# Tagged Questions

**3**

votes

**1**answer

68 views

### Diagonalization via the Toda flow

According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...

**1**

vote

**1**answer

102 views

### Global version of the Picard-Lindelöf theorem [closed]

Let $I\subseteq \mathbb{R}^{n}$ be an arbitrary (not necessarily closed) intervall and $f:I\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ a continuous function such that in $I\times \mathbb{R}^{n}$ ...

**1**

vote

**0**answers

188 views

### The integral of torsion

I found the following * exercise(exercise *9) in page 407 of the book of Do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...

**0**

votes

**0**answers

94 views

### projecting Laplacian onto tangent and normal bundles of submanifold

If I have a simple linear differential equation involving covariant derivatives such as $\nabla^2 g_{\mu\nu}+ 2g_{\mu\nu}=0$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) ...

**4**

votes

**1**answer

395 views

### Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange.
My question is where the following differential equation arises naturally and where it ...

**1**

vote

**0**answers

118 views

### Reference request: has this semilinear version of Navier Stokes been studied?

I have noticed that the
Navier Stokes equations can be written as a semilinear symmetric first
order system
$$
u_t+A_1u_{x_1}+A_2u_{x_2}+A_3u_{x_3} = f(u)
$$
for a 9 by 1 vector $u$ containing the ...

**2**

votes

**0**answers

172 views

### J-function of cotangent bundle of complete flag variety

Givental and Kim showed that the $J$-function of the complete flag variety $Fl_n=SL_{n}/B$ becomes an eigenfunction of the Toda Hamiltonian. How about the $J$-function of the cotangent bundle ...

**3**

votes

**0**answers

106 views

### Nonexistence of Limit Cycle

Consider a planar dynamical system described in polar coordinates as
$$
\left\{
\begin{array}{ll}
\dot{\theta}=\Delta - r \sin \theta,\\
\dot{r} = - r + 1 + \cos \theta,
\end{array}
\right.
$$
where ...

**3**

votes

**2**answers

251 views

### existence and uniqueness of solutions for ODEs in formal power series?

I came across this question and it looked like something that is likely to have been looked into, but I couldn't find a reference.
Let $k$ be some (algebraically closed, if needed) field. There is a ...

**4**

votes

**1**answer

137 views

### Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics
http://www.encyclopediaofmath.org/index.php/Differential_inequality
the following result is due to Chaplygin ...

**1**

vote

**1**answer

171 views

### Number of solutions of a system of equation!

Let $\Theta =(\theta_1,\ldots, \theta_n)\in {\mathbb T}^n$. I want to show that the system of equations
$$
\sum_j 2\sin(\theta_i -\theta_j)+\sin(2\theta_i -2\theta_j) =0,\ \ i=1,\ldots, n,
$$
has ...

**2**

votes

**1**answer

138 views

### Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...

**1**

vote

**0**answers

53 views

### On global attraction of a stable node for a four dimensional nonlinear system

Consider the dynamical system on ${\mathbb R}^2\times{\mathbb I}^2$ (or ${\mathbb T}^2\times{\mathbb I}^2$) described by
$$\left\{
\begin{array}{l}
\dot{\theta}_1 = \omega_1 - ...

**2**

votes

**1**answer

208 views

### Curve on a surface defined by its geodesic curvature

Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...

**4**

votes

**1**answer

220 views

### Exact Differential Equations of Order n via Pfaffian Differential Equations?

I'm wondering if somebody could both shed some light on, & offer references for more details about, this interesting quote:
The derivation of the conditions of exact integrability of an ...

**8**

votes

**2**answers

769 views

### Reference for a nice proof of “undetermined coefficients”

I'm teaching an honors differential equations class and have been using linear algebra heavily. I thought it would be interesting to include a proof of the method of undetermined coefficients along ...

**5**

votes

**1**answer

172 views

### Periodic Holomorphic ODE

Suppose I have an annulus $U\subset \mathbb{C}$ and a single-valued holomorphic function $V:U\to \mathbb{C}$.
I would like to know if there are (tractable) conditions on $V$ that ensure that the ...

