Let $K$ be the group generated by four elements $x_1,\cdots,x_4$ with relations that each generator commutes with all its conjugates. (An equivalent relation is, any simple commutator with repeated generator is trivial; for example, $[[x_2,[x_1,x_3]],x_3]=1$.) It can be proved that $K$ is finitely presented. Let $A$ be the subgroup of $K$ generated by the following […]

I have the following question(s): I have an “Algebra-With-One” $R$ as a subalgebra of a full matrix algebra in GAP. Furthermore, I have 5 primitive orthogonal idempotents $e_1,…,e_5$, which sum up to $1_R$ (the identity matrix). I would like to compute the projective indecomposable modules $P_1=e_1R,…,P_5=e_5R$ with GAP (or another computer programm (e.g. SAGE) which […]

I. The first solution to, $$\sum^6_{n=1} a_n^9 =\sum^6_{n=1} b_n^9$$ $$13^9+18^9+23^9-5^9-10^9-15^9 = 9^9+21^9+22^9-1^9-13^9-14^9$$ was found in 1967 by computer search by Lander et al. It stood for 40 years as a “numerical curiosity” until Bremner and Delorme discovered it had the highly structured form, $$\small(u + 9)^k + (u + 14)^k + (u + 19)^k + […]

I had a look at Knuth’s The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the following: L1: [Initialize] Set $y\leftarrow0$, $z\leftarrow x/2$, $k\leftarrow1$. L2: [Test for end] If $x=1$, stop. L3: [Compare] If $x-z<1$, go to L5. L4: […]

So I’ve done some hands-on work with converting integers from one base to another using the well-known method of division and taking the remainder. The most generic algorithm involves dividing the number recursively, and looking up the digits of the target base using the remainder. For example, let’s say I wanted to convert 72310 from […]

Assume $A$ is a $n \times n$ matrix, and $rank(A)<n$. For $b \in \mathbb{R}^n$, assume $AX=b$ has a solution $X=(x_1, \cdots, x_n)$, then clearly there exist infinitely many solutions. By the structure of the solutions, we may assume $\sum_{i=1}^n x_i=1$. Now my question is, for any $\epsilon>0$, does there exist invertible $n \times n$ matrix […]

The graph canonical labelling package nauty is widely regarded as one of the best (if not the best) around. Unfortunately, it’s quite a large package, and making a GPU version seems to be a highly nontrivial task. In my research into algorithms for network motif detection, we often require an effective solution to the problem […]

Say I have a given numerical velocity field in two dimensions, (u,v). I am trying to find the streamlines from this data set at a particular contour level and I thus have to solve the differential equation $$ dy/dx = v/u=g(x,y) $$ I can rewrite the equation to $$ dy = g(x_i,y_i)dx $$ The subscript […]

I know this question has been asked before; I just want to enquire if anybody has any suggestions to learn how to compute finite element problems, including plenty of examples. The topics I would like to focus in are as follows: Introduction to finite elements for 1D and 2D problems covering: weak formulation Galerkin approximation […]

I’m creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I’m at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the formulas I have found to describe the derivative have an $1 – x^2$ in the denominator, and I […]

Intereting Posts

List of explicit enumerations of rational numbers
Radius of convergence of entire function
Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution
Proof that $A \subseteq B \Leftrightarrow A \cup B = B$
Fascinating induction problem with numerous interpretations
Subgroup generated by $1 – \sqrt{2}$, $2 – \sqrt{3}$, $\sqrt{3} – \sqrt{2}$
Graph of the function $x^y = y^x$, and $e$ (Euler's number).
What are some classic fallacious proofs?
How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?
Stirling numbers of the second kind on Multiset
How to prove $\dfrac{1+\sin{6^\circ}+\cos{12^\circ}}{\cos{6^\circ}+\sin{12^\circ}}=\sqrt{3}$
$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
Evaluating $\lim_{n\rightarrow\infty}x_{n+1}-x_n$
Sur- in- bijections and cardinality.
Proving $\frac{d}{dx}x^2=2x$ by definition