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Questions tagged [differential-operators]

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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Grothdieck differential operators - are they all compositions of first order operators?

Let $A$ be a (unital, commutative, associative) algebra over a field $\mathbb{k}$. Then the ring of Grothendieck differential operators on A is defined as: $$ \operatorname{Diff}(A) = \bigcup_{\ell \...
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cocycle condition of stratification

Let $R,A$ be commutative rings, and $A$ is an $R$-algebra, $E$ is an $A$-module, $P = A \otimes_R A$, and $P^n$ is the $n$-th infinitesimal neighborhood of $A$ in $P$. A stratification on $E$ is a ...
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Taylor expansion for a differential operator

I'm struggling with a problem in which I need to expand a differential operator near a known function. The problem is the following: $L(u)$ is a differential operator, in my problem: $L(u)$ is defined ...
Luca Javier Gomez Bachar's user avatar
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Disagreement between two methods of computing the determinant of a differential operator

I want to compute the determinant of the following operator over the interval $[0,L]$: \begin{equation} A=-\frac{d^2}{dx^2}+\omega^2. \end{equation} I imposed the boundary conditions $\phi(0)=\phi(L)=...
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Which differential equations are invariant under change of camera projection?

For background, I am working in the plane $\mathbb{R}^2$. I know that the derivatives $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ are invariant under translation. I know that the ...
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Equality between rings of invariant differential operators

Let $G$ be a real connected Lie group in the Harish-Chandra class [GV - Definition 2.1.1]. Let $\mathfrak{g}$ denote its Lie algebra. This is a reductive Lie algebra and we have $\mathfrak{g} = \...
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Question about the index of two elliptic operators over a 4-dimensional Riemannian manifold

Let $X$ be a compact Riemannian 4-manifold, $P$ a principal $G$-bundle over $X$, and $\mathfrak{g}$ be its adjoint bundle. Let $\omega$ be a self-dual conneciton on $P$ (i.e. its curvature $\Omega \in ...
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What's the product rule for the exponential differential operator?

So I was thinking say you have a linear differential operator such as the exponential differential one which is renown in some fields in physics: $$e^{\mathrm D_x}\equiv\sum_{n=0}^\infty\frac{\mathrm ...
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How does the boundary term appear when taking the transposed form of the inner product with a linear operator

I'm trying to figure out how the second term appears here: $$\int_\Omega \mathcal{L}(u) w \;d\Omega = \int_\Omega u \mathcal{L}^*(w) \; d\Omega + \int_\Gamma \left[S^*(w) G(u) - G^*(w)S(u)\right] d\...
Cedric Martens's user avatar
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Must an operator $\mathcal{O}$ satisfying $\mathcal{O} (f + c) = \mathcal{O} (f)$ for any constant function $c$ be a differential operator?

Consider an operator (not necessarily linear) $\mathcal{O}: C^{\infty} (\mathbb{R}) \to C^\infty(\mathbb{R})$ which satisfies the following property for any function $f \in C^\infty(\mathbb{R})$ and ...
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History of the Leray Projection [closed]

I am hoping to cite the first use of the Leray projection, $\mathbb{P}$, onto the divergence-free vector-field. I tried scanning Leray's famous 1934 paper, 'Sur le mouvement...,' but couldn't find it, ...
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Index theorem example

I understand characteristic classes, elliptic operators, vector bundles and the final statement of index theorem. I am desperately trying to work out an example of index theorem however almost all the ...
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Definition of differential operators in Heat Kernels and Dirac Operators

From page $64$ of Heat Kernels and Dirac Operators: Let $E$ be a vector bundle over $M$. The filtered algebra of differential operators on $E$, denoted by $D(M, E)$, is the subalgebra of $\mathrm{End}...
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What does this differential form mean?

On a comment to my this answer, there is a comment where this expression is written: $$\frac{1}{n^n} \left(p\frac{{\rm d}}{{\rm d}p}\right)^k \left(q\frac{{\rm d}}{{\rm d}q}\right)^{n-k} (p+q)^n \...
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derivative in two variables.

Suppose we are doing two-variable calculus in $x,y$. Let $t=x+y$, and I would like to know how to calculate $\partial/\partial t$. We know that $\partial t/\partial x=\partial t/\partial y = 1$. Using ...
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Interesting link between a Maclurin Series and an Integral Operator

Recently I came across a rather interesting equivalence to finding the Maclurin Expansion of $e^x$ . The Solution goes as follows; Let $\mathbb{D^{-1}_{m}}$ be the antiderivative operator , where $m\...
Amy Skinner's user avatar
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Map between rings of invariant differential operators

Suppose we have a reductive group $H$ and a representation $V$. Let $G$ be a group containing $H$ as a closed subgroup and let $W=G\times_H V$. The rings of differential operators $D(V)$ and $D(W)$ ...
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Functional Analysis - Finding the eigenvalues and corresponding eigenfunctions of a linear differential operator

I've been on this question for ages now, and eventually I came to this solution, however I'm very unsure on if I'm correct or if my approach is right. Question: Let $\hat{H}=\frac{d^2}{dx^2}-2\frac{d}...
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Einstein notation and differential operators.

