If I had some expression `x^2 + x^3 - 31` and I want to apply a previously defined function `f` to it, is there a quicker way of doing `f(x^2 + x^3 - 31)`? I'm thinking along the lines of Mathematica's `x^2 + x^3 - 31 // f`, where `//` applies the function to the expression that precedes it ..read more

Hello, I'm trying to understand step 3 in the algorithm 1 provided in the paper "Algebraic Reduction of Hidden Markov Models" by Tommaso Grigoletto and Francesco Ticozzi. It requires me to compute alg($\mathcal{X}$) and it is defined in the paper like so: > Let alg($\mathcal{X}$) denote the minimal sub-algebra of $\mathbb{R}^n$ containing the set $\mathcal{X}$. In example 1 in the paper, the authors give a set $\mathcal{R}$ and the resulting alg($\mathcal{R}$) like this: $$ \begin{aligned} & \mathcal{R} = \text{span}\\{\begin{bmatrix} 1/5 & 1/5 & 3/5 \end{bmatrix}\\}\\\\ & ..read more

Is there any implemented algorithm [or any algortihm which is not too slow or hard to implement] to find the ***sup***groups of a given group? More precisely, say I have a permutation group $H$ which is given as a subgroup of $S_n$ (the permutation group on $n$ elements) and some integer $k$. Is there a way to find supgroups K of H so that the index of $H$ in $K$ is equal to (or is at most) some number $k ..read more

There is a note in the reference manual about Links that even states that "The strands of the braid that have no crossings at all are removed". This is not the expected behavior and can cause trouble if you generate links using braids. I also don't know an easy workaround. While conjugating the braids works most of the time, if you conjugate a trivial braid it does not. Is there an easy way to circumvent this problem ..read more

In fact, I can use the following code (in conjunction with Erdős and Gallai's theorem) where the number of edges is less than or equal to 38. num_vertices = 12 selected_graphs = [] for g in graphs.nauty_geng(f'{num_vertices} -c 11:38'): if g.matching(value_only=True) == 4: selected_graphs.append(g) len(selected_graphs) I am wondering if it's possible to combine the structure of such graphs to improve efficiency. For example, Let $G$ be a connected graph with 12 vertices and maximum matchings with size of 4. Let $M=\\{u_1u_2,u_3u_4,u_5u_6,u_7u_8\\}$ be a matching. Let $S=V(G)-\\{u_1,u_2\dots u_ ..read more

SageMath version 9.5 release 30_1_2022 , using Python 3.11.6, everything seems to work but Jupyter stays sme time trying to connect to Sage Kernel without success ..read more

Example: F. = FreeGroup(); H = [a^3, a*b*a*b^(-1)*a^(-1)*b^(-1), b*c*b*c^(-1)*b^(-1)*c^(-1), c*d*c*d^(-1)*c^(-1)*d^(-1), d*e*d*e^(-1)*d^(-1)*e^(-1), a*c*a^(-1)*c^(-1), a*d*a^(-1)*d^(-1), a*e*a^(-1)*e^(-1), b*d*b^(-1)*d^(-1), b*e*b^(-1)*e^(-1), c*e*c^(-1)*e^(-1),]; G = F/H G is a factor group of the braidgroup on 6 strands. By a result from Coxeter (since 1/6 + 1/3 ..read more

I've had this problem like several times today, doing integrals of the form I=1/sqrt(1+(z-z1)^2) from sage.symbolic.integration.integral import indefinite_integral indefinite_integral(I,z) Long error follows, ending with ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(z1>0)', see `assume?` for more details) Is z1 zero or nonzero? So I get that Maxima might want to know if z1 happens to be zero, but unfortunately: ", ".join(map(str, maxima("features")._sage_())) 'integer ..read more

Let's say we have 2 equations and 3 variables (N equations and M variables, where M > N), for example: 0.3*a+0.1*b+0.2*c=10 + E 0.1*a+0.2*b+0.3*c=11 + E We want to find a, b, c that are in the range [0.0, 1.0] and we want to minimize the error E. We want a smooth solutions that in this context means that if we chart a, b, c in a chart, the chart is smooth. How this kind of problems can be approached with sagemath ..read more

I would like to construct a free module over QQ (for simplicity) whose basis is index by finite sequences of strictly decreasing positive half odd integers (so odd numbers divided by 2). `CombinatorialFreeModule()` seems like the tool to use for this, if I can define the class of sequences mentioned in the previous sentence. I understand that these sequences are in bijection with integer partitions, so I could just use `Partitions()` as in `CombinatorialFreeModule(QQ,Partitions())`. However, I'm trying to improve my understanding of Sage and so I'm trying to program an appropriate class of seq ..read more