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
3
votes
1
answer
152
views
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 ...
- 912
4
votes
1
answer
199
views
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. ...
- 91
5
votes
1
answer
264
views
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 ...
- 53
9
votes
8
answers
1k
views
$n$-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 ...
- 393
6
votes
1
answer
154
views
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 $...
- 99
3
votes
0
answers
394
views
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 ...
- 3,144
6
votes
2
answers
409
views
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 ...
4
votes
1
answer
186
views
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
$$\...
- 930
5
votes
0
answers
125
views
Lie algebra cohomology of formal non-commutative vector fields
Let $k$ be a field of characteristic $0$ and $A=k\langle\langle x_1,\dotsc,x_n\rangle\rangle$ be a free completed associative algebra. The space of continuous derivations $\mathrm{Der}(A)$ is ...
- 985
2
votes
1
answer
196
views
Hausdorff dimension of the curve of a continuous nowhere differentiable function
It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the ...
- 16.2k
9
votes
1
answer
444
views
Hochschild cohomology of a group algebra
Let $K$ be a field and $G=\pi_1(\Sigma_g)$ the surface group of genus $\geq 2$. I want to know the Hochschild cohomology of the group algebra $A=K[G]$ with coefficients in $A$ and $A\otimes A$, namely,...
- 985
0
votes
1
answer
174
views
Differentiation of a norm
I am interested in determining whether the function $F:L^{\infty}(0,\infty;L^{\infty}(0,1)) \to \mathbb R$ defined by
$$u \mapsto F(u)=\int_0^\infty \int_0^1 u(x,t)^2 \ \mathrm dx \, \mathrm dt $$
is ...
- 55
2
votes
0
answers
152
views
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 ...
- 1,347
3
votes
2
answers
351
views
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 ...
- 33
21
votes
4
answers
2k
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 ...
- 241
4
votes
1
answer
313
views
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$....
3
votes
1
answer
378
views
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.
$\...
- 11.8k
3
votes
0
answers
159
views
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 ...
- 893
1
vote
1
answer
126
views
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 ...
- 55
2
votes
0
answers
199
views
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).$$
The use of derivations is of paramount importance in mathematics. I think ...
0
votes
1
answer
84
views
Derivative in Sobolev space extended by zero
Let $u(x) \in H^1_0$ is a complex function in sobolev space extension by zero.
How to find $J'(u)$ for
$$
J(u)= \int\limits_0^l |u(x)|^2\operatorname{d\!}x\;??
$$
In $L_2$ it's easy:
$$
J'(u) = \left(\...
- 3
7
votes
3
answers
2k
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 ...
user168611
7
votes
0
answers
234
views
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) = \...
- 3,730
14
votes
2
answers
748
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 ...
- 21.8k
5
votes
1
answer
345
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 ...
- 6,406
4
votes
0
answers
148
views
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 ...
- 329
1
vote
1
answer
293
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 ...
- 11
5
votes
1
answer
175
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 ...
- 1,507
17
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$, ...
- 48.9k
11
votes
2
answers
1k
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 ...
- 546
2
votes
0
answers
107
views
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 $, ...
- 61
1
vote
2
answers
313
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 $...
- 366
1
vote
0
answers
42
views
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_{...
- 199
0
votes
1
answer
171
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, $...
0
votes
0
answers
39
views
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 ...
2
votes
0
answers
162
views
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{...
- 83
5
votes
0
answers
164
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 ...
1
vote
1
answer
389
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 ...
- 313
-1
votes
4
answers
3k
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?
- 77
2
votes
1
answer
186
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). ...
- 243
1
vote
0
answers
245
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:
$$I(f)(x)=\int_0^x f(...
- 377
4
votes
1
answer
260
views
Left- (right-) multiplications of an algebra that are derivations
Let us say that $A$ is a (finite-dimensional) algebra over a field of characteristic zero. We can assume commutativity
but not associativity, if that makes it easier. Indeed, I am mostly interested in ...
- 4,729
1
vote
0
answers
146
views
Simple proof of the equivalence between two definitions of étale
This question shouldn't be too hard to answer, but I'm looking for the most streamlined approach.
Let $K$ be a field and let $L$ be a finite dimensional field extension of $K$. I am interested in two ...
user30211
3
votes
1
answer
293
views
Derivative of an integral of a Gaussian
I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian:
$ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
- 43
1
vote
1
answer
107
views
Derivatives and exponential derivatives quotient operators on two variables
I consider for example the following function of two variables given by
$$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}...
- 159
2
votes
1
answer
297
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 ...
- 13.6k
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]...
- 4,073
2
votes
1
answer
108
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 ...
- 686
6
votes
0
answers
490
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 ...
- 445
0
votes
1
answer
335
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'(...
- 11