**-1**

votes

**1**answer

92 views

### Oddify an even function and vice versa: need a Fourier transform-based formula [closed]

I have constructed an operator that applied to an odd function will give its even counterpart and also an inverse operator that would transform an even function into an odd one. I need a ...

**1**

vote

**0**answers

29 views

### Uniform upper bound for dim of kernel and codimension of range of certain familly of PDE

A polynomial vector field of degree $n$ on $S^{2}$ is the Poincare compactification of a $n$ degree polynomial vector field on $\mathbb{R}^{2}$.
According to the comment of Joonas on this ...

**3**

votes

**1**answer

71 views

### Horizontal lift of differential operator

On a Riemannian manifold $M$, there is a canonical horizontal lift $X^{\mathrm{hor}}$ of vector fields $X$ to $TM$, which is characterized by the two properties that
$X^{\mathrm{hor}}$ is a ...

**0**

votes

**0**answers

64 views

### Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...

**2**

votes

**0**answers

97 views

### Is Laplacian a surjective operator?

For a closed manifold the laplacian is almost surjective operator since the index of $\Delta$ is zero and there is no a non constant harmonic function. So the codimension of the image ...

**1**

vote

**0**answers

109 views

### The “Rolle theorem” for sections of a vector bundle

1)Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...

**0**

votes

**1**answer

150 views

### elliptic operators corresponds to non vanishing vector fields

Let $X$ be a non vanishing vector field on a compact manifold $M$. The only differential operator associated with $X$ which I am aware of, is the derivational operator $D(g)=X.g$. Unfortunately ...

**5**

votes

**0**answers

174 views

### Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...

**4**

votes

**1**answer

96 views

### Euclidean Algorithm for differential operators

While perusing through the article "Algorithms for solving linear ordinary differential equations" by Winfried Fakler (a pdf can be found through a google search), I noticed Faker mentioning on page 2 ...

**0**

votes

**1**answer

22 views

### Pullback via flow as operator group

Let $X$ be a vector field on a manifold $M$ that induces a complete flow $\Theta_t$. Then the operator family $\Theta_t^*$,
$$\Theta_t^*u(x) = u(\Theta_t(x))$$
is a strongly continuous semigroup of ...

**1**

vote

**0**answers

20 views

### Differential operator with codimension 2 singularity in the domain

The soft version of the question is as follows: suppose I have a linear operator, and I know it is a 'nice' differential operator on its domain minus a singular set of codimension two. Does the ...

**3**

votes

**1**answer

362 views

### Helmholtz equation Poynting vector integral

The Maxwell's equation for harmonic time dependent field in vacuum is
\begin{align}
\nabla \times B + i\omega E &= 0\\
\nabla \times E - i\omega B &= 0 \\
\nabla \cdot B &= 0 \\
\nabla ...

**0**

votes

**0**answers

104 views

### Existence of the Dirichlet heat kernel for arbitrary open subsets?

consider first of all an open and bounded subset $\Omega\subset\mathbb{R}^n$, s.t. the boundary $\partial \Omega$ is a manifold of class $C^2$. Then I know that there exists a Dirichlet heat kernel, ...

**4**

votes

**1**answer

157 views

### Vector Laplace Beltrami operator of the Gauss map

Consider an abstract surface $(M,g)$ embedded into $\mathbb{R}^3$ via $f:M \to \mathbb{R}^3$. Denote by $N:M \to \mathbb{R}^3$ the Gauss map (normal field) of the surface. Write the Laplace Beltrami ...

**0**

votes

**1**answer

268 views

### Yang-Mills equations are not elliptic [closed]

How does one prove that the Yang-Mills equations (from classical Yang-Mills theory) are not elliptic?
Alternatively, how does one calculate the principal symbol of the Yang-Mills equations?
Can ...

**-3**

votes

**1**answer

149 views

### Derivatives of infinite order [closed]

Is there any sense of taking an infinite number of derivatives? Is it discussed in the literature?
For example, can one make sense of
$$\frac{\partial^{\infty}f(x_1,x_2,\cdots)}{\partial x_1 ...

**4**

votes

**1**answer

120 views

### Differential Operator Simplification

Does anyone know the explicit formulation for the $q_k$'s in, $$(x+D)^n=\sum_{k=0}^n q_k(x)D^k\ \ \ \ ?$$
I know that $e^{-x^2/2+x}$ is a fixed point of $(x+D)$. I also, know that ...

**0**

votes

**0**answers

101 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) ...

**1**

vote

**0**answers

116 views

### The Moyal action of a planar vector field

Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be a polynomial vector field on $\mathbb{R}^{2}$. Consider the following (Moyal) derivation on $\mathbb{C}[x,y]$:
...

**1**

vote

**1**answer

115 views

### Laplacian on space of measures

Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with ...

**2**

votes

**1**answer

82 views

### Eigenvalue problem of an operator involving the exterior derivative of differential forms

Consider two functions $\alpha,\beta: \mathbb{R}^2 \to \mathbb{R}$, where $\alpha$ is given and we look for solutions $\beta$ such that
$$*(d\alpha \wedge d\beta) = \lambda \beta$$
for some $\lambda ...

**1**

vote

**1**answer

152 views

### de Rahm Laplace operator on forms bounded

Let $M$ be a closed differentiable manifold. Let $E^{p}(M)$ be the vector space of $p$-forms on $M$ equipped with the $L^{2}$-inner product $(\alpha, \beta) = \int_{M}\alpha \wedge \star \beta$. The ...

**0**

votes

**0**answers

283 views

### A Generalized De Rham cohomology

Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by ...

**6**

votes

**3**answers

226 views

### On formal solutions to differential equations

Let $k$ be a field of characteristic zero. Put $K=k[\![ t]\!]$ and $W=k\langle t,\partial\rangle / ([\partial,t]=1)$. Then $W$ operates on $K$ in the obvious way ($\partial f = \frac{d f}{dt}$), and ...

**10**

votes

**2**answers

411 views

### Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map
$D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...

**7**

votes

**0**answers

80 views

### Euler number of the complex of basic forms

Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ...

**0**

votes

**0**answers

53 views

### Parabolic partial differential equation, initial conditions

Let $U\subset\mathbb{R}^n$ be open bounded, $T>0$.
Given the parabolic PDE $$\partial_tf+a\partial_xf+b\partial_{xx}f = g \qquad (1)$$ I'm interested in the initial and boundary conditions that ...

**13**

votes

**1**answer

433 views

### Derivation on real analytic manifolds

Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...

**7**

votes

**0**answers

317 views

### Why should the Laplacian in $\mathbb{C}^n$ act on a specific line bundle over the quadric $x^2=0$ in $\mathbb{P}^{n-1}$?

I recently encountered the following nice fact, and I'm wondering if it's part of a more general story.
Let $x\in \mathbb{C}^n$ satisfy
$$x^2:=\sum_i x_i^2 = 0,$$
and consider functions $f(x)$ ...

**1**

vote

**1**answer

62 views

### Estimate the analytical wavefront set $WF_A(u)$ given $WF_A(A_K u)$

Let $X$ and $Y$ be two real-analytic manifolds and $Z \subset X \times Y$ be a real-analytic embedded closed submanifold. Suppose now that $K \in \mathscr D'(X \times Y)$ is a distribution conormal to ...

**4**

votes

**1**answer

127 views

### Practical way to check whether a distribution is conormal

Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. HÃ¶rmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
L_1 ...

**1**

vote

**0**answers

85 views

### Action of Landweber-Novikov algebra on infinite polynomial ring

Denote $\text{Aut}(\hat{{\mathbb A}}^1)$ be the affine group over ${\mathbb Z}$ that sends some ring $R$ to the strict automorphisms of $R[[t]]$, i.e. those of the form $X\mapsto X + r_1 X^2 + r_2 X^3 ...

**2**

votes

**0**answers

186 views

### Radon-Nikodym derivatives as limits of ratios

Let $\mu_1$ and $\mu_2$ be measures with $\mu_1 \ll \mu_2$. Suppose we can characterize (a version of) their Radon-Nikodym derivative this way:
$$\frac{d\mu_1}{d\mu_2}(x) = \lim_{n \to \infty} ...

