**9**

votes

**1**answer

327 views

### An algebraic Hamiltonian vector field with a finite number of periodic orbits(1)

Edit: The previous version of this question contained 2 part. In this new version, I deleted the first part and move it to a new question.
Is There a polynomial Hamiltonian ...

**0**

votes

**0**answers

10 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**-1**

votes

**0**answers

21 views

### Solve Higher ODE with boundary condition at infinity [on hold]

Given below is a second-order linear differential,
$y''+Ay'+By=0$
The boundary conditions are:
(a) $y(t=0)=1+H\frac{dy}{dt}$
(b) $y(t=\inf)=0$
Does a solution exist for this problem?

**3**

votes

**1**answer

200 views

### Find a maximizing solution to an ODE which depends on a paramater function

(For the physical meaning of this problem see http://physics.stackexchange.com/questions/122818/how-should-i-throttle-my-rocket-to-reach-highest-altitude).
Given $g \in (0,\infty), k \in C^1( [0, ...

**-4**

votes

**0**answers

41 views

### Gamma Functions [on hold]

Writing the Integral Equation of the gamma function
I(n)=n-1*I(n)
is there a way to prove, if possible, that there exists only one gamma function?
Please Help!

**15**

votes

**1**answer

732 views

### Kontsevich's flow on the space of Poisson structures

In §5.3 of Kontsevich's Formality Conjecture he writes:
This (...) gives a remarkable vector field on the space of bi-vector fields on $\mathbf{R}^d$. The evolution with respect to the time $t$ is ...

**3**

votes

**2**answers

73 views

### Given $x$ in a path-connected open set $S$ on the plane, are there non-crossing paths from $x$ to every point in $\partial S$?

A have a (non-simply) bounded path-connected open set $S\subset\mathbb{R}^2$.
Given $x\in S$, there are paths in $S$ from $x$ to any point in the boundary $\partial S$.
However, can all these paths ...

**1**

vote

**1**answer

46 views

### Bounds for smallest eigenvalue of Schrödinger equation with a superpotential and periodic boundary conditions

A very specific question, but posted on the off-chance that someone may be able to help. If we have a Schrödinger equation with arbitrary "superpotential"
\begin{equation}
-\frac{d^2 \psi}{dx^2} ...

**3**

votes

**1**answer

318 views

### Monge–Ampère with drift

Let $I\subseteq \mathbb{R}$ be an interval.
Let smooth $M(x,y):I\times(0,\infty) \to \mathbb{R}$ satisfies PDE:
$$
M_{xx}M_{yy}-M_{xy}^{2}+\frac{M_{y}M_{yy}}{y}=0.
$$
My question is to ...

**1**

vote

**0**answers

42 views

### Singularity Confinement For Differential-Difference Systems

This is a follow-up question to an old question on this site (link) which has a solution that describes the singularity confinement property for discrete systems.
Are there any papers or books that ...

**0**

votes

**0**answers

31 views

### Questions about the definition of ``stabilization entropy" for dynamical systems

Let $\phi (t,x,u)$ be the solution to the differential equation, $\dot{x}(t) = f(x(t),u(t))$ where $x(t) \in \mathbb{R}^d$, $u : [0,\infty) \rightarrow \mathbb{R}^m$ and $f: \mathbb{R}^d \times ...

**3**

votes

**1**answer

99 views

### A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation
$$ \frac{d x (t)}{dt} = f(x(t)) $$
with some initial condition $x(0)=x_0$ has no solution?

**1**

vote

**0**answers

104 views

### Ricci flow in complex analysis [closed]

Occasionally, I find a paper http://arxiv.org/abs/math/0505163 written by Chen, Lu and Tian. In this paper, the uniformalization theorem was proved by Ricci flow. I think it is a very interesting ...

**0**

votes

**0**answers

45 views

### Extracting information from a differential equation if the zero eigenvalue eigen-function is known

Given the second order linear homogeneous differential equation
$$
-\dfrac{d^2}{dx^2}\psi_m(x) + V(x)\psi_m(x)=E_m\psi_m(x)
$$
with eigen-functions $\psi_m(x)$ and eigenvalues $E_m$, what information ...

**30**

votes

**2**answers

2k views

### Why is differential Galois theory not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation.(e.g. ...

**4**

votes

**1**answer

113 views

### Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...

**0**

votes

**0**answers

55 views

### Proving a differential inequality without performing iteration [migrated]

I'm seeking a better proof of the following fact: If $g$ is a non-negative bounded function, $g(0)=0$ and $g'(t)\leq \sqrt{g(t)}$ for all $t>0$, then $g(t)\leq t^2/4$.
The upper bound $t^2/4$ is ...

**1**

vote

**1**answer

72 views

### First order ODE from Michaelis–Menten kinetics

From Michaelis-Menten kinetics one could easily derive scalar first-order differential equation
$$\frac{dx}{dy} = A_0 + A_1 x + A_2 y + B \frac{y}{x},$$
where
$A_0, A_1, A_2, B$ are some constants, ...

**6**

votes

**1**answer

1k views

### Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...

**2**

votes

**0**answers

63 views

### [This might be a easy question]: A possible trace (inequality) defined under negative Sobolev scale

I have a stupid question:
Why is that not possible (if it is not) to define the trace of a function in a very weak regularity space $H^{-s}(\partial \Omega)$? We usually encounter trace theorem as
...

**1**

vote

**1**answer

169 views

### Is regularity closed under products?

Let $G \colon [0,1] \to [0,1]$ be a differentiable cumulative distribution function (monotonically non-decreasing function with $G(0) = 0$ and $G(1) = 1$). We say that $G$ is regular if $$ x - ...

**0**

votes

**0**answers

20 views

### Finding incomplete geodesics [migrated]

I have a problem with the notion of incomplete geodesics. Can someone give me a minimal example for such a geodesic?
In particular, I am trying to solve the following exercise:
Consider the upper ...

**0**

votes

**0**answers

61 views

### Energy Oscillations in a One Dimensional Crystal

Good day!
Can anyone help me find articles on similar topics "Energy Oscillations in a One Dimensional Crystal" (I have links to one article on this subject)?
article, that I have
Especially ...

**11**

votes

**1**answer

192 views

### Does Peano's existence theorem admits a constructive proof?

$$y(t)=y_0+\int_0^t b(y(s))ds$$ $b\in C(R^d)\cap L^\infty(R^d)$
The classical proof for Peano's existence theorem in ODE need use the Ascoli's theorem, so it's not constructive. When $d=1$, in the ...

**12**

votes

**4**answers

4k views

### Can an integral equation always be rewritten as a differential equation?

Given an integral equation is there always a differential equation which has the same (say smooth) solutions?
It seems like not but can one prove this in some example?
Edit: Naively I'm hoping for ...

**4**

votes

**2**answers

127 views

### How to find an ODE with prescribed terminal values?

Let us consider an ODE
$$\frac{dx_t^y}{dt}=g(x_t^y),$$
where y is the initial condition i.e. $x_0^y=y$.
Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...

**1**

vote

**1**answer

89 views

### Is the endpoint map smooth

Given $a,b \in \mathfrak{su}(n)$ and (with $U_0 = I$ taken) the following ODE:
$\frac{d U_t}{dt} = (a + w(t)b)U_t$
consider the "fixed time" endpoint map $V_T: L^2([0,T]) \rightarrow SU(n)$ for ...

**1**

vote

**2**answers

93 views

### Nonlinear matrix differential equation

I want to solve the equilibrium of the following differential equation:
$\dot{x_i} = \sum_j A_{ij} x_j + x_i \sum_j B_{ij}x_j$
which is essentiall in matrix notation:
$\dot{\mathbf{x}} = ...

**0**

votes

**0**answers

43 views

### Quadratic stability of linear time varying system

(This question was originally asked at Math.SE, where it didn't receive any answers.)
Consider the linear time-varying system
$$ \dot{x} = A(t) x, $$
where $x \in \mathbb{R}^n$ and $A: [0,+\infty) ...

**8**

votes

**1**answer

345 views

### Two surfaces with zero gaussian curvature

There is classical result of Hartman and Nirenberg:
Theorem. Every point of a $C^2$ surface of Gauss curvature zero has a neighborhood which admits a parameterization of the form $x=a(u)v+b(u)$ where ...

**-1**

votes

**1**answer

242 views

### Which operators other than self-adjoint operators have no purely imaginary eigenvalues? [closed]

Given an operator mapping between suitable spaces, what is the condition that guarantees all eigenvalues have nonzero real part? Obviously self-adjointness implies all eigenvalues are real, but how ...

**0**

votes

**2**answers

261 views

### General description of surface with zero gaussian curvature

Let suppose function $g(s,t)$ satisfies partial differential equations:
$g_{ss} g_{tt} - g_{st}^2=0$. It may be treated as the surface has zero gaussian curvature. I am searching for general solution ...