So I am dealing with the differential operator $\mathbf{D} = \mathbf{r} \times \boldsymbol{\nabla}$ where $r = x_i \mathbf{e}_i$. We then introduce $(\mathbf{D}f)_i$, which can be expressed as $\...
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Exponential map of an "imaginary" differential operator acting on a real valued function

The shift operator $T^{t}$ (where $t \in \mathbf{R}$ ) takes a function $f$ on $\mathbf{R}$ to its translation $f_{t}$, $T^{t} f(x)=f_{t}(x)=f(x+t)$. A practical operational calculus representation of ...
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dd^c of log of a meromorphic function [closed]

In Atsushi Moriwaki's Arakelov Geometry, page 100, in a discussion about Greens functions, the proof of proposition 4.13 claims that $$dd^c(\log|\phi|)=0$$ for a non-zero rational function $\phi$ on a ...
wei's user avatar
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Proving Lagrange's Identity with Differential Operators

Consider the second order linear differential operators $L$ and $\bar{L}$ given by $$ L = p(x) \frac{d^2}{dx^2} + q(x) \frac{d}{dx} + r(x), \ \ \bar{L} = \frac{d^2}{dx^2}p(x) - \frac{d}{dx} q(x) + r(...
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Convergence of Lie Series

In the answer to this question the Lie Series result $$ \boxed{ \; e^{t \phi(x) \frac{d}{dx}} f(x) = f\left(e^{t \phi(x) \frac{d}{dx}} x\right) \; } $$ was mentioned. Now for $f(x)$ analytic and ...
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How to compute exterior derivative of generalized angular form in $\mathbb R^n$

On $\mathbb R^n$, with $m\in\mathbb Z$, I want to calculate the exterior derivative of the angular form \begin{align} w=\sum_{i=1}^n(-1)^{i-1}\frac{x_i}{\|x\|^m}dx_1\cdots dx_{i-1}\hspace{0.03cm}\...
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Performing Coordinate Change for For PDE Using Differential Operator Form

so lately I've been working with The PDE: $2u_{xx}+3u_{yy}+2\sqrt{2}u_{xy}=0$, I know so far that this can be written as $\begin{pmatrix}\partial_{x}&\partial_{y}\end{pmatrix} \begin{pmatrix}2&...
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Functional Application of the Differential Operator: Can the Order of Differentiation be a Function?

I've been contemplating the traditional differential operator ( D ) used in calculus, and I'm interested in a potentially broader application. Instead of having a fixed real or fractional order for ...
Wade Hunter's user avatar
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Where does the gradient of a Lipschitz function end up in?

Let $X = \mathsf{Lip(\mathbb{R}^n;\mathbb{R}^n)}$ be the space of (globally, or perhaps locally) Lipschitz functions, say from $\mathbb{R}^n$ to $\mathbb{R}^n$. From Rademacher's theorem we know that $...
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Is a differential operator $\frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx}$ well-known?

I would like to know if the following differential operator $L$ on $(0,\infty)$ is well-known or derived from such one: \begin{align} L := \frac{1}{2}\frac{d^{2}}{dx^{2}} - a x^{b} \frac{d}{dx} \quad (...
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Properties of the inverse Laplacian operator

The inverse of the Laplacian is given by $$(-\Delta)^{-1} u(x) = C \int_{\mathbb{R}^n} u(x-y) \frac{1}{|y|^{n-2}} dy$$ where $n$ is the dimension of $\mathbb{R}^n$. I would like to learn more about ...
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Solution of linear ODE

I am worried about use of differential operator method to solve linear ODE. In this method, we unknowingly treat L(D), Polynomial operator, as a real polynomial in variable x. And, we factorize it ...
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Closed Form for Geometric-like Finite sum of Bell Polynomials

I'm trying to see if there's a nice closed form expression for the following sum: $\sum_{k=0}^{M} \cos(\pi k t) B_k(x)$ where $M \in \mathbb{N}$, $t \in (0,1)$, and $x \in \mathbb{R}^+$. Notation: ...
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Can the units of an arbitrary differential operator be completely arbitrary on a smooth manifold?

On a smooth manifold, given any two derivative operators $\tilde{\nabla}_{a}$ and $\nabla_{a}$, there exists a connection $C^{c}_{ab}$ such that, acting on a metric tensor $g_{bc}$, we have: \begin{...
B K's user avatar
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Why is determinant present in the definition of the metric tensor version of the Laplace-Beltrami operator?

Suppose that our ambient space if $\mathbb{R}^n$. Then the metric tensor version for the Laplace-Beltrami operator is given by $$ \begin{align}\tag{$\ast$} \Delta_{LB}\, u = \dfrac{1}{\sqrt{\left\...
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The Principal Symbol- Why are those Two Definitions Equivalent?

This discussion is taken from P. Topping's "Lectures on the Ricci Flow". The author defined the differential operator $L:C^\infty (M) \to C^\infty (M)$ as: $$ L(u) = a_{ij}\partial_i \...
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Am I allowed to add a total divergence inside an integral with no boundaries?

