Related papers: A Finite Soluble Quotient Algorithm
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its…
Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form…
We describe an algorithm that computes the index of a finitely generated subgroup in a finitely $L$-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in…
By combining well-known techniques from both noncommutative algebra and computational commutative algebra, we observe that an algorithmic approach can be applied to the study of irreducible representations of finitely presented algebras. In…
Every mathematician is familiar with the beautiful structure of finite commutative groups. What is less well known is that finite commutative semigroups also have a neat and well-described structure. We prove this in an efficient fashion.…
Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum of isomorphic simple $\mathbb{Z}_p H$-modules such…
In this paper, using some properties of fundamental groups and covering spaces of connected polyhedra and CW-complexes, we present topological proof for some famous theorems about finitely presented groups.
Several structural properties of a universal algebra can be seen from the higher commutators of its congruences. Even on a finite algebra, the sequence of higher commutator operations is an infinite object. In the present paper, we exhibit…
We present two uncountable families of finitely generated residually finite groups all having the same profinite completion. One consists of soluble groups, the other of branch groups.
The concept of quantum representation of finite groups (QRFG) has been a fundamental aspect of quantum computing for quite some time, playing a role in every corner, from elementary quantum logic gates to the famous Shor's and Grover's…
The dominant theme of this thesis is the construction of matrix representations of finite solvable groups using a suitable system of generators. For a finite solvable group $G$ of order $N = p_{1}p_{2}\dots p_{n}$, where $p_{i}$'s are…
A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum. This is proved for the general case and the connection between the parameters in…
We introduce the subgroup identification problem, and show that there is a finitely presented group G for which it is unsolvable, and that it is uniformly solvable in the class of finitely presented locally Hopfian groups. This is done as…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative…
We determine a reasonable upper bound for the complexity of collection from the left to multiply two elements of a finite soluble, or polycyclic, group by restricting attention to certain polycyclic presentations of the group.
The Qth-power algorithm produces a useful canonical P-module presentation for the integral closures of certain integral extensions of $P:=\mathbf{F}[x_n,...,x_1]$, a polyonomial ring over the finite field $\mathbf{F}:=\mathbf{Z}_q$ of $q$…
We present an algorithm to compute the number of solutions of the (constrained) number partitioning problem. A concrete implementation of the algorithm on an Ising-type quantum computer is given.
In this introductory review, we focus on applications of quantum computation to problems of interest in physics and chemistry. We describe quantum simulation algorithms that have been developed for electronic-structure problems,…