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

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18 views

Generalizing approximate $\mathbb{Z}$-equivariance of a simple function

Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. http://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
0
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0answers
59 views

An innocuous second order linear ODE [on hold]

Is there much work done on equations of the form $$ y'' + \alpha(t)y = 0,$$ where $\alpha(t) \in C^\infty([0,\infty))$ and $\alpha(t) > 0$? In particular, I am looking for some blow-up results. I ...
1
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1answer
82 views

Can this equality hold for a nonzero $b$?

Please may you kindly assist me on this integration exercise: For real $a, b$ with $a \neq 0$, consider the equality $$\int_1^\infty f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^\infty f(x)\sin(a\...
3
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2answers
63 views

Is there a full-rank map with connected graph and simply connected image that is not injective?

I want to find a continuously differentiable function $F:X\to Y$, where $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$ are open ($n\le m$) with ${\rm rk}\, \frac{\partial F}{\partial x}(x) = ...
5
votes
1answer
158 views

A two-point inequality

Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...
-1
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0answers
94 views

Summing a series of integrals [closed]

EDIT: This IS related to my research (investigating representations of the harmonic mean)and I gave the wrong formula for the sum the first time around. The sum formula has been amended below. I ...
4
votes
1answer
68 views

uniform one-sided van der Corput inequality

Is the following true (and if yes, where the best proof is written?)? For any $c>0$ for large enough positive integers $N$ we have $\sum_{k=0}^{N-1} \cos(k^2t)\geqslant -cN$ for all real $t$? Hm,...
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0answers
13 views

Conditions for convergence to non-isolated fixed points

Consider a dynamical system of the form $$ \dot{x}=f(x), \quad x\in X, $$ and assume that the system possesses a set of non-isolated fixed points. Suppose moreover that there exists a Lyapunov $V(x)$ ...
5
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99 views

For $f$ a polynomial, does strict convexity of $\log f(e^s)$ imply that the second derviative of $\log f(e^s)$ has no zeros?

Let $f(t)$ be a monic real polynomial such that $f(t) > 0$ for all $t \ge 0$. Suppose that $\log f(e^x)$ is strictly convex on $\mathbb{R}$, i.e. $f(s^2) \cdot f(t^2) > f(st)^2$ for all $s, t \...
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0answers
38 views

References for the Sturm oscillation theorem

What is the most general form of the Sturm oscillation theorem? So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...
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0answers
44 views

Continuity and divergence

Put \begin{equation} F(s) = \sum_{n \geq 1}a_{n}(s), \end{equation} where $a_{n}(s) \geq 0$ for all $s \in \mathbb{R}$ and the series converges for $\text{Re}(s) > 1$. Suppose that $F(s)$ is ...
2
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0answers
73 views

Imposing boundary conditions and self-similarity on a PDE

This question is an exact duplicate of the question Imposing boundary conditions AND self-similarity on a PDE posted by Stan Corey Carter on math.stackexchange.com. I have a PDE in the ...
6
votes
1answer
123 views

L1 analog of Bernstein's inequality

Let $p(x)$ be a degree $n$ polynomial over $[-1, 1]$, and let $q(x) = p'(x) \sqrt{1-x^2}$. Is it true that $$ \|q\|_1 \leq O(n) \|p\|_1 $$ where we define $\|f\|_p := \left(\int_{-1}^1 |f(x)|^pdx\...
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0answers
73 views

some strange sums ramanujan type

i found sum as $$\sum _{k=0}^{\infty } \frac{e^{k x} \left(-\frac{1}{y}\right)^k}{p-e^{k x}}=\sum _{n=1}^{\infty } \left(\frac{-1+\, _2F_1\left(1,\frac{2 i n \pi -\log \left(-\frac{1}{y}\right)}{x};\...
2
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0answers
105 views

Decay of the Fourier transform of a surface area measure

Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
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77 views

Can we always extract a proper Hausdorff measurable subset from a Hausdorff measurable set?

I also put this question on MSE here Let $\Gamma\subset \Omega\subset \mathbb R^N$ be such that $\mathcal H^{N-1}(\Gamma)<+\infty$ (this also implise that $\Gamma$ is Hausdorff measurable). Let $\...
0
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1answer
210 views

The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]

In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
5
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1answer
68 views

Optimal constant for a Sobolev-type inequality

Let $\overset{\circ}{H^s}(\mathbb T)$, where $s\ge 0$, be the space zero average of $2\pi$-periodic functions $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx},$ such that $$ \lvert u\rvert_s = \...
2
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0answers
122 views

Parametric normalized Fermat curves (Fermat functions)

The normalized Fermat curve is $X^n+Y^n=1$. We have of course infinite possibilities for parametrisation, but the periodicity is a special characteristic here. E.g. $\cos^2x+\sin^2x=1$ has ...
13
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1answer
434 views

Why sum of three squares of real polynomials is a sum of two squares?

