Questions tagged [differential-algebra]

Differential algebra is the study of differential rings and fields and related structures.

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How can one formalize elementary-ness?

I'm trying to decipher what general principals generalize all elementary functions. Does the following hold?: All elementary functions $y = f(x)$ are a solution an equation of the form $P(x,e^x,\ln(x),...
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Is the eigen-composite under differentiation unique, at least up to a scalar addition or coefficient?

Let's suppose you have two differentiable functions $f(x)$ and $g(x).$ First of all, is there a name to a situation where $\frac{d}{dx}[ f \circ g(x)] = g(x)$, or, are there any articles that go into ...
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Can a differential algebraist corroborate what this article means?

I'm reading through this article on Liouville's differential algebra theorem which is very tedious and advanced with respect to my background. I'm looking at Propositions 1.13 and 4.2 trying to break ...
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34 views

Is the differential operator the only operator that satisfies the product rule?

Sorry, I'm not an advanced abstract algebra person, so I haven't seen this situation addressed before. Suppose you abstract the differential operator to just any ol' linear operator $L$ over a field ...
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1answer
26 views

Example of how to classify a function's field in Liouville's theorem?

In building off of this question Can anyone show an example of going through Liouville's differential algebra theorem? I'm slowly starting to understand more components of Liouville's theorem ...
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How can one generalize the differential operator to a space linear transforms of smooth function space?

I'm new to the theory of unbounded operators. I haven't studied it formally, I've only come across it in conversation and reading on my own. Specifically I'd like to study smooth functions. Let's say ...
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42 views

Is the abstraction of continuity in a general differential field implied?

I'll preface this by saying I don't have an advanced background in abstract algebra, so I'm sorry if these concepts already exist but I just have never heard of them. So, in my study of real analysis, ...
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177 views

Can anyone show an example of going through Liouville's differential algebra theorem?

WARNING: This is long and layman-like, you may have a difficult time withstanding reading this if you consider yourself a seasoned mathematician. At one point I came across Liouville's theorem of ...
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78 views

Showing that exponential of a derivation is automorphism on formal power series ring

Let $R$ be a $\mathbb{Q}-$algebra and let $D_0: R \to R$ be some non-trivial derivation on $R$. Moreover let $R[[T]]$ be formal power series ring. We define $D$ - extension of $D_0$ on $R[[T]]$ as $D\...
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21 views

C-planes and associated valuations in the paper of James Ax titled "On Schanuel's conjecture"

I am currently trying to understand Proposition 2. (page 257 of the JSTOR version) in the above cited paper of Ax. It says the following: Let $F\supseteq C \supseteq \mathbb{Q}$ be a tower of fields. ...
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86 views

The field of constants of a differential ring. Derivative of real and complex numbers.

Let $D$ be a derivation operator over a ring $R$: $$D(a + b) = D(a) + D(b) \\ D(ab) = D(a)b + aD(b)$$ for all elements $a,b\in R$. If the ring is the field $\mathbb{Q}$, all derivatives should be ...
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If C is a coalgebra with counit € and D=ker(€), how define a coalgebra structure on tensor algebra T(D) so that C to T(D) is inclusion coalgebra map

Let C be a coalgebra with comultiplication N and counit € and let D = ker(€). Consider the tensor algebra on D , denoted by T(D). My question is : how one can define a coalgebra structure on T(D) so ...
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What is "Cohen’s book of the twenties" about differential algebra Gian-Carlo Rota mentions in his TEN LESSONS?

I am reading "Ten lessons I wish I had learned before I started teaching differential equations" by Gian-Carlo Rota. On item 4, he says: Should we, then, let the students remain blissfully ...
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derivations of the ring of germs of $C^{\infty}$ functions

Let $\mathcal{O}_{\mathbb{R},0}$ be the ring of germs of $C^{\infty}$ funcitons on the real line. A derivation of $\mathcal{O}_{\mathbb{R},0}$ is a $\mathbb{R}$-linear map $\partial:\mathcal{O}_{\...
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When does $\sqrt{f(x)}\exp{g(x)}$ have an elementary antiderivative?

Liouville's original criterion for elementary anti-derivatives states: If $f,g$ are rational, nonconstant functions, then the antiderivative of $f(x)\exp{g(x)}$ can be expressed in terms of ...
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Is there a solvable differential equation with a nonsolvable lie group of symmetries?

