Related papers: Generic-case complexity, decision problems in grou…
Phase transitions in combinatorial problems have recently been shown to be useful in locating "hard" instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been…
Complexity is an interdisciplinary concept which, first of all, addresses the question of how order emerges out of randomness. For many reasons matrices provide a very practical and powerful tool in approaching and quantifying the related…
We study a number of local and global classification problems in generalized complex geometry. In the first topic, we characterize the local structure of generalized complex manifolds by proving that a generalized complex structure near a…
We study random walks in a random environment on a regular, rooted, coloured tree. The asymptotic behaviour of the walks is classified for ergodicity/transience in terms of the geometric properties of the matrix describing the random…
Simple random walks are a basic staple of the foundation of probability theory and form the building block of many useful and complex stochastic processes. In this paper we study a natural generalization of the random walk to a process in…
Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability.…
We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G^n is already the whole of G^n , for various equationally defined subsets. We link this with the inner measure of…
The cover-time problem, i.e., time to visit every site in a system, is one of the key issues of random walks with wide applications in natural, social, and engineered systems. Addressing the full distribution of cover times for random walk…
Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our…
We consider random walk on a finite group $G$ as follows. We can consider $G$ as a group of substitutions. Randomly (i.e. with probability $U(g)=|G|^{-1}$ ) we choose a substitution $g \in G$ and execute it twice in a row, i.e. execute a…
We analyse the computational complexity of three problems in judgment aggregation: (1) computing a collective judgment from a profile of individual judgments (the winner determination problem); (2) deciding whether a given agent can…
In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.
Given a graph G, we investigate the question of determining the parity of the number of homomorphisms from G to some other fixed graph H. We conjecture that this problem exhibits a complexity dichotomy, such that all parity graph…
We consider invariants of a finite group related to the number of random (independent, uniformly distributed) conjugacy classes which are required to generate it. These invariants are intuitively related to problems of Galois theory. We…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
In this article we consider transient random walks on HNN extensions of finitely generated groups. We prove that the rate of escape w.r.t. some generalised word length exists. Moreover, a central limit theorem with respect to the…
A general simplicity problem in category theory is proposed. A particular example, the simplest choice of generators of an algebra is specified and illustrated by an example.
The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing…
We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form…
Many dynamical phenomena display a cyclic behavior, in the sense that time can be partitioned into units within which distributional aspects of a process are homogeneous. In this paper, we introduce a class of models - called conjugate…