**1**

vote

**1**answer

116 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

298 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

153 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

279 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]:
...

**6**

votes

**4**answers

374 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

163 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

212 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 ...

**1**

vote

**1**answer

88 views

### Inversion of incomplete elliptic integral of third kind

I would like to know whether there is any solution available on the inversion of elliptic integrals of the third kind (incomplete)?
That means that given $\Pi(n,u,m) = f(x)$, I would like to obtain ...

**3**

votes

**2**answers

344 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

**1**answer

137 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

408 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

287 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

163 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

207 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

238 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

89 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

839 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

125 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

304 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

208 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

220 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

128 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

120 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

84 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

96 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

255 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

238 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

278 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

289 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

65 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

216 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

58 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

192 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

360 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

479 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

540 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

185 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
...

**4**

votes

**2**answers

112 views

### Reference for the fact that the images of the narrow and wide Denjoy integrals are respectively $ACG_\ast$ and $ACG$?

The narrow Denjoy integral (which also goes by the names Henstock-Kurzweil integral, Perron integral, and Lusin integral) is a transfinite integration process defined by Denjoy in the early 20th ...

**26**

votes

**0**answers

960 views

### Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least n+1 zeros on $(-1,1)$

Let $f\in C^{\infty}(\mathbb{R},\mathbb{R})$ such that $f(x)=0$ on $\mathbb{R}\setminus (-1,1)$. Prove that there exists $n\in\mathbb{N}$ such that $f^{(n)}$ has at least $n+1$ zeros on $(-1,1)$
...

**0**

votes

**1**answer

315 views

### Pros and cons of probability model for permutations

I am studying probability model of random permetuation
Let $b(n; k)$ denote the number of permutations of {1,...,n} with precisely k
inversions ($inv(\pi)$). The analytic approach was considered by ...

**4**

votes

**1**answer

106 views

### Delay Differential Equations Numerical methods

I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay ...

**1**

vote

**1**answer

72 views

### Implicit function theorem for boundary value problems

I have a nonlinear, two point boundary value problem of the form
$F(x, y(x), y'(x); \Omega ) = y''$ along with some boundary conditions of the form $y_\Omega(0) = a_\Omega, y_{\Omega}(1) = b_\Omega$. ...

**1**

vote

**1**answer

126 views

### Absolute convergence of multi-dimensional Fourier series

For a Lipschitz function $f$ defined in $[0,2\pi]^d$ for $d>1$, is that true
that the multi-dimensional Fourier series converges absolutely?
In other words, $\sum_{k\in ...

**1**

vote

**1**answer

132 views

### Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...

**0**

votes

**0**answers

79 views

### Detailed Taxonomy of Multivariate Real Functions: $\mathbb{R}^n\rightarrow\mathbb{R}$

I want to classify the following functions:
$$ f:(x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\sqrt{(x+1)^2+y^2}+\sqrt{(x-1)^2+y^2}-2$$
$$ ...

**0**

votes

**1**answer

77 views

### Examples of Bivariate Analytic, Globally Continuous Functions, whose Set of Zeros are Boundaries of Polygons

Are there any known examples of analytic, globally continuous functions $f(x,y): (x,y)\in\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ such that
$$f(x,y)\lt 0\Leftrightarrow ...