# 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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### functions from a higher dimension to lower can be injective? [migrated]

of course not.But how to prove it? assuming f is of class C^1 I tried to use implicit function theorem and some others,, but i couldnt make it.
110 views

### 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 ...
141 views

### 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 ...
1 vote
88 views

### 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 ...
74 views

159 views

### 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 ...
20 views

### About extending maps to semi derivations

Let consider a map $d: X \to P$, where $X$ is a generating set of $P$ and $(P,\cdot, \{,\})$ is a free Poisson algebra. How can we extend $d$ to a semi derivation map from a subalgebra $P$ to $P$? By ...
30 views

### 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 ...
1 vote
34 views

1 vote