4-Holes in point sets

We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft

The Shadows of a Cycle Cannot All Be Paths

A "shadow" of a subset $S$ of Euclidean space is an orthogonal projection of $S$ into one of the coordinate hyperplanes. In this paper we show that it is not possible for all three shadows of a cycle (i.e., a simple closed curve) in $\mathbb R^3$ to be paths (i.e., simple open curves). We also show two contrasting results: the three shadows of a path in $\mathbb R^3$ can all be cycles (although not all convex) and, for every $d\geq 1$, there exists a $d$-sphere embedded in $\mathbb R^{d+2}$ whose $d+2$ shadows have no holes (i.e., they deformation-retract onto a point).Comment: 6 pages, 10 figure

A contingency framework of enterprise governance in the UK: A value-based management approach

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Corporate governance (CG) has recently received much attention because of the wave of financial scandals in the early 2000s and the more recent global financial crisis. CG reforms, including laws, codes and listing rules have been established to protect shareholders’ rights and restore investors’ confidence in the capital market. These reforms have largely contributed to the evolution of internal and external governance mechanisms that are aimed at mitigating agency conflicts between managers and shareholders. However, overemphasis has been placed on the monitoring and control dimensions of governance, which may hinder entrepreneurial activities, obscure business prosperity and contribute to a narrow perspective on CG. It has been argued that there is a need to broaden CG beyond compliance (conformance) to a set of rules and laws, to include the performance aspects of governance that focus on strategy and value creation. In other words, governance should not only focus on monitoring managerial performance to ensure accountability to shareholders, but also on mechanisms that motivate management to optimise shareholders’ wealth. Enterprise governance (EG) framework has been introduced to keep the balance between the conformance and performance dimensions of governance. However, few studies address the possible tension between conformance and performance. Moreover, there is no agreement among these studies on the relationship between conformance and performance in the governance context. Arguably, Value-based Management (VBM) is an appropriate approach to address the issue of EG. VBM adopts value creation as an overall objective, develops a strategy that contributes to value creation and integrates it into decision-making. In this way, VBM can act as an effective mechanism for motivating management to maximise shareholder wealth, which works in parallel with other CG mechanisms, to mitigate agency conflicts resulting from the separation between ownership and management. This study aims to develop a contingency framework of EG through operationalising the conformance using CG and performance using corporate entrepreneurship (CE). This framework examines the inter-relationships between VBM, compliance with the Combined Code on Corporate Governance (CCCG), CE and the ultimate effect on organisational performance. More specifically, the study empirically examines the effect of compliance with the CCCG on CE, and whether VBM can achieve a balance between compliance with the CCCG and CE, should a conflict exist. The study also examines whether a fit between contingency variables (company size, agency conflicts, uncertainty, strategy and decentralisation), VBM, compliance with the CCCG codes and CE is associated with organisational performance. To achieve the aim of this study a cross-sectional survey, based on a questionnaire, is conducted to identify the level of VBM implementation, contextual and organisational factors in the large and medium quoted companies in the UK. The questionnaire targets the Chief Financial Officers (CFOs) in these companies as key informants. In addition, a content analysis of the annual reports of the sampled companies is undertaken to measure the level of compliance with the CCCG. Financial data (e.g. organisational performance) have been obtained from the DataStream, Fame and Thomson One Banker databases. Partial Least Squares Structural Equation Modelling (PLS-SEM) is adopted for data analysis and hypotheses testing. The results suggest that VBM implementation is positively associated with agency conflicts, low cost strategies and decentralisation. Compliance with the CCCG is positively associated with agency conflicts and company size. CE is positively associated with company size, uncertainty and differentiation strategies. In addition, the fit between compliance with the CCCG and contingency factors significantly predicts the marketbased performance. The fit between CE and the contingency factors significantly predict the perceived performance. However, the results regarding the effect of VBM on organisational performance are mixed. While VBM has no significant direct effect on the market-based performance, VBM has indirect positive effect on the market-based performance acting through compliance with the CCCG as an intervening variable. VBM is significantly associated with compliance with the CCCG but not with CE. No evidence is found for negative association between compliance with the CCCG and CE. The results support a large number of the proposed relationships between the contingency factors, VBM, compliance with the CCCG and CE. The results also suggest that using both compliance with the CCCG and CE as intervening variables in the relationship between VBM and organisational performance contributes to explaining the mixed results in the VBM literature. In terms of the EG framework, VBM does not keep a balance between conformance and performance. VBM emphasises the compliance with the CCCG (conformance) at the expense of CE (performance). The results did not provide significant evidence of a conflict between compliance with the CCCG and CE, the area which lacks empirical evidence. This study contributes to the literature at different levels. At the theoretical level, this study develops a theoretical model that links a performance management system (PMS), i.e. VBM, to CG practices and CE. This model attempts to bridge the gap between different disciplines, including management accounting, CG and entrepreneurship. Furthermore, combining both the contingency theory and the agency theory lenses contributes to the development of a comprehensive model of EG. At the methodological level, unlike previous studies, this study measures VBM practices on a continuum, rather than categories. Multiple data collection methods are used, and a powerful statistical technique (PLS-SEM) is adopted for data analysis. At the empirical level, the study is conducted in the UK. Though it is different from the US in many aspects, very few studies have been conducted in this context in many research areas such as VBM, CG and CE.This study is supported by the Egyptian Ministry of Higher Education and the Egyptian Cultural Bureau in London

The opaque square

The problem of finding small sets that block every line passing through a unit square was first considered by Mazurkiewicz in 1916. We call such a set {\em opaque} or a {\em barrier} for the square. The shortest known barrier has length $\sqrt{2}+ \frac{\sqrt{6}}{2}= 2.6389\ldots$. The current best lower bound for the length of a (not necessarily connected) barrier is $2$, as established by Jones about 50 years ago. No better lower bound is known even if the barrier is restricted to lie in the square or in its close vicinity. Under a suitable locality assumption, we replace this lower bound by $2+10^{-12}$, which represents the first, albeit small, step in a long time toward finding the length of the shortest barrier. A sharper bound is obtained for interior barriers: the length of any interior barrier for the unit square is at least $2 + 10^{-5}$. Two of the key elements in our proofs are: (i) formulas established by Sylvester for the measure of all lines that meet two disjoint planar convex bodies, and (ii) a procedure for detecting lines that are witness to the invalidity of a short bogus barrier for the square.Comment: 23 pages, 8 figure