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2
votes
1answer
150 views

Confusion about commutative differential algebra

In the article "Sullivan minimal model" of the Encyclopedia of Math, we read the following: Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ ...
0
votes
0answers
26 views

Solvability of linear homogeneous differential equation in terms of lower order

Let $ L(y)=0$ be a linear homogeneous differential equation with coefficients in a differential field $k$. We say that a differential field $K$ containing $k$ is a Picard-Vessiot extension of $k$ for ...
1
vote
0answers
22 views

Differential operator and multi-index

By induction it can prove Leibnitz rules $\displaystyle D^\alpha(fg)=\sum_{|\beta| \leq |\alpha|} \binom{\alpha}{\beta} D^\beta f D^{\alpha - \beta} g$ from the book where I'm studying, it says that ...
1
vote
1answer
62 views

How to decide whether it is a system of Differential Algebraic Equations or a System of Ordinary Differential Equations?

I am struggling to name some of my dynamic models right. To be specific, I am not sure whether I should call it a system of Differential Algebraic Equations (DAEs) or a System of Ordinary Differential ...
1
vote
1answer
122 views

May the integral $\int\root 3 \of{\cos(x)^2}\,dx$ be expressible by elementary functions?

I would like to decide by methods of differential algebra whether the integral $\int\root 3 \of{\cos(x)^2}\,dx$ might be contrary to the output of CAS Mathematica Online Integrator expressible by ...
1
vote
1answer
42 views

How can I solve this difference equation, if $p=q$?

I'm trying to solve the difference equation $$ pE_{k+1} - E_{k} + qE_{k-1} = -1 $$ given the boundary conditions $E_{0} = 0, \; E_{a} = 0$, if $p=q=\frac{1}{2}$ To attempt this, I first found and ...
5
votes
0answers
124 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring ...
1
vote
0answers
30 views

Index reduction for DAE

I have to simulate a set of DAE's. Therefore I have to reduce the index for this problem: $ (ms+mb)*\ddot z + mb*ls* \ddot \phi s + mb*lg* \ddot \phi b = -(ms+mb)*g - \lambda2$ $ (mb*ls)*\ddot z + ...
8
votes
1answer
603 views

Proof that the solutions are algebraic functions

I am looking at the following: $$$$ $$$$ I haven't really understood the proof... Why do we consider the differential equation $y'=P(x)y$ ? Why does the sentence: "If ...
5
votes
2answers
63 views

What is a constant field?

I am looking at the following: Could you explain to me what a constant field is? $$$$ P.S. I found this in the paper of T. Honda, "Algebraic differential equation" (pages 170-176).
2
votes
1answer
60 views

Reduction modulo a prime ideal

I am looking at the following part of a paper: $$$$ $$$$ When we reduce the differential equation $(1)$ modulo the prime $p$ we do the following: $$\alpha_i \equiv \tilde{\alpha}_i ...
3
votes
1answer
79 views

Proof that that $K=\mathbb{Q}$

I am looking at the following part: $$$$ $$$$ $$$$ $$$$ $$$$ I haven't really understood the proof... We suppose that Grothendieck's problem stand and that almost all ...
4
votes
1answer
55 views

Algebraic solutions of a differential equation.

Given a differential equation $y' = (1/x)(y^2 + y^3)$. My question is how does one go about finding the solutions of this differential equation which are algebraic over the field $\Bbb{C}(x)$,if ...
1
vote
0answers
19 views

Adjoining a derivative

Let $u$ be an algebraic solution of $y'=(1/x)(y^2 + y^3)$ other than $-1$ and $0$ over $\Bbb{C}(x)$ (the quotient field of $\Bbb{C}[x]$). So, $u$ is some fractional power series. Suppose, we adjoin ...
1
vote
1answer
113 views

Proof of nonexistence of a solution to an equation in terms of elementary functions?

In a numerical methods class I'm taking, it was claimed that the equation $A = \frac{R^2}{2} \left(\theta - \sin\theta \right)$ cannot be analytically solved for $\theta$. I don't doubt that this is ...
0
votes
0answers
17 views

Determine a change of variables to transform one DE to another

Given two ODE's, is it possible to determine if one can be obtained from the other via a change of variables? In particular, I have the two ODEs: $$ \begin{split} ...
3
votes
1answer
68 views

A tricky Differential Equation

How do you solve $$\frac{dy}{dx} = \frac{y^3}{e^{2x} + y^2}$$ I just need a hint. Its not an exact differential nor a linear D. E which I can solve...
1
vote
1answer
100 views

The solutions are linearly independent and algebraic

The Grothendieck problem for differential equations (Grothendieck-Katz conjecture) is the following: $$\alpha_n(x)y^{(n)}(x)+\dots +a_1 (x)y'(x)+a_0(x)y(x)=0, a_i \in \mathbb{Z}[x]\ \ \ \ (*)$$ We ...
4
votes
3answers
279 views

Differential algebra and differential-algebraic equations

Could you give me some information about differential algebra? What is it about? Differential-algebraic equations (DAEs) are polynomials with complex coefficients and the unknown variables are $z, ...
1
vote
1answer
29 views

Joint Continuity implies being a Hausdorff space(in the given case)?

If we consider G to be the group of non-singular n by n matrices carrying Zariski topology. Then the joint continuity of the multiplication map from G x G to G would make G Hausdorff.(We consider the ...
3
votes
2answers
73 views

Does there exist higher degree graded derivations on $\Omega(M) $

Does there exist any other graded derivation on $\Omega(M)$ other than the one of degree one which is the exterior derivative (i.e. maps such as $d: \Omega^p(M) \rightarrow \Omega^{(p+r)}(M) $, where ...
0
votes
0answers
14 views

Finite extension of a closed differential field is closed.

