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 ...
5
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1answer
64 views

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 ...
0
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1answer
134 views

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'(...
1
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1answer
87 views

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 ...
6
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3answers
1k views

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 ...
2
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1answer
224 views

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 ...
4
votes
1answer
167 views

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 ...
6
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0answers
117 views

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 ...
2
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0answers
54 views

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....
0
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0answers
35 views

Using Taylor formula to prove Integral representation of higher order derivatives

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be even smooth function. Let $w(x)=f(\sqrt x)$ for $x>0$. I found the following integral formula $$w^{(k)}(x^2)=\frac{(2x)^{-2k+1}}{(k-1)!} \int_0^x (x^2-...
0
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1answer
181 views

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 ...
4
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0answers
164 views

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 $\...
2
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1answer
158 views

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 ...
1
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0answers
89 views

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). ...
4
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1answer
128 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: $$...
10
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2answers
445 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 ...
6
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0answers
78 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
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2answers
650 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
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1answer
150 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
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1answer
266 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
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1answer
165 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
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1answer
154 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
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1answer
276 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
4answers
635 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
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0answers
59 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
7
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1answer
155 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 ...
5
votes
3answers
722 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
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1answer
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
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1answer
117 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
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0answers
91 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
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0answers
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
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1answer
104 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 ...
2
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1answer
164 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
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2answers
961 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
2answers
718 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
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1answer
567 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}...
12
votes
3answers
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
1answer
2k 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
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1answer
104 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
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0answers
59 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
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0answers
138 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
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2answers
206 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
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0answers
208 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
1answer
619 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
1answer
281 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
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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 ...
2
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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 $u$ ...
0
votes
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 = \...
3
votes
1answer
273 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 ...