**10**

votes

**1**answer

242 views

### smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...

**3**

votes

**1**answer

125 views

### Generalization of the Hermite-Bielher-Kakeya Theorem

A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a ...

**1**

vote

**0**answers

73 views

### Estimating decay of certain trigonometric polynomials

For $p=0,1,2,\dots$ and $n=0,1,2,\dots,$, let $f_{n,p}(z)=\sum_{k=0}^n k^p z^k$ be a sequence of polynomials. Restricted to the unit circle, the functions $g_{n,p}(t):=f_{n,p}(e^{it})$ are ...

**2**

votes

**0**answers

182 views

### More information on Kruskal's treatment of Surreal numbers as an asymptotic behavior of a real valued function

The only way that I could think about Surreal numbers is how Conway defined them inductively, with the two axioms and so on. I can't find any information about Kruskal's point of view and would very ...

**-2**

votes

**1**answer

104 views

### a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...

**2**

votes

**0**answers

80 views

### Diffusion equation on mixing of diffusing particles

I am trying to study mixing of diffusing particles like it was done by E. Ben-Naim On the Mixing of Diﬀusing Particles.
The picture below shows the idea how permutations and inversion numbers reflect ...

**4**

votes

**2**answers

285 views

### Abstract ODE; PDE; uniqueness of solution

I have a somewhat vague question regarding an abstract ODE in a Banach space.
Suppose $A:D(A) \subset X \rightarrow X$ is some linear operator (let's assume it's closed) and maybe add some other ...

**17**

votes

**2**answers

611 views

### What can be said about the Fourier transforms of characteristic functions?

What can be said about the Fourier transform of the characteristic function $1_A$, where $A\subset \mathbb{R}^n$ is of finite Lebesgue measure? In particular,
What properties are common to ...

**0**

votes

**1**answer

203 views

### Theorem with an example [closed]

i have this theorem
in the paper they gives an example:
but here $H_1$ is not satisfied !
How to correct it please?

**1**

vote

**0**answers

212 views

### Error term in formula for products of necklaces

Let us consider the $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$, where the number of fixed necklaces of length n composed of a types of beads $N(n,a)$ can be calculated via totient function: ...

**1**

vote

**1**answer

112 views

### On the geometry of roots of a sum of complex linear fractions

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form
$$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$
where the $a_j$'s are nonzero complex ...

**8**

votes

**2**answers

277 views

### On the convexity of certain integrals involving Bessel functions

Let $n\geq 0$ be an integer and let $J_n=J_n(r)$ denote the usual Bessel function (of the first kind) of order $n$ i.e. one of the solutions to Bessel's differential equation
...

**4**

votes

**1**answer

148 views

### Asymptotic expansion of the Mordell integral

my question concerns the Mordell integral
$$h(z;\tau):=\int_{-\infty}^\infty \frac{e^{\pi i\tau w^2-2\pi zw}}{\cosh(\pi w)}dw,\qquad \Im(\tau)>0,\quad z\in\mathbb{C},$$
which frequently occurs in ...

**2**

votes

**0**answers

83 views

### Quick estimate of attractor of non-linear dynamical system [closed]

Say I have a system of form
$$
\frac{dy}{dt} = f(y),
$$
and it is know this system has an attractor. Can I quickly for given $\varepsilon$ guess some point, such in its $\varepsilon$- neighbourhood ...

**4**

votes

**1**answer

271 views

### solution of functional equation $f^{\circ k}(x) = x$

The equation $f^{\circ k}(x) = \mathrm{Id}$ for $x\in E$ is called the Babbage equation and the general solution is given in the following way [M. Kuczma, Functional equations in a single variable]:
...

**5**

votes

**4**answers

331 views

### NP-hard problems in linear algebra and real analysis [closed]

I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of ...

**3**

votes

**2**answers

151 views

### Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form
$$\frac{du}{dt} + Au = f$$
where $A$ is an accretive nonlinear operator under some ...

**1**

vote

**2**answers

210 views

### Looking for a limit related to the series in a previous post

Can any one show that the following limit?
$$
\lim_{z\rightarrow \infty} \sqrt{z} \: e^{-z}\sum_{k=1}^\infty \frac{z^k}{k! \sqrt{k}} \quad \stackrel{?}{=} \quad\sqrt{2}-1.
$$
If one uses the ...

