Properties of Steiner triple systems of order 21

Properties of the 62,336,617 Steiner triple systems of order 21 with a non-trivial automorphism group are examined. In particular, there are 28 which have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20 that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and two that avoid the prism. All systems contain the grid. None have a block intersection graph that is 3-existentially closed.Comment: 12 page

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)

Inference and Mutual Information on Random Factor Graphs

Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created; in the stochastic block model, this would be the planted partition. The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs we verify a formula for the mutual information predicted by physics techniques. As an application we prove a conjecture about low-density generator matrix codes from [Montanari: IEEE Transactions on Information Theory 2005]. Further applications include phase transitions of the stochastic block model and the mixed k-spin model from physics

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

Overview of Turbomachinery for Super-Critical CO2 Applications

TutorialsCycles involving super critical carbon dioxide (sCO2) have the potential to increase system efficiencies well beyond current industry norms. Research on advanced direct and indirect cycles is ongoing in national labs and major companies. sCO2 machinery tends to have small foot print sizes making for excellent applications in marine, or space limited, environments. As scCO2 turbomachinery gains acceptance in various industries the need to understand the applications, potential, and limits is paramount. Discussed in this tutorial are 1) various direct and indirect cycles 2) various applications, and 3) specific impact to turbomachinery design. Specific applications are described in detail including waste heat recovery, power generation, concentrated solar power, and marine applications. Discussed are transient, thermalmechanical, material, rotordynamic and many other factors affecting the turbomachinery

Spartan Daily, October 6, 1986

Volume 87, Issue 27https://scholarworks.sjsu.edu/spartandaily/7484/thumbnail.jp

Critical sets of full Latin squares

This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes. In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs. Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest. Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible. Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists. In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable. As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8