# Questions tagged [derivations]

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### Validity of PBW theorem for Poisson algebras and extending derivation map

I am studying Poisson algebras. Recently I studied a very interesting paper on validity of PBW theorem for Poisson algebras entitled Poisson enveloping algebras and the Poincaré–Birkhoﬀ–Witt theorem ...
109 views

### Generalized commutator

A well-known generalization of the commutator for operators is the so-called q-commutator defined as $$[A,B]_q=AB-qBA.$$ I was wondering if the case where $q$ is not a number but other operator has ...
122 views

### Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)? I'm under the impression that when ...
1 vote
104 views

### Hessian matrix of vectorized matrix product

I need to find the Hessian Matrix of $f(X,Y) = C \operatorname{vec} (A X^{-1} Y)$ where $C$ and $A$ are constant matrices and $X$ and $Y$ are the variable matrices. This would be a vector function of ...
67 views

### Subgradient of a convex integral

I have an integral to minimize that writes like $$F: \mathbb R^d \to \mathbb R: \theta \mapsto \int_{[0,1]^d} f(\langle x,\theta\rangle) dx$$. The function $f$ is a convex function, which makes $F$ a ...
1 vote
90 views

### Simple proof of the equivalence between two definitions of étale

This question shouldn't be too hard to answer, but I'm looking for the most streamlined approach. Let $K$ be a field and let $L$ be a finite dimensional field extension of $K$. I am interested in two ...
97 views

### Derivative of an integral of a Gaussian

I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian: $Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx$, then $\frac{dZ}{dK}=\text{??}$ ...
74 views

### Lie derivations of algebra of smooth functions in a symplectic manifold

Let $(M,\omega)$ be a finite-dimensional symplectic manifold. The algebra $C^\infty(M)$ of smooth functions is a Poisson algebra. Derivations $D : C^\infty(M) \to C^\infty(M)$ of the Poisson ...
1 vote
126 views

### Partial derivative of the heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can ...
123 views

1 vote
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1 vote
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### Derivation in Sobolev space [closed]

Let $f\in W^{1,\infty}(]0,T[)$ $(0<T\le\infty)$ such that $f(x)>0$ a.e. $x\in\mathopen]0,T[$ and let $$g(x)=e^{-\int_0^x \frac{ds}{f(s)}}$$ Formally $g' = -\frac{1}{f}g$. How can I justify this ...
69 views

### Reference for a certain derivation on the ring of ordered series over a free monoid

Let $R$ be a (commutative or non-commutative) unital ring, $X$ be a non-empty set, and $R \langle\! \langle X \rangle\! \rangle$ be the ordered series ring (in fact, a ring of formal power series over ...
159 views

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### Left- (right-) multiplications of an algebra that are derivations

Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity but not associativity, if that makes it easier. Indeed, I am mostly interested in ...
177 views

### Derivations for symmetric functions

A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra ...
184 views

### Is it true that $g-t$ is divisible by $f$?

Assume $f\in k[x_1,\ldots, x_n]$ is irreducible. Let for $g\in k[x_1,\ldots, x_n]$, $\partial(g)$ is divisible by $f$ for each derivation $\partial$ with $f\in\ker\partial$. Is it true that $g-t$ is ...
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119 views

### Continuity of the derivations from semisimple Banach algebras

Let $A$ be a Banach algebra and $X$ a Banach $A$-bimodule. It is known that if $A$ is a $C^*$-algebra, then by Ringrose theorem every derivation $D:A\rightarrow X$ is continuous. Also, a famous ...
219 views

### Locally nilpotent derivations on rings with zero divisors

Almost all books that I have found deal with derivation on several types of rings (or algebras) (for instance, commutative, noncommutative, domains, non-domains etc). However, each paper about ...
### "Insanely increasing" $C^\infty$ function with upper bound
Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set $f^{(0)} = f$, ...