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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Locally nilpotent derivations and triangularizability
If $ k $ is a field of characteristic zero and $ \delta \in T_{\mathbb{A}^{n}_{k}/k} $, then $ \delta $ is triangular if $ \delta = \sum_{i=2}^{n} f_{i}(x_{1},\dots,x_{i-1}) \frac{\partial}{\partial ...
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First derivative of $f(A) = \frac{1}{\lambda_{\min}(A)}$ for perturbed matrix
I am working with the matrix function
$$
f(A) = \frac{1}{\lambda_{\min}(A)},
$$ where $A \in \mathbb{R}^{n \times n}$ is a positive definite matrix and $\lambda_{\min}(A)$ is its smallest eigenvalue. ...
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Counter example for Hadamard Differentiability
I am having a hard time while trying to fully understand Hadamard differentiability.
I use the following definition taken from a German source ( Martin Brokate, "Konvexe Analysis und ...
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Derivations and central extensions of some infinite dimensional simple Lie algebras in characteristic zero
Let $F$ be a field of characteristic zero and consider the polynomial algebra $A=F[x_1,x_2,\ldots,x_n]$ in $n$ indeterminates over $F$. I recall that the derivations of $A$ form a simple Lie algebra $...
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Derivate involving Bessel function of second type
Let.
$$f := (x, y) \mapsto \text{BesselK}(1, c \cdot (a - b \cdot (x + y))) \cdot \exp(c \cdot b \cdot (y - x))$$
Is there a close formula for this $$\frac{\partial^{m+n}}{\partial y^m \partial x^n} f(...
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Why should we study derivations of algebras?
Some authors have written that derivation of an algebra is an important tools for studying its structure. Could you give me a specific example of how a derivation gives insight into an algebra's ...
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$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$
Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:
$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right).$$
I am struggling to find ...
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Some specific case of the abc-conjecture related to the Mason-Stothers proof of Serge Lang for polynomials?
I stumbled upon a version of the Mason-Stothers theorem, which can be modified to make a similar statement for natural numbers, which I think I can prove here. It is not the abc-conjecture, but it has ...
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Lie algebra cohomology of formal non-commutative vector fields
Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
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Hausdorff dimension of the curve of a continuous nowhere differentiable function
It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the ...
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Taylor expansion theorem for Gateaux differentiable functions
I am having a hard time studying Gateaux derivatives (see https://en.wikipedia.org/wiki/Gateaux_derivative), it seems that every author mentions the concept but only as a cliffhanger to study Fréchet ...
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Are the two notions of free $\mathbb{G}_a$-actions equivalent?
Consider a finitely generated integral $\mathbb{C}$-domain $B$. An algebraic $\mathbb{G}_a$-action on $X:=\mathrm{Spec}(\mathcal{O}(X))$ is equivalent to a locally nilpotent $\mathbb{C}$-derivation
$$\...
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Differentiation of a norm
I am interested in determining whether the function $F:L^{\infty}(0,\infty;L^{\infty}(0,1)) \to \mathbb R$ defined by
$$u \mapsto F(u)=\int_0^\infty \int_0^1 u(x,t)^2 \ \mathrm dx \, \mathrm dt $$
is ...
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Trying to decode a module functor
This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten.
Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...
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Hochschild cohomology of a group algebra
Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
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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). ...
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Subdifferential of a convex function admits a continuous selection
Let $F$ be a continuous convex function on $\mathbb{R}^n$.
If the subdifferential $\partial F(x)$ of $F(x)$ admits a continuous selection, for every $x \in \mathbb{R}^n$, does it mean that $F$ is ...
CommunityBot
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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 ...
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Applications of the theory of derivators to constructing cone functors
One of main reasons that the theory of derivators was introduced is to fix the non-functoriality of the cone construction of triangulated categories. I know that today derivator theory is broad and ...
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Adjoint sensitivity analysis for a cost functional under an ODE constraint
I am trying to recover the result given by equation 10 in the article here. I am unable to get rid of the integral, any help would be much appreciated. To keep the description as self contained as ...
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Derivative in Sobolev space extended by zero
Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero.
How to find $J'(u)$ for
$$
J(u)= \int\limits_0^l |u(x)|^2\operatorname{d\!}x\;??
$$
In $L_2$ it's easy:
$$
J'(u) = \left(\...
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What is the relationship between higher-order derivations (in the sense of Hasse-Schmidt) and differential operators?
Let $A$ and $B$ be $R$-algebras. A Hasse-Schmidt $m$-derivation $D : A \to B$ is a tuple $(D_0, D_1, \dots, D_m)$ of $R$-linear maps $A \to B$ satisfying the generalized Leibniz law,
$$ D_k(xy) = \...
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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 ...