For a polynomial equation in one variable over $\mathbb{Q}$, it is well known that the equation is solvable by radicals if and only if the equation's Galois group (which is a finite group) is solvable....
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71 views

Loewy decomposition of differential operators

The paper by Fritz Schwarz, "Loewy decomposition of linear differential equations", contains the following lemma, which I try to prove in order to understand the algorithm which Schwarz ...
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Derivation on rings of sequences

Let $R$ be a commutative ring with unity and let $X$ be a set. Are there natural derivations on the ring $R^X$ of functions $X\rightarrow R$ with pointwise sum and product? What about the particular ...
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100 views

What are the factors of the exterior product of this exterior differential algebra?

I have questions about [Risch 1979] (see the citation and reference below). 1.) What are the factors of the exterior product of the exterior differential algebra in the citation below? I already know ...
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102 views

How to apply Risch's algorithm to $\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x$?

It is known that $$\int\frac{x}{\sqrt{x^4-2x^3+3x^2+4x+1}}\,\mathrm{d}x=$$ $$-\frac{1}{6}\log\Big((2x^4-10x^3+24x^2-28x+14)\sqrt{x^4-2x^3+3x^2+4x+1}-2x^6+12x^5-36x^4+56x^3-42x^2+13\Big)+C.$$ To get ...
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197 views

Prove that $\int \sin(x^2)dx$ is not elementary

See edit It is known that the anti derivative of $\sin(x^2)$ is not an elementary function, and one can represent it using a power series by term-by-term integration of its Taylor series. However, is ...
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156 views

Can we find a simple basis for the cokernel of this derivation?

Let $K$ be a field of characteristic zero. Let $R$ be the $K$-algebra $K[x_0,x_1,\ldots]$ of polynomials in countably infinitely many variables. Consider the $K$-linear derivation $\delta:R\to R$ ...
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44 views

Bounded function and Derivatives [closed]

Does eventually all the successive derivatives of a bounded functions become bounded if one of them becomes bounded?(for entire number line case)
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A priori structural identifiability analysis of a system of ordinary differential equation using differential algebra

Consider an ordinary differential equation model of a dynamic system: $\dot{x} = f(x,u,p)$ $y = g(x,p)$ where $x$ is the n-dimensional state vector, $u$ is the r-dimensional input vector, $p$ is ...
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22 views

What the orbits of the following DS look like?

Let us consider the following DS: $$x'(t) = y(t)$$ $$y'(t) = −ax(t) + bx^2(t)$$ I have to describe the orbits of the system when $a>0$ and $b=0$. The eigenvalues of the matrix associated to the ...
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116 views

Find the equation of the orbit of following ODE

We call orbit associated to the initial datum $(t_0,x_0)$, the set of points: $C=\{$x$(t;t_0,x_0), t \in T\}$. Given the Matrix A: \begin{pmatrix}  0 & 1 \\ -1 & 0 \end{pmatrix} Write the ...
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45 views

Differential algebra in Wikipedia and nlab

The difference between the two definitions is clear in Wikipedia and nlab regarding the definition of a graduated algebra. How to explain this nuance? From Wikipedia: A differential ring is a ...
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186 views

Definition of $\mathfrak{g}$-differential graded algebra

I am reading Group actions on manifolds by Eckhard Meinrenken (Lecture Notes, University of Toronto, Spring 2003). In page $45$, definition $5.2$, author introduce the notion of $\mathfrak{g}$-...
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Are all unital endormorphisms of a Weyl algebra automorphisms?

Given $k$ a field of characteristic $0$, let $A_n(k)$ be the $n$-th Weyl algebra over $k$ – i.e. the unital algebra generated by elements $p_1, ..., p_n$ and $q_1, ..., q_n$ modulo the relations $[p_i,...
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74 views

Accessible first book/resource on differential algebra

I am very interested in learning about the conditions under which a function can be integrated in elementary terms, a topic that I understand falls within the purview of differential algebra. Having ...
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333 views

Intuition of Liouville's Theorem (differential algebra) Proof

At the end of my abstract algebra class this spring, we were given an overview of differential algebra and some differential Galois theory. We went too fast to prove anything nontrivial, but I found ...
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1answer
66 views

Solved inverse Galois problem for $\mathbb{C}(z)$ seems to contradict the theory about Liouvillian extensions.

The theory about Liouvillian extensions tells us that a Picard-Vessiot extension $L \supset k$ is Liouvillian if and only if the identity component $G^°$ of $G = Gal(L / k)$ is solvable. I think I ...
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1answer
40 views

Need help with the proposition 4.2 in R. C. Churchill's paper on Liouville's Theorem in differential algebra

I'm currently studying Liouville's theorem in differential algebra from this paper. I'm stuck on the proof of the propositions 4.2. In the chapter 4, this "logarithmic derivative identity" is ...
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114 views

Which kinds of equations of elementary functions can have elementary solutions?

