The purpose of this note is to give an accessible proof of Moliens Theorem in Invariant Theory, in the language of today's Linear Algebra and Group Theory, in order to prevent this beautiful theorem from being forgotten.

1 The bias and a direct link to codes We study certain classes of structures, which are being widely used in computer science and cryptography. These structures comprise ?biased arrays , t-wise ?biased arrays and t-wise ?dependent arrays. A link between ?biased arrays and error-correcting codes will be exhibited. This paper is not designed to be published. Rather we make an eeort to collect all the pertinent theory, including loose ends.

A practical implementation of a genetic algorithm for routing a real autonomous robot through a changing environment is described. Moving around in a production plant the robot collects information about its environment and stores it in a temporal map, which is virtually a square grid, taking account of changing obstacles. The evolutional optimizer continuously searches for short paths in this map using string representations of paths as chromosomes. The main features of the implementation include physical realization, random walk exploration, temporal mapping, and dedicated genetic operators.

We propose an aperiodic tiling suitable for a floor pattern to provide an autonomous robot with position information. The most direct approach to this problem would be to equip the robot with an optical sensor to detect and count its traversals across the lines of a rectangular grid painted on the floor. However, dirt and other sensing errors would result in missing counts and imprecise information of its absolute position. Our approach replaces the rectangular grid by aperiodic tilings similar to Penrose tilings. These patterns may have rotational symmetries, but no nontrivial translational symmetries. By scanning a small area the robot can gather information about its position, where the precision of the information is increasing as the scanned area increases, using an optical sensor. The position information is encoded into the local configuration of the tiles, i.e. into the way the tiles are put together locally. Scanning a larger neighbourhood will increase the precision of the coordinate information. Advantages of this approach include scalability and low cost. Further we discuss a practical implementation, an application to 3-D object model recovery from 2-D images using structured light, and an extension to three dimensional position detection, where the two dimensional tiles are replaced by three dimensional analogs

Motivated by a construction of highly expanding simple Cayley graphs of dihedral groups derived from-or induced by-highly expanding Cayley digraphs fo cyclic groups presented by F. R. K. chung, constructions of simple Cayley graphs on a semidirect product of groups and Cayley digraphs of one of its factors are suggested. In case of the non-normal factor being the cyclic group of order 2. a condition is given to derive spectral bounds of the Cayley graph of the product from those of the Cayley graph of the normal factor.