CommunityBot
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A zoo of derivations
Recall that given a $k$-algebra $A$, a derivation on $A$ is a $k$-linear morphism $d:A\to A$ such that $$d(ab)=d(a)b+ad(b).$$
The use of derivations is of paramount importance in mathematics. I think ...
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Confusion in notation of representation of Bastiani derivative
In the paper "Properties of field functionals and characterization of local functionals" at page 5 the Authors give the following definitions
Definition II.2. Let $U$ be an open subset of a ...
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Differential of tensor product of maps
Let $G$ be a Poisson-Lie group with Poisson bivector field $\pi.$ Let $\pi^{R}$ be the trivialization of $\pi$ with respect to the right translations i.e. $$\pi^{R} (g) = (d_{g} R_{g^{-1}} \otimes d_{...
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Why Gateaux derivative is a distribution?
Thanks to Jan Bohr answer and comment I edited this question.
Let $E$ be a vector bundle , $E^*$ the dual bundle and $D$ a density bundle. Denote by $\Gamma(E)$ the space of section of the bundle $E$....
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Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?
Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles.
$\...
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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 ...
CommunityBot
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Gradient of QZ decomposition
Let $A$ and $B$ be an $m \times n$ matrix of rank $ k_1 \le \min(m,n) $ and $ k_2 \le \min(m,n) $. Then the QZ decomposition or the generalized Schur decomposition is $A = USV^T$ and $B = UTV^T $, ...
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Jacobi formula for matrices: variations
Jacobi’s formula says: $\frac{d}{dt}\text{det}(A(t))=\text{det}(A(t)) \cdot \text{tr}(\text{Ad}(A(t))\cdot\frac{d}{dt}(A(t))$.
Exists maybe a variation of the Jacobi’s formula where $\text{det}(\frac{...
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Jacobian-like conjecture about the derivations of a polynomial algebra
Let $A = k[x_1,\ldots, x_n]$ be a polynomial algebra over a field of characteristic $p$.
Let $Der_k(A)$ denote the Lie algebra of derivations of $A$.
As we know, the Jacobian conjecture provides a ...
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Generalized commutator
A well-known generalization of the commutator for operators is the so-called q-commutator defined as
$$[A,B]_q=AB-qBA.$$
I was wondering if the case where $q$ is not a number but other operator has ...
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Finding non-inner derivations of simple $\mathbb Q$-algebras
What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when ...
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Subgradient of a convex integral
I have an integral to minimize that writes like $$F: \mathbb R^d \to \mathbb R: \theta \mapsto \int_{[0,1]^d} f(\langle x,\theta\rangle) dx$$.
The function $f$ is a convex function, which makes $F$ a ...
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A characterization of the integral
Let $I(f)$ be an endomorphism of the smooth functions with zero value in zero such that:
$$\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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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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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 ...
CommunityBot
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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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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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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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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, $...
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Proving convexity of total distance between two parties with one meeting point [closed]
Apologies if I´m not using the correct mathematical wording or notation. I´ll try my best in the following problem
Suppose you have a coordinate system and 5 coordinates ($A1$, $A2$, $B1$, $B2$, $M_{x,...
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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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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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1
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Derivatives and exponential derivatives quotient operators on two variables
I consider for example the following function of two variables given by
$$f(x,y)=\sum_{n=0}^{+\infty}\frac{\delta_{x}^{-m} \exp(\delta_{x})}{\delta_{y}^{-m}\exp(\delta_{y})}\left(\frac{x}{y}\right)^{n}...
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Global sections of canonical line bundle on projective curve with everywhere vanishing derivative
Let $k$ be an algebraically closed field of positive characteristic $p$, $C$ be a curve (projective, non-singular, connected) of genus $g\geq 2$ over $k$ and $\omega \in H^0(C, \Omega_C)$ be a regular ...
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What's the relation between Lipschitz constant and the determinant of Jacobian matrix? [closed]
For a vector fuction $y=f(x)$ where $x, y$ are vectors. What's the relation between Lipschitz constant and the determinant of Jacobian matrix?
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2
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On solvability of equation $D(x)=1$ where $D:A\to A$ is a bounded outer derivation on a $C^*$ algebra
Let $A$ be a unital $C^*$ algebra. Assume that $D:A\to A$ is a bounded derivation.
Can one say that $1$ can not be in the image of $D$?
If the answer is no:
What is a counter example? What kind of $...
- 114k
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Derivation in Sobolev space [closed]
Let $f\in W^{1,\infty}(]0,T[)$ $(0<T\le\infty)$ such that
$f(x)>0$ a.e. $x\in\mathopen]0,T[$ and let
$$g(x)=e^{-\int_0^x \frac{ds}{f(s)}}$$
Formally $g' = -\frac{1}{f}g$.
How can I justify this ...
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