**12**

votes

**0**answers

189 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

69 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

53 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

184 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

497 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

140 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

136 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

214 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

94 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

59 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

164 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

149 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

67 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 ...

**10**

votes

**6**answers

689 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

665 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

34 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

91 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

138 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

497 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

39 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

105 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

80 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

253 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

41 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

159 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 ...

**1**

vote

**0**answers

49 views

### Most general hypotheses to have matrix-exponential as solution of linear matrix-ODE? [closed]

What are the most general hypotheses under which the exponential function $\exp(-B(t))$, with $B(t)$ the indefinite integral of a real Lebesgue-integrable $d\times d$-matrix function $A(t)$, solves ...

**0**

votes

**2**answers

108 views

### Class of analytically-integrable divergence-free vector fields?

Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$?
That is, I'm looking for a large class of vector fields given by $...

**7**

votes

**2**answers

172 views

### Integral inequality similar to Hardy's inequality

I am not very sure if that's research level, I hope you don't find it too elementary for this place.
I am trying to solve the following puzzle:
We are given a real function $f$, where $f(x) \geq 0$ ...

**0**

votes

**0**answers

59 views

### When is the discrete logarithmic energy not approximable by its ostensibly more general counterparts?

In my answer at Maximum of the Vandermonde determinant / minimum of the logarithmic energy it is shown that
that for each large enough natural $n$ there is some $a=(a_0,\dots,a_{n-1})$ with $0=a_0<...

**14**

votes

**4**answers

479 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 ...

**2**

votes

**1**answer

208 views

### An elementary functional inequality

Let $g$ be a $C^1$ function with $g(0)=0$ and $g(t)>0$ for all $t>0$. I am surprised that for all such $g$ the following seems to hold
$\frac{\int_0^t(g'(s))^2ds}{g^2(t)}\geq \frac{1}{t}$ for ...

**1**

vote

**1**answer

88 views

### Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field.
Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials $P_n:...

**5**

votes

**1**answer

177 views

### Unicity of additive, $(-1)$-homogeneous, and shift invariant probability measures on $\mathbf N^+$

Let $\mathcal D$ be the set of all (finitely) additive probability measures $\mu^\ast: \mathcal P(\mathbf N^+) \to [0,\infty[$ such that $\mu^\ast(k \cdot X + h) = \frac{1}{k} \mu^\ast(X)$ for all $X \...

**2**

votes

**0**answers

73 views

### Does there exist $k\ge2$ s.t. $X \subseteq \mathbf N^+$ has positive upper Banach density if the counting function of $X$ is $\gg n/\log^{[k]}(n)$?

Does there exist an integer $k \ge 1$ such that ${\sf bd}^\ast(X) > 0$ whenever $X \subseteq \mathbf N^+$ and $\pi_X(n) \gg \frac{n}{\log^{[k]}(n)}$ as $n \to \infty$? Here, ${\sf bd}^\ast$ is the ...

**0**

votes

**0**answers

95 views

### The following ODE global existence theorem reference?

There is an ODE existence theorem of the form:
Let $f:[a,b]\times \mathbb{R}^n \to \mathbb{R}^n$ be a Caratheodory function.
Suppose that there is a constant $c$ such that if $y$ is a ...