Questions tagged [derivations]
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41 questions
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Derivations of perfect Lie algebras
Assume that a perfect Lie algebra \mathfrak{g} admits an outer derivation. Then does Out(\mathfrak{g}) contain a diagonal derivation? Is it always the case?
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$\exp(\mathrm{Der}(L)) \subseteq \mathrm{Aut}(L)$ implies $\mathrm{Der}(L) \subseteq \mathrm{Lie}(\mathrm{Aut}(L))$
Let $L$ be a (finite-dimensional real) Lie algebra. Let $\mathrm{Aut}(L) \subseteq \mathrm{GL}(L)$ be the set of all Lie algebra automorphisms on $L$, and let $\mathrm{Der}(L) \subseteq \mathfrak{gl}(...
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Why is Lie algebra $[L_{\lambda}, L_{\mu}]$ contained in $L_{\lambda+\mu}$
This is for the 2nd part of question 9.8 in the book "Introduction to Lie Algebras" by Karin Erdmann and Mark J. Wildon.
A generalized eigenspace of derivation $\delta$ is defined as as $L_\...
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Why are kinematic tangent vectors continuous in the convenient vector space topology?
In the following excerpt from Michor's The Convenient Setting of Global Analysis, why is $X_a$ continuous in the topology of $E$? $\def\RR{\mathbb{R}}$ No explanation is given in the text.
28.1. The ...
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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 ...
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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, ...
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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 ...
user528789
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Why don't I see "vector-valued vector fields"?
Let $P\xrightarrow{\pi}M$ be a principal $G$-bundle with connection $\omega\in \Omega^1(P;\mathfrak{g})$.
When I am studying these things, there are Lie-algebra valued differential forms all over. ...
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Derivation of two equivalent functions yields different derivation result. Why?
$f(x)=x+\frac{2x^3}{3-2x}-x^2-1 \\
g(x)=\frac{x(3-2x)+2x^3-x^2(3-2x)-3+2x}{3-2x}$
$f'(x)=\frac{-16x^3 + 46x^2 -30x+9 }{(3-2x)^2}\\
g'(x)=\frac{-16x^3 + 50x^2 -34x+9 }{(3-2x)^2}$
Why $f'(x)$ and $g'(x)$...
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Derivation of Inequality 3 from Inequality 4 Using Bayes' Theorem in "Scheduling Multithreaded Computations by Work Stealing"
In the paper "Scheduling Multithreaded Computations by Work Stealing" under the section "Atomic accesses and the recycling game", it is mentioned that inequality 4:
$$ \Pr \left\{ ...
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How was this $y\left(y’’+\frac{1-p}vy’\right)-(y+1-p)y’^2=0$ power series recurrence derived?
In György Steinbrecher’s and William Shaw’s Quantile mechanics $(47)$ to $(51)$, it can be found that:
$$\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=\...
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Why is a derivation always defined on a Ring of functions?
In many places on the internet, you can find that a derivation is defined as a linear operator on a ring or algebra, satisfying the Leibniz rule.
I'm trying to understand the algebraic meaning of this ...
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Size of the pusdown stack of NPDA
Assume that a (nondeterministic) pushdown automaton $A$ is given. Given a word $w \in L(A)$, one would like to know how a derivation of $w$ in $A$ might look like. One way to formalise this question ...
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Complex function completely simplified and i cant figure out how?
I've been working through some homework on derivatives, particularly complex derivatives, with the following question:
calculate the derivative of function f defined on ]-∞ ; -1]U[1;+∞[ by:
f(x) = $(...
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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 ...
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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 ...
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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 ...
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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, ...
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Pushforward of a derivation
I am trying to compute the pushforward of a tangent vector as a derivation. Here are my definitions:
Let $v\in \left.(TM)\right|_U$ be a tangent vector at the point $p$. Given some coordinates $(\phi,...
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Quaternion derivation and integration for attitude representation in common simulators
I have been checking the derivation and integration of quaternions to implement in my simple drone simulator. The problem is that I'm not sure if I can use a simplification for my case or not. The ...
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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\...
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Prove that the composition of Lie algebra derivations need not be a derivation. [duplicate]
In case the definition is not standard, my book defines a derivation $D \in \text{End}(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ as a linear map such that:
$D([X,Y]) = [D(X), Y]\ +\ [X, D(Y)]$ ...
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Distinct derivations of polynomial over finite field
I am a student studying algebra and cryptography.