Let's say that I am trying to find the solution to a complicated linear PDE on an infinite 3D domain for a scalar field $\Phi$. $$ \mathcal{L}_1[\Phi]=0 $$ I can integrate over the volume on both ...
space-guy's user avatar
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How does the canonical commutator works on differential operators?

I'm having trouble understanding how the canonical commutator works on differential operators: in this article, page 2, they define the multiplication-by-$x$ operator, that is a linear operator acting ...
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Integral representation for the operator $\frac{\log{\Delta}}{\Delta}$ in two dimensions

Recently, I have stumbled upon the following non-local operator in $\mathbb{R}^{2}$ $$ \frac{\log{\Delta}}{\Delta} \ , $$ where $\Delta=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^...
E. Marc.'s user avatar
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Willmore extension theorem for tensor derivations

I came across the following theorem in Foundations of mechanics by Ralph Abraham   Let $M$ be a smooth manifold. Suppose for each open $U\subset M $, we have maps $E_U: C^\infty(U) \to C^\infty(U)$ ...
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Understanding $a\vec b * \nabla c$ [closed]

I have some difficulties when working with nabla-operators and I might need advice. Is it correct to say that $$a\vec b \cdot \nabla c = a\nabla \cdot (\vec bc) + ac\nabla \cdot \vec b~~~?$$ If so, ...
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1 answer
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Induced differential operator in long exact sequence of cohomology groups of differential complexes is well defined

In Differential Forms in Algebraic Topology, Bott & Tu they claim that given a short exact sequence of differential complexes: $$0\rightarrow A \overset{f} \rightarrow B \overset{g} \rightarrow C \...
Ofek Aman's user avatar
3 votes
2 answers
107 views

Differential Operator squared

If I have the differential operator $$L = \dfrac{d}{dx} + c v(x)$$ what will be equal to $L^2$? To $L^2 v(x)?$ What is the meaning of $L^2$? What are my issues to compute it? In my task I have the ...
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Is it always okay to substitute $a^2$ in place of $D^2$ in the inverse differential operator of cosh ax?

In class, we were told that it was okay to make the following substitution while trying to solve a particular integral of a nonhomogeneous linear ODE, if $f(D)$ contains only even powers of $D$, and $...
keska_learning's user avatar
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Can the eigenvalue of an differential operator be a function itself?

My issue is this: Let's say I have the operator $\hat{X} = \frac{d}{dx}$ and I want to find complex-valued functions $f:\mathbb{C}\rightarrow\mathbb{C}$ which satisfy $\hat{X} f = \lambda f,$ with $\...
Space junk's user avatar
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1 answer
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What does $e^{\cos(D)}$ do?

We know for example, $$e^D f(x) = f(x+1)$$ and for example, $$e^{-D^2} x^n = H_n(x/2)$$ But what if the exponent is also an exponential, like $$e^{-\cos(D)}$$ or $$e^{-e^D}\ \ \ ?$$ Do we at least ...
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Why does this substitution give an incomplete particular integral solution for this linear ODE?

Trying to solve the following linear nonhomogeneous ODE $$ (D + 2)^2y = \cosh 2x $$ We can easily find the complementary function $$ y_c = c_1 e^{-2x} + c_2 {x} e^{-2x} $$ Using the inverse ...
keska_learning's user avatar
1 vote
1 answer
103 views

Finding characteristic curves for second order differential linear operator.

I've been reading Shubin's "Invitation to partial differential equations" and I've recently struck out on section $1.6$. We start with a differential operator of the form $$A = a\frac{\...
Lucas Giraldi's user avatar
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Functional determinant inconsistency

While trying to compute functional determinants, I faced an inconsistency, which I can exemplify by defining the following matrix $$ M = \begin{pmatrix} i \frac{d}{dt} + t & 0 \\\ 0 & i \frac{...
Adrien Martina's user avatar
2 votes
2 answers
107 views

Kernel and Differential Operator

I hope your day is great so far. I have a question. I am given a Cauchy-Euler ODE $$\big((x+2)^3\mathbf{D}^3+4(x+2)^2\mathbf{D}^2+3(x+2)\mathbf{D}+1\big)y=\frac{1}{x+2},$$ where $\mathbf{D}^n=d^n/dx^n$...
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2 answers
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Norm of the gradient of a function between Riemannian manifolds.

Let $(M, g), (N, h)$ be two Riemannian manifolds and $u: M \to N$ a smooth function. I would like to know how to show that, for $x \in M$, $$|\nabla u|^2(x) = g^{\alpha \beta}(x)h_{ij}(u(x)) \frac{\...
Falcon's user avatar
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When considering matrices (or differential operators), $A > B \implies \exp(A) > \exp (B)$?

First, I would like to ask what is the conventionnal definition of $ A > B $ for matrices (or differential operators in my case of interest). I guess that a natural definition would be that $A-B$ ...
Adrien Martina's user avatar

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