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.
89 questions
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1
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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 ...
- 329
4
votes
0
answers
178
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Question about basis of $\text{Der}_{k}(k[X])$
Let $k[X] = k[x_1,\ldots, x_n]$ be the polynomial ring over a field of characteristic zero.
Assume that $(D_1,\ldots, D_n)$ is a $k[X]$-basis of $\text{Der}_k(k[X])$. Suppose that the vector space $\...
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votes
1
answer
345
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Sufficient conditions for $\mathrm{Der}_k(A)$ to be f.g. projective
Let $k$ be a field and $A$ a commutative $k$-algebra. What are sufficient conditions for the module of derivations $\mathrm{Der}_k(A)$ to be finitely generated projective?
I'm looking for conditions ...
- 6,406
2
votes
1
answer
186
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Derivations of differential operators
For a smooth affine variety $\operatorname{Spec} A$ over a ring $R$ we have an algebra of differential operators $\mathcal{D}_A$ (here I mean not the Grothendieck differential operators but PD-ones). ...
- 243
4
votes
1
answer
180
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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:
$$...
- 1,516
11
votes
2
answers
1k
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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 ...
- 546
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0
answers
97
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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$ ...
- 1,269
14
votes
2
answers
748
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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 ...
- 21.8k
4
votes
1
answer
264
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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 ...
- 1,090
2
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1
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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 ...
- 339
1
vote
1
answer
235
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The derivative of a filter with respect to a output signal [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)
$$
where $*$ denotes convolution.The impulse response $w(t)$ may be ...
- 13
3
votes
1
answer
193
views
How to prove monotonicity of such function?
Let $0<a \le 1, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\...
- 395
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votes
1
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Why this function is monotonic?
Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...
- 395
-1
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4
answers
3k
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What's the relation between Lipschitz constant and the determinant of Jacobian matrix? [closed]
For a vector fuction $y=f(x)$ where $x, y$ are vectors. What's the relation between Lipschitz constant and the determinant of Jacobian matrix?
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0
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62
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References: Lie derivations of Full matrix algebra [closed]
I want, if possible a list of references that trait Lie derivations of Full matrix algebras. Other than Lie derivations of generalized matrix algebras
.
Thanks
- 51
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1
answer
273
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Structure of the module of derivations on the space of Holomorphic functions
Maybe this is well-known, maybe not.
Let
$\Omega\subset \mathbb{C}$ be connected open and non-empty.
It can be shown that if
$d\in\mathfrak{Der}(\mathcal{H}(\Omega))$
(i.e. $d$ is a derivation of ...
- 3,738
5
votes
3
answers
762
views
Does there exist another form of the derivative for polynomials?
Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...
- 4,074
2
votes
1
answer
221
views
When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]
Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure
$$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$
- 311
4
votes
1
answer
148
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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 ...
- 5,878
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0
answers
250
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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 set ...
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votes
0
answers
75
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})...
- 21
2
votes
1
answer
143
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 ...
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votes
1
answer
262
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 ...
- 31
17
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2
answers
1k
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
$f^{(0)} = f$, ...
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2
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2
answers
1k
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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 $k=0:\quad g(n,k) = 1$
$...
- 123
16
votes
1
answer
662
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}...
- 4,074
21
votes
4
answers
2k
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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 ...
- 241
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votes
1
answer
5k
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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?
- 343
3
votes
1
answer
142
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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 ...
user111524
2
votes
0
answers
77
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 ${\rm Aut}_{L_1}(L)$ ...
- 81
1
vote
0
answers
189
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[...
user111524
1
vote
2
answers
435
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 ...
- 159
2
votes
0
answers
283
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, $v_i\in V$; $O$...
- 21
8
votes
1
answer
826
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 ...
- 2,130
11
votes
1
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334
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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,...
- 630
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0
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112
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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 ...
- 5,878
2
votes
1
answer
405
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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 $u$ ...
- 7,127
0
votes
1
answer
379
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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
410
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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 ...
- 5,878