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Elliptic, parabolic and hyperbolic operators. Laplace, Laplace-Beltrami, Schrödinger, Dirac. Exterior derivative and Lie derivative operators.

0 votes
0 answers
126 views

The tensor product of two Fredholm operators

What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of operat …
Ali Taghavi's user avatar
1 vote
0 answers
93 views

The asymptotic growth of codimension of range of polynomial differential equation on finite ...

Inspired by the seminal paper of Andre Weil on the number of solutions of equations on finite fields we would like to present the following question: Let $P(x,y), Q(x,y)$ be two polynomials of deg …
Ali Taghavi's user avatar
1 vote
0 answers
97 views

The module generated by kernel of an elliptic differential operator

Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\i …
Ali Taghavi's user avatar
2 votes
1 answer
206 views

On which subspace $W\subset C^{\infty}[0,1]$ is $(Df)(x)=xf'(x)$ a bounded operator provided...

I have already asked this question on MSE; now I repeat it on MO. https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator First we introd …
Ali Taghavi's user avatar
4 votes
1 answer
373 views

Differential inequalities under which a flat function must be identically zero

Let $f:\mathbb{R}\to \mathbb{R}$ be a smooth function which is flat at $0\in \mathbb{R}$. That is $f^{(k)}(0)=0,\; k=0,1,2,\ldots $. Assume that $|f''(x)|\leq M|f(x)|\quad \forall x\in \mathbb{R}$ …
Ali Taghavi's user avatar
2 votes
1 answer
235 views

Elliptic operators and Leibniz rule

Let $M$ be a manifold. Does it necessarily admit an elliptic operator on $C^{\infty}(M)$ which satisfy Leibniz rule? Let $M$ be a symplectic manifold with the standard Poisson structure on $C^{\inf …
Ali Taghavi's user avatar
3 votes
1 answer
363 views

When is the exterior derivation $d$ a Lie algebra morphism?

In this question we search for some conditions under which the exterior derivation $d:\Omega^i(M)\to \Omega^{i+1}(M)$ on a differentiable manifold $M$ is a Lie algebra morphism in a certain sense. We …
Ali Taghavi's user avatar
6 votes
1 answer
340 views

Fredholm theory of non elliptic operators

In this question we search for a big list of non elliptic operators whose Fredholm index is finite or whose Fredholm theory is extensively discussed. The main motovation is the conference linked in th …
2 votes
1 answer
123 views

Keeping track of limit cycles via certain second order differential operator

Inspired by the two posts which are linked bellow we ask the following question: Question: For a vector field $X$ on the plane we define the differential operator $D$ on $C^{\infty}(\mathbb{R}^2)$ wi …
Ali Taghavi's user avatar
0 votes
0 answers
62 views

A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact paralleliza...

Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2} …
Ali Taghavi's user avatar
0 votes
0 answers
76 views

A kind of heat equation on a foliated 3D manifold whose leaves are invariant under the flow ...

Assume that $\mathcal{F}$ is a foliation of a $3$ dimensional compact Riemannian manifold $M$ which is invariant under flow of a transversal non vanishing vector field $X$. Let $\Delta_{\ …
Ali Taghavi's user avatar
5 votes
1 answer
450 views

An alternative representation of the principal symbol of the Laplace operator

Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent? First condition …
Ali Taghavi's user avatar
0 votes

Derivations of $\chi^{\infty}(M)$ which are elliptic operator

I would like to apply the hints presented in the answer by Bazin to give the following answer. Every derivation of $\chi^{\infty}(M)$ is inner and it is obvious that every inner derivation is a non …
Ali Taghavi's user avatar
0 votes
2 answers
284 views

Derivations of $\chi^{\infty}(M)$ which are elliptic operator

What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such th …
Ali Taghavi's user avatar
5 votes
0 answers
217 views

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 $s\in \Gamma(E)$. Ass …
Ali Taghavi's user avatar

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