Steiner triple systems with transrotational automorphisms

AbstractA Steiner triple system of order v is said to be k-transrotational if it admits an automorphism consisting of a fixed point, a transposition, and k cycles of length (v−3)k. Necessary and sufficient conditions are given for the existence of 1- and 2-transrotational Steiner triple systems

Configurations and colouring problems in block designs

A Steiner triple system of order v (STS(v)) is called x-chromatic if x is the smallest number of colours needed to avoid monochromatic blocks. Amongst our results on colour class structures we show that every STS (19) is 3- or 4-chromatic, that every 3-chromatic STS(19) has an equitable 3-colouring (meaning that the colours are as uniformly distributed as possible), and that for all admissible v > 25 there exists a 3-chromatic STS(v) which does not admit an equitable 3-colouring. We obtain a formula for the number of independent sets in an STS(v) and use it to show that an STS(21) must contain eight independent points. This leads to a simple proof that every STS(21) is 3- or 4-chromatic. Substantially extending existing tabulations, we provide an enumeration of STS trades of up to 12 blocks, and as an application we show that any pair of STS(15)s must be 3-1-isomorphic. We prove a general theorem that enables us to obtain formulae for the frequencies of occurrence of configurations in triple systems. Some of these are used in our proof that for v > 25 no STS(u) has a 3-existentially closed block intersection graph. Of specific interest in connection with a conjecture of Erdos are 6-sparse and perfect Steiner triple systems, characterized by the avoidance of specific configurations. We describe two direct constructions that produce 6-sparse STS(v)s and we give a recursive construction that preserves 6-sparseness. Also we settle an old question concerning the occurrence of perfect block transitive Steiner triple systems. Finally, we consider Steiner 5(2,4, v) designs that are built from collections of Steiner triple systems. We solve a longstanding problem by constructing such systems with v = 61 (Zoe’s design) and v = 100 (the design of the century)

Generating Uniformly-Distributed Random Generalised 2-designs with Block Size 3

PhDGeneralised t-designs, defined by Cameron, describe a generalisation of many combinatorial objects including: Latin squares, 1-factorisations of K2n (the complete graph on 2n vertices), and classical t-designs. This new relationship raises the question of how their respective theory would fare in a more general setting. In 1991, Jacobson and Matthews published an algorithm for generating uniformly distributed random Latin squares and Cameron conjectures that this work extends to other generalised 2-designs with block size 3. In this thesis, we divide Cameron’s conjecture into three parts. Firstly, for constants RC, RS and CS, we study a generalisation of Latin squares, which are (r c) grids whose cells each contain RC symbols from the set f1;2; : : : ; sg such that each symbol occurs RS times in each column and CS times in each row. We give fundamental theory about these objects, including an enumeration for small parameter values. Further, we prove that Cameron’s conjecture is true for these designs, for all admissible parameter values, which provides the first method for generating them uniformly at random. Secondly, we look at a generalisation of 1-factorisations of the complete graph. For constants NN and NC, these graphs have n vertices, each incident with NN coloured edges, such that each colour appears at each vertex NC times. We successfully show how to generate these designs uniformly at random when NC 0 (mod 2) and NN NC. Finally, we observe the difficulties that arise when trying to apply Jacobson and Matthews’ theory to the classical triple systems. Cameron’s conjecture remains open for these designs, however, there is mounting evidence which suggests an affirmative result. A function reference for DesignMC, the bespoke software that was used during this research, is provided in an appendix

An Epistemic Structuralist Account of Mathematical Knowledge

This thesis aims to explain the nature and justification of mathematical knowledge using an epistemic version of mathematical structuralism, that is a hybrid of Aristotelian structuralism and Hellman’s modal structuralism. Structuralism, the theory that mathematical entities are recurring structures or patterns, has become an increasingly prominent theory of mathematical ontology in the later decades of the twentieth century. The epistemically driven version of structuralism that is advocated in this thesis takes structures to be primarily physical, rather than Platonically abstract entities. A fundamental benefit of epistemic structuralism is that this account, unlike other accounts, can be integrated into a naturalistic epistemology, as well as being congruent with mathematical practice. In justifying mathematical knowledge, two levels of abstraction are introduced. Abstraction by simplification is how we extract mathematical structures from our experience of the physical world. Then, abstraction by extension, simplification or recombination are used to acquire concepts of derivative mathematical structures. It is argued that mathematical theories, like all other formal systems, do not completely capture everything about those aspects of the world they describe. This is made evident by exploring the implications of Skolem’s paradox, Gödel’s second incompleteness theorem and other limitative results. It is argued that these results demonstrate the relativity and theory-dependence of mathematical truths, rather than posing a serious threat to moderate realism. Since mathematics studies structures that originate in the physical world, mathematical knowledge is not significantly distinct from other kinds of scientific knowledge. A consequence of this view about mathematical knowledge is that we can never have absolute certainty, even in mathematics. Even so, by refining and improving mathematical concepts, our knowledge of mathematics becomes increasingly powerful and accurate

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

On the relationship between plane and solid geometry

Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned area

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum