The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]: ...
6
votes
4answers
433 views

NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent. I would thus like to collect in this thread a list of ...
3
votes
2answers
169 views

Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some ...
1
vote
2answers
212 views

Looking for a limit related to the series in a previous post

Can any one show that the following limit? $$ \lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1. $$ If one uses the ...
1
vote
1answer
92 views

Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)? That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain ...
3
votes
2answers
356 views

How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration? $$ \int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0. $$ This post is related to my previous question here , ...
1
vote
1answer
147 views

Intermediate value theorem for the Jacobian determinant restricted to a curve

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian determinant. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. Is it right that on every continous curve connecting ...
14
votes
2answers
410 views

What geometric information is carried by the Fourier coefficients of the components of a closed curve?

Let $\gamma$ be a smooth closed curve in the plane and let $(x(t), y(t))$ be a parametrization. The functions $x(t)$ and $y(t)$ are smooth and periodic, so each has a uniformly convergent Fourier ...
1
vote
1answer
291 views

Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...
3
votes
1answer
165 views

A differentiable version of the Michael selection theorem

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map. Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?
1
vote
2answers
213 views

Jacobian of an injective mapping

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. I want to prove two things that seem intuitively right: 1) $f$ is ...
3
votes
1answer
245 views

A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...
1
vote
0answers
92 views

Periodic solution of first order ODE

There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system $$ \left\{ \begin{array}{l} \dot{x}=f(x), \\ x(t_0)=x_0, ...
2
votes
0answers
95 views

Stationary Distribution for Markov-like system?

Let \begin{equation} A= \begin{pmatrix} 0 & a_{1,2} & a_{1,3} \\ a_{2,1} & 0 & a_{2,3} \\ a_{3,1} & a_{3,2} & 0 \end{pmatrix}, \end{equation} \begin{equation} B= ...
9
votes
1answer
849 views

Has anyone seen this series?

I come across the following infinite series. $$ \sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}. $$ In particular, I am interested in the case where $a=1/4$. ...
2
votes
1answer
128 views

Global Solutions of Ordinary Differential Equations

Background Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying, $f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$, for every ...
1
vote
0answers
304 views

Presence of singular points in the trajectory of a double pendulum

Watching the trajectory of a double pendulum, I caught myself wondering if it would be possible to prove that the path the second pendulum makes contains "cusps" or singular points. Upon investigating ...
1
vote
0answers
115 views

Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...
3
votes
1answer
218 views

About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality $f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...
4
votes
2answers
238 views

Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows: $V_0(x) = 1-a^2$ $V_1(x) = 1-a^2 - x$ $V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$ ...
0
votes
1answer
130 views

Solution of an Ordinary Differential Equation

I'd like to know if there is an analytical solution of the following Ordinary Differential Equation : $\displaystyle{\dfrac{d}{dt}x_i(t) = D_i + \sum_{j=1}^{n}L_{ij}x_j(t) + ...
-1
votes
1answer
120 views

Method of steepest descents for $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt\sim\frac{i}{z}$

Through some calculations I ended up with the integral $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt$. I would like to obtain the result that the behaviour of the integral is $\sim\frac{i}{z}$ as ...
5
votes
0answers
86 views

Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$. By this I mean that there exists a polynomial $P$ with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!) such that $P(\phi) = 0$ Let ...
-1
votes
1answer
97 views

A proof for $[(f^k)^{(n)}]^2 \geq (f^{k-1})^{(n)} (f^{k+1})^{(n)}$

I want to show $[(f^k)^{(n)}]^2 \geq (f^{k-1})^{(n)} (f^{k+1})^{(n)}$, where $f$ satisfies $f^{(n)} \geq 0$ for all integer $n$ and $f^k$ denotes the $k$-th power of $f$. I believe it's right but I ...
2
votes
2answers
274 views

What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?

There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point ...
3
votes
1answer
259 views

Asymptotic expansion of modified Bessel function $K_\alpha$

An integral representation for the Bessel function $K_\alpha$ for real $x>0$ is given by $$K_\alpha(x)=\frac{1}{2}\int_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where ...
27
votes
3answers
1k views

A translation of the Cantor set contained in the irrationals

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would ...
1
vote
0answers
307 views

Inverse Transpose of Jacobian Matrix

Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by \begin{equation} f(x)\approx ...
3
votes
2answers
293 views

An integral related to the Euler Gamma function

The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral $$\eta=2\int\limits_0^1 ...