Let G be the differential Galois group of a differential field extension M of K. Then any finite dimensional extension of a closed intermediate differential field is closed. I can not realise this ...
2
votes
0answers
167 views

What are all types of elementary second order ordinary differential equation that can not be expressed in closed form?

Can we define all types of elementary second order ordinary differential equation that can not be expressed in closed form as opposed to the one that we can solve? In differential algebra, ...
10
votes
0answers
255 views

Why differential Galois theory is not widely used?

E.R.Kolchin has developed the differential Galois theory in 1950s. And it seems powerful a tool which can decide the solvability and the form of solutions to a given differential equation. My ...
8
votes
1answer
262 views

Prerequisites for Differential Galois theory

I would like to know the prerequisites for Differential Galois theory. I have taken Rings, Fields, Groups, Galois theory, and Algebraic Geometry + Commutative Algebra. Looking at the wikipedia page, ...
1
vote
0answers
40 views

Differentiating both sides of a DE

In general if you have a differential equation with two variables such that: $$L(x,y)=h_1[f(x),f'(x),f^{(2)}(x),...,f^{(n)}(x),g(y),g'(y),g^{(2)}(y),...,g^{(n)}(y)]\\ ...
4
votes
0answers
265 views

chain rule for derivations

Off we go. So let $b:X\rightarrow Y$ be a function from $X$ to $Y$ endowed with as much structure as it needs to make sense of the question :) and $a:Y\rightarrow \mathbb R$ a function into the reals. ...
2
votes
1answer
63 views

What are differential algebras?

On the bottom of page 10 of this paper (/!\ It's in french !) they talk about the 'differential $\mathbb{C}$-algebra of convergent series $\{C(x);x^2d/dx\}$' and about the 'differential ...
3
votes
1answer
58 views

Asymptotic elementary expression for the antiderivative of $x^x$

It is well known that there exists no elementary function $f$ with $$\int x^x\,dx \quad = \quad f$$ Is there an elementary function $g$ such that $$\int x^x\,dx \quad \tilde{} \quad g$$ in the ...
3
votes
0answers
28 views

The spectral transfinite open spaces with quintic characteristics of second kind

Context: Beginning with the formal definition of transfinite spaces together with the Picker-Hansel theorem, we obviously get a relation $$ \bigcap\xi_{|\sigma|\mapsto \theta^*} \oplus_\psi ...
157
votes
6answers
13k views

How can you prove that a function has no closed form integral?

I've come across statements in the past along the lines of "function $f(x)$ has no closed form integral", which I assume means that there is no combination of the operations: addition/subtraction ...
2
votes
1answer
82 views

What are elementary field extensions?

While reading about symbolic integration I encountered some concepts of Differential Algebra. I do not know much of D.A and Fields in general also I have encountered as an extension of Rings. I ...
4
votes
1answer
170 views

Is tetration a transcendental function?

Is tetration a transcendental function? If so are there any papers with a proof? I suspect that it is because I have not seen any algebraic situations where tetration is the answer and the fact that ...
1
vote
1answer
80 views

Controllability on nonlinear systems

Dynamic system in the book (chapter 6, page 67 in http://www.me.berkeley.edu/ME237/6_cont_obs.pdf) $$ \begin{cases} \dot{x}_{1}=x_{2}^{2}\\ \dot{x}_{2}=u\end{cases} $$ so $$ ...
9
votes
1answer
146 views

Theorem: Anti-differentiation is harder than differentiation

The question of why anti-differentiation is "harder" than differentiation was the topic of an earlier question, and some of the answers are interesting, but I'm not sure they fully answer it, and this ...
4
votes
0answers
43 views

If $D:A\to A$ is a derivation, what can be said about the range of $D$?

What can be said about the relation between the domain and range of a derivation as a function? If $A$ is the domain, any space of functions, what does $D(A)$ look like, where $D$ is a derivation? ...
3
votes
1answer
128 views

Trace and Norm maps on differential extensions

I'm working through a proof which is rather algebraic, and my abstract algebra is probably only basic to intermediate. I have a differential extension $E/K$ of a differential field $K$, and the proof ...
2
votes
1answer
201 views

Introduction to Elementary Functions

I'm looking for an introductory text on algebraic treatment of elementary functions. Really short and easy-going. Video lectures are even better. I want to learn basic ideas (i.e. definitions) behind ...
0
votes
1answer
90 views

“fluent” functions

In an old mathematics book (Ritt, 1948, p.5) I have come across the notion of "monogenic analytic" and "fluent" functions. These are complex valued functions. Has anyone heard of these terms before? ...
0
votes
3answers
93 views

What is $\lim\limits_{x\to 0}x^2\cos(2/x)$?

I have to evaluate $$\lim_{x\to 0}x^2\cos(2/x)$$ using one or more of the limit laws. I am using the multiplication law and I am wondering if I am on the right track here? I have split it up to: ...
1
vote
0answers
78 views

Algebras vs. rings in algebraic differential calculus

Vector fields (and differential operators of higher order) on a real manifold are often defined in terms of $\mathbb{R}$-algebra $C^\infty(M)$. However, it is not clear to me why is the ...
3
votes
3answers
640 views

What is the connection between Grothendieck's Differential Operators and Hochschild Cohomology

For a given commutative algebra $A$ over a field $\mathbb{K}$(with char=0) the algebra of differential operators on $A$ is the set of endomorphism $D$ of $A$ such for some $n$ we have that for any ...
4
votes
2answers
333 views

Determination of inverse laplace transform using primitive functions

In How can you prove that a function has no closed form integral?, the accepted answer points to http://www.sci.ccny.cuny.edu/~ksda/PostedPapers/liouv06.pdf where one can find a corollary by Liouville ...