Related papers: A combinatorial problem arising in group theory
We construct a finitely presented (two-sided) totally orderable group with insoluble word problem.
We develop the representation theory of a finite semigroup over an arbitrary commutative semiring with unit, in particular classifying the irreducible and minimal representations. The results for an arbitrary semiring are as good as the…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable…
In this paper we show that certain special cases of the hidden subgroup problem can be solved in polynomial time by a quantum algorithm. These special cases involve finding hidden normal subgroups of solvable groups and permutation groups,…
We present an algorithm to solve a system of diagonal polynomial equations over finite fields when the number of variables is greater than some fixed polynomial of the number of equations whose degree depends only on the degree of the…
A Dirichlet-type problem is studied for an equation of even order with variable coefficients. A criterion for the uniqueness of a solution is given. The solution is built in the form of a Fourier series. When justifying the convergence of…
It is shown that there exists a finitely generated infinite simple group of infinite commutator width, and that the commutator width of a finitely generated infinite boundedly simple group can be arbitrarily large. Besides, such groups can…
The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an…
Combinatorial $t$-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a $t$-design. Till now only a small amount of…
We consider the problem of intruder deduction in security protocol analysis: that is, deciding whether a given message $M$ can be deduced from a set of messages $\Gamma$ under the theory of blind signatures and arbitrary convergent…
We present a polynomial-time quantum algorithm for the Hidden Subgroup Problem over $\mathbb{D}_{2^n}$. The usual approach to the Hidden Subgroup Problem relies on harmonic analysis in the domain of the problem, and the best known algorithm…
In this study, we investigate a mixed problem linked to a second-order parabolic equation, characterized by temporal dependencies and variable~coefficients, and constrained by non-local, non-self-adjoint boundary conditions. By defining…
In 2005 Wolfgang Willems put forward a conjecture proposing a lower bound for the sum of squares of the degrees of the irreducible $p$-Brauer characters of a finite group $G$. We prove this conjecture for the prime $p=2$. For this we rely…
A survey of problems, conjectures, and theorems about quasi-isometric classification and rigidity for finitely generated solvable groups.
Computational problems concerning the orbit of a point under the action of a matrix group occur throughout computer science, including in program analysis, complexity theory, quantum computation, and automata theory. In many cases the focus…
We study irreducible restrictions from modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the…
This paper concerns the characterisation of second order marginals for random sets in a discrete setting. Under the instance of unit covariances, this problem possesses a combinatorial symmetry, exploited jointly in the companion paper to…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
We develop and study a Lefschetz theory in a combinatorial category associated to a root system and derive an upper bound on the exceptional characteristics for Lusztig's formula for the simple rational characters of a reductive algebraic…