**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

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{...

**2**

votes

**1**answer

74 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

127 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

287 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

133 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

113 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 ...

**12**

votes

**0**answers

191 views

### Surprising approximate identity

While answering this MO question Connection between Bernoulli numbers and Riemann-Siegel theta function? Dan Romik
found the following surprising approximate identity:
$$\ln{8\pi}\approx \pi\left[ 2\...

**2**

votes

**0**answers

71 views

### Optimal condition for the weak convergence of the jacobian determinant

Whenever $n<q,$ it is known that given a sequence $\{ u_{k} \}$ which is weakly convergent in $W^{1,q}(U)$ one has that the Jacobian determinants $\text{det} Du_{k}$ converge weakly in $L^{q/n}(U).$...

**0**

votes

**0**answers

57 views

### A question about the approximation of convex cones

I have the following question which maybe is too naive.
Let $K$ be a convex cone on $\mathbf{R}^n$. Can we approximate $K$ by a sequence of polyhedral convex cone $K_i$ such that for any compactly ...

**3**

votes

**2**answers

170 views

### Name for an orthogonal decomposition of $L^2 (\mathbb{R}^2; \mathbb{C})$

The space $L^2 (\mathbb{R}^2; \mathbb{C})$ can be decomposed as
$$
L^2 (\mathbb{R}^2; \mathbb{C}) = \bigoplus_{k \in \mathbb{Z}} L^2_k (\mathbb{R}^2; \mathbb{C}),
$$
where
$$
L^2_k (\mathbb{R}^2; \...

**5**

votes

**0**answers

64 views

### How to derive explicit bound for the solution of following equation?

Let's have equation
$$
y''(t) + \left(\frac{3}{16t^{2}} + \frac{a}{t} -\frac{b}{t^{\frac{5}{4}}}cos(2t)\right)y(t) = 0, \quad t \in (1, \infty), \quad a, b > 0
$$
How to derive explicit upper bound ...

**2**

votes

**0**answers

188 views

### Hartogs's extension theorem

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$, and
let $f:H\to \mathbb{C}^n$ be a holomorphic injective map. Then we know that $f$ extends holomorphically to the polydisc $P$, i.e. there ...

**1**

vote

**0**answers

64 views

### Derivatives of $O$-regular varying functions are $O$-regular varying functions?

The Monotone Density Theorem for regularly varying functions says, in essence:
Theorem (Monotone Density Theorem). Let $f$ be a differentiable regularly varying real-function of index $\rho$ well-...

**1**

vote

**0**answers

34 views

### How to treat equation with alternating square of frequency?

Let's have equation
$$
\tag 1 \frac{d^{2}y(t)}{dt^{2}} +\omega^{2}(t)y(t) = 0, \quad t \in (t_{\text{in}}, \infty)
$$
Here
$$
\omega^{2}(t) = A(t) - B(t)cos(2t),
$$
and functions $A(t), B(t)$ have ...

**6**

votes

**1**answer

501 views

### An elementary lower bound on the number of primes

Recall the second Chebyshev function: $$\psi(x) = \sum_{p \leq x} \lfloor \log_p x \rfloor \log p$$ where $x$ is a positive integer, and $p$ runs over all primes $\leq x$.
In a hunt for an "...

**1**

vote

**1**answer

141 views

### Is this function concave or convex? [closed]

let $g_{n,\gamma}(\sigma)$ be the function defined as the following
$$
g_{n,\gamma}(\sigma)= \left(\frac{(\sigma-1)^2 +\gamma^2}{\sigma^2
+\gamma^2} \right)^{n/2} T_n\left( \frac{\sigma(\sigma-1)
+\...

**0**

votes

**0**answers

41 views

### Mathieu equation instability

Let's have Mathieu equation:
$$
\tag 1 \frac{d^2y(x)}{dx^2} +(A -2q\times cos(2x))y(x) = 0, \quad q\geqslant 1, \quad A > 2q
$$
This case is different from the famous case $q << 1, A> 2*q$,...

**6**

votes

**1**answer

140 views

### Asymptotic behaviour of an integral

For $k\in\mathbb{N}_{0}$ and $x\in\mathbb{R}$, define
$$I_{k}(x):=\int_{0}^{\pi/2}\cos(xg(\theta))\sin^{2k}\theta\,\mathrm{d}\theta$$
where
$$g(\theta)=\int_{\sin\theta}^{1}\frac{\mathrm{d}t}{\sqrt{(1-...

**3**

votes

**0**answers

154 views

### Dynamics of an inequality

The dynamics $D\ni(r_i,r_{i+1})\mapsto(r_{i+1},r_{i+2})\in D$ on the set $D:=\{(x,y)\in\mathbb{R}^2\colon x>0,y>x^2/2\}$ is given by the recurrence
$$r_{i+2}=\frac{r_{i+1}^2}2+\frac1{r_{i+1}^3}
...

**7**

votes

**2**answers

218 views

### differential equation of conics

Function $y(x)$ on the plane defines a conic (strictly speaking, at most second order algebraic curve) if and only if $\frac{d^3}{dx^3} (y'')^{-2/3}=0$. What is intuition behind this? How may I see ...

**5**

votes

**0**answers

95 views

### Functions on [0,1] with positive fractional series coefficients and symmetric under x->1-x

Suppose a function $g(x)$ has a convergent power series expansion with real (not necessarily integer or rational) exponents on $[0,1]$ of the form
$$
g(x)=\frac{1}{x^\delta}(1+\sum_{i=1}^\infty c_i x^{...

