Related papers: Algorithmic and combinatorial methods for enumerat…
In this article we discuss the presentation of a random binary matrix using sequence of whole nonnegative numbers. We examine some advantages and disadvantages of this presentation as an alternative of the standard presentation using…
An equivalence relation in the symmetric group, where is a positive integer has been considered. An algorithm for calculation of the number of the equivalence classes by this relation for arbitrary integer has been described.
We give a conjectural presentation of the infinitely generated group PGL(2,Q) with an infinite list of relators.
We generalize the Plesken-Fabia\'nska $\mathrm{L}_2$-quotient algorithm for finitely presented groups on two or three generators to allow an arbitrary number of generators. The main difficulty lies in a constructive description of the…
We construct examples of finitely generated decidable group presentations that satisfy certain combinations of solvability for the word problem, solvability for the bounded word problem, and computablity for the Dehn function. We prove that…
The braid group appears in many scientific fields and its representations are instrumental in understanding topological quantum algorithms, topological entropy, classification of manifolds and so on. In this work, we study planer diagrams…
We construct a group K_n with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes and simplicial chambers. The generators of K_n are given by 2-element subsets of {0, .., n}.…
A generalization of recent group-theoretic matrix multiplication algorithms to an analogue of the theory of partial matrix multiplication is presented. We demonstrate that the added flexibility of this approach can in some cases improve…
We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for practical computation in…
We give an efficient algorithm to enumerate all sets of $r\ge 1$ quadratic polynomials over a finite field, which remain irreducible under iterations and compositions.
This habilitation thesis is intended to be a good introduction to enumeration, the problem of listing solutions. It focuses on the different ways of measuring complexity in enumeration, with a particular emphasis on my contributions to the…
Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group. In the course of the proof of this theorem, we obtain some other results of independent interest. For instance,…
Generalizing classical extension theory, we solve a Schreier-type extension problem for polygroups by groups. As a consequence, we obtain a method for computing a presentation for a group from its action on a set. The usefulness of this…
The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…
In this paper, we examine the general algorithm for class group computations, when we do not have a small defining polynomial for the number field. Based on a result of Biasse and Fieker, we simplify their algorithm, improve the complexity…
We present an algorithm that unconditionally computes a representation of the unit group of a number field of discriminant $\Delta_K$, given a full-rank subgroup as input, in asymptotically fewer bit operations than the baby-step giant-step…
The aim of this thesis is to determine classes of NP relations for which random generation and approximate counting problems admit an efficient solution. Since efficient rank implies efficient random generation, we first investigate some…
We give an efficient algorithm to enumerate all elements of a Pareto front in a multi-objective optimization problem in which the space of values is finite for all objectives. Our algorithm uses a feasibility check for a search space…
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups…
We completely generalize previous results related to the counting of connected Feynman diagrams. We use a generating function approach, which encodes the Wick contraction combinatorics of the respective connected diagrams. Exact solutions…