Theta Graph Designs

We solve the design spectrum problem for all theta graphs with 10, 11, 12, 13, 14 and 15 edges

Completing the design spectra for graphs with six vertices and eight edges

Apart from two possible exceptions, the design spectrum has been determined for every graph with six vertices and at most eight edges. The purpose of this note is to establish the existence of the two missing designs, both of order 32

Literacy and numeracy skills and labour market outcomes in Australia

Australian adults are above the OECD average in literacy but only average in numeracy, according to a staff paper released by the Productivity Commission. The paper analyses the profile of adult literacy and numeracy skills in Australia, and how important those skills are for labour market outcomes. Key points: Adult literacy and numeracy skills contribute to wellbeing in many ways. At an individual level, they are central to social and economic participation. Literacy and numeracy skills are a core part of a person\u27s human capital. They also support the development of other forms of human capital, including knowledge, other skills and health. Some Australians have low (level 1 or below) literacy and numeracy skills. In 2011–12: 14 per cent of Australians could, at best, read only relatively short texts from which they were able to locate only a single piece of information. 22 per cent could only carry out one-step or simple processes such as counting where the mathematical content is explicit with little or no text or distractors. At the other end of the skill distribution, 16 per cent of Australians had high (level 4/5) literacy skills and 12 per cent had high numeracy skills in 2011–12. People with high literacy skills can make complex inferences and evaluate subtle truth claims or arguments in lengthy or multiple texts. People with high numeracy skills can understand a broad range of mathematical information that may be complex, abstract or embedded in unfamiliar contexts. Most Australians have skills somewhere between these levels. Groups with relatively low literacy and numeracy skills include: people with low levels of education; older persons; people not working; and immigrants with a non-English speaking background. Compared with other countries in the OECD, Australia performs above average on literacy but average in numeracy. Higher literacy and numeracy skills are associated with better labour market outcomes (employment and wages). Econometric modelling shows that: an increase in literacy and numeracy by one skill level is associated with an increased likelihood of employment of 2.4 and 4.3 percentage points for men and women, respectively an increase in literacy and numeracy skills is associated with a similar increase in the probability of employment, whether a person had a degree, diploma/certificate or Year 12 education an increase in literacy and numeracy by one skill level is associated with about a 10 per cent increase in wages for both men and women. This positive association is equivalent to that of increasing educational attainment from Year 11 to Year 12 or to a diploma/certificate up to 40 per cent of the association between education and employment is attributable to literacy and numeracy skills. These results are consistent with education providing many other attributes of human capital that are valued in the workplace more than half of the \u27penalty\u27 that affects the wages of people with a non-English speaking background is explained by their lower literacy and numeracy skills. • Staff working papers are not formal publications of the Commission. They have been prepared and are authored by individual staff to advance understanding of issues on the Commission’s supporting research program

Snark Designs

The main aim of this paper is to solve the design spectrum problem for Tietze's graph, the two 18-vertex Blanusa snarks, the six snarks on 20 vertices (including the flower snark J5), the twenty snarks on 22 vertices (including the two Loupekine snarks) and Goldberg's snark 3. Together with the Petersen graph (for which the spectrum has already been computed) this list includes all non-trivial snarks of up to 22 vertices. We also give partial results for a selection of larger graphs: the two Celmins-Swart snarks, the 26- and 34-vertex Blanusa snarks, the flower snark J7, the double star snark, Zamfirescu's graph, Goldberg's snark 5, the Szekeres snark and the Watkins snark.Comment: 92 pages. References updated, pictures added, typos corrected. Abridged version omitting all design details accepted by Utilitas Mathematic

Group divisible designs with block size five: direct constructions

We give direct constructions for 226 group divisible designs with block size five, mostly of type $g^u m^1$, $m > 0$.Comment: 54 pages. This is intended as a supplement to a paper currently under revie

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)

Designs for graphs with six vertices and ten edges - II

The design spectrum has been determined for ten of the 15 graphs with six vertices and ten edges. In this paper, we solve the design spectrum problem for the remaining five graphs with three possible exceptions

Ulcer syndrome in tropical Africa

I. The exact relationship of Bacillus fusi - formis and Spironema to Ulcus tropicum remains unproved, but the evidence suggests that these organisms are causal.II. Ulcer, whatever the clinical type, is a physical sign, and is of much greater significance than the one or two page description, usually given in books, would indicate.III. Ulcer is a class disease, and is not found among the better classes or in institutions.IV. Ulcer seems to be closely related to calcium and/or parathyroid deficiency, and may be a manifestation of latent tetany.V. The calcium deficiency does not appear to be in the diet, but seems to be due to the diet, faulty absorption probably being a factor.VI. The exact connection of the parathyroid with the condition is not clear, but the gland appears to have factors controlling calcium regulation and skin vitality and résistance, and it may be affected by dietary deficiency.VII. One of the most important considerations is the quality of the protein in the diet. Ulcers are much more common in vegetarians than in those who include animal substance in their diet. It is essential that a fair proportion of the protein should be of animal origin,as even nuts and beans have the disadvantage of being covered with resistant fibre.VIII. The vegetarian diet, as consumed by the African native, is much too) bulky. Digestion of protein, fat and calcium may be greatly interfered with. The most important prophylactic measure, which would go far to prevent the appearance of the ulcer syndrome, is to include meat, eggs, fish, or fowl in the diet, and to reduce the amount of vegetable matter; the consumption of three smaller meals instead of two larger, would favour optimal utilisation