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Questions tagged [differential-algebra]

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

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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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1answer
28 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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Hodge star operator on differential algebras

Given an arbitrary differential graded algebra $(\Omega,d)$ (over the field $\mathbb{R}$ or $\mathbb{C}$ ), it is posible to define an operator that acts like the Hodge star operator on differential ...
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1answer
37 views

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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Which kinds of simply composited elementary functions have elementary antiderivatives?

The elementary functions are defined in differential algebra. That are the functions $X\in\mathbb{C}\to Y\in\mathbb{C}$ that are composed of $\exp$, $\ln$ and/or unary or multiary univalued algebraic ...
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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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Graphing Lie transport of a function

I am relatively new to differential geometry. I am studying it from Fecko Textbook on differential geometry. As soon as he introduces the concept of lie derivative,he asks to do exercise 4.2.2 in ...
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1answer
52 views

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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1answer
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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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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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(Sub)gradient of a function with vector dot products

I am trying to understand the subgradient of a function for a time series, where for each time instance t: $f_t: p \in \Delta_K \mapsto \ell_t(\mathbf{p}\cdot \mathbf{x_t} )\in \mathbb{R}_+ $ the ...
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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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1answer
42 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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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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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)=(\...
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How to obtain Appell's PDEs from GKZ-equations?

In F. Beukers' Notes on A-Hypergeometric functions on p. 17, the Appell hypergeometric PDEs are derived from the GKZ-Equations \begin{align*} \partial_{1}\partial_{2}\Phi-\partial_{4}\partial_{5}\Phi &...
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Use Liouville-Ostrowsi theorem

I don't know how to answer this questions : 1) $e^x/x, exp(exp(x))$ as no elementary primitive. 2) $1/(1+x^2)$ as no elementary primitive in $R(x)$ but have one in $C(x)$. If someone could help me i ...
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Are all solutions to an ordinary differential equation continuous solutions to the corresponding implied differential equation and vice versa? [duplicate]

Regarding the duplicate. Yes, I know the other one has a lot of shared text, but those were just definitions/setup and I was being lazy. The core questions are still different unless you believe ...
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Differentiation, logarithmic fucntion [closed]

The Derivative of $\log_{10} x$ with respect to $x^2$ is? The Answer is having loge(base10)
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1answer
330 views

Tangent vectors as derivations

We have such a definition: Given an algebra $A$ over the field $K$, and $M$ is an $A$-bimodule. Derivation is a linear map: $$D:A \to M$$ That satisfies: $D(a*b)=D(a)\bullet b + a\bullet D(b)$ (...
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Inconsistencies in the definition of derivative of a polynomial over a field

A problem I came across defines a particular differentiation operator $D$ over the set of polynomials $\{P\}$ over a field $F$ with "the normal formula; that is $D(\sum_{i=0}^n a_nx^i) = \sum_{i=1}^n ...
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Closed forms of $\int_A^B \sin(\sin(ax))dx$ and $\int_A^B \sin(\sin(ax)) \cdot \sin(\sin(bx))dx$

I would like to know if: $\int_A^B \sin(\sin(ax))dx$ has a closed-form? The solution of Maple requires the presence of Struve functions in its expression. But at least Maple is able to solve it, so ...
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1answer
112 views

Constants in localized differential rings

Fix a differential commutative ring $(A, d_A)$, and a multiplicative set $S$ in $A$. Suppose that $C_A = \{a \in A \mid d_A(a) = 0\}$. Let $B = S^{-1}A$ and transfer the derivation $d_A$ to $B$, ...
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3answers
90 views

Please solve the differential equation in its simplest form.

$$sec^2(y) \frac{dy}{dx} + 2x \tan(y) = x^3$$ I am not able to get rid of that $dy/dx$ . Please help. PS: I need the answer as $\tan(y) =\frac{x^2 - 1}{2}$
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1answer
114 views

Is the Antiderivative of an Elementary Function Being Nonelementary Generic?

Many elementary functions, like $e^{-x^2}$ and $\frac{\sin(x)}{x}$ have antiderivatives that are are nonelementary; is this property generic? That is, does the set of all elementary functions whose ...
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0answers
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Are computer able to implement a algorithm theoretically to determine if a single variable integrals have closed form? [duplicate]

Are computer able to implement a algorithm theoretically to determine if a single variable integrals have closed form? There are Risch's algorithms already could solved any single variable integrals ...
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3answers
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Constants in localizations of a differential ring

Fix a commutative ring $A$, a derivation $\partial_A$ on $A$, and a multiplicative set $S$ in $A$. Then let $B = S^{-1}A$, $f : A \to B$ be the natural map, and $\partial_B$ be the natural extension ...
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1answer
388 views

Existence of homotopy operator is equivalent to zero homology

Let $(E,\partial)$ be a differential space. A linear map $h:E\to E$ is a homotopy operator if $h\partial+\partial h = {\rm Id}$. Then, there is a homotopy operator in $E$ if and only if $H(E)=0$. ...
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1answer
121 views

Are all derivations of real-valued functions derivatives?

To me it was obvious that all derivatives are derivations, since one can show that they are linear and satisfy the product rule. However, I was very surprised when I learned that every derivation of $...
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1answer
565 views

Possible Research Topics [closed]

I am a mathematics undergraduate in the last semester of my junior year. I am looking for research projects involving differential equations, topology, and abstract algebra and any combination thereof ...
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Do derivations over a commutative ring form a ring?

Given a commutative ring $R$ and a derivation $D : R \rightarrow R$, we might consider a ring built over "differential operators" built over this operator, with the elements of the ring being spans of ...
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2answers
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Why is $e^g$ transcendental over $\mathbb{C}(z)$ for $g$ rational?

In the 1972 paper Integration in Finite Terms by M Rosenlicht he says: we note that if $g(z)$ is is a non-constant rational function of the complex variable $z$ then $e^g$ is not algebraic over $\...
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Does the “Leibniz multicategory over $R$” have an accepted name?

Let $R$ denote a commutative ring. Definition. The "Leibniz multicategory" over $R$ is given as follows: Objects. $R[D]$-modules (where $D$ is a formal symbol; an 'indeterminate'). ...
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Differentiating Query

Does the following make logical mathematical sense: $$x^2=t$$ $$\frac{d} {dy} (x^2)=\frac{d} {dy} (t)$$ $$2x\cdot\frac{dx}{dy}=\frac{dt} {dy} $$ $\mathbf{\therefore \frac{dy} {dx} =2x \cdot\frac{dy}{...
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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 ...
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1answer
131 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 ...
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1answer
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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 ...
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108 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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2answers
144 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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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 $(3)_{\mathfrak{p}}$ ...
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1answer
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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 prime ideals ...
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1answer
432 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 \pmod ...
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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
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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 any....
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1answer
512 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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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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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, x,...
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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 $\mathbb{R}[x,y]...