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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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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Example of a differentiable function optimization where derivative free methods are used
While preparing a workshop on the derivative free methods, and fminsearch in MATLAB, I found an example function where fminsearch converges better and in less iterations than fmincon with calculated ...
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Prove or disprove: A differentiable function $f$ is always non-negative with this condition
I want to prove that a differentiable function $f: [0,\infty) \rightarrow \mathbb{R} $, which satisfies the following condition is always non-negative:
Assume $f(0)=0$ and whenever $f(a)=0$, then $f'(...
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A question on fractional derivatives
I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.
I just wanted to ask if there is a notion of ...
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What is the derivative of $1/g$ in a differential semiring?
Let $(S,+,\cdot)$ be a semiring; a derivation on $S$ is a map $\partial : S \to S$ that is linear and Leibniz, in the sense that
It is a semigroup homomorphismm with respect to $+$;
$\partial(a\cdot ...
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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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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 ...
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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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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:
$$...
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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 ...
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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$ ...
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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 ...
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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 ...
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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 ...
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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}...
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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,...
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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,\...
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Why this function is monotonic?
Let $a> 0, \alpha<0$ and $\beta>0$. How to prove that the function:
$$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \...
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References: Lie derivations of Full matrix algebra [closed]
I want, if possible a list of references that trait Lie derivations of Full matrix algebras. Other than Lie derivations of generalized matrix algebras
.
Thanks
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Structure of the module of derivations on the space of Holomorphic functions
Maybe this is well-known, maybe not.
Let
$\Omega\subset \mathbb{C}$ be connected open and non-empty.
It can be shown that if
$d\in\mathfrak{Der}(\mathcal{H}(\Omega))$
(i.e. $d$ is a derivation of ...
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Does there exist another form of the derivative for polynomials?
Let $F : \mathbb{R}[X] \rightarrow \mathbb R[X]$ be a linear map and let $H \in \mathbb{R}[u,x,y,z]$ be a polynomial. Suppose that
$$ F(P \cdot Q) = H(F(P),F(Q),P,Q)$$
for all $P, Q \in \mathbb{R}[X]...
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1
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When $\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)$? [closed]
Let $\phi:G\rightarrow H$ be a morphism of (linear or not?) algebraic groups. What are, in general, the conditions to assure
$$\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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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})...
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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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"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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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 $k=0:\quad g(n,k) = 1$
$...
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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?
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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 ...
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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 ${\rm Aut}_{L_1}(L)$ ...
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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[...
CommunityBot
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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 ...
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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 ...
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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 ...
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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 $u$ ...
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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 = \...
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