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

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674 views

How to “fill in” 3-dimensional Laplacian kernels

The other day, I asked this question 3x3x3 Laplace Kernel?, regarding what the 3x3x3 kernel was for applying a Laplacian convolution. On that page, it mentions the kernels were "deduced by using ...
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1answer
2k views

3x3x3 Laplace Kernel?

Does anyone know what a 3x3x3 Laplacian kernel looks like? I realize that might be an open-ended question, but I need to apply a Laplacian convolution using a 3x3x3 Laplacian kernel, and frankly I ...
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1answer
379 views

Does the image of a differential operator always contain an ideal?

Let $\delta$ denote a non-zero complex algebraic differential operator in a single variable x. That is, it can be written as a sum $$ \delta = \sum_i f_i\partial_x^i$$ where there $f_i$ are complex ...
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3answers
730 views

Is there a good account of D-affinity and localization theorem for partial flag varieties?

Recall that a topological space is called $A$-affine for a sheaf of algebras $A$ if taking global sections of coherent sheaves of $A$-modules is an equivalence of categories to finitely generated ...
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0answers
280 views

Why should we consider D-module on flag variety of Lie algebra?

Why don't we stay at D-module on base affine space but go to study flag variety of Lie algebra? I remembered there are nice papers of Bernstein-Gelfand-Gelfand and Gelfand-Kirillov discussing the ...
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1answer
198 views

The space of spectral sections and connections to K-theory

We look at a familiy $D_\alpha$ of Dirac operators over a (compact) base space B. The projection $\Pi^+_\alpha$ onto the positive eigenspaces of $D_\alpha$ is usually not continuous in α. ...
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1answer
579 views

Self-adjoint extension of locally defined differential operators

The following is well known. Given a symmetric differential operator, like $\partial_x^2$, defined on smooth functions of compact support on $\mathbb{R}$, $C_0^\infty(\mathbb{R})$, one can count the ...
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4answers
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Have people successfully worked with the full ring of diferential operators in characteristic p?

This question is inspired by an earlier one about the possibility of using the full ring of differential operators on a flag variety to develop a theory of localization in characteristic $p$. (Here ...
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0answers
525 views

How hard is it to make a differential operator Hermitian?

Let $M$ be a closed finite-dimensional smooth manifold (over $\mathbb R$). Let $C^\infty(M) = C^\infty(M,\mathbb C)$ be the algebra of smooth complex-valued functions on $M$, with the natural complex ...