Which kinds of zeroing equations of elementary functions can have solutions which are elementary numbers or explicit elementary numbers? Take the equation $$F(x)=0,$$ wherein $F$ is an elementary ...
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101 views

Do these differential operators commute?

Let $D$ denote the differentiation operator on the real functions in $C^1$. For $\alpha\in \Bbb{R}$ and $f\in C^1$ define ${1\over D-\alpha}$ as $${1\over D-\alpha} \,f(t)=e^{\alpha t}\Bigl( \int_0^tf(...
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154 views

Integration of the Riccati equation

Simply speaking my goal is to integrate the differential Riccati equation $$x'(t)=a(t)x^2+b(t)x+c(t)$$ I know that this is impossible to express the solution by the means of the elementary functions....
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29 views

German for "Liouvillian extension"

How do I correctly translate "Liouvillian extension" to german, especially "Liouvillian"? "Liouvillsche Erweiterung" sounds rather strange, but might be correct. Anyone knows if this is correct?
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Finding the solution to differential equation dilemma.

Suppose $\ddot{x_1} =\frac{k(x_2-2x_1)}{3m}$ and $\ddot{x_2}=\frac{k(x_1-x_2)}{2m}$ now how do I solve for $x_1 $ and $x_2$ where $x_2 $& $x_1$ aren't independent of each other ie can't be held ...
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118 views

Cancelling differential terms

Can we cancel two differential terms while they are in a ratio. For example if we have (dx/dt) / (dy/dt), can we just directly cancel dt by dt and write it as dx/dy. I mean is is this step allowed?
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Which kinds of compositions of invertible elementary and nonelementary functions are elementary?

Let $f$ be a bijective elementary function, elementary invertible or not. Let $h$ be a bijective nonelementary function, elementary invertible or not. Which of the compositions $h(f(x))$ and $f(h(x))$ ...
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169 views

Explain Application of Risch's Structure Theorem for Elementary Functions

Could you please explain the application of Risch's structure theorem for elementary functions and give some detailed examples? The Structure Theorem for Elementary Functions: "Let $(\mathfrak{E},Y)$...
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Mistake (?) in differential Galois theory

I have found the following exercise in the book of Crespo and Hajto “Algebraic groups and differential Galois theory”: Let $$\mathcal{L}(Y):=Y^{(n)}+a_{n-1}Y^{(n-1)}+\dots+a_1Y’+a_0=0$$ and let $W$ ...
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Showing that a differential Ideal is prime

Consider $\mathbb{R}$ with trivial derivation, $\mathbb{R}\{x\}$ the ring of differential polynomials in $x,$ and let $J$ be the differential ideal generated by $x''+4x.$ In the quotient $\mathbb{R}\{...
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40 views

Proof that the universal first order calculus satisfies its' universal property in the noncommutative case.

Recently I've been reading this paper, and in the first section on (differential calculus on associative algebras) they reference a theorem from Bourbaki's Algebra I Chapter 3. I have a problem with ...
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1answer
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Problem with a proof where algebraic extensions are assumed to be finite extensions

I'm reading the article "Integration in Finite Terms" by Maxwell Rosenlicht and I have a problem with one step in a proof. Rosenlicht wants to prove the following: If $F$ is a differential field of ...
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Creating new constants in differential field extensions via superfluous solutions to a D.E.

I am reading the lecture series book Lectures on Differential Galois Theory by Andy Magid. I came across an example, and I am wondering if it generalizes. The example Let $F_1=\mathbb{C}(z)$ be ...
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121 views

How to use differentiation to show that a curve is symmetric above x axis

If $x^2+y^2=9$, how can I show that this curve is symmetric above $x$-axis using differentiation. Am I correct? Clearly $\frac{dy}{dx} =-\frac{x}{y}$ . So for all $x$ in the domain, $y=+$ or $-$. ...
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38 views

Using $\text{SL}(2)$ to solve $2\times2$ linear system of ODEs of order 1 with variable coefficients

Define the matrix $A(t)=\begin{pmatrix} - p(t) & 1 \\ -1 & p(t) \end{pmatrix}$ where $t$ is a real variable and $p$ a function of $t$. Looking for solutions $u=(u_1(t),u_2(t))$ to the ...
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48 views

Fixed differential field by a subgroup of a differential Galois group

I am currently reading the book "Differential Galois Theory" by Springer and Van der Put. In the process of establishing that the field fixed by the differential galois group of a picard vessiot ...
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Linear Approximations from Differential Algebras

Suppose we have a differential ring $(R,+,\cdot)$ with derivation $\partial: R\to R$ which is linear $$\partial(f+g)=\partial f+\partial g$$ and obeys the Leibniz rule: $$\partial(f\cdot g)=(\...