**5**

votes

**0**answers

160 views

### Faà di Bruno formula

Let $f,g$ be smooth functions from $\mathbb R$ to $\mathbb R$. Then
$$
\frac{(g\circ f)^{(n)}}{n!}=\sum_{1\le r\le n}\frac{(g^{(r)}\circ f)}{r!}
\sum_{\substack{(n_1,\dots, n_r)\in {\mathbb N^*}^r\\...

**2**

votes

**0**answers

100 views

### 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 ...

**3**

votes

**2**answers

197 views

### functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...

**8**

votes

**2**answers

300 views

### Is the Fourier transform of $e^{-|x|^n}$ positive?

Let
$$\Phi(x) = \int_{\mathbf{R}^n} e^{-|y|^n +i (x,y)} dy.$$
Is $\Phi$ positive everywhere in $\mathbf{R}^n$?
Could someone helps me answer this question or gives a reference for it? Thanks.

**4**

votes

**1**answer

110 views

### Hellinger integral for the Student/Cauchy family

Let $p$ and $q$ be probability densities on $\mathbb R$, with respect to the Lebesgue measure $dx$. The corresponding Hellinger integral is
$H(p,q):=\int_{\mathbb R}\sqrt{pq}\,dx$.
Let now $p$ be ...

**1**

vote

**0**answers

77 views

### A limsup representation for the upper Buck density

The upper Buck density, introduced by R.C.Buck in 1946, is defined to be the function
$$
\mathfrak{b}^\star: \mathcal{P}(\mathbf{N}^+) \to \mathbf{R}: X\mapsto \inf_{X\subseteq A:A\in \mathscr{A}}\...

**1**

vote

**0**answers

45 views

### Bounds on $\sum_{j=1}^m\frac{\pi^j}{\Gamma(j)(x^2+(j+1/4)^2)}$

During our search of real rooted entire function approximations to Riemann $\Xi$ function, we need to calculate the upper and lower bounds of
$$f_m(x):=\sum_{j=1}^m b_j(x):=\sum_{j=1}^m\frac{\pi^j}{\...

**3**

votes

**1**answer

58 views

### Redundancy in transseries representation of functions?

"Transseries" are a kind of generalized power series that allow things like fractional exponents and exponentials (with another transseries as the exponent). I know very little about them but I have ...

**1**

vote

**1**answer

61 views

### ODE estimate for boundary value problem

Let $X$ be a solution to the boundary value problem
$$ X^{\prime\prime}(s) = A(s)X(s), \quad X(0) = X_0, ~~X(t) = X_1,$$
where $0 < t \leq 1$, $A$ is some matrix-valued function, defined on $[0, 1]$...

**-1**

votes

**1**answer

54 views

### Idempotent solutions to the implict function theorem other than the identity?

I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...

**9**

votes

**2**answers

748 views

### Algebraic independance of exponentials

First of all, a happy new year. Be it better than 2015,
healthy, wealthy, fruitful and cross-fertilizing
for you, familly and friends.
In order to cope with families of solutions of evolution ...

**8**

votes

**1**answer

143 views

### How to eliminate secular terms for perturbed non-oscillatory equations?

Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely
$$x(t)=a_0+b_0e^{-t}+\epsilon(...

**9**

votes

**1**answer

368 views

### Number of real roots of an exponential polynomial

Let $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ be real numbers, and assume that $\{a_i\} \neq \{b_i\}$. Can the equation
$$ e^{a_1 x} + e^{a_2 x} + \dots + e^{a_n x} = e^{b_1 x} + e^{b_2 x} + \...

**3**

votes

**2**answers

166 views

### Roots of the Chebyshev polynomials of the second kind

It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of $(...

**1**

vote

**2**answers

131 views

### 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 ...

**1**

vote

**0**answers

49 views

### “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 ...

**6**

votes

**1**answer

111 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) $f(\...

**9**

votes

**3**answers

293 views

### Decay of real continuous algebraic functions at infinity

Let $f$ be a real valued continuous algebraic function on $\mathbb R^n$. Suppose the zero set of $f$ is bounded, i.e., if $|x|$ is large enough, $f(x)\neq 0$. Is there any estimate of the sort $|f(x)|\...

**2**

votes

**0**answers

59 views

### 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 ...

**12**

votes

**1**answer

550 views

### 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 ...

**4**

votes

**3**answers

554 views

### 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 ...

**5**

votes

**0**answers

70 views

### Some questions about the Lévy monoid of certain densities

Let $\bf H$ be a set, $f: \mathcal P({\bf H}) \rightharpoonup \bf R$ a partial function, and $\mathcal{D}$ the domain of $f$.
Next, denote by $\mathcal M(f)$ the set of all (total) functions $\theta: ...

**1**

vote

**0**answers

53 views

### Complexity theory and closed form formulas in analysis

My question concerns definitions of "closed form" solutions. In hamiltonian systems this is closely related to complete integrability. In this context closed form can refers to having $(q(t),p(t))$ ...

**2**

votes

**1**answer

189 views

### General solution to system of stochastic linear differential equations

Assume we are given the system of linear stochastic differential equations
$$dx_i = \sum_{j=1}^n a_{ij}(t) \cdot x_j \cdot dt + \sum_{j=1}^n \sigma_{ij}(t) \cdot x_j \cdot dB_{ij,t} + b_j(t)\cdot dt+\...

