# Questions tagged [derivations]

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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 ...
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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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### Question about derivations of fraction field

For each $D\in\text{Der}_k(k(x_1,\ldots, x_n))$ denote by $\text{Nil}(\text{ad}D)$ the set of derivations $U$ of $\text{Der}_k(k(x_1,\ldots, x_n))$ such that $0=(\text{ad}D)^N(U)$ for $N$ large enough....
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### When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]

Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure $$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$
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### First adjoint cohomology space of simple Lie algebras

Let $L$ be a central extension of a simple Lie algebra $\mathfrak{g}$ such that $L=[L,L]$. It is not difficult to see that if $H^1(\mathfrak{g}, \mathfrak{g})=0$ then $H^1(L,L)=0$. In other words, if ...
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### Derivations of algebras graded by a group

Let $A$ be an algebra. A derivation of $A$ is a linear map $d:A \rightarrow A$ such that $d(ab)=d(a)b + a d(b)$ for $a, b \in A$. If $A$ is a $\mathbb{Z}-$graded algebra, where $\mathbb{Z}$ is the ...
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### Prove a $C^{\infty}$ multivariable function is lipchitz via Jacobian matrix [closed]

I would like to prove a real $C^{\infty}$(polynomial) multivariable function $F : (a_1,a_2,...a_n) \rightarrow (b_1,b_2,...b_n)$ is lipchitz of parameter $l$ is it sufficient to prove the norm of ...
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### Derivative with multiple summation operators

I have a defined utility function as Eq.(1), and I am seeking the minimized utility subjects to some constraints. The notation used is as following: \linebreak $V$ is the set of nodes, $v_i\in V$; $O$...
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### Semantics of derivations as derivatives

My understanding of how derivations on commutative rings are like derivatives is that a derivation on $R$ is differentiation with respect to a vector field on $\text{Spec}(R)$. But derivations are ...
Let $L$ be a Lie algebra over a field of characteristic $p>0$ and $D$ a derivation of $L$. For every $x\in L$ denote by $\mathrm{ad} x$ the adjoint map $\mathrm{ad}x: L \rightarrow L, a\mapsto [x,... 0answers 96 views ### Restricted universal extensions and lifting of derivations Let$L$be a perfect Lie algebra. Then it is well-known that$L$has a universal central extension$\hat{L}$and every derivation of$L$can be lifted to a derivation of$\hat{L}$. (See e.g. Section 2 ... 1answer 271 views ### Formal Cauchy-Riemann equations for formal power series without complex analysis Consider the ring$\mathbb{C}[[X,Y]]$and its subring$\mathbb{C}[[X+iY]]$, where$i=\sqrt{-1}$. One can show that$f(X,Y):=u(X,Y)+iv(X,Y)\in \mathbb{C}[[X,Y]]$lies in$\mathbb{C}[[X+iY]]$iff$u$... 1answer 275 views ### The limitation of derivation of modified Bessel function of second kind The final result I draw is related to the integral of modified Bessel function of the second kind. But I can not solve it, and I need a explicit solution Are you willing to help me? Thank all$I = \...
Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized ...