Some z_n-1 terraces from z_n power-sequences, n being an odd prime power

A terrace for Zm is a particular type of sequence formed from the m elements of Zm. For m odd, many procedures are available for constructing power-sequence terraces for Zm; each terrace of this sort may be partitioned into segments, of which one contains merely the zero element of Zm, whereas every other segment is either a sequence of successive powers of an element of Zm or such a sequence multiplied throughout by a constant. We now refine this idea to show that, for m=n−1, where n is an odd prime power, there are many ways in which power-sequences in Zn can be used to arrange the elements of Zn \ {0} in a sequence of distinct entries i, 1 ≤ i ≤ m, usually in two or more segments, which becomes a terrace for Zm when interpreted modulo m instead of modulo n. Our constructions provide terraces for Zn-1 for all prime powers n satisfying 0 < n < 300 except for n = 125, 127 and 257

Narcissistic half-and-half power-sequence terraces for Zn with n=pqt

AbstractA power-sequence terrace for Zn is a Zn terrace that can be partitioned into segments one of which contains merely the zero element of Zn whilst each other segment is either (a) a sequence of successive powers of an element of Zn, or (b) such a sequence multiplied throughout by a constant. If n is odd, a Zn terrace (a1,a2,…,an) is a narcissistic half-and-half terrace if ai−ai−1=an+2−i−an+1−i for i=2,3,…,(n+1)/2. Constructions are provided for narcissistic half-and-half power-sequence terraces for Zn with n=pqt where p and q are distinct odd primes and t is a positive integer. All the constructions are for terraces with as few segments as possible. Attention is restricted to constructions covering values of n with n=pqt and n<300; terraces are provided for all such values except n=189. Particularly elegant constructions are available for n=275

Sectionable terraces and the (generalised) Oberwolfach problem

AbstractThe generalised Oberwolfach problem requires v people to sit at s round tables of sizes l1,l2,…,ls (where l1+l2+⋯+ls=v) for successive meals in such a way that each pair of people are neighbours exactly λ times. The problem is denoted OP(λ;l1,l2,…,ls) and if λ=1, which is the original problem, this is abbreviated to OP(l1,l2,…,ls). It was known in 1892, though different terminology was then used, that a directed terrace with a symmetric sequencing for the cyclic group of order 2n can be used to solve OP(2n+1). We show how terraces with special properties can be used to solve OP(2;l1,l2) and OP(l1,l1,l2) for a wide selection of values of l1, l2 and v. We also give a new solution to OP(2;l,l) that is based on Z2l−1. Solutions to the problem are also of use in the design of experiments, where solutions for tables of equal size are called resolvable balanced circuit Rees neighbour designs

The Gordon game

In 1992, about 30 years after Gordon introduced group sequencings to construct row-complete Latin squares, John Isbell introduced the idea of competitive sequencing, the Gordon Game. Isbell investigated the Gordon Game and found solutions for groups of small order. The purpose of this thesis is to analyze the Gordon Game and develop a brute force method of determining solutions to the game for all groups of order 12 (up to isomorphism) as well as for abelian groups of order less than 20. The method used will be a depth first search program written in MATLAB. Consequently, group representation using matrices will be studied within the thesis --Document

Some constructions of combinatorial designs

The objects of study of this thesis are combinatorial designs. Chapters 2 and 3 deal with various refinements of whist tournament, while Chapters 3 and 4 focus on terraces. Chapter 2 is devoted to the investigation of Z-cyclic ordered triplewhist tournaments on p elements, where p = 5 (mod 8); Z-cyclic ordered triplewhist and directed triplewhist tournaments on p elements, where p = 9 (mod 16); and Z-cyclic directed moore (2,6) generalised whist tournament designs on p elements, where p = 7 (mod 12). In each of these cases, p is prime. In an effort to prove the existence of an infinite family of each of these tournaments, constructions are introduced and the conditions under which they give the initial round of a tournament of the kind we desire are found. A bound above which these conditions are always satisfied is then obtained, and we try to fill in the appropriate gaps below that bound. In Chapter 3 we investigate the existence of tournaments of the type seen in Chapter 2 which involve four players per game, with an additional property. This is known as the three person property and is defined in Chapter 1. Here, we focus on one of the constructions introduced in Chapter 2 for each type of tournament. Then we find a new bound using only that construction with the additional conditions introduced by the three person property, and again try to fill in the appropriate gaps below the bound. Chapter 4 is an investigation of logarithmic terraces and their properties. Very little work has been done on them previously, so this was really an opportunity to look at them more closely in an effort to find as many interesting properties as possible. Some general results and examples are given, with the focal point of the chapter being the study of terraces which are simultaneously logarithmic for two different primitive roots. In Chapter 5, a more specific problem is addressed which involves training schedules for athletes. Here we want n(n - 1) athletes to carry out n tasks in some order, then keep repeating them in different orders in blocks of n as many times as possible so that certain conditions are satisfied. These conditions are listed in Chapter 5. We make use of the Williams terrace and the Owens terrace in our attempt to find a general method which allows the given conditions to be satisfied and gets as close as possible to the theoretical limit where each athlete carries out the n tasks n - 1 times

Nonlinear dynamics in one dimension: On a criterion for coarsening and its temporal law

We develop a general criterion about coarsening for a class of nonlinear evolution equations describing one dimensional pattern-forming systems. This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time. An intermediate scenario may occur, namely `interrupted coarsening'. The power of the criterion lies in the fact that the statement about the occurrence of coarsening, or selection of a length scale, can be made by only inspecting the behavior of the branch of steady state periodic solutions. The criterion states that coarsening occurs if lambda'(A)>0 while a length scale selection prevails if lambda'(A)<0, where $lambda$ is the wavelength of the pattern and A is the amplitude of the profile. This criterion is established thanks to the analysis of the phase diffusion equation of the pattern. We connect the phase diffusion coefficient D(lambda) (which carries a kinetic information) to lambda'(A), which refers to a pure steady state property. The relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically for several classes of equations. Another important and new result which emerges here is that the exploitation of the phase diffusion coefficient enables us to determine in a rather straightforward manner the dynamical coarsening exponent. Our calculation, based on the idea that |D(lambda)|=lambda^2/t, is exemplified on several nonlinear equations, showing that the exact exponent is captured. Some speculations about the extension of the present results to higher dimension are outlined.Comment: 16 pages. Only a few minor changes. Accepted for publication in Physical Review