Boolean algebras and Lubell functions

Let $2^{[n]}$ denote the power set of $[n]:=\{1,2,..., n\}$. A collection \B\subset 2^{[n]} forms a $d$-dimensional {\em Boolean algebra} if there exist pairwise disjoint sets $X_0, X_1,..., X_d \subseteq [n]$, all non-empty with perhaps the exception of $X_0$, so that \B={X_0\cup \bigcup_{i\in I} X_i\colon I\subseteq [d]}. Let $b(n,d)$ be the maximum cardinality of a family \F\subset 2^X that does not contain a $d$-dimensional Boolean algebra. Gunderson, R\"odl, and Sidorenko proved that $b(n,d) \leq c_d n^{-1/2^d} \cdot 2^n$ where $c_d= 10^d 2^{-2^{1-d}}d^{d-2^{-d}}$. In this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets \F with \F\subseteq \tsupn is defined by h_n(\F):=\sum_{F\in \F}1/{{n\choose |F|}}. We prove the following Tur\'an type theorem. If \F\subseteq 2^{[n]} contains no $d$-dimensional Boolean algebra, then h_n(\F)\leq 2(n+1)^{1-2^{1-d}} for sufficiently large $n$. This results implies $b(n,d) \leq C n^{-1/2^d} \cdot 2^n$, where $C$ is an absolute constant independent of $n$ and $d$. As a consequence, we improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras.Comment: 10 page

Life After Debt in Puerto Rico: How Many More Lost Decades?

As Puerto Rico enters the legal debt restructuring process, this paper examines the future prospects for an economy that has had no growth over the last ten years — a lost decade. The facts indicate that the fiscal plan approved by Puerto Rico's Financial Oversight Board will not lead to an economic recovery in the foreseeable future and that another lost decade, or worse, is a much more likely outcome. Nor will the proposed restructuring satisfy creditors, who may then further impede economic recovery by taking legal action in an attempt to collect the full value of the bonds they hold. The paper also notes the recent historical and structural causes of the economy's decline, which led to the debt crisis; and that these, too, will need to be addressed if Puerto Rico is ever to have a sustainable recovery

Methodological Developments in the Use of Stable Isotope Analysis for Reconstructing Paleodiets

Understanding the composition of paleodiets is crucially important for the archaeological investigation of lifeways and subsistence practices within ancient societies. Among the numerous research methods capable of investigating various elements of paleodiets, the use of stable isotope analysis has become increasingly popular. Advantages of this research technique include its ability to recognize multiple types of dietary inputs through the examination of stable carbon and stable nitrogen isotopes, and to investigate dietary composition and variation at an individual level. This research paper examines how this methodology has developed through time. Specific attention will be directed toward both the procedural advances that continue to reshape the utility of this method, as well as changes in the theoretical assumptions that have guided its application

An IS perspective on managing change in a university

This thesis aimed firstly to explore what is happening with respect to change in a South African university, and how this change affects those involved, viewed from an IS standpoint. The focus is on determining the essence and nature of organisational change in its usual situation

Information technology policy : first steps for the new South Africa

Bibliography: pages 183-200.The fundamental question is not whether South Africa should have a National Information Technology Policy to exploit Information Technology (IT), but more fundamentally what kind of government policies and programs for IT make sense. In today's competitive world what kind of policies and programs will be for the betterment of South Africa's economic and social structures? How can the South African government use IT to increase international competitiveness, and at the same time enhance economic and technological progress and create social improvement in the country? These and many other fundamental questions are the basis for this investigation. The starting point of this research was to compare South African IT to that in other countries; Expert opinion was obtained to identify specific issues, goals and technologies, and to propose ideas for government policies to: i) increase the country's competitiveness, ii) enhance its economic and technological progress, and iii) uplift its people. The main points emerging from this research are that the South African government needs to formulate a National IT Policy in order to direct the development of IT. Competitiveness, progress and social upliftment are inseparable and must be balanced when positioning such a policy. The policy should be directed from a high level to ensure that its recommendations are followed

An IS perspective on managing change in a university

This thesis aimed firstly to explore what is happening with respect to change in a South African university, and how this change affects those involved, viewed from an IS standpoint. The focus is on determining the essence and nature of organisational change in its usual situation

Mechanical strength of silica fiber splices after exposure to extreme temperatures

By using a combination of type-I and regenerated gratings, the mechanical strength of optical fiber splices after exposure to temperatures over 1300 C was characterized. Splice strength was found to decrease with temperature with a secondorder polynomial dependence after exposure to environments hotter than 500 C. Splices exposed to temperatures above 1300 C were 80% more fragile than non-exposed splices. The lack of optical attenuation and the narrowing distribution of breaking strengths for higher temperatures suggest surface damage mechanisms, such as hydrolysis, play a key role in weakening post-heating and that damage mechanisms dominate over strengthening induced by crack melting

Certified Operators: Does Certification Provide Significant Results in Real-World Pool & Spa Chemistry?

This study statistically compared operators certified as YMCA pool operator on location (POOL)) with noncertified operators managing YMCA aquatic facilities with respect to water chemistry (pH and Langelier Saturation Index [LSI]) and chlorine levels (free-available and combined-chlorine levels). The study used a convenience sample of 572 pools and spas located at approximately 250 YMCAs. Certification was used as the factor, and the responses included pH, compliance with pH, free chlorine, compliance with free chlorine, combined chlorine, compliance with combined chlorine, LSI, and compliance with LSI. The results indicate that the POOL certification program does provide some significant results related to the proper care of pools and spas at YMCAs. Significant differences (p \u3c .05) were found between certified and noncertified operators in the level of pH, the level of combined chlorine, and compliance with combined-chlorine standards. The findings of this study underscore the need for increased training for pool and spa operators and greater responsibility on their behalf in maintaining safe chemical conditions in the water. This research supports the YMCA and the aquatic profession that POOL-certified operators do make a difference in maintaining water quality and safer water-chemistry standards

Critical insights into the design of big data analytics research: How Twitter “moods” predict stock exchange index movement

The research explored whether one or more of the South African Twitter moods could be used to predict the movement of the Johannesburg Stock Exchange (JSE) All Share Index (ALSI). This is a proof of principle study in the field of big data analytic research in South Africa, which is at a relatively early stage of development. The research methods used secondary data from Twitter’s application programming interfaces (APIs), and formulated a model to extract public mood data and search for a causal effect of the mood on the closing values of the JSE ALSI. Over three million tweets were gathered and analysed over a 55-day period, with data collected from the JSE for 39 weekdays, from which only one variable (mood states) was considered. Four of the South African Twitter mood states did not produce any correlation with the movement of the JSE ALSI. The mood Depression had a significant negative correlation with the same day’s JSE ALSI values. The major finding was that there was a highly significant positive correlation between the Fatigue mood and the next day’s closing value of the JSE ALSI, and a significant causality correlation from the Fatigue mood to the JSE ALSI values. The findings support the behavioural finance theory (Wang, Lin & Lin, 2012), which states that public mood can influence the stock market. Organisations and governments could use Twitter data to gauge public mood and to ascertain the influence of public mood on particular issues. However, very large data sets are required for analytical purposes, possibly five to ten years of data, without which predictability is likely to be low.Department of Information Systems, University of Cape Town, South Afric