**2**

votes

**2**answers

327 views

### How to integrate an exponential function of an exponential function?

Does any one know how to calculate the following integration?
$$
\int_{\mathbb{R}} \left(\exp(z \: e^{-y^2})-1\right)^2 dy=?,\quad z>0.
$$
This post is related to my previous question here , ...

**1**

vote

**0**answers

102 views

### Intermediate value theorem for the Jacobian determinant restricted to a curve

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian determinant. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. Is it right that on every continous curve connecting ...

**14**

votes

**2**answers

403 views

### What geometric information is carried by the Fourier coefficients of the components of a closed curve?

Let $\gamma$ be a smooth closed curve in the plane and let $(x(t), y(t))$ be a parametrization. The functions $x(t)$ and $y(t)$ are smooth and periodic, so each has a uniformly convergent Fourier ...

**1**

vote

**1**answer

252 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...

**3**

votes

**1**answer

160 views

### A differentiable version of the Michael selection theorem

Assume that $X$ and $Y$ are Banach spaces and $T:X\to Y$ is a bounded surjective linear map.
Is there a Gateaux differentiable function $g:Y\to X$ such that $T\circ g=Id_{Y}$?

**1**

vote

**2**answers

192 views

### Jacobian of an injective mapping

Let $f:R^N \to R^N$ be a differentiable mapping, and $J_f$ its Jacobian. Suppose that $\exists a,b \in R^N : J_f(a)<0,J_f(b)>0$. I want to prove two things that seem intuitively right: 1) $f$ is ...

**3**

votes

**1**answer

228 views

### A (non trivial) continuous map on a Banach space which is nowhere Frechet differentiable

Assume that $X$ is a Banach space. Is there a continuous map $f:X\to X$ such that $f$ is nowhere Frechet differentiable, but its restriction to every finite dimensional subspace is every where Frechet ...

**1**

vote

**0**answers

80 views

### Periodic solution of first order ODE

There is a famous result shows that for every continuous function $f:{\mathbb R}\rightarrow {\mathbb R}$, the first order autonomous system
$$
\left\{
\begin{array}{l}
\dot{x}=f(x), \\
x(t_0)=x_0,
...

**2**

votes

**0**answers

95 views

### Stationary Distribution for Markov-like system?

Let
\begin{equation}
A=
\begin{pmatrix}
0 & a_{1,2} & a_{1,3} \\
a_{2,1} & 0 & a_{2,3} \\
a_{3,1} & a_{3,2} & 0
\end{pmatrix},
\end{equation}
\begin{equation}
B=
...

**9**

votes

**1**answer

817 views

### Has anyone seen this series?

I come across the following infinite series.
$$
\sum_{n=1}^{\infty} \frac{t^n}{n!\: n^{a}}, \quad\text{for $t>0$ and $a>0$}.
$$
In particular, I am interested in the case where $a=1/4$.
...

**2**

votes

**1**answer

115 views

### Global Solutions of Ordinary Differential Equations

Background
Let $f: [0, \infty) \times {\mathbb R}^n \rightarrow {\mathbb R}^n$ be a jointly measurable function satisfying,
$f(t, \cdot)$ is locally Lipschitz for every $t \geqslant 0$,
for every ...

**1**

vote

**0**answers

302 views

### Presence of singular points in the trajectory of a double pendulum

Watching the trajectory of a double pendulum, I caught myself wondering if it would be possible to prove that the path the second pendulum makes contains "cusps" or singular points. Upon investigating ...

**1**

vote

**0**answers

113 views

### Bound for a certain integral expression

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...

**3**

votes

**1**answer

193 views

### About generalized Minkowski inequality

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality
$f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + ...

**4**

votes

**2**answers

206 views

### Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix

I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...

**0**

votes

**1**answer

122 views

### Solution of an Ordinary Differential Equation

I'd like to know if there is an analytical solution of the following Ordinary Differential Equation :
$\displaystyle{\dfrac{d}{dt}x_i(t) = D_i + \sum_{j=1}^{n}L_{ij}x_j(t) + ...

