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

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27
votes
1answer
1k views

Geometric interpretation of the half-derivative?

For $f(x)=x$, the half-derivative of $f$ is $$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$ Is there some geometric interpretation of (Q1) this specific derivative, and, ...
3
votes
2answers
161 views

The epigraph of a semi-convex function has positive reach

I've been trying to prove the following theorem for several hours with no result so far. Claim. Let $f:\mathbb{R} \to \mathbb{R}$ be a semi-convex function, i.e. there exists a constant $C > 0$ ...
6
votes
1answer
341 views

Find functions such that f(f(x))=f(x)e^x

Are there monotonically increasing functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = e^x f(x)$?
2
votes
1answer
124 views

Is Poisson's kernel integrable?

Let $E$ be a smooth domain. Green's function is defined as $G(x,y)=F(x,y)-\Phi(x,y)$ where $F$ is the fundamental solution to the Laplace equation. For a fixed $x\in E$, $\Phi(x,\cdot)$ is a harmonic ...
10
votes
3answers
513 views

Relating the roots of polynomials to the solution sets of certain functional equations

Consider a functional equation of the following form: $$\sum_{k=0}^n a_k\,\underbrace{f(f(\cdots f}_{k}(x)\cdots )=0\quad \big(f:\,\mathbb{R}\to\mathbb{R},\;a_i\in ...
42
votes
2answers
1k views

Is a function with nowhere vanishing derivatives analytic?

My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic? EDIT. According to a ...
14
votes
2answers
942 views

Generalization of Darboux's Theorem

Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$. Does any of the following generalizations Let ...
6
votes
1answer
223 views

Alternative proof of Lojasiewicz inequality

is there a "brute force proof" of the Lojasiewicz inequality? By "brute force" i mean without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e. standard ...
2
votes
0answers
95 views

A two dimensional integral equation

I have the following integral equation: $\phi(x, y) = \frac{a}{x-y} \int_y^x \phi(s, y) ds + \frac{b}{x-y} \int_y^x \phi(x, s) ds$ where $a > 1$ and $b> 1$ are constants, and $x \geq y$. The ...
1
vote
1answer
97 views

Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
0
votes
1answer
129 views

Lipschitz boundary vs rectifiable curve boundary

I was looking at an old paper about domains with Lipschitz boundary. I am wondering, suppose that the boundary of a compact domain homeomorphic to a disk is a rectifiable injective curve : is this ...
1
vote
1answer
172 views

Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations $$ \sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n, $$ has ...
2
votes
1answer
66 views

Expressions in “continued” monotone functions

Recall continued fractions: http://en.wikipedia.org/wiki/Continued_fraction Now take a look at this question: ...
-5
votes
1answer
120 views

Solution of $y'' - ax y' - y = 0$ [closed]

I am looking for solutions $y(x)$, $x$ positive, for the ODE $y''(x) - a x y'(x) - y(x) = 0$, where $a$ is any real number. It seems not to be any of the usual ODEs (i.e. those listet on Wikipedia), ...
2
votes
0answers
74 views

From Selberg integral to Dyson integral

My question is from the drivation from Slberg integral to Dyson integral in this paper: Selberg integral : $$ S_n(\alpha,\beta,\gamma) = \int_0 ^1 \cdots \int_0 ^1 \prod_{i=1}^n ...
7
votes
2answers
88 views

Bounds on coefficients of factors of a multivariate polynomial

Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the ...
0
votes
1answer
158 views

Problem with understanding an equation

I have read the article Short-wavelength Spectral Properties of the Gravity Field from a Range of Regional Data Sets and I don't know how to interpret Equation (10) on page 630, because this equation ...
4
votes
1answer
282 views

Is there a differentiable but nonsmooth version of the continuous Implicit Function Theorem?

From the result discussed in Does the inverse function theorem hold for everywhere differentiable maps? (which I'll call the differentiable nonsmooth Inverse Function Theorem) one can obtain a ...
5
votes
2answers
293 views

Thurston-Cannon $S^2$-filling curves

I have been looking into equivariant space-filling curves $S^1\to S^2$ as discussed by Cannon & Thurston in these two papers: Three-Dimensional Manifolds, Kleinian Groups and Hyperbolic Geometry ...
2
votes
1answer
141 views

Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
2
votes
1answer
71 views

Gradient of Ronkin function

I have a complex curve $P(z,w)=z+w-1=0$. I get the amoeba map $$(z,w)\rightarrow (\log|z|,\log|w|)$$ of this curve. It's look like this http://en.wikipedia.org/wiki/Amoeba_(mathematics) (the first ...
4
votes
1answer
300 views

Integral of a product of Laguerre polynomials

In order to estimate the non linear term in a particular PDE, I have to decompose $L_k^\alpha(x)^3\cdot x^{-\delta}$ (with $0<\delta<\alpha+1$) into a basis consisting of Laguerre polynomials ...
2
votes
1answer
202 views

A special case of the Divergence theorem

I am interested in the following statement: Let $F$ be a vector field in $\mathbb{R}^n$ that is $C^1$-smooth in a domain $U$, continuous up to the boundary $\partial U$, and vanishing on ...
3
votes
1answer
46 views

Hopf bifurcation for systems where the dynamics is homogeneous of degree 1

Consider dynamical system in dimension 3 $$x'(t)=f(x(t),d)$$ where the dynamics f is homogeneous of degree 1 and there is exactly one line of equilibrium points. This line is independent of the ...
2
votes
0answers
124 views

How many ways we have to prove that a topologically (or analytically) nice mapping is injective?

I would like to know what are the methods people have used to prove that a topologically (or analytically) nice mapping $f: B\to \Omega$ is injective? Above, $B$ is the unit ball in $\Bbb R^n$ and ...
1
vote
1answer
70 views

variational problem related to an integral

Recently I came up with a type of variational problem in stochastic process. It can be stated in the following way: Given $a$ and $b$ positive, and an increasing function $f$ on $(0,1)$ (may be not ...
1
vote
0answers
153 views

matrix Khintchine inequality

The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then \begin{equation*} \left( ...
3
votes
1answer
196 views

Asymptotic behavior for the solution of a nonlinear ODE

In a nutshell, if $u$ is a solution to $$ \partial_r^2 u(r)+ \frac{1}{r} \partial_r u(r) - u(r) ( 1- u(r)) = 0, \quad \text{for} \; r>r_0>0\\ \lim_{r \to \infty} u(r) = 0, \quad \text{and} ...
8
votes
1answer
836 views

If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
1
vote
0answers
56 views

Sufficient conditions for sums of Laguerre polynomials to be non-negative

I am interested in sufficient conditions on non-negative sequences of coefficients $\{c_{2n}\}_{n\ge 0}$ guaranteeing that $$%\begin{equation}\label{cond} \sum_{n=0}^\infty c_{2n} L_{2n}^{(1)}(x)\ge ...
3
votes
1answer
187 views

Non-linear first order ODE

This is a two part question. On one hand, I am trying to find positive solutions of the following equation: $$1-\frac{d}{dx}f=\frac{f\log(f)}{x}$$ for $x>1$. If that is not possible, I would at ...
3
votes
1answer
144 views

Infinite series - analytical solution

Analytical Solution is required for: $$\sum_{n=0}^\infty (2n+1)\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty (2n+1)^2\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty n(n+1)(2n+1)\exp(-n(n+1)x),$$ $$\sum_{n=0}^\infty ...
1
vote
0answers
206 views

Proof of an inequality for a linear combination of three trigonometric functions

Given a function $$f(t) = k_{1} \sin(t+\alpha) + k_{3} \sin(3t+\beta) + k_{5} \sin(5t+\gamma)$$ where $k_{1}, k_{3}$ and $k_{5}$ are all positive parameters, and the three phase angles, $ ...
6
votes
0answers
818 views

