All Questions
Tagged with differential-algebra differential-equations
10 questions
15
votes
2
answers
2k
views
Solvability in differential Galois theory
It is well known that the function $f(x) = e^{-x^2}$ has no elementary anti-derivative.
The proof I know goes as follows:
Let $F = \mathbb{C}(X)$. Let $F \subseteq E$ be the Picard-Vessiot extension ...
- 1,214
9
votes
0
answers
89
views
When could a diligent calculus student compute all Picard iterates algebraically?
As is well known, in the typical proof of the Picard–Lindelöf theorem, one shows the existence of a solution of the initial value problem $y'(t) = f(t,y(t))$, $y(t_0) = y_0$ by considering the Picard ...
- 13k
6
votes
1
answer
418
views
Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)
Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying
$$
D(a.v)=(t\frac{d}{dt}a)....
- 2,751
5
votes
0
answers
99
views
Differential equations analogue of fundamental theorem of symmetric functions
In Gian-Carlo Rota's article "Ten Lessons I Wish I Had Learned Before I Started Teaching Differential Equations", at the end of the third lesson he states a theorem:
"Every differential ...
- 226
4
votes
0
answers
81
views
Reference request, or maybe not really a reference request, on differential algebra
Of differential algebra, Gian-Carlo Rota wrote:
No elementary presentation of this beautiful subject has ever been attempted, to the best of my knowledge; Cohen’s book of the twenties is the closest, ...
3
votes
1
answer
227
views
"Canonical" form for gauge equivalence classes of matrices in $\mathfrak{gl}_n(x)$
Let $\mathfrak{gl}_n(x)= \mathfrak{gl}_n \otimes_\mathbb{C}\mathbb{C}(x)$ be the algebra of matrices taking values in rational functions.
Definition: Two matrices $A, B \in \mathfrak{gl}_n(x)$ are ...
- 7,789
2
votes
1
answer
141
views
Is there a bound on the number of connected components of a zero set of an integrable function?
If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it ...
- 21
2
votes
1
answer
2k
views
What is the index of a given DAE system of equations?
I have very simple multi-body dynamic system from which I have to solve following DAE:
$ \textbf{q}(t) - 3 \times1 \text{ vector of known state variables} $
$ \phi(\textbf{q}(t))=0 - 2 \times 1 - \...
- 131
2
votes
0
answers
36
views
Is there an extension of the Kovacic algorithm to handle algebraic coefficients?
Kovacic's algorithm solves second-order linear homogeneous differential equations with rational function coefficients.
I'm wondering if anybody has extended this algorithm to handle algebraic ...
- 415
1
vote
0
answers
1k
views
Are all solutions to an ordinary differential equation continuous solutions to the associated implied differential equation and vice versa?
Now I have to heavily emphasize the fact that I have never studied differential algebra or the concept of other types of differentiation (which is what I believe is the concept behind a differential ...
- 111