**-1**

votes

**1**answer

118 views

### Method of steepest descents for $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt\sim\frac{i}{z}$

Through some calculations I ended up with the integral $\int_0^\infty\exp(iz(\frac{1}{3}t^3+t))dt$. I would like to obtain the result that the behaviour of the integral is $\sim\frac{i}{z}$ as ...

**5**

votes

**0**answers

81 views

### Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$.
By this I mean that there exists a polynomial $P$
with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!)
such that $P(\phi) = 0$
Let ...

**-1**

votes

**1**answer

93 views

### A proof for $[(f^k)^{(n)}]^2 \geq (f^{k-1})^{(n)} (f^{k+1})^{(n)}$

I want to show $[(f^k)^{(n)}]^2 \geq (f^{k-1})^{(n)} (f^{k+1})^{(n)}$, where $f$ satisfies $f^{(n)} \geq 0$ for all integer $n$ and $f^k$ denotes the $k$-th power of $f$.
I believe it's right but I ...

**2**

votes

**2**answers

209 views

### What are the difference between modeling with stochastic differential equations (SDE) and ordinary differential equations (ODE) with a random force?

There are lots of differences between SDE and ODE. From the theoretical point of view an also from the numerical algorithms used for simulations. But I am interested in knowing if there is a point ...

**3**

votes

**1**answer

187 views

### Asymptotic expansion of modified Bessel function $K_\alpha$

An integral representation for the Bessel function $K_\alpha$ for real $x>0$ is given by $$K_\alpha(x)=\frac{1}{2}\int_{-\infty}^{\infty}e^{\alpha h(t)}dt$$ where ...

**27**

votes

**3**answers

1k views

### A translation of the Cantor set contained in the irrationals

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here.
What would ...

**1**

vote

**0**answers

213 views

### Inverse Transpose of Jacobian Matrix

Let $f:\mathbb{R}^n\mapsto\mathbb{R}^n$ be a bijective function. Fixed $a\in\mathbb{R}^n$. For any $x$ closes to $a$, using Taylor's series we can approximate $f(x)$ by
\begin{equation}
f(x)\approx ...

**3**

votes

**2**answers

279 views

### An integral related to the Euler Gamma function

The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral
$$\eta=2\int\limits_0^1 ...

**2**

votes

**1**answer

63 views

### radius of the ball where the inverse of Lipschitz maps exists

I am aware of the inverse function theorem for Lipschitz maps, which uses the notion of generalised derivative $δ_{x_0} f$ of a Lipschitz map $f$, due to F.H.Clarke:
Teorem. Let $f : U ⊂ \mathbb{R}^n ...

**3**

votes

**1**answer

210 views

### Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...

**0**

votes

**0**answers

56 views

### How to choose negative definite function $\lambda (x)$, so that $\lambda^{-1} \in L^{1}(\mathbb R)$?

We define,
$$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$
where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...

**5**

votes

**2**answers

190 views

### Algebraic characterization of real differentiation

I've seen some previous questions that show that the derivative operator on the set of smooth functions can be given by the Leibniz rule and/or chain rule and some other axioms.
Is there a similar ...

**3**

votes

**1**answer

315 views

### The Weyl algebra modules which are also rings

Consider the space $C^\infty(\mathbb{R})$. We have on it a natural action of the Weyl algebra generated by $x$ and $d/dx$, where $x$ is the coordinate on $\mathbb{R}$, and there is plenty of Weyl ...

**7**

votes

**1**answer

473 views

### Ramanujan's problem 754 still open?

In addition to the MO question The Ramanujan Problems. , I would like to ask the following.
Problem 754 from the list of the Ramanujan's problems ( ...

**9**

votes

**1**answer

532 views

### Real-rootedness, interlacing, root-bounds of a sequence of polynomials

Problem: the number $a(n,k)$ is defined by the following recurrence
\begin{equation}
a(n,k)=(k+1)(k+2)\, a(n-1, k)+\frac{(k+1)(k+2)(k+3)}{k} \,a(n-1, k-1),
\end{equation}
with $a(1,1)=1$ and ...

**1**

vote

**1**answer

184 views

### Find a bijection near to a given surjection

Let $X$, $Y$ be Banach spaces with the same cardinality and $f: X \rightarrow Y$ be a surjection. Is it possible for $\varepsilon >0$ to find a bijection $g: X \rightarrow Y$ such that
...