**1**

vote

**3**answers

840 views

### book on PDE on manifolds

let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...

**1**

vote

**0**answers

272 views

### Archimedes’ and Galileo’s spirals in one equation.

The differential equation in polar coordinates
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const, for large $t$ presents Archimedes’ Spiral and Galileo's spiral for $t \to 0$.
I find it surprisingly, however ...

**1**

vote

**0**answers

232 views

### Linear differential equations with regular singularities

Consider a differential equation
$$
\delta^n+b_1(z)\delta^{n-1}+\ldots+b_n(z)
$$
with a regular singular point zero. (Here $\delta=z\frac\partial{\partial z}$). Assume that its indicial polynomial ...

**7**

votes

**1**answer

484 views

### Proof of the “Neo-classical Inequality”

I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1, n$:
$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{j/p}b^{(n-j)/p}}{(j/p)!((n-j)/p)!}\leq ...

**4**

votes

**1**answer

470 views

### Regularity for transport equation?

In the book of Evans the transport equation,
$$\frac{d}{dt} u + b\cdot \nabla u = 0, \quad u(t=0)=u_0,$$
is solved by the method of charateristics for $b$ and $u_0$ smooth enogh (in terms of ...

**3**

votes

**2**answers

286 views

### parallel transport along $W^{1,2}$-curves

Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t_0\in I$ and $\xi_0\in T_{c(t_0)}Q$ be arbitrary. I am looking for a citeable reference ...

**3**

votes

**5**answers

445 views

### Analytic hypoellipticity of linear ordinary differential operators

Let $P = a_n(x) D_x^n + a_{n-1}(x) D_x^{n-1} + \ldots + a_0(x)$ be a linear ordinary differential operator with polynomial (or real analytic) coefficients $a_j(x)$. Suppose that $a_n(x)$ doesn't ...

**4**

votes

**1**answer

682 views

### Christodoulou's paper on naked singularities in inhomogeneous dust collapse

I have been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is ...

**12**

votes

**5**answers

1k views

### 2- and 3-body problems when gravity is not inverse-square

Suppose that gravity did not follow an inverse-square law, but was instead a central force diminishing
as $1/d^p$ for distance separation $d$ and some power $p$.
Two questions:
Presumably the 2-body ...

**3**

votes

**1**answer

640 views

### Limit of a discrete time dynamical system

I have the following discrete time dynamical system
$$ y(t+1) = y(t) + \frac{1}{1+\exp(z+ u f y(t))} ,\quad y(0)=0,$$
where $z$ is a real number $f$ and $u$ are non-negative reals. I know I have ...

**23**

votes

**13**answers

20k views

### Good differential equations text for undergraduates who want to become pure mathematicians

Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate ...

**3**

votes

**0**answers

337 views

### Infinite System of Stochastic Ordinary Differential Equations Coupled by Infinite Graphs

$\ \ \ $ In Time Evolution of Infinite Anharmonic Systems,
Lanford,Lebowitz and Lieb, roughly speaking, proved that for some families of functions $F_v$ $(v\in\mathbb Z^d)$
and a large set of initial ...

**2**

votes

**1**answer

620 views

### A formula for the Jacobian of a flow

Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation ...

**3**

votes

**2**answers

569 views

### Good references for analytic solutions to nonlinear ordinary differential equations?

I am faced with a non-autonomous initial value problem for a function $x:[0,\infty) \to \mathbb{R}^2$ of the form
$$ x'(t) = f(t,x(t)) $$
for $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}^2$ with ...

**3**

votes

**2**answers

559 views

### Analyzing the solution to a second-order, non-linear ODE

Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...

**2**

votes

**4**answers

3k views

### Undergraduate Derivation of Fundamental Solution to Heat Equation

It is well known that the 1-dimensional heat equation $$\frac{\partial}{\partial t} u(x,t)=a\cdot\frac{\partial^2}{\partial x^2} {u(x,t)}$$ has the fundamental solution $$K(x,t)=\frac{1}{\sqrt{4\pi a ...