I wonder below question is possible.
Can I make some polynomials $f(x)$ over finite field that all derivations $f^{(k)}(x)$ are distinct when x is ...
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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 \...
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Proving universal property in Kähler differentials
I have recently been introduced to Kähler differentials. Our lecturer gave a sketch of the proof of how to construct such object. He said that if $A$ is a $k$-algebra, then we can construct the ...
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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
$$\...
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Very complicated differentiation
I am trying to solve all steps in a economics paper , but after spending two days with the same differentiation Im losing faith. Can someone out there help me? The problem:
Differentiate:
$$
\begin{...
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Why is the Discrete Fourier series a sum from $0$ to $N - 1$?
I want to derive the Discrete Fourier series:
$$f(x) = \sum_{k=0}^{N-1} X_k e^{i2\pi xk}$$
from the Continuous Fourier Series formula:
$$f(x) = \sum_{k=-\infty}^{\infty} X_k e^{i2\pi xk}$$
...
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How is the equation for this virtual site derived?
I am trying to understand virtual sites in MD simulations, and I came across this configuration:
Here, coordinate $\mathbf{s}$ represents the virtual site, which is formed by three other atoms $\...
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Mock scalar product's nonlinearity as a derivation
Context. I was reading about the converse of the parallelogram law (a norm satisfying the law defines an inner product), and in an answer on MO it is stated that this is impossible without the norm's ...
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intuition behind deriving the equation of a double-napped cone
I know the equation of a double-napped cone is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{z^2}{c^2}$ but don't fully understand how this is derived. For a right circular cone centered at the origin, ...
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Understanding proof of the Eisenbud, Commutative Algebra, Theorem 16.24 ($\Omega_{S/R} \cong I/I^2 $ , where $I = \ker (\mu : S \otimes_R S \to S)$).
I am reading the Eisenbud, commutative algebra, proof of Theorem 16.24 and some question arises.
First, let me note some associated definition and theorem.
Definiton. If $S$ is a ring and $M$ is an $S$...
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How to derive: $\sin(x-a)\sin(x+a) = \sin^2(x) - \sin^2(a)?$
I was looking at the solution of
$$\int\frac{\sqrt{\sin(x−a)}}{\sqrt{\sin(x+a)}}dx.$$
Where the formula,
$$\sin(x-a)\sin(x+a) = \sin^2x - \sin^2a = \cos^2 a - \cos^2 x$$
is used, I want to know how it ...
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Correspondence: Derivations and vector fields
Let $D$ be a derivation on the $\mathbb{R}$-algebra $C^\infty(\mathbb{R})$ of smooth functions on $\mathbb{R}$. Let $f\in C^\infty(\mathbb{R})$. By the Hadamard lemma we find a smooth function $g\in C^...
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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 ...
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On the free resolution of module of Kahler differentials of a hypersurface
Let $k$ be an algebraically closed field of characteristic $0$. Consider the local ring $S=k[x_1,...,x_d]_{(x_1,...,x_d)}$, and call its maximal ideal $\mathfrak m$. Let $f\in k[x_1,...,x_d]$ be such ...
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How can I understand the definition of a derivation in differential geometry?
Let me define the space $C^\infty_p:=\{f:U\rightarrow \Bbb{R}:\text{ U is a neighbourhood of p}\}$. Then we define a derivation to be the linear map $X:C^\infty_p\rightarrow \Bbb{R}$ s.t. it satisfies ...
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Trying to come up with matrix inverses algebraically — how does one do this?
I've been trying to find the inverse of a matrix through this sort of roundabout manner but to no avail. Sure, I could just use the standard way, but I figured working through this particular process ...
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Relation between algebraic and analytic derivations in $\mathcal C(\mathbb R)$
Let $A,B$ be two $\mathbb R$-algebras. By an $\mathbb R$-linear (algebraic) derivation from $A$ to $B$, we mean a group homomorphism $D:A\rightarrow B$ which contains $\mathbb R$ in its kernel and ...
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How to prove vacuous quantifier
$ \exists x (Fx \rightarrow \forall x Fx) $
I am trying to construct a derivation for this problem which can be found in the UCLA logic software program under chapter 3 of the derivation section.
Up ...
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Noether Normalization and Lie Algebra of derivations on a commutative algebra $A$
In my Lie algebra course, the professor asked us to use Noether's Normalization Theorem and use it to say something about the Lie algebra of derivations over a commutative, associative algebra. Now, ...
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