Questions tagged [derivations]
38 questions
17
votes
2 answers
A non-trivial derivation on $C^{k}(\mathbb{R})$ for $k\geqslant 1$?
Recall that a derivation on a commutative algebra $A$ is a linear operator $D:A\to A$ which satisfies the Leibniz rule for products $D(fg)=fDg+gDf$. The standard differentiation $f\mapsto f'$ is certainly a derivation on $C^{\infty }(\mathbb{R})$…
Udo Zerwas
- 569
6
votes
1 answer
Soft Question - Generalizations of the Derivative
This is a soft question. I'm asking for any interesting and rather unknown generlizations of the derivative. I know it is generalized through derivations which are functions $\delta$ satisfying
$$\delta(uv) = v\delta(u) + u\delta(v)$$
Some of them…
Mako
- 718
6
votes
0 answers
Is there a combinatorial interpretation of the arithmetic derivative?
The arithmetic derivative is a derivation on $\mathbb{Z}$ that is $1$ for all prime numbers. On positive integers other than 1, this always returns a positive integer. My question is, is there a way to interpret the arithmetic derivative as…
Darryl Johnson
- 115
5
votes
1 answer
Existence of $\mathbb{N}$-grading compatible with LNDs.
Let $B$ be a finitely generated integral $\mathbb{C}$-domain. Let $\partial:B\to B$ be a LND, locally nilpotent derivation, i.e. a $\mathbb{C}$-linear map satisfying
Leibniz rule: $\partial(fg)=f\partial(g)+\partial(f)g$ for $f,g\in B$;
locally…
Yikun Qiao
- 1,529
4
votes
1 answer
On a Universal Property for the Tangent Space.
A time ago, I was asking if there exists an universal property of the tangent space and what it says about any construction of it. I've found the definition maded in Tammo Dieck's book of Algebraic Topology: A tangent space at $p$ is a vector space…
Pauli
- 867
3
votes
1 answer
Are there non-trivial $\mathbb{Z}$-linear derivations over the real numbers?
Given the algebraic definition of a $\mathbb{Z}$-linear derivation over a commutative ring
$D:R\to R$ with $D(a+b)=D(a)+D(b)$ and $D(a \cdot b)=D(a) \cdot b+a \cdot D(b),$
there is always the trivial derivation where $\forall_{a\in R} \,D(a)=0$.
I…
Gyro Gearloose
- 1,384
3
votes
0 answers
Tangent sheaf and Tangent space
A few days ago, I came across the notion of the tangent sheaf of an affine $k$-scheme $X = \operatorname{Spec} A $, with $k$ an arbitrary field. The natural question that arose was: what is the relationship between this and the tangent…
3
votes
2 answers
How to derive an answer from Implicit Differentiation to another answer?
When we find $dy \over dx$ of the equation ${1 \over x} + {1\over y} = x - y$,
we can differentiate both sides to obtain:
${dy \over dx} = {y^2(x^2 + 1)\over x^2(y^2-1)}$ ...(1)
On the other hand, we can first transform the equation into $y + x =…
user528789
2
votes
2 answers
Confusion on Notations of Partial Derivatives on Manifolds
I'm confused with the notations of partial derivative on manifolds in Tu's An Introduction to Manifolds..
Just to make clear the notations I'm using, what I've known and which part I'm confusing, please let me to restate the question formally. Let…
TheHan6edMan
- 21
2
votes
1 answer
How was this $y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0$ power series recurrence derived?
In György Steinbrecher’s and William Shaw’s Quantile mechanics $(47)$ to $(51)$, it can be found that:
$$\begin{aligned}y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0,y(0)=0,y
(0)=1\iff (1-p)y’^2=\frac{1-p}vyy’+yy’’-yy’^2\\\implies…
Тyma Gaidash
- 13,576
2
votes
0 answers
An adjunction involving connections by analogy with derivations
I am trying to figure out whether the concepts of connection and flat connection can be defined in a way analogous to how derivations are defined here by Akhil Mathew.
As is discussed at the link, derivations can be understood using free abelian…
user900250
2
votes
1 answer
Calculating the derivatives in a linear layer in NN
There is the following puzzle that stems from Neural networks:
I have a matrix $\mathbf{Y} = \mathbf{X}\mathbf{W}^{T} + \mathbf{B}$ where $\mathbf{Y} \in \mathbb{R}^{S \times N}$, $\mathbf{B} \in \mathbb{R}^{S \times N}$, $\mathbf{X} \in…
Jose Ramon
- 71
2
votes
1 answer
Very complicated differentiation
I am trying to solve all steps in a economics paper , but after spending two days with the same differentiation Im losing faith. Can someone out there help me? The problem:
Differentiate:
$$
\begin{align}
&R_{pg}(w) =
\left(\frac{r + \mu +…
Johan
- 23
1
vote
1 answer
Canonical isomorphism between tangent space and translation vector space of affine space
I've a question about the following. Take an affine space $(E,V)$ and consider the tangent space $T_aE$ at a point $a$.
From John Lee book "Introduction to smooth manifolds" chapter 3 there exist a canonical isomorphism between the "geometric…
CarloC
- 125
- 6
1
vote
1 answer
Derivation of an epitrochoid
I was working on an assignment and had a good idea to have a water jet model an epitrochoid to create a fountain. While the overall idea was approved by my teachers, I was told that I need to show a proof of the derivation of the parametric…
