# Questions tagged [derivations]

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37
questions

**4**

votes

**1**answer

108 views

### Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor:
$$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$
left-adjoint to the (forgetful) embedding:
$$...

**-4**

votes

**0**answers

51 views

### partial differential inequality [duplicate]

I want to prove that if $f\leq l$ and $\lim_{a\to \infty}\frac{f(a)}{l(a)}=1$ (we can also suppose that $\lim_{a\to \infty}f(a)=\lim_{a\to \infty}l(a)=1$) where $f$ is a positive continuous bounded ...

**9**

votes

**2**answers

253 views

### The relation between t-structures and derived category

Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...

**5**

votes

**0**answers

74 views

### Kac-Moody Lie algebra as derivations of associative algebras

The set of derivations of an algebra $\Bbb A$ forms a Lie algebra. This is one aspect of why Lie algebras are interesting. When $\Bbb A$ is polynomial algebra in $n$ variable then $\text{Der } \Bbb A$ ...

**13**

votes

**2**answers

619 views

### Does any derivation of commutative algebra preserve its nil-radical?

Given a commutative associative unital algebra over a field of characteristic zero.
Is it true that any derivation of it preseves its nil-radical?
More explicitly, let $D$ be a derivation of an ...

**4**

votes

**1**answer

120 views

### Characterizing the Haagerup property of finite von Neumann algebras via unbounded derivations

A correspondence $_{N} H_{N}$ is a Hilbert space with two normal commuting left/right representations of $N$ in $B(H)$. The following characterization of property (T) for finite von Neumann algebras ...

**1**

vote

**1**answer

213 views

### Katz's paper on $p$-curvature – help with proof understanding

I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures.
I don't understand the ...

**1**

vote

**1**answer

145 views

### The derivative of a filter with respect to a output singal [closed]

I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e.
$$
d(t)*w(t)=p(t)
$$
where $*$ denotes convolution.The impulse response $w(t)$ may be ...

**3**

votes

**1**answer

151 views

### How to prove monotonicity of such function?

Let $0<a \le 1, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\...

**-2**

votes

**1**answer

270 views

### Why this function is monotonic?

Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...

**-2**

votes

**3**answers

205 views

### What's the relation between Lipschitz constant and the determinant of Jacobian matrix?

For a vector fuction $y=f(x)$ where $x, y$ are vectors. What's the relation between Lipschitz constant and the determinant of Jacobian matrix?

**1**

vote

**0**answers

58 views

### References: Lie derivations of Full matrix algebra [closed]

I want, if possible a list of references that trait Lie derivations of Full matrix algebras. Other than Lie derivations of generalized matrix algebras
.
Thanks

**0**

votes

**0**answers

47 views

### Generalized derivations in the sense of Brešar and Gölbaşı

When I search about the definition of generalised derivations, I found there are two definitions. The first definition by Brešar On the distance of the composition of two derivations to the ...

**7**

votes

**1**answer

131 views

### Structure of the module of derivations on the space of Holomorphic functions

Maybe this is well-known, maybe not.
Let
$\Omega\subset \mathbb{C}$ be connected open and non-empty.
It can be shown that if
$d\in\mathfrak{Der}(\mathcal{H}(\Omega))$
(i.e. $d$ is a derivation of ...

**4**

votes

**3**answers

703 views

### Does there exist another form of the derivative for polynomials?

Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...

**2**

votes

**1**answer

191 views

### 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
$$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$

**4**

votes

**1**answer

109 views

### 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 ...

**2**

votes

**0**answers

74 views

### 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 ...

**2**

votes

**0**answers

72 views

### How can I prove that the following function is increasing according to x1?

Suppose that
$0 \le {X_1} < {X_2} < {X_3}$
.
How is it possible to prove the following function is increasing based on
${X_1}$
in the range of
$0 \le {X_1} < {X_2}$ ?
$f({X_1},{X_2},{X_3})...

**2**

votes

**1**answer

98 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 ...

**1**

vote

**1**answer

110 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 ...

**16**

votes

**2**answers

849 views

### “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
$f^{(0)} = f$, ...

**2**

votes

**2**answers

471 views

### Find formula for recurrence relation with two function and two variables

$f(n,k) = 2g(n-2,k-1)+f(n-1,k)$
$g(n,k) = g(n-1,k-1)+f(n,k)$
when $n\le0$ or $k\le0: \quad f(n,k) = 0$
when $n < k:\quad f(n,k) = 0$
when $n-k<-1:\quad g(n,k) = 0$
when $k=0:\quad g(n,k) = 1$
$...

**16**

votes

**1**answer

542 views

### Does every real function have this weak derivation property?

After this question : Does every real function have this weak continuity property?
Natrualy there are an other (more difficult) :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...

**11**

votes

**3**answers

1k views

### Why should we study derivations of algebras?

Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...

**2**

votes

**1**answer

1k views

### Partial derivatives of spherical harmonics

Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic?

**3**

votes

**1**answer

93 views

### Locally nilpotent derivation on $A[X,Y]$ whose kernel is $A$; where $A$ is an affine $k$ domain, $char k=0$

Let $k$ be a field of characteristic zero and $A$ be a $k$-algebra. A derivation on $A$ is a $k$-linear map $D: A \to A$ such that $D(ab)=aD(b)+bD(a), \forall a,b \in A$. A derivation is called ...

**2**

votes

**0**answers

58 views

### Lie algebra of derivations for a transcendental field extension and intersection fields

Suppose that $L$ is a finite Galois extension of the field $K$.
If $L_1$ and $L_2$ are subfields of $L$ containing $K$ then $L_1\cap L_2=L^H$
where $H$ is the group generated by ${\rm Aut}_{L_1}(L)$ ...

**1**

vote

**0**answers

131 views

### Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...

**1**

vote

**2**answers

156 views

### 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 ...

**2**

votes

**0**answers

184 views

### 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, $v_i\in V$; $O$...

**6**

votes

**1**answer

560 views

### 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 ...

**10**

votes

**1**answer

268 views

### An identity in Lie algebras over fields of positive characteristic

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,...

**4**

votes

**0**answers

93 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 ...

**2**

votes

**1**answer

253 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 $u$ ...

**0**

votes

**1**answer

267 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 = \...

**3**

votes

**1**answer

242 views

### Derivations of central extensions of simple Lie algebras

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 ...