Naming in Multichannel with Beeps in the Strong Model

In this paper, a system of anonymous processes is considered that communicates with beeps through multiple channels in a synchronous communication model. In beeping channels, processes are limited to hearing either a beep or a silence from the channel with no collision detection. A strong model is assumed in which a process can beep on any single channel and listen on any specific channel during a single round. The goal is to develop distributed naming algorithms for two models where the number of processes is either known or unknown. A Las Vegas algorithm was developed for naming anonymous processes when the number of processes is known. This algorithm has an optimal time complexity of O(nlogn) rounds and uses O(nlogn) random bits, where n is the number of processes for the largest group. For the model with an unknown number of processes, a Monte Carlo algorithm was developed, which has an optimal running time of O(nlogn) rounds and a probability of success that is at least 1−12Ω(logn). The algorithms solve the naming problem in new models where processes communicate through multiple channels.

On Random Symmetric Bimatrix Games

An experiment was conducted on a sample of [Formula: see text] randomly generated symmetric bimatrix games with size [Formula: see text] and [Formula: see text]. Distribution of support sizes and Nash equilibria are used to formulate a conjecture: for finding a symmetric NEP it is enough to check supports up to size [Formula: see text] whereas for nonsymmetric and all NEPs this number is [Formula: see text] and [Formula: see text], respectively. If true, this enables us to use a Las Vegas algorithm that finds a Nash equilibrium in polynomial time with high probability.

Champ: a Cherednik algebraMagmapackage

We present a computer algebra package based onMagmafor performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded$G$-module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.Supplementary materials are available with this article.

Computing Vf modulo p

This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing modular representations. An algorithm in this chapter usually means a probabilistic Las Vegas algorithm. In some places it gives deterministic or probabilistic Monte Carlo algorithms, but this will be stated explicitly. The main reason for using probabilistic Turing machines is that there is a need to construct generating sets for the Picard group of curves over finite fields. Solving such a problem in the deterministic world is out of reach at this time. The unique goal is to prove, as quickly as possible, that the problems studied in this chapter can be solved in probabilistic polynomial time.

A Reduction Algorithm for Large-Base Primitive Permutation Groups

The authors present a nearly linear-time Las Vegas algorithm that, given a large-base primitive permutation group, constructs its natural imprimitive representation. A large-base primitive permutation group is a subgroup of a wreath product of symmetric groups Sn and Sr in product action on r-tuples of k-element subsets of {1, …, n}, containing Anr. The algorithm is a randomised speed-up of a deterministic algorithm of Babai, Luks, and Seress.