Questions on de Branges' work on the Riemann hypothesis

According to Wikipedia, Louis de Branges de Bourcia has obtained some notable results, such as a proof of the Bieberbach conjecture in 1985, which is now known as de Branges' theorem. Initially, his ...
2
votes
1answer
38 views

Prescribing finitely many unparameterised planar geodesics

Given a finite collection of embedded $C^\infty$ curves which pass through the origin in $\mathbb{R}^2$ with different tangent directions and never again intersect, is there a clean way of prescribing ...
3
votes
2answers
154 views

Identifying a special function from its power series

Here is a power series, which looks a bit like a Hypergeometric function series, but I don't think that it is. Has anyone any idea what it is? Here $n,p,r$ are integers with $n\ge 0$ and $p\ge r\ge ...
5
votes
2answers
416 views

What is the translation in Fourier transform for a function to have exp. decay at $x\to -\infty$

It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if ...
1
vote
0answers
78 views

Representing quasianalytic functions in several variables

For functions in a quasianalytic Denjoy-Carleman class we have the property that their Taylor expansions at a point (the origin) determines the function. For classes that don't only contain analytic ...
27
votes
2answers
2k views

Dynamical properties of injective continuous functions on $\mathbb{R}^d$

Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. $\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all ...
1
vote
3answers
256 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
5
votes
1answer
191 views

Integral Identity Involving Bell Numbers

Is the following identity true ? $$\int_0^\infty \frac{b(x)}{B(x)} dx \quad \overset{?}{=} \quad \int_0^\infty \frac{x!}{x^x} dx$$ where $$b(x) = \sum_{n=1}^\infty \frac{n^x}{n^n} \qquad \text{and} ...
2
votes
2answers
161 views

Regularity of a nonlinear ODE [Traveling wave solutions of parabolic systems]

In the book of Volpert on Traveling wave solutions of Parabolic Systems (AMS), one reads "the following assertion is readily proved and we shall not discuss it in detail". The same result is tacitely ...
7
votes
1answer
410 views

Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?

Is the mapping $$ f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n} $$ surjective? If not, what is its image? If yes, what can be said about ...
4
votes
1answer
230 views

Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote: The derivation of the conditions of exact integrability of an ...
1
vote
0answers
81 views

Does the difference quotient of an absolut cont. funct. converge in L^1?

Assume that $\mu$ is a finite Radon measure on the real line and $f$ is integrable wrt. $\mu$. Define $F(x)=\int_{]\infty;t]}f(y)d\mu(y) $ Is the following statement true? The functions ...
0
votes
1answer
118 views

Stability analysis of ODE

My questions concerns the stability analysis of the following dynamical system : $\dfrac{d}{dt} a_{i}(t) = D_{i} + \displaystyle{\sum_{j=1}^{n}L_{ij}a_{j}(t) + \sum_{j=1}^{n}\sum_{k=1}^{n} C_{ijk} ...
11
votes
1answer
985 views

Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$ runs over the integers? The existence of the limes inferior follows from Dirichlet's approximation theorem, but the ...
2
votes
0answers
60 views

Closed form for a simple hypergeometric q series

I've run across an interesting hypergeometric q-series that I feel must have been found before: $\sum_{n=0}^{\infty}(-1)^n$$\frac{e^{n b y}}{\prod_{k=1}^{n}(e^{\pi k b^2}-e^{\pi k b^{-2}})} = ...
10
votes
2answers
287 views

Simultaneous zero set of two equations in $\mathbb R^3$

Can we have positive reals $x,y,z$ with $$ x^{\left( y^z \right)} = y^{\left( z^x \right)} = z^{\left( x^y \right)} $$ in cyclic permutation, other than the line $x=y=z$? I put this at ...
2
votes
2answers
137 views

Estimate of a ratio of two incomplete gamma functions

I would like to bound from above the expression $$ \frac{\Gamma(\alpha,x)-\Gamma(\alpha,y)}{\Gamma(\beta,x)-\Gamma(\beta,y)} $$ for $x>y>0$. By plotting the above expression I have found that ...