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13 views

Soft Question : Help required regarding a paper by Michael Singer

I am reading a paper by Michael Singer titled "Algebraic Relations Among Solutions of Linear Differential Equations : Fano's Theorem". I have few doubts which I am stuck at and would be really ...
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0answers
120 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 ...
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27 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 + ...
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1answer
593 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
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2answers
62 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).
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1answer
43 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 ...
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1answer
78 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 ...
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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 ...
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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 ...
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1answer
109 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 ...
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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
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1answer
60 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...
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1answer
97 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 ...
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3answers
244 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, ...
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1answer
25 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 ...
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2answers
72 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 ...
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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 ...
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0answers
164 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, ...
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232 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 ...
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1answer
245 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, ...
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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)]\\ ...
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253 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. ...
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1answer
62 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
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1answer
56 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 ...
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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 ...
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12k 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 ...
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0answers
101 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 ...
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1answer
75 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 ...
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1answer
163 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
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1answer
79 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
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1answer
142 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 ...
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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? ...
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1answer
130 views

Confusion about commutative differential algebra

Here we read the following Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a ...
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1answer
122 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
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1answer
199 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 ...
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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? ...
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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: ...
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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 ...
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3answers
623 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 ...
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332 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 ...