**5**

votes

**0**answers

117 views

### Is there a Lie II theorem for monoids?

Lie's second theorem says that if $G$ is a connected simply connected Lie group with Lie algebra $\mathfrak g$, then the functor of "differentiation" from the category $\mathrm{Rep}^f(G)$ of ...

**7**

votes

**1**answer

362 views

### Proving that a space is Hilbert

Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...

**1**

vote

**1**answer

187 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**

vote

**1**answer

117 views

### How to Separate Charpit Equations

I've been attempting to solve this non-linear PDE
$$4\Omega x^2 y^2 \frac{\partial z}{\partial y} -x^2 y (\frac{\partial z}{\partial x})^2 + 2x^2 y^2 E-N^2=0$$
using Charpit's method. The ...

**1**

vote

**0**answers

376 views

### Partial feedback linearization (Control theory)

I'm trying to understand a theorem about partial feedback linearization from the paper "On the largest feedback linearizable subsystem" by R. Marino (published in: Systems & Control Letters, ...

**2**

votes

**0**answers

87 views

### Motivation for the existence of periodic solutions [closed]

I have been reading the book Critical Point Theory and Hamiltonian System by Mawhin and Willem, as well as several other papers on the existence of periodic solutions for equations of the form
...

**8**

votes

**3**answers

514 views

### Rigorous justification that overdetermined systems do not have a solution

There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions ...

**0**

votes

**0**answers

51 views

### smoothness of green's function in wave equation

I have a linear acoustic wave propagation originated from a monopole source, written as
\begin{align}
\mathcal{L}p(\mathbf{x},t) = S_m(\mathbf{x},t), \quad \mbox{in } \Omega
\end{align}
where the ...

**2**

votes

**1**answer

132 views

### acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation:
\begin{align}
\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...

**2**

votes

**0**answers

54 views

### About the space of function used in the harmonic extensions for elliptic problems involving the fractional laplacian

recently i am studying the following paper:
A concave–convex elliptic problem involving the fractional Laplacian -
C. Brandle, E. Colorado, A. de Pablo and U. Sánchez.
At the Pgs 41, 42, the ...

**0**

votes

**0**answers

50 views

### regularity of non-local linear elliptic equation

$\alpha\in (0,1)$, $u$ satisfies:
\begin{equation*}
b\cdot \nabla u(x)+\sum_{i=1}^d \int_{R} \left[u(x+se_i)-u(x)-s\mathbb{I}_{\{|s|\leq ...

**7**

votes

**0**answers

237 views

### Mass Transportation Through Wonderful Roller

There is a wonderful roller that transports mass from A to B and there is a pile of sand with weight $W$ in point A and we want to transport it to B.
Wonderfulness of roller comes from this property ...

**1**

vote

**1**answer

89 views

### A differential inequality and a special value

Let $G \colon [0,1] \to [0,1]$ be a monotonically decreasing function with $G(0) = 1$ and $G(1) = 0$. Suppose that $G$ is differentiable infinitely many times, and that: $$G(x)G''(X) \leq ...

**5**

votes

**1**answer

166 views

### Are all rational exactly solvable differential equations known?

Are there known necessary and sufficient conditions that specify in terms of an algorithm in a real arithmetic model (where real operations, elementary functions, and comparisons are elementary steps) ...

**3**

votes

**1**answer

547 views

### Short time existence on Hyperbolic Ricci flow in non-compact case

We know
Laplace equation (elliptic equations)
$ Δ u = 0$
Heat equation (parabolic equations)
$u_t − Δu = 0$
Wave equation (hyperbolic equations)
$u_{tt} − Δu = 0$
we have
- Hyperbolic geometric ...

**1**

vote

**0**answers

14 views

### solution to Helmholtz equation with non circular boundary

I have 2D homogeneous domain $D$ with non circular boundary $\partial D$ and I am trying to solve the Helmhotz equation
$\nabla^2 u(r, \varphi) + k^2 u(r, \varphi) = - f(r, \varphi)$
in which $k$ ...

**0**

votes

**0**answers

87 views

### Is the trivial solution the unique solution to the following initial value problem?

This question is a duplicate one asked by myself elsewhere. But there are no answers or comments so far. The initial value problem that I am considering is:
$$ y'' (3y+2x)^2=3(3y'-1)(9yy' + 4xy' -y), ...

**29**

votes

**1**answer

2k views

### D-modules, deRham spaces and microlocalization

Given a variety (or scheme, or stack, or presheaf on the category of rings), some geometers, myself included, like to study D-modules. The usual definition of a D-module is as sheaves of modules over ...