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

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2
votes
3answers
184 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 ...
2
votes
0answers
73 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
33 views

convergence of ODE [on hold]

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
2answers
82 views

Analytic continuation of a specific integral with respect to a parameter

The following integral absolutely converges for $Re(z)<0$ and is analytic in this domain: $$F(z):=\int_{0}^{+\infty}\frac{\sin r}{r}e^{\sqrt{r^2+m^2}z}dr ,$$ where $m>0$ is fixed. Question. To ...
8
votes
2answers
900 views

Wasserstein distance in R^d from one dimensional marginals

This question occurred to me while I was reading Klartag's papers on central limit theorems for convex bodies. Given probability measures $\mu$, $\nu$ on (the Borel $\sigma$-field of) $R^d$ with ...
2
votes
1answer
69 views

Challenging problems concerning Jacobian elliptic functions with complex modulus

I study some qualitative properties of Jacobian elliptic functions. Consider, for example, function $sn(u,k)$. In most applications, modulus $k\in(0,1)$ and then everything is very clear, since ...
0
votes
0answers
126 views

What is the value of this infinite product? [closed]

$\prod^{\infty}_{n=3}\cos\frac{\pi}{n!}=?$ I can calculate the product, $\prod^{\infty}_{n=3}\cos\frac{\pi}{2^n}$, but I don't know how to calculate the above product. Could you help me? Thanks a ...
1
vote
1answer
98 views

What are the best known bounds on the Hermite polynomials?

The best I could find on the net is this paper, http://arxiv.org/pdf/math/0401310.pdf Has this been improved?
14
votes
4answers
474 views

Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant $$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i) $$ over all $a_0,\dots,a_{n-1}$ such ...
1
vote
1answer
107 views

Extremal problem for sequences

Let $a_n$ be a sequence of positive numbers and define $$A_n=\sum_{k=1}^{n-1} a_k a_{n-k}.$$ I am interested in the supremum of the following quantity $X/Y$ where $$X=\sum _{i=1}^{\infty } \left(\sum ...
2
votes
0answers
39 views

Harmonic functions in tempered distribution sense

Suppose $g$ is a metric on $\mathbb{R}^3$ and $\Omega \subset\subset \mathbb{R}^3$. We assume that $g$ is euclidean outside $\Omega$. My question concerns solutions to $\triangle_g u =0$ that are say ...
5
votes
2answers
122 views

Integral over the Cantor's set Hausdorff dimension

As can be seen in the David Morin's Classical Mechanics, there are some scaling strategies in order to calculate the moments of inertia of certain fractals, for example, the Cantor's set has a moment ...
0
votes
0answers
26 views

Uniqueness and Properties of Nonlinear 2nd Order ODE with Asymptotically Constant Coefficients

I have the following differential equation: $$ V(z) = e^z [1-\varphi(\gamma)]+(\gamma-g)V'(z)+\frac{1}{2}\kappa^2 V''(z)$$ with $$ e^z \varphi'(\gamma) = V'(z)$$ where $\varphi(\cdot)$ is a ...
21
votes
4answers
697 views

When are some products of gamma functions algebraic numbers?

I want to know when certain expressions of the form $ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $ are algebraic numbers. These ...
5
votes
4answers
374 views

How do the roots of a polynomial change when another polynomial is added?

I need to obtain an analytical solution to an equation of the following form: $$ (x-a)(x-b)(x-c)=d(x-e)(x-f), $$ where $a$, $b$, $c$, $d$, $e$, and $f$ are known numbers and $x$ is the variable. Of ...
2
votes
0answers
39 views

1D inhomogeneous linear Schrodinger equation

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm ...
3
votes
1answer
155 views

On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...
1
vote
1answer
147 views

find solution of complex number recurrence equation

I have the following recurrence equation: $$(\mu\ n + \nu) f_{n} + J\Phi^{*} \sqrt{n+1}f_{n+1} + J\Phi\ \sqrt{n}f_{n-1} = 0$$ for complex numbers $f_{n}$ where $n = 0,1,2,3,...,\infty$ and complex ...
0
votes
0answers
84 views

Does this set of (structured) equations always have a solution?

