相关论文: Generic-case complexity, decision problems in grou…

Myasnikov et al. have introduced the knapsack problem for arbitrary finitely generated groups. In previous work, the authors proved that for each graph group, the knapsack problem can be solved in $\mathsf{NP}$. Here, we determine the exact…

Initial steps are presented towards understanding which finitely generated groups are almost surely generated as semigroups by the path of a random walk on the group.

Using generating functions, we enumerate regular semisimple conjugacy classes in the finite classical groups. For the general linear, unitary, and symplectic groups this gives a different approach to known results; for the special…

In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.

We introduce the framework of general probabilistic theories (GPTs for short). GPTs are a class of operational theories that generalize both finite-dimensional classical and quantum theory, but they also include other, more exotic theories,…

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichm\"uller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the…

We establish a connection between the generalized conjugacy problem for a $G$-by-$\mathbb{Z}$ group, $GCP(G \rtimes \mathbb{Z})$, and two algorithmic problems for $G$: the generalized Brinkmann's conjugacy problem, $GBrCP(G)$, and the…

We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.

By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_\infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study…

Complex numbers define the relationship between entities in many situations. A canonical example would be the off-diagonal terms in a Hamiltonian matrix in quantum physics. Recent years have seen an increasing interest to extend the tools…

We consider the probability theory, and in particular the moment problem and universality theorems, for random groups of the sort of that arise or are conjectured to arise in number theory, and in related situations in topology and…

In this work, we will introduce the notion of generalized topological groups using generalized topological structure and generalized continuity defined by ?A. Cs?asz?ar [2]. We will discuss some basic properties of this kind of structures…

Weanalyzethecomputationalcomplexityofanalgorithmtosolve the conjugacy search problem in a certain family of metabelian groups. We prove that in general the time complexity of the conjugacy search problem for these groups is at most…

We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain…

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key…

In the world, in which acceptance and the identification with social communities are highly desired, the ability to predict evolution of groups over time appears to be a vital but very complex research problem. Therefore, we propose a new,…

A number of papers have examined various aspects of "random random" walks on finite groups; the purpose of this article is to provide a survey of this work and to show, bring together, and discuss some of the arguments and results in this…

Random walks (or Markov chains) are models extensively used in theoretical computer science. Several tools, including analysis of quantities such as hitting and mixing times, are helpful for devising randomized algorithms. A notable example…

This article concerns a class of generalized linear mixed models for clustered data, where the random effects are mapped uniquely onto the grouping structure and are independent between groups. We derive necessary and sufficient conditions…

We introduce and study Polish topologies on various spaces of countable enumerated groups, where an enumerated group is simply a group whose underlying set is the set of natural numbers. Using elementary tools and well known examples from…