47
votes
Accepted
Does there exist a complete implementation of the Risch algorithm?
No computer algebra system implements a complete decision process for the integration of mixed transcendental and algebraic functions.
The integral from the excellent paper of Schultz may be solved by ...
- 586
33
votes
Is it always possible to calculate the limit of an elementary function?
EDITED. I use the definition of "elementary function" of Liouville and Ritt (also repeated in Wikipedia). See Ritt's papers in TAMS 25 (1923) 211-222, and
TAMS 27 (1925) 68-90. This definition ...
- 91.8k
30
votes
When can an invertible function be inverted in closed form?
I recommend the following paper:
MR1501299
Ritt, J. F.
Elementary functions and their inverses.
Trans. Amer. Math. Soc. 27 (1925), no. 1, 68–90.
(freely available on the web). It indeed gives a short ...
- 91.8k
27
votes
Does there exist a complete implementation of the Risch algorithm?
Fricas, an open-source clone of Axiom, implements a considerable chunk of Risch, see
http://fricas-wiki.math.uni.wroc.pl/RischImplementationStatus
Fricas is also available as a optional package of ...
- 14k
12
votes
Accepted
Semantics of derivations as derivatives
In all of these contexts, derivations are infinitesimal automorphisms, in the sense that $D$ is a derivation on $A$ (an algebra, a Lie algebra, etc.) iff $\exp(Dt)$ is an automorphism of $A \otimes k[...
- 118k
11
votes
When can an invertible function be inverted in closed form?
Closed-form functions need the definition which set of functions is allowed to represent the function. Take e.g. the algebraic definition of the Elementary functions by Liouville and Ritt (Wikipedia: ...
- 1,053
8
votes
Accepted
Which CAS can do basic non-commutative differential algebra?
Does this fit your desires? This is Mathematica code:
D[(1 + f[t]) ** (1 + f[t]), {t, 2}]
Out:=(1 + f[t]) ** f''[t] + 2 f'[t] ** f'[t] + f''[t] ** (1 + f[t])
...
- 10.1k
8
votes
Accepted
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
Revamped Feb. 12, 2022:
I posted an answer to this (perennial) question in detail in the old MO-Q "Formula for n-th iteration of dx/dt=B(x)" and pointed out a common conflation of related ...
- 10.5k
7
votes
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
Suppose that we don't reorder the parts within terms and that we always post-multiply the $f$. So e.g. $$\begin{eqnarray*}f_1 &=& f \\
f_2 &=& (Df)f \\
f_3 &=& (D^2f)ff + (Df)(...
- 7,226
7
votes
Does a theory of stochastic differential algebras exist?
Yes. A systematic study of stochastic (differential) algebra could be found in
Grenander, Ulf. Probabilities on algebraic structures. Dover Books, 1981.
Grenander studied the operation of ...
- 8,071
6
votes
Is it always possible to calculate the limit of an elementary function?
In fact, zero-equivalence for combinations of polynomial and sine functions is undecidable, which means that it is undecidable whether the limit is zero in that setting (This goes back to at least ...
- 96.4k
5
votes
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
In the paper 'Nested Derivatives:' A simple method for computing series expansions of inverse function' by D. Dominici, with arXiv version
https://arxiv.org/pdf/math/0501052.pdf
has something very ...
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4
votes
"Canonical" form for gauge equivalence classes of matrices in $\mathfrak{gl}_n(x)$
For your second question, you may want to look at "Jordan decomposition for a class of singular differential operators" by A. H. M. Levelt, https://projecteuclid.org/download/pdf_1/euclid.afm/...
- 149k
4
votes
Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative
I'm not sure if this is exactly what you're looking for, but my go-to volume for these kinds of question is Symbolic Integration I by Manuel Bronstein. Risch's original treatment is sketchy in many ...
- 82.7k
3
votes
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
This, too, is not an answer, just a preview of a paper I probably won't be writing for a while (one of only 9 papers on my immediate to-do list).
It so happens I derived a related formula a few weeks ...
- 33.8k
3
votes
References on function fields over imperfect fields in positive characteristic
Not sure if this is what you have in mind, but the paper
Michael Szydlo, Elliptic fibers over non-perfect residue fields, J. Number Theory 104 (2004), no. 1, 75-99 (MR2021627)
is a detailed study of ...
- 47.4k
2
votes
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
While not pretending to answer the OP, the following is too long to fit in a comment while it might contain elements of interest to the poster.
If $f$ is a convergent object (smooth or analytic), then ...
- 5,428
2
votes
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
It also related to the so-called elementary differentials which appear in the algebraic setting for Runge-Kutta methods. See for example the related chapter in the book Hairer, Wanner and Lubich.
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1
vote
Is there a bound on the number of connected components of a zero set of an integrable function?
In your example, you can eliminate $\alpha$ by one more differentiation, and obtain that $f=\cos\alpha x$ satisfies
$$f'''f-f''f'=0,$$
so the number of components of the zero set cannot be estimated ...
- 91.8k
1
vote
Accepted
The combinatorics of $(f \partial)^n$ in the noncommutative setting?
In fact, Comtet's formula works almost directly in noncommutative case under an appropriate ordering of the products. Here is just a bit deeper look under the hood.
In umbral form $(R_f \partial)^n f$ ...
- 34.3k
1
vote
Reference request for results that involve the transcendence degree
For the second result, you can reduce to the same question with $A = \mathbb{Q}[x_1,\ldots,x_n]$. It is known that this polynomial ring is regular, which means that its localisation $A_Q$ at the prime ...
- 20.8k
1
vote
Books/Lecture notes which contrast Risch algorithm with basic standard procedure of finding an antiderivative
J. H. Davenport, On the integration of algebraic functions. Lecture Notes in Computer Science, 102. Springer-Verlag, Berlin-New York, 1981.
J. H. Davenport, Integration in closed form. Computers in ...
- 91.8k
1
vote
What is the index of a given DAE system of equations?
Having investigated Scholarpedia, I am able to respond to first question. We can distinguish certian class of DAEs called semi-explicit, such as:
\begin{array}{ccc}
y' & = & f(t,y,z) \\
0 &...
- 131
1
vote
Complications barring differential rings with an infinite number of derivations
I do not know which results proofs you refer to. So I cannot say whether these results hold for an infinite number of variables. But let me mention the following the following generalization.
You can ...
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