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.
99 questions
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Extending the "infinitesimal automorphism" construction of derivations to algebraic geometry
In the context of smooth manifolds $M$, there is a well-known correspondence between the "infinitesimal" versions of certain objects and derivations of the algebra $C^\infty(M, \mathbb{R})$. ...
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Derivation in the sense of distributions
I consider the Bessel kernel $$ K(x) = \frac{1}{(4 \pi)^{\frac{N}{2}}} \int_0^{+ \infty} e^{-t} e^{- \frac{\left|x\right|^2}{4t}} t^{- \frac{N}{2}} dt,\ x \in \mathbb{R}^N, x \neq0. $$ We can assume ...
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Existence of a derivation from a condition on the inner product
I am looking for a proof or reference of the following claim (that I suspect to be true):
Let $D$ be a derivation and $(\cdot,\cdot)$ be an inner product on a pre-Hilbert space $X$. Suppose $g\in X$ ...
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When is $f(x) - g(x)$ strongly convex? ($f$ exponential, $g$ Beta/Gamma-based)
Let $r \in (-1,1)$ be fixed. Consider the two functions:
$$
f(x) := (1 + e^x + e^{-x}) \ln 2 - e^x \ln(1 + r) - e^{-x} \ln(1 - r),
$$
and
$$
g(x) := -\ln B(1 + e^x,\ 1 + e^{-x}) = -\ln \Gamma(\alpha(x)...
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Who first defined vector fields as derivations?
A vector field on a smooth manifold can be defined as a derivation on smooth scalar fields, ie. a linear map $D:C^\infty(M,\mathbb{R})\rightarrow C^\infty(M,\mathbb{R})$ such that $D(ab)=(Da)b+a(Db)$.
...
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How should one generalize Mori's conjecture on uniruledness to log pairs?
A variety (integral, separated scheme of finite type over an algebraically closed field $ k $) $ Z $ is uniruled if there is a dominant, generically finite, rational map $ \psi: \mathbb{P}^{1}_{k} \...
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Derivations relative to a pair of morphisms of $R$-algebras
Given an $R$-algebra $A$, one defines the $R$-module $\mathrm{Der}_R(A,A)$ of derivations of $A$ as
$$\mathrm{Der}_R(A,A)\mathbin{\overset{\small\mathrm{def}}{=}}\left\{D\in\mathrm{Mod}_R(A,A)\ \...
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If an element of $ \operatorname{Der}_{k}(A,A) $ is locally nilpotent on a transcendence base then is it locally nilpotent?
Let $ k $ be a field of characteristic zero and let $ A $ be a nonsingular $ k $-algebra. A derivation $ \gamma \in \operatorname{Der}_{k}(A,A) $ is locally nilpotent if for every $ a \in A $, there ...
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Differential calculus on Frölicher Groups
Is there a general definition of differential for mappings between Frölicher groups?
user522448
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Local criteria for local nilpotency of derivations in $ \operatorname{Der}_{k}(k[X],k[X]) $
If $ \operatorname{Spec}(A) $ is a variety over a field $ k $ of characteristic zero, then there exists an action of $ \mathbb{G}_{a} $ on $ \operatorname{Spec}(A) $ if and only if there is a locally ...
- 1,215
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Locally nilpotent derivations and triangularizability
If $ k $ is a field of characteristic zero and $ \delta \in T_{\mathbb{A}^{n}_{k}/k} $, then $ \delta $ is triangular if $ \delta = \sum_{i=2}^{n} f_{i}(x_{1},\dots,x_{i-1}) \frac{\partial}{\partial ...
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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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Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero
Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $...
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$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$
Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:
$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right).$$
I am struggling to find ...
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