**1**

vote

**1**answer

64 views

### A Statement from Brauer and Nohel's book on stability of time-depending linear systems

On page 158 (The qualitative theory of differential equations; an introduction) the authors cite a $2 \times 2$ counterexample by Vinograd to the system $y'=A(t)y$, where
$$
A(t) =
\left(
\begin{...

**7**

votes

**2**answers

419 views

### The sum of a series, continued

In this question the OP asks whether the sum
$$
f(q, \alpha) = \sum _{k=1}^{\infty } \frac{q^k \left(q^k-1\right)^\alpha}{(q;q)_k}
$$
is ever zero. An experiment with Mathematica indicates, to any ...

**12**

votes

**1**answer

165 views

### Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]+[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as $$\{a\}...

**4**

votes

**1**answer

151 views

### Hyperfunctions supported at a point

Is it true that the space of hyperfunctions on $\mathbb{R}^n$ supported at 0 coincides with the space of Schwartz distributions supported at 0?
More explicitly, is it true that any hyperfunction ...

**4**

votes

**0**answers

70 views

### Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions.
I am wondering whether hyperfunctions have any advantages over ...

**11**

votes

**6**answers

705 views

### Identities and inequalities in analysis and probability

Usually, at the heart of a good limit theorem in probability theory is at least one good inequality – because, in applications, a topological neighborhood is usually defined by inequalities. Of course,...

**1**

vote

**0**answers

101 views

### Algorithm for finding eigenfunctions

I have an $ L^2(\mathbb{R}) $ operator that looks like
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |,
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...

**54**

votes

**6**answers

2k views

### Why does $d^n \exp(-x-x^{-1})/(dx)^n$ only have $n$ positive real zeroes?

Set $f(x) = \exp(-x-x^{-1})$. An easy induction shows that
$$\frac{d^n}{(dx)^n} f(x) = \phi_n(x^{-1}) f(x)$$
for $\phi_n$ a polynomial of degree $2n$. Clearly, the roots of $\phi_n(x^{-1})$ are the ...

**10**

votes

**2**answers

690 views

### What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory.
Any recommendations?

**0**

votes

**1**answer

87 views

### Fit a system of linear ODEs from several experiments

Assume we are given several initial vectors $x^{(1)},\ldots,x^{(r)} \in \mathbb{R}^n$, where the dimension $n=6$ (in any event a number below 10) , and the number of initial vectors $r$ is in the ...

**0**

votes

**0**answers

35 views

### Explicit solution for a variational inequality?

Assume $\eta_t \in \mathbb{R}$ is a given continuous function depending on time.
It is known that if we look for a continuous solution $\zeta_t \in \mathbb{R}$ depending on time of the following ...

**27**

votes

**5**answers

1k views

### How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question.
For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...

**1**

vote

**0**answers

29 views

### Error bounds for approximation with dyadic sums of polynomials

Are there any bounds known for approximating a genuine multidimensional polynomial function with a sum one-dimensional polynomials over the independent variables?
In the 2-dimensional case the ...

**0**

votes

**0**answers

94 views

### Integral inequality with Hardy's inequality

Let $f\in C_{0}^{\infty}((-1,1))$. Prove that for any $t\in (-1,1)$ we have
$$(f(t))^4\le \left(\int_{-1}^{1}\dfrac{[2(1-|x|)f'(x)-f(x)][2(1-|x|)f'(x)+f(x)]}{4(1-|x|^2)}dx\right)\cdot\left(\int_{-1}^{...

**2**

votes

**0**answers

70 views

### Point Spectrum of a Second Order System of Differential Equations

Consider the following operator acting on $H^1(\mathbb{R})$
$$
\mathcal{L} \left(\begin{array}{c} \phi \\ \psi \end{array}\right) = -\left(\begin{array}{c} \phi \\ \psi \end{array}\right)'' + V(x) \...

**3**

votes

**1**answer

139 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**18**

votes

**1**answer

500 views

### Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...

**0**

votes

**0**answers

40 views

### Distance minimization and submodular functions

I have a question about square distance minimization and submodular functions. Suppose I have a random variable $X = (X_1,...,X_n)$ over $\mathbb{R}^n$. Let $x$ be a realization of this random ...

**2**

votes

**1**answer

106 views

### Two ODEs, why is one the solution of the other? (Caratheodory ODE)

This question is based on Zeidler II/B, Problem 30.2.
Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t.
$$u'(t) = F(t,u(t))$$
$$u(0) = u_0$$
given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ ...

**0**

votes

**0**answers

70 views

### Is there a space in which the $\vec a$ in $\sin(a_1\cdot x)+\sin(a_2\cdot x)$ is linear?

Suppose one has equations of the form $\sin(a_1\cdot x_i)+\sin(a_2\cdot x_i)=y_i$ for $i = 1, \dots, n$ (there are also amplitudes and phase shifts, but let's ignore these for now). I want to solve ...

**1**

vote

**1**answer

82 views

### Runge-Kutta convergence [closed]

I am facing a problem solving a ODE with a Runge-Kutta 4th order method:
The expression in order to solve is :
\begin{equation}
Ay^{''}+By^{'}+Cy= Cu
\end{equation}
\begin{equation}
y =OUTPUT
\end{...

**5**

votes

**1**answer

256 views

### Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...

**0**

votes

**0**answers

44 views

### Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients.
...

**2**

votes

**1**answer

160 views

### Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.
The background
So he wants to show that any symplectic form is ...

**2**

votes

**2**answers

250 views

### Reference request: using integral equations to study asymptotics of ODEs

I was told by my supervisor that one way to study the asymptotic behaviour of solutions to ODEs is "to reformulate them as integral equations, and use fixed-point kind theorems on the resulting ...