Questions tagged [derivations]

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Validity of PBW theorem for Poisson algebras and extending derivation map

I am studying Poisson algebras. Recently I studied a very interesting paper on validity of PBW theorem for Poisson algebras entitled Poisson enveloping algebras and the Poincaré–Birkhoff–Witt theorem ...
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5 votes
0 answers
109 views

Generalized commutator

A well-known generalization of the commutator for operators is the so-called q-commutator defined as $$[A,B]_q=AB-qBA.$$ I was wondering if the case where $q$ is not a number but other operator has ...
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5 votes
1 answer
122 views

Finding non-inner derivations of simple $\mathbb Q$-algebras

What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)? I'm under the impression that when ...
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1 vote
1 answer
104 views

Hessian matrix of vectorized matrix product

I need to find the Hessian Matrix of $f(X,Y) = C \operatorname{vec} (A X^{-1} Y)$ where $C$ and $A$ are constant matrices and $X$ and $Y$ are the variable matrices. This would be a vector function of ...
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  • 11
2 votes
1 answer
67 views

Subgradient of a convex integral

I have an integral to minimize that writes like $$F: \mathbb R^d \to \mathbb R: \theta \mapsto \int_{[0,1]^d} f(\langle x,\theta\rangle) dx$$. The function $f$ is a convex function, which makes $F$ a ...
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  • 623
1 vote
0 answers
90 views

Simple proof of the equivalence between two definitions of étale

This question shouldn't be too hard to answer, but I'm looking for the most streamlined approach. Let $K$ be a field and let $L$ be a finite dimensional field extension of $K$. I am interested in two ...
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  • 4,293
3 votes
1 answer
97 views

Derivative of an integral of a Gaussian

I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian: $ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
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  • 43
2 votes
0 answers
74 views

Lie derivations of algebra of smooth functions in a symplectic manifold

Let $(M,\omega)$ be a finite-dimensional symplectic manifold. The algebra $C^\infty(M)$ of smooth functions is a Poisson algebra. Derivations $D : C^\infty(M) \to C^\infty(M)$ of the Poisson ...
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1 vote
1 answer
126 views

Partial derivative of the heat kernel

I happen to have the heat kernel on the two-dimensional hyperbolic space and I need to take partial derivatives in order to check that it satisfies the heat equation as expected. The problem is I can ...
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0 votes
1 answer
123 views

Solution of this differential equation [closed]

I wonder if it is possible to solve analytically the following equation $$ \dot{\alpha}_t = -\frac{2}{m} \alpha^2_t + \frac{1}{2m} (\alpha_t - \alpha_t^*)^2 $$ Where $\alpha_t$ is a complex function, $...
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1 vote
0 answers
47 views

Proving convexity of total distance between two parties with one meeting point [closed]

Apologies if I´m not using the correct mathematical wording or notation. I´ll try my best in the following problem Suppose you have a coordinate system and 5 coordinates ($A1$, $A2$, $B1$, $B2$, $M_{x,...
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1 vote
0 answers
57 views

Derivative of a function of ordered variables

Can I differentiate $$(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)^\top(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)$$ with respect to $\pmb{a}$? (I want to minimize the expression with respect to $\pmb{a}$.) Here, $...
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  • 133
1 vote
1 answer
95 views

Derivatives and exponential derivatives quotient operators on two variables

I consider for example the following function of two variables given by $$f(x,y)=\sum_{n=0}^{+\infty}\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}\left(\frac{x}{y}\right)^{n}...
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1 vote
0 answers
228 views

A characterization of the integral

Let $I(f)$ be an endomorphism of the smooth functions with zero value in zero such that: $$\ln[1+I(f)]=I\left(\frac{f}{1+I(f)}\right). $$ Then, does it exist $g$ smooth such that: $$I(f)(x)=\int_0^x f(...
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6 votes
0 answers
229 views

Global sections of canonical line bundle on projective curve with everywhere vanishing derivative

Let $k$ be an algebraically closed field of positive characteristic $p$, $C$ be a curve (projective, non-singular, connected) of genus $g\geq 2$ over $k$ and $\omega \in H^0(C, \Omega_C)$ be a regular ...
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1 vote
2 answers
308 views

On solvability of equation $D(x)=1$ where $D:A\to A$ is a bounded outer derivation on a $C^*$ algebra

Let $A$ be a unital $C^*$ algebra. Assume that $D:A\to A$ is a bounded derivation. Can one say that $1$ can not be in the image of $D$? If the answer is no: What is a counter example? What kind of $...
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1 vote
0 answers
83 views

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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5 votes
1 answer
69 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 ...
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0 votes
1 answer
159 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'(...
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1 vote
1 answer
183 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 ...
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6 votes
3 answers
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 ...
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2 votes
1 answer
243 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 ...
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4 votes
1 answer
185 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 ...
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6 votes
0 answers
177 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 ...
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0 votes
1 answer
184 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 ...
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4 votes
0 answers
166 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 $\...
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3 votes
1 answer
212 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 ...
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1 vote
0 answers
101 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). ...
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  • 213
4 votes
1 answer
153 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: $$...
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  • 1,241
10 votes
2 answers
657 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 ...
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6 votes
0 answers
82 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 $\text{Der } \Bbb A$ ...
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  • 1,129
14 votes
2 answers
686 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 ...
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  • 19k
4 votes
1 answer
182 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 ...
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2 votes
1 answer
324 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 ...
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1 vote
1 answer
195 views

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 ...
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3 votes
1 answer
157 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,\...
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  • 373
-2 votes
1 answer
280 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 \...
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  • 373
-1 votes
4 answers
1k views

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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1 vote
0 answers
60 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
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8 votes
1 answer
204 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 ...
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5 votes
3 answers
741 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]...
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  • 3,011
2 votes
1 answer
195 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 $$\operatorname{Lie}(\ker(\phi))=\ker(d\phi_e)\text{?}$$
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  • 271
4 votes
1 answer
123 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 ...
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3 votes
0 answers
167 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 set ...
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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})...
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  • 21
2 votes
1 answer
119 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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  • 167
3 votes
1 answer
219 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 ...
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  • 31
16 votes
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 votes
2 answers
1k 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 $k=0:\quad g(n,k) = 1$ $...
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16 votes
1 answer
607 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}...
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