**1**

vote

**0**answers

72 views

### Asymptotics of a elliptic pde when exponent gets large

I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...

**1**

vote

**0**answers

98 views

### Integrating a series expansion of $\mbox{frac}(x)\lfloor x\rfloor$ coming from Fourier series of sawtooth function

Let me preface this question by saying that I am not exactly sure it counts as research level. It is crossposted on mathstackexchange: http://math.stackexchange.com/questions/1519724/integrating-a-...

**2**

votes

**0**answers

72 views

### Non-existence for a sort of probability measures

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$.
$W_{t}$ is standard Wiener.
This solution is ...

**2**

votes

**0**answers

127 views

### An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as
$$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx
+ \frac{f(n-1) + f(0)}2
+
\sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - 1)}(n-1)...

**6**

votes

**1**answer

345 views

### A property of the derivatives of a function

Suppose that $f,g_1,g_2,\dots$ are functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f'=f\,g_1$ and $g'_j=g_j^2-g_j g_{j+1}$. Here and in what follows, $j$ is any natural number. Then, by ...

**1**

vote

**0**answers

68 views

### Recurrence sequence

Is it possible to find a Recurrence sequence that Satisfying the following inequality
$ d_{n+k}\geq \alpha ^k d_n +\beta^k \delta(A,B),$
where $0<\alpha<1, \alpha ^k+\beta^k\geq ...

**2**

votes

**1**answer

103 views

### Lower bounds from Fourier dimension?

According to Mattila, Geometry of sets and measures in Euclidean spaces, p. 168, the Fourier dimension $\text{dim}_F(A)$ of $A\subseteq \mathbb R^n$ is the unique number in $[0,n]$ such that for any $...

**2**

votes

**0**answers

38 views

### Asymptotic expansion of Mellin transform of products of modified Bessel function K

Let $n\ge 1$ be an integer, let
$$F(x,y)=\int_0^\infty u^{n(x+y)} (K_{x-y}(u))^n du$$
for $x,y\ge 0$.
When $n=1$, this is just Mellin transform of the Bessel K function. When $n=2$, $F(x,y)$ has ...

**10**

votes

**1**answer

491 views

### A tricky tractrix question about vertical tangents

This is raised by a recent question occurring in combinatorial geometry.
It is about a sort of tractrix, but instead of a line, the pulling end moves along a circle of radius $r>\frac12$ (...

**2**

votes

**0**answers

28 views

### Uniform convergence of the best $L_1$ approximations by polynomials

Let $P_n$ be the vector space of real multivariate polynomials in $d$ variables of total degree no more than $n$ and let $f \in C(X)$, where $X \subset \mathbb{R}^d$ is compact. Let $(p_n)_{n=1}^\...

**6**

votes

**2**answers

75 views

### 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 ...

**7**

votes

**2**answers

439 views

### 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 $X \in \Sigma$ ...

**1**

vote

**0**answers

44 views

### 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 ...

**8**

votes

**2**answers

259 views

### 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 ...

**0**

votes

**1**answer

115 views

### 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 $f:[a,\infty)\to \mathbb{R}$ ...

**2**

votes

**1**answer

172 views

### comparing Laplacian and gradient of function on boundary

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 ...

**1**

vote

**0**answers

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 ...

**1**

vote

**1**answer

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 ...

**16**

votes

**1**answer

666 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{...

**2**

votes

**1**answer

71 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 $sn(u,...

**3**

votes

**1**answer

126 views

### Generalized Theorem of Laguerre

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 ...

**2**

votes

**1**answer

124 views

### An extreme of Jacobi elliptic function on an interval

Consider the Jacobi elliptic function $sn(\cdot,k)$ restricted to the interval $(0,2K)$, where $K=K(k)$ is complete elliptic integral of the first kind. If $0<k<1$, then it is well known the ...

**10**

votes

**1**answer

272 views

### Normal approximation of tail probability in binomial distribution

My problem: From the Berry--Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|=O\left(\frac 1{\sqrt n}\right),$$ where $B_n$ has the standardized binomial distribution and $\Phi$ ...

**4**

votes

**1**answer

71 views

### Getting out a system of linear ODEs by knowing the Magnus expansion

Assume we are given for a transition between two time points $t_0 = 0$ and $t_1$ a matrix relationship, eventually describing the solution of a system of linear with non-constant coefficients,
$$Y(t_1)...

**2**

votes

**1**answer

130 views

### A question about viscosity solutions

Let $A:=[a,b]$ be a closed interval in $\mathbb{R}$. Let $F(x,p,q,r)$ be a function from $[a,b]\times \mathbb{R} \times \mathbb{R} \times\mathbb{R} \rightarrow \mathbb{R}$ describing a second order ...

**2**

votes

**1**answer

112 views

### Rate of convergence of ergodic averages related to irrational rotation

Let $\alpha$ be an irrational number, consider the basic dynamical system $T^{n}(0) = \{n \alpha\}$ where $\{.\}$ denotes the fractional part.
Let $a < b$ be two numbers in $[0, 1]$. Then by ...