If $f(x),g(x)$ are real polynomials, then $f^2+g^2+1$ is a sum of two squares of polynomials. This easily follows from Fundamental Theorem of Algebra, but is there an argument avoiding it? What are ...
0
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1answer
57 views

characterization of normality by selection theorem

The Urysohn's extension theorem states that a space $X$ is normal iff every continuous function $f:A \rightarrow \mathbb{R}$, with $A$ a closed subset of $X$, can be extended to a continuous function $...
6
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1answer
127 views

The “Peano phenomenon” for differential equations

Consider the following statement: If $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, for the autonomous equation $$x' = f (x)$$ the "Peano phenomenon" can arise only at those values ...
5
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3answers
284 views

Nonlinear ODE: $y'=(1+axy)/(1+bxy)$

Consider the first order nonlinear ODE problem: $$ y'(x)=\frac{1+ay(x)x}{1+by(x)x}, \quad x>0 $$ where $a, b>0$ are some constants. I would like to know if these kind of equations were ...
3
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2answers
359 views

An integral identity evaluating the gamma function

While reading a number theory paper I encountered the identity $$ \int_{- \infty}^{\infty} (1 + x^2)^{ - \frac{z}{2} - 1} dx = \sqrt{\pi} \frac{ \Gamma(\frac{z + 1}{2}) }{\Gamma(\frac{z}{2} + 1)},$$ ...
3
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0answers
87 views

Optimal Kakeya Maximal Bound for Bushes

Let $\{T_{\alpha}\}$ be a collection of $1\times\cdots\times 1\times N$ tubes, where $N\gg 1$, with maximal $1/N$-separated directions, which all are centered at the origin (i.e. they form a bush). In ...
0
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1answer
77 views

Behavior of a Solution of a Nonlinear ODE

In my work, I encountered the following equation: $$ (a'(x)+1)^2+k^2(x) a^2(x)=1,\;\;k(x)=2 {\mbox{sech}}(x). $$ I would like to know as much as possible about the solution. More particularly, I would ...
14
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2answers
313 views

Needing proof of convergence for a sequence

Let $\left\{u_i\right\}_{i=1}^\infty$ be a sequence of real vectors (i.e. $u_i\in R^n, i=1,2,... $) and $m$ an integer large enough such that $\sum_{i=1}^m u_i u_i^T$ is a positive definite matrix. ...
5
votes
1answer
153 views

Domain of Laplacian

Let $L$ be an operator on $C^2(\mathbb R)$, defined by $$L \phi (x) = \int_{|y|<1} (\phi(x+y) - \phi(x) - \phi'(x) \ y)\ \nu(dy), \text{ for all } x\in \mathbb R$$ for a measure $\nu(dy) = |y|^{-2} ...
1
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1answer
99 views

Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
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24 views

diffusion and potentials in several dimensions

In a Lagrangian field theory there are conserved quantities, like total energy. This allows geometric arguments to be made about the behaviour of a system with known potential. (E.g. on asymptotic ...
2
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1answer
94 views

Smooth dependence on the initial condition of the integral of an ODE

I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi_t x$ be the value of the solution at time $t$. I assume that my ODE ...
2
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1answer
52 views

Differential inequalities for a strictly diagonal dominant system of linear ODEs

Let $A$ be a real $d\times d$ matrix. The diagonal elements are strictly negative ($a_{ii}<0$) and the off-diagonal elements are non-negative ($a_{ij}\geq 0$ for $i\neq j$). $A$ is strictly column ...
2
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2answers
382 views

Generalizations of the Euler-Maclaurin Summation Formula

I'm using the Euler-Maclaurin formula in a research I'm working on. However brilliant is the elementary proof found here, I need and want to know more about it. Namely Specifically, I would like to ...
2
votes
1answer
179 views

Is there a matrix that converts the gradient of every possible function to gradient of other function?

I have already asked this question on math.stackexchange.com http://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe Now I ...
6
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1answer
214 views

Is the exponent $2$ sharp in the Balog-Szemerédi-Gowers Theorem?

