# Questions tagged [derivations]

A derivation on a ring 𝑅 is a map 𝐷:𝑅→𝑅 satisfying 𝐷(𝑎+𝑏)=𝐷(𝑎)+𝐷(𝑏) and 𝐷(𝑎𝑏)=𝑎𝐷(𝑏)+𝐷(𝑎)𝑏.

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### First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix

I am working with the matrix function $$f(A) = \frac{1}{\lambda_{\min}(A)},$$ where $A \in \mathbb{R}^{n \times n}$ is a positive definite matrix and $\lambda_{\min}(A)$ is its smallest eigenvalue. ...
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### Counter example for Hadamard Differentiability

I am having a hard time while trying to fully understand Hadamard differentiability. I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
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### Differentiation of a norm

I am interested in determining whether the function $F:L^{\infty}(0,\infty;L^{\infty}(0,1)) \to \mathbb R$ defined by $$u \mapsto F(u)=\int_0^\infty \int_0^1 u(x,t)^2 \ \mathrm dx \, \mathrm dt$$ is ...
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### Trying to decode a module functor

This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten. Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...
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### Hausdorff dimension of the curve of a continuous nowhere differentiable function

It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the ...
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### Hochschild cohomology of a group algebra

Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
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### Subdifferential of a convex function admits a continuous selection

Let $F$ be a continuous convex function on $\mathbb{R}^n$. If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...
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### Applications of the theory of derivators to constructing cone functors

One of main reasons that the theory of derivators was introduced is to fix the non-functoriality of the cone construction of triangulated categories. I know that today derivator theory is broad and ...
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### Adjoint sensitivity analysis for a cost functional under an ODE constraint

I am trying to recover the result given by equation 10 in the article here. I am unable to get rid of the integral, any help would be much appreciated. To keep the description as self contained as ...
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### A zoo of derivations

Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$ The use of derivations is of paramount importance in mathematics. I think ...
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### Confusion in notation of representation of Bastiani derivative

In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions Definition II.2. Let $U$ be an open subset of a ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...