If G is a semidirect product N oH with N finitely generated then G has the property that every finite group is a quotient of some finite index subgroup of G if and only if one of N and H has this property. This has applications to 3-manifolds; for instance for any fibred hyperbolic 3-manifold M and any finite simple group S, there is a finite cyclic cover of M whose fundamental group surjects t...

We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two...

We show that if a finite simple group G isomorphic to PSLn(q) or PSUn(q), where either n 6= 4, or q is prime or even, acts on a vector space over a field of the defining characteristic of G, then the corresponding semidirect product contains an element whose order is distinct from every element order of G. As a consequence, we prove that the group PSLn(q), n 6= 4 or q prime or even, is recogniz...

This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic constraints, mechanical systems with symmetry and mechanical systems evolving on s...

In this paper we present a cohomological description of the equivariant Brauer group relative to a Galois finite extension of fields endowed with the action of a group of operators. This description is a natural generalization of the classic Brauer–Hasse–Noether’s theorem, and it is established by means of a three-term exact sequence linking the relative equivariant Brauer group, the 2nd cohomo...

Tropical linear algebra has been recently put forward by Grigoriev and Shpilrain as a promising platform for implementation of protocols Diffie-Hellman Stickel type. Based on the CSR expansion tropical matrix powers, we suggest simple algorithm following discrete logarithm problem: “Given that A=V?F?t unique t matrices A, V, F appropriate dimensions, find this t.” We then use to attack protocol...

A Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce different weights for distinct nuclei. The aim of this paper is to compute the automorphism group of the Euclide...

We present the cyclic coordinate descent (CCD) algorithm for optimizing quadratic objective functions on (3), and apply it to a class of robot sensor calibration problems. Exploiting the fact that (3) is the semidirect product of (3) and , we show that by cyclically optimizing between these two spaces, global convergence can be assured under a mild set of assumptions. The CCD algorithm is also ...

The purpose of this paper is to classify invariant hypercomplex structures on a 4-dimensional real Lie group G. It is shown that the 4dimensional simply connected Lie groups which admit invariant hypercomplex structures are the additive group H of the quaternions, the multiplicative group H∗ of nonzero quaternions, the solvable Lie groups acting simply transitively on the real and complex hyper...

This is a sequel to our paper (Lett. Math. Phys. (2000)), triggered from a question posed by Marcel, Ovsienko, and Roger in their paper (1997). In this paper, we show that the multicomponent (or vector) Ito equation, modified dispersive water wave equation, and modified dispersionless long wave equation are the geodesic flows with respect to an L2 metric on the semidirect product space ̂ Diffs(S...