Let $r_1,\ldots,r_K$ be arbitrary positive numbers. Does $$|\mathcal{A}|\log\left(1+\frac{1}{|\mathcal{A}|}\left(\sum_{n\in \mathcal{A}} \sqrt{x_n(\exp(r_n)-1)}\right)^2\right)\leq \sum_{n\in ...
2
votes
0answers
75 views

Is a one-dimensional unstable manifold of an ODE a union of the associated equilibrium point and two full orbits? [closed]

Consider an ordinary differential equation (ODE) system \begin{align} \frac{dx}{dt} = f(x) \end{align} where $x \in \mathbb{R}^n$ ($n \geq 2$) and the vector field $f$ is defined on an open subset $X$ ...
2
votes
0answers
27 views

A question about Carleman linearization

Carleman linearization is a technique used to embed a finite dimensional system of analytic ordinary differential equations into an infinite system of linear differential equations:¹⁻² Let $f$ be ...
2
votes
1answer
129 views

Proof of Euler's reflection formula via rapidly decreasing Fourier series

Story I want to prove Euler's reflection formula by showing that \begin{equation*} f(s) = \sin(\pi s) \Gamma(s) \Gamma(1 - s) \end{equation*} is constant, where $s = \sigma + it$. It's easy to see ...
5
votes
1answer
600 views

Is the following integral nonzero?

Recently I met an integral as follow: $$\int_0^{2\pi}\cdots\int_0^{2\pi}\left(\prod\limits_{1\leq ...
8
votes
2answers
2k views

roots of sum of two polynomials

I believe that there is no common theory for finding roots of polynomial sum. In my case I have $$P_{n}(x)+AQ_{n}(x)$$. I am wondering how roots of this sum depend on $A$?
2
votes
0answers
366 views

What is the Fourier transform of this function?

Consider the function $$ f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du. $$ It is known that $f(x_1,x_2)\in ...
6
votes
1answer
301 views

Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
1
vote
1answer
80 views

Estimate a Fourier Transform [closed]

I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...
2
votes
1answer
97 views

π based on the perimeter of inscribed polygons [closed]

So, last year I got obsessed with the idea of finding a way to calculate π that wasn't already done. After reading some history, the Greek idea of measuring polygons inscribed within circles and ...
4
votes
2answers
117 views

An English version Borok's work on finite-infinite systems of ordinary differential equations

I am looking for the English translation of the paper by V. M. Borok (originally in Russian) The Cauchy problem for finite-infinite systems of linear differential equations. This work is about the ...
21
votes
8answers
9k views

Does the exponential function have a square root?

(asked by Nathaniel Hellerstein on the Q&A board at JMM) Is there a "half-exponential" function $h(x)$ such that $h(h(x))=e^x$? Is it unique? Is it analytic? Related question: Is there an ...
2
votes
0answers
54 views

What does the square root sign tells us in the wave equation? [closed]

I have been reading the paper on wave equations, and I have some confusion in notations. Consider the initial value problem(IVP)(Wave equation): $\frac{\partial ^2 u } {\partial t^2}(x,t) = ...
48
votes
1answer
2k views

Square root of dirac delta function

Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
8
votes
2answers
2k views

Where was/is Compensated Compactness used?

This last summer, I read up on Tartar's so called Method of Compensated Compactness (or at least how it applied to scalar conservation laws). I used this theory to prove the existence of $L^{\infty}$ ...
35
votes
5answers
1k views

Undergraduate ODE textbook following Rota

I imagine many people are familiar with the extremely entertaining article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations" by Gian-Carlo Rota. (If you're not, do ...
6
votes
1answer
102 views

Are the extremal points of a certain set of functions $\mathcal P(\mathbf N) \to \bf R$ weakly additive?

Let an upper density (on $\mathbf N$) be a (set) function $f: \mathcal P(\mathbf N) \to \mathbf R$ such that, for all $X, Y \subseteq \bf N$ and $h,k \in \mathbf N^+$, the following hold: (F1) ...
6
votes
0answers
214 views

An inequality which involves a sum of integrals

Please help me to prove $$ \sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad ...
1
vote
0answers
69 views

Finding a closed form for a certain double integral

I am working with bivariate and accurate Birnbaum-Saunders distribution to find the probability density function of a particular model for this, I would like to find a closed form for the full below: ...
2
votes
0answers
108 views

The boundedness of an entire function along the imaginary axis

I am looking for an answer in the affirmative or the negative concerning the asymptotics of an entire function. The question, relatively simple to state, has proven very taxing to solve and requires ...
5
votes
1answer
131 views

Multidimensional integrals that diverge by oscillation

It's not hard to extend the theory of integration over ${\bf R}$ so that the integral of any compactly supported function is its usual value, while the integral of $f(t) = \cos (at+b)$ (with $a \neq ...
1
vote
1answer
251 views

Ordinal of injectivity for a smooth regular curve with a finite arc-length

Let $\gamma: [a,b]\to\mathbb{R}^d$ defined by $$\gamma(t)=(\gamma_1(t),\dots,\gamma_d(t)) $$ be a smooth (i.e., $\gamma\in C^\infty (\mathbb{R}))$ and regular ($\gamma^\prime(t)\neq \vec 0$) curve ...
2
votes
1answer
49 views

How to relate this summation to standard discrete cosine transformation?

The standard type III discrete cosine transformation (DCT) is defined as follows: $${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...
-1
votes
0answers
34 views

Metrics for a Wireless Sensor Network coherent model

If we describe the signal of a Wireless Sensor Network through the following coherent Gaussian model $$ S(x,y; \boldsymbol{r}_i) = \sum_{i=1}^N \exp\left[-\frac{(x-x_i)^2+(y-y_i)^2}{2 ...
2
votes
1answer
85 views

Lebesgue measurability of singular set

Let $\Omega$ be a bounded domain in $\mathbb{R}^{d}$ and $f:Q\to\mathbb{R}$ be continuous function. Define a superdifferential of $f$ at $x\in Q$ by $$ D^{+}f(x)=\{p\in\mathbb{R}^{d} \mid ...
0
votes
2answers
76 views

Root of a special rational function with positive coefficients

During my research I came across the following problem: I need to find a root of the following function: $$\Gamma_{N}(x) = \sum\limits_{i=0}^{M}\left(\frac{\sum\limits_{n=0}^{n_F}n\ \alpha_{i,n} ...
-1
votes
0answers
25 views

Convexity of two variables function depending in a parameter

Let $~f_V : \Omega \subset R^2 \to R$ be a function that associates to $(x\in R^2$ the value $~f_V(x)$ (which is non linear and there is no explicit formula for $f_V(x)$). $V \in R^M$ is a parameter ...
5
votes
0answers
184 views

Nonzero solutions to the functional ODE $f'(x)=f(f(x))$

Does $\frac{df}{dx}=f(f(x))$ have nonzero solutions? And if so, what analytic/numerical methods can be used to characterize them?
0
votes
0answers
29 views

Interpolation functional for BV spaces?

Recently, while trying to tackle a problem, I found that it would be convenient that I could find some sort of 'interpolation theorem' for Bounded Variation Spaces. More specifically, let's define, ...
3
votes
2answers
70 views

Sequence of subharmonic functions on shrinking domains

Set $G_\eta:=\{(x,y)\in \mathbb{R}^2|-\eta<x<\eta, 0<y<1\}$. If $u_\eta\geq 0$ is a sequence of subharmonic functions defined on $G_\eta$ such that $$ \int_{G_\eta}|u_\eta|^2dx\wedge ...
1
vote
1answer
70 views

On Wazewski's theorem on system of differential inequalities

According to Springer's Encyclopedia of Math entry on differential inequalities, T. Wazewski proved in 1950 the following theorem: Consider the system of differential inequalities given by $$ ...
0
votes
1answer
381 views

A polynomial inequality

Given $x_1,x_2, \ldots, x_n \ge 0, \alpha \ge 1$, show that $\sum_{i}\alpha x_i(\sum_{j \le i}{x_j})^{\alpha-1} \ge (\sum_{i}x_i)^\alpha$ We're pretty sure the inequality holds for the given ...