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20 views

Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
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2answers
167 views

Differentation Operator

having trouble completing the proof for this question Let $D:\mathbb{R}[X] \to \mathbb{R}[X]$ be the differentiation operator $D(f(X))=f'(X) .$ Prove that $e^{tD}(f(X)) = f(X+t)$ for $t \in ...
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0answers
116 views

Eigenvalues of differential operator

If $L : C^2[a,b] \rightarrow C^0[a,b] $ is s.t. $L y(t) = \ddot y(t) +p \dot y + q y(t) $ and $L$ is invertible then $L^{-1}$ has at most countable eigenvalues and they accumulate in $0$. Why ...
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1answer
136 views

Prove that D (the differential operator) maps V (a vector space) into V.

I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space. Here's some of the ...
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2answers
291 views

Transpose of the differential operator

Is the differential operator ${d\over dx}$ antisymmetric? If so, what does it even mean to take it's transpose? Thank you.
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1answer
233 views

Differentiation operator applying on matrix

I need to apply a differential operator (nabla) on a matrix. Problem is, that I don't know how to calculate that. Do I treat nabla as a column vector and simply multiply vector with the matrix? Or is ...