21
votes
Accepted
How does Sage order the elements of the symmetric group?
I disagree with the question, starting with the first sentence. In Sage, the symmetric group is not a list. In fact clearly it isn't, because I can ask it for ...
- 9,637
14
votes
Accepted
Genus of the graph $K_{4,2,2,2}$
I understand it's been years since this question was asked, but I figured I should give an answer for anyone still interested.
From the Euler equation, if $K_{4,2,2,2}$ has genus 2, it would actually ...
- 296
14
votes
13
votes
Number of collinear ways to fill a grid
By the method I used to solve
Counting "connected" edge orderings (shellings) of the complete graph,
I can show the following:
$$ g(m,n) = (mn-1)!\,m!\,n!\sum \frac{b_1 b_2\cdots b_{m+n-...
- 50.8k
13
votes
Is there any fast implementation of four color theorem in Python?
If you have have some specific, moderately large graphs that you want to color with four colors, you could try using a SAT solver. For each vertex $v$ and each integer $i\in \{1,2,3,4\}$, let $x_{v,i}...
- 82.6k
12
votes
Accepted
Number of collinear ways to fill a grid
We managed to obtain a solution via Stanley's comment above and some manipulations of binomial coefficients. See https://arxiv.org/abs/1809.10263
- 356
11
votes
Is there any fast implementation of four color theorem in Python?
Robertson, Sanders, Seymour and Thomas, who produced a more streamlined proof of the 4-colour theorem, also addressed the algorithmic question in the paper
https://dl.acm.org/doi/pdf/10.1145/237814....
- 12.7k
11
votes
Accepted
Discrepancy in Magma's calculation and Sage's of elliptic curve?
Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage.
In fact, when executed with higher values of the parameters so that the ...
- 8,945
10
votes
Accepted
Computer program for counting graph homomorphisms
EDIT: I have since found that the Digraphs package sometimes counts homomorphisms incorrectly. Perhaps this problem has been fixed in more recent versions though. I use Minion for counting ...
- 2,339
10
votes
Accepted
Computation of a minimal polynomial
To compute the minimal polynomial of integer multiple of an algebraic integer is easy, so the only thing you need for linear combinations is the minimal polynomials of sums. Now, note that if $A$ is ...
- 96.4k
10
votes
Accepted
How to get the dimension of Atkin-Lehner eigenspace or do you have any data already obtained?
You can compute these dimensions using modular symbols (an auxiliary space which has the same Hecke action as modular forms, but is easier to compute). Here's a Sage example for weight 4 cusp forms of ...
- 37k
9
votes
Computation of a minimal polynomial
Using resultants can greatly help. For example, knowing that $\sqrt[5]{2}$ is a zero of $P(x)=x^5-2$ and $1-\exp\frac{2\pi i}{5}$ is a zero of $Q(x)=(x-1)^5+1$, we conclude $\sqrt[5]{2}\cdot (1-\exp\...
- 34.3k
9
votes
Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$
A summation method for this...
$$
F(s) = -\sum_{n=2}^\infty \frac{(-1)^n}{n^s} = -(2^{1-s}-1)\zeta(s)
\qquad\text{for $s>0$}
$$
Differentiate:
$$
\sum_{n=2}^\infty \frac{(-1)^n\log n}{n^s} = F'(s)
...
- 41.1k
8
votes
Accepted
How do computer algebra packages like Sagemath implement rank of a matrix
I don't know what algorithm Sage actually uses, but computing rank over the integers is fun and easy: Complexity of computing matrix rank over integers . It is NOT so easy if you want good running ...
- 96.4k
8
votes
How to get the dimension of Atkin-Lehner eigenspace or do you have any data already obtained?
Exact formulas for dimensions of Atkin-Lehner eigenspaces follow from trace formulas of Yamauchi and Skoruppa-Zagier. Skoruppa-Zagier corrected some clerical errors in Yamauchi's paper. See:
Nils-...
- 6,039
7
votes
Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$
As shown in my previous answer, the value of the sum that you see is
$$\lim_{t\uparrow1}\sum_{n=2}^\infty (-t)^n \ln n. $$
Here is a "manual" way to show this. Writing
\begin{equation}
\...
- 128k
7
votes
Accepted
Computations of half-integer forms in SAGE/Magma
This can be done using PARI/GP, which can deal with spaces of modular forms of half-integral weight. Given a modular form $f$ of weight $k$ (possibly half-integral), the command mfslashexpansion can ...
- 20.8k
6
votes
Sage: Evaluation precision for elliptic curves over p-adic fields
If you are willing to use another CAS system then
PARI/GP can do the job:
...
- 2,784
6
votes
Is there any fast implementation of four color theorem in Python?
Here is a greedy algorithm by Febi Mudiyantoto solve the four-color problem in Python.
And here is another Python algorithm that also uses Sage.
If you wish to rely on a program with a more formal (...
- 188k
6
votes
Accepted
Construction of skew-Hadamard matrix of order 292
The skew Hadamard matrix of order 292 can be constructed from skew supplementary difference sets (kindly supplied to us by Prof. Djokovic). This same construction is used, for example, in Djokovic - ...
- 261
5
votes
Accepted
GAP versus SageMath for branching to Lie subgroups
I don't know GAP but Sage has a nice tutorial for branching and is quite usable. It is however, slower than LiE which is on the other hand quite "basic" i.e. it requires you to write the branching ...
- 8,597
5
votes
Number of collinear ways to fill a grid
Let me give an alternative proof with only finitely many variables used. It also gives some formulae for the following numbers: "when the filling becomes not collinear for the first time, there ...
- 109k
4
votes
Computation of a minimal polynomial
Although exact calculations are to be preferred where possible, if all you have is a high-accuracy numeric value of a number which you know to be algebraic then you can use an integer relation finding ...
- 7,226
4
votes
Existence of a non-Eulerian atomistic lattice with this property on the Möbius function
I found the following example with SageMath. Below is the Hasse diagram and verification in SageMath that the poset is a lattice which is not Eulerian (in fact not graded) that satisfies the desired ...
- 7,875
4
votes
All rational periodic points
Your polynomial has good reduction outside of 2, so for all primes $p\ge3$, any rational preperiodic point will have period $n=m_p\cdot r_p\cdot p^{e_p}$, where $m_p$ is its period modulo $p$ and $r_p$...
- 47.4k
4
votes
Computer program for counting graph homomorphisms
In the past few months, we have been working towards a SageMath library for exact counting graph homomorphisms, based on the proof of Prop. 1.6 from "Homomorphisms Are a Good Basis for Counting ...
- 75
3
votes
Accepted
Branching to Levi subgroups in SAGE and the circle action
Branching to Levi subalgebras should really take into account the central part of the Levi subalgebra but it is not the case. The problem is that the ...
- 8,597
3
votes
GAP versus SageMath for branching to Lie subgroups
I recommend LiE, which is a specialized software for computations in finite dimensional representations of semisimple Lie algebras. There is an online interface.
See http://www-math.univ-poitiers.fr/~...
- 2,449
3
votes
Accepted
Mistake in SageMathCell code, finding integral points on elliptic curves
After about an hour and a half running SageMath 9.0.beta7 on my computer, I see this:
...
- 4,254
3
votes
Accepted
Software for $S$-unit equation
This SageMath implementation promises the full generality you are seeking:
A robust implementation for solving the S-unit equation and several applications
See also this Phys.Org announcement.
- 188k
Only top scored, non community-wiki answers of a minimum length are eligible