Related papers: Quasirandomness in additive groups and hypergraphs
We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a…
It is proposed to define "quantumness" of a system (micro or macroscopic, physical, biological, social, political) by starting with understanding that quantum mechanics is a statistical theory. It says us only about probability…
Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement'', a property exhibited by ensembles of…
In this paper, we present a notion of quasiconvexity in the setting of finitely-generated groups with hyperbolically embedded subgroups. Our main result shows that this notion yields uniform quasiconvex constants in the setting of coned-off…
In contrast with software-generated randomness (called pseudo-randomness), quantum randomness is provable incomputable, i.e.\ it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability --- an…
A notion of a quantum automorphism group of a finite quantum group, generalising that of a classical automorphism group of a finite group, is proposed and a corresponding existence result proved.
This introductory survey deals with mathematical and physical properties of discrete structures such as point sets and tilings. The emphasis is on proper generalizations of concepts and ideas from classical crystallography. In particular,…
A rigorous general definition of quantum probability is given, which is valid for elementary events and for composite events, for operationally testable measurements as well as for inconclusive measurements, and also for non-commuting…
We discuss the notion of the universal relatively hyperbolic structure on a group which is used in order to characterize relatively hyperbolic structures on the group. We also study relations between relatively hyperbolic structures on a…
The basic idea of quantum complexity geometry is to endow the space of unitary matrices with a metric, engineered to make complex operators far from the origin, and simple operators near. By restricting our attention to a finite subgroup of…
The purpose of this chapter is to provide an overview of the exciting field of black hole quasi-normal modes and its capabilities to test general relativity in the 21st century. After motivating this line of research, we provide a…
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase…
A semi-projective representation is a homomorphism of a finite group into the group of semi-projective transformations of a finite dimensional vector space over a field. Schur's concept of a representation group for projective…
Many learning machines such as normal mixtures and layered neural networks are not regular but singular statistical models, because the map from a parameter to a probability distribution is not one-to-one. The conventional statistical…
Motivated by recent appearance of multivalued structures in categorification, tropical geometry and other areas, we study basic properties of abstract multisemigroups. We give many new and old examples and general constructions for…
Random intersection graphs model networks with communities, assuming an underlying bipartite structure of groups and individuals, where these groups may overlap. Group memberships are generated through the bipartite configuration model.…
It has been experimentally demonstrated that quantum coherence can persist in macroscopic phenomena [J.R. Friedman et al.,Nature, 406 (2000) 43]. To face the challenge of this new fact, in this article QM in its standard form is assumed to…
We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely…
A group is coherent if all its finitely generated subgroups are finitely presented. In this article we provide a criterion for positively determining the coherence of a group. This criterion is based upon the notion of the perimeter of a…
The form factor of a quantum graph is a function measuring correlations within the spectrum of the graph. It can be expressed as a double sum over the periodic orbits on the graph. We propose a scheme which allows one to evaluate the…