# Tagged Questions

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

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### Log-concave polynomial is a log-concave function?

A polynomial $\sum\limits_{k=0}^n a_kx^k$ is log-concave if $a_0,\ldots,a_n$ constitute a log-concave sequence. I wonder whether the log-concave polynomial is also a log-concave function with respect ...
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### Convergence of Sobolev functions near the boundary

Let $B_0(1)$ be the unit ball in $\mathbb R^n$, $n\geq2$. Let $f\in W_0^{1,2}(B_0(1))$, and $W^{1,2}(B_0(1))\ni f_i\to f$ in the sense of $L^2(B_0(1))$-norm, as $i\to \infty$. Question 1: Can we ...
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### “Harmonic oscillator” with $p$-Laplacian

I wonder if there is any literature on the eigenvalue problem for the "$p$-harmonic oscillator" $$-(|u'|^{p-2}u')'(x)+(x^2-\lambda) |u(x)|^{p-2} u(x)=0$$ in $L^p(\mathbb R)$, $p\in(1,\infty)$. Are ...
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### Reference request on a notion of independence for families of [real-valued] functions

This is basically another reference request. Let $X$ be a set, and $\mathscr{F} = (f_i)_{i \in I}$ an indexed family of functions $X \to \bf R$. If $\preceq$ is a partial order on $I$, we say that ...
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### How to prove that a monotone function is differentiable at some point?

This fact, which eventually belongs to Lebesgue, is usually proved with some measure theory (and we prove that the function is differentiable a.e.). Is there a significantly different approach? Let me ...
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### Exact formula for $\sum_{n=0}^{+\infty}\frac1{n^2+1}$

Actually, as the corresponding integral $\frac{\ln(\cos x)}{1-x}$ or something like that) cannot be expressed in closed form by Liouville theorem, this shouldn't exist, but I believe I have seen it ...
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### Closed field lines in the plane

A dipole in the plane consists of a positive charge P and an equal and opposite negative charge N separated by a fixed distance . Almost all of the resulting electric field lines (which fill the ...
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### A result of Sierpiński on non-atomic measures

There is a classical result commonly attributed to W. Sierpiński that reads as follows: Theorem 1. If $f: \Sigma \to \bf R$ is a non-atomic (*) measure on a set $S$, then for every $X \in \Sigma$ ...
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### Triangle inequality for nonconvex functions of singular value vector

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave ...
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### A specific linear differential equation on $\mathbb{C}-\{0,1\}$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$

Is there a linear differential equation on $\mathbb{C}$ with singularities at $0$ and $1$ whose monodromy group represents the fundamental group of $\mathbb{C}-\{0,1\}$? If so, can someone give a ...
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### growth of derivative

(Alexandre Eremenko gave the answer to the question I posted earlier. Then I realised the question didn't give what I wanted. Here is the updated question.) Suppose that $f:[a,\infty)\to \mathbb{R}$ ...
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Consider $E(x)$ some smooth function on $\Omega$ (some smooth bounded domain in $R^N$) and suppose $E=0$ on $\partial \Omega$. Suppose one knows that there is some $C_1,C_2 \in R$ such that $x ... 0answers 57 views ### Log convexity for the norm of a vector-valued function Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory. Given$v \in \mathbb{C}^n$,$D$a diagonal matrix with ... 1answer 102 views ### Can a (non-measurable) autonomous flow have a non-trivial periodic orbit without a minimal period? Given a set$X$, a function$x \colon \mathbb{R} \to X$is periodic if there exists$\tau>0$such that$x(t+\tau)=x(t)$for all$t \in \mathbb{R}$; and if$\tau$is the smallest positive number ... 1answer 669 views ### Are the algebraic numbers dense everywhere on the boundary of the Mandelbrot set? Let$\mathcal{B}$denote the boundary of the Mandelbrot set, and let$\overline{\mathbb{Q}}$denote the algebraic closure of the rationals. Further put$\mathcal{B}_{\overline{\mathbb{Q}}} := \mathcal{...
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 \$sn(u,...
There is known theorem of Laguerre, that every linear ordinary differential equation of second order $$y''+A(t)y'+B(t)y=0$$ by point transformation could be mapped into $$y'' = 0,$$ that in few words ...