The Balog-Szemerédi-Gowers theorem can be stated in the following form: let $A,B$ be subsets of $\mathbb{Z}/n\mathbb{Z}$ (say) with equal cardinality, such that $$ \|1_A*1_B\|_2 \ge K^{-1} \|1_A\|_1 \|...
2
votes
1answer
153 views

Ramanujan-type sum

Could you show that $$\sum _{k=0}^{\infty } \frac{k}{e^{\frac{\pi k}{2}}+1}=\frac{7 \pi ^2+6 \left(\psi _{e^{2 \pi }}^{(1)}(1)+\psi _{e^{2 \pi }}^{(1)}\left(1-\frac{i}{2}\right)-\psi _{e^{2 \pi }}^{(...
1
vote
0answers
104 views

Ramanujan sum type

I try to show $$\sum _{k=1}^{\infty } \frac{e^{-2 k} k}{e^{-2 k}+1}=\frac{\pi ^2}{48}-\frac{\pi ^2-6 \left(\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}(1)+\psi _{e^{\frac{\pi ^2}{2}}}^{(1)}\left(\frac{-2 i+\pi }...
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0answers
48 views

Dirichlet series decomposition of arbitrary function

Originally asked on MSE here: http://math.stackexchange.com/q/1780149/52694 Analytic functions can be decomposed into a Taylor series, and furthermore the Taylor series converges back to the original ...
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0answers
60 views

History of Cauchy-Euler Equations

As I teach a class in ODE, and following this post and Rota's paper, I wandered what is the history of the research of - $\sum\limits_{k=0}^n a_k x^k y^{(k)}(x) = g(x),\quad \forall k=0,\...
3
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1answer
78 views

Algorithm for definite integral of rational functions of x and exp(-x)

I'd like to find the class of functions that may appear as definite integrals of a rational function of $x$ and $\exp(-x)$ from $0$ to $\infty$. For that I imagine there might be an algorithm to ...
3
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0answers
60 views

Factoring Bessel functions into an amplitude and a phase

Take some $\nu>0$. Let $J_\nu(x)$ be the Bessel function of the first kind. Let's restrict its domain to $\mathbb R^+$. Is it possible to find a pair of functions $A_\nu(x), \phi_\nu(x):\mathbb R^+\...
7
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1answer
151 views

$C^{k,\alpha}$ diffeomorphisms and vector fields

This feels like something I should know, but I have a hard time finding a definite reference. Let $M$ be a compact (Riemannian) manifold, $k\ge 1$ be an integer and $\alpha\in(0,1)$. When v is a $C^k$...
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1answer
78 views

Question on Littlewood-Paley trichotomy

In proving the product estimate, we need the Littlewood-Paley trichotomy. See http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps. In the decomposition $$P_k (fg)=\sum_{k',k''\in Z} P_k (P_{k'} f P_{k'...
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2answers
60 views

Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix. I've seen a similar ...
0
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1answer
83 views

Branches of the tetration function

Letting $\eta = e^{1/e}$ where $e$ is Euler's constant, there exists a function $F(z)=\, ^z \eta$ with the following relevant properties. (I won't bother showing the existence of this function, or the ...
7
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1answer
288 views

Rotation invariance of an integral

Consider the integral depending on 2 parameters $$f(\tau,x):=\int_{-\infty}^{+\infty}\frac{dp}{\sqrt{p^2+1}}e^{-\sqrt{p^2+1}\tau+ipx},$$ where $\tau >0,x\in \mathbb{R}$. This integral absolutely ...
3
votes
2answers
238 views

Nice way to express $H^{-1}(\mathbb{S}^1)$

I am looking for a good way to write down what $H^{-1}(\mathbb{S}^1)$ is. What do I mean by this: Well, certainly one can define this space via charts that map from the sphere into $\mathbb{R}^2$ and ...
3
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0answers
175 views

Modified Jacobi’s theta function

Be $t\in\mathbb{R}_0^+$. Jacobi’s theta function is $$\Theta(t):=\sum\limits_{k=-\infty}^{+\infty} e^{-\pi k^2 t}$$ with $$\Theta(\frac{1}{t})=\sqrt{t}\Theta(t)$$ Therefore $$\sum\limits_{k=1}^\infty ...
1
vote
0answers
36 views

convergence of ODE [closed]

I have 2 coupled linear ODEs. I used Mathematica to solve for analytical solution. But the analytical solution looks too complicated. I only need to derive some monotonicity property of the solution. ...
2
votes
3answers
241 views

Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$. Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$ $$f(a;b):=\prod\limits_{k=1}^\infty ...