**4**

votes

**1**answer

217 views

### Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form
$L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$
for ...

**2**

votes

**2**answers

215 views

### Schrodinger's equation via Spectral Theorem

How do you prove basic facts on the Schrodinger equation using the spectral theorem? More precisely, here is what I have in mind.
The version of the Spectral Theorem I am familiar with is the ...

**3**

votes

**1**answer

135 views

### Is the ideal membership problem solvable for differential ideals? Is there a good notion of a Gröbner basis?

Let $K$ be a field of characteristic zero. Let $\Omega = K[x_1, \dots, x_n, dx_1, \dots, dx_n]$ be the differential ring of algebraic differential forms over $K[X_1, \dots, X_n]$.
Is there an ...

**11**

votes

**1**answer

356 views

### Atiyah-Singer for pseudodifferential operators via heat kernel?

The Atiyah-Singer theorem for Dirac-type operators can be proved using the heat kernel and this proof has an advantage over the proof via K-theory, because the first is local but the latter is not. ...

**3**

votes

**2**answers

136 views

### Space of differential operators

Let $A$, $B$ be two smooth vector bundles of finite rank over a smooth manifold $M$. Let $Diff(A,B)$ be the space of differential operators from $A$ to $B$. Can I talk about "the space of smooth maps ...

**1**

vote

**1**answer

433 views

### Precise versions of “differential operators are unbounded but closed linear operators”

I am trying to understand to what extent the following result of Hille is an extension of the usual theorems on differentiation under the integral sign.
Theorem (Hille). Let $(\Omega,\Sigma,\mu)$ ...

**-1**

votes

**1**answer

138 views

### Stone Cech compactification for exponential map

Recently I met with a problem related to Stone-Cech Compactification theorem
in Furstenberg's famous paper "non-commuting product."
I try my best to understand Stone-Cech compactification theorem by ...

**4**

votes

**1**answer

267 views

### Poisson structure on the cotangent bundle

Let $X$ be a smooth variety over $\mathbb{C}$ and $\mathscr{A}$ a sheaf of twisted differential operators on $X$. The latter comes equipped with a natural filtration and the associated graded algebra ...

**13**

votes

**5**answers

1k views

### Book Recommendation - PDE's for geometricians / topologists

I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...

**2**

votes

**1**answer

305 views

### Mellin transform between heat kernel and zeta-function

For some notion of a "positive operator" $D$ of "Laplacian type" one seems to be able to define a notion of a zeta-function as $\xi(s,f,D) = Tr_{L^2}(f D^{-s})$ where $f \in L^2$ (the space of ...

**1**

vote

**0**answers

217 views

### Formula for the curvature of an induced connection

Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...

**7**

votes

**2**answers

258 views

### Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...

**4**

votes

**1**answer

168 views

### Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...

**4**

votes

**1**answer

444 views

### Algorithm to find exponential map of differential operators acting on function

I am trying to write a computer program which computes the action of the exponential of a differential operator on a function, for any given differential operator.
Examples:
$\exp(\varepsilon ...

**0**

votes

**0**answers

146 views

### Equivariant integration (localization formula)

We consider the action of $S^{1}$ on $S^{2}$ by rotation respect the vertical axes. We want to integrate the $2-$ equivariant form
$$\alpha(X)= -X\cos(\phi) +\sin(\phi)\,d\phi\,d\theta.$$
We have ...

**1**

vote

**1**answer

283 views

### Combinatorics: Product Rules.

I couldn't find a way to figure this out, though it is a somewhat basic question that came up when studying the stationary phase expansion of an integral. The abstract version is the following:
I ...

**1**

vote

**1**answer

166 views

### Symmetric Operators Robin Boundary Conditions

How can you show that an operator is symmetric with robin boundary conditions?
I know I need to show that < Tf,g > = < f,Tg >; however, the robin boundary conditions are throwing me off.
This ...