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5 votes
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A differential operator analogy of certain fact in real analysis of smooth functions

Let $E\to M$ be a smooth vector bundle over a smooth manifold $M$. Let $D$ be a differential operator defined on the space $\Gamma(E)$ of smooth sections of $E$. We fix a section $s\in \Gamma(E)$. ...
Ali Taghavi's user avatar
5 votes
0 answers
273 views

Is there any geometrical/homological intuition behind symmetrized gradient?

The gradient/differential/exterior differential/divergence/curl are all strictly related first order differential operators. As far as I understood, they are the base of (co)homological theories in ...
Romeo's user avatar
  • 980
5 votes
0 answers
270 views

Differential operators that preserve real-rootedness

Is there some description of polynomial differential operators, $\mathcal{D}=\sum f_i(x) D_x^i$ such that, if $h$ is a polynomial all of whose roots are in $[0,1]$, then so are all the roots of $\...
David E Speyer's user avatar
3 votes
0 answers
53 views

Controlling a Schwartz kernel near the diagonal

Let $D$ be a first-order elliptic differential operator that is essentially self-adjoint on $L^2(\mathbb{R}^n)$. Consider the operator $(D+i)^q$ acting on $L^2(\mathbb{R}^n)$ with domain $C_c^\infty(\...
geometricK's user avatar
  • 1,903
2 votes
0 answers
131 views

Taylor series with less than differentiability

I have a function $f^0\colon (0;\infty) \to \mathbb R$ with the property that the following limit exists and is finite $$ F^1 := \lim_{x\to \infty} x \cdot f^0(x) $$ Then I consider $f^1(x) := x \cdot ...
Kolodez's user avatar
  • 335
2 votes
0 answers
42 views

Analysis of coefficients for quickly vanishing analytic vector field

Let $u = (u_1, u_2, u_3): \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a divergence-free analytic vector field for $n =3$ or $n =4$, i.e., $u_i : \mathbb{R}^n \rightarrow \mathbb{R}$ are analytic ...
tobias's user avatar
  • 749
0 votes
0 answers
87 views

Curl-Div equation with singular matrix

I want to solve the equation: $$ \begin{cases} \nabla \times (A \mathbf v)=f, \quad x\in \Omega \\ \operatorname{div}(\mathbf v)=0, \end{cases} $$ where $\Omega \subset\mathbb{R}^n$, is an open set, $...
Gustave's user avatar
  • 617
0 votes
0 answers
63 views

Computing the eta invariant of a rather contrived operator on the circle

For physical reasons, I am interested in computing the eta invariant of the following Hermitian operator acting on complex valued functions on the circle with circumference 1. I define the operator ...
Blind Miner's user avatar
0 votes
0 answers
76 views

Linear dependence of the derivatives of a vector valued function

Let $f:\mathbb{R}\rightarrow\mathbb{R}^5$ be an injective smooth function, and consider the function $$ g:\mathbb{R}^5\rightarrow\mathbb{R}^5 $$ given by $$ g(t_1,t_2,t_3,a,b) = f(t_1)+a(f(t_2)-f(t_1))...
Puzzled's user avatar
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