We summarize some of the recent developments of our research group and of other groups in the design and analysis of pseudorandom number generators for massively parallel computers. The three parallelization techniques we will consider in detail for mapping pseudorandom streams onto distinct parallel processes are: 1. Splitting maximal-period generators' full period into nonoverlapping subse-qu...

Currently, the way we represent stabilizer states is by hand. However, this method is neither efficient nor productive. Several algorithms have been proposed to address the issue of efficient simulation of stabilizer states, but no one has implemented them yet. This paper discusses how the graph state approach method is used to develop a software application that will make quantum state visuali...

The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [3] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. For a recent survey, see Mühlherr [10...

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from Cn, or complex hyperbolic space CHn, or the Hermitian symmetric space for O(2, n), and then takes the quotient by a discrete group PΓ. The classical example is the braid group, but there are many similar “braid-like” groups that arise in topol...

We show that the normal closure of any periodic element mapping class group a nonorientable surface whose order is greater than 2 contains commutator subgroup, which for g≥7 equal to twist and provide necessary sufficient conditions closures involutions contain subgroup. Finally, we criterion normally generate M(Ng) give examples.

Recent work has used deep learning to derive symmetry transformations, which preserve conserved quantities, and obtain the corresponding algebras of generators. In this letter, we extend technique sparse representations arbitrary Lie algebras. We show that our method reproduces canonical (sparse) generators Lorentz group, as well U(n) SU(n) families groups. This approach is completely general c...

The Nielsen-Schreier Theorem states that every subgroup of a free group is free. To formalize this theorem in weak subsystems of second order arithmetic, one has to choose between defining a subgroup in terms of a set of group elements and defining it in terms of a set of generators. We show that if subgroups are defined by sets, then the Nielsen-Schreier Theorem is provable in RCA0, while if s...

A semi-presentation for a group G is a set of relations which characterises a set of generators of G up to automorphism. We discuss some techniques for finding semi-presentations and illustrate them by exhibiting semi-presentations on standard generators for the 26 sporadic simple groups and their automorphism groups. We then show how these semipresentations were used to check the data in the W...

We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative.