From symmetry breaking to Poisson Point Process in 2D Voronoi Tessellations: the generic nature of hexagons

We bridge the properties of the regular triangular, square, and hexagonal honeycomb Voronoi tessellations of the plane to the Poisson-Voronoi case, thus analyzing in a common framework symmetry breaking processes and the approach to uniform random distributions of tessellation-generating points. We resort to ensemble simulations of tessellations generated by points whose regular positions are perturbed through a Gaussian noise, whose variance is given by the parameter α2 times the square of the inverse of the average density of points. We analyze the number of sides, the area, and the perimeter of the Voronoi cells. For all valuesα >0, hexagons constitute the most common class of cells, and 2-parameter gamma distributions provide an efficient description of the statistical properties of the analyzed geometrical characteristics. The introduction of noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α = 0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise withα <0.12. For all tessellations and for small values of α, we observe a linear dependence on α of the ensemble mean of the standard deviation of the area and perimeter of the cells. Already for a moderate amount of Gaussian noise (α >0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α >2, results converge to those of Poisson-Voronoi tessellations. The geometrical properties of n-sided cells change with α until the Poisson- Voronoi limit is reached for α > 2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established. This law allows for an easy link to the Lewis law for areas and agrees with exact asymptotic results. Finally, for α >1, the ensemble mean of the cells area and perimeter restricted to the hexagonal cells agree remarkably well with the full ensemble mean; this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be interpreted as typical polygons in 2D Voronoi tessellations.

Agricultural tenancy reform: the end of law; or a new popular culture?

This paper applies a reading of the postmodernisation of law to the incremental reform of agricultural holdings legislation over the last century. In charting the shifting legal basis of agricultural tenancies, from ‘black letter’ positivism to the cultural contextuality of sumptuary law, the paper theorises that the underlying political imperative has been allied to the changing significance of property ownership and use. Rather than reflecting the long-term official desire to maintain the let sector in British agriculture, however, the paper argues that this process has had other aims. In particular, it has been about an annexation of law to legitimise the retention of landowner power while presenting a rhetorical ‘democratisation’ of farming, away from its plutocratic associations and towards a new narrative of ‘depersonalised’ business.

Curve crossing for the reflected Levy process at zero and infinity

Let $R_{t}=\sup_{0\leq s\leq t}X_{s}-X_{t}$ be a Levy process reflected in its maximum. We give necessary and sufficient conditions for finiteness of passage times above power law boundaries at infinity. Information as to when the expected passage time for $R_{t}$ is finite, is given. We also discuss the almost sure finiteness of $\limsup_{t\to 0}R_{t}/t^{\kappa}$, for each $\kappa\geq 0$.

Online data processing consent under EU law: a theoretical framework and empirical evidence from the UK

This article analyses the results of an empirical study on the 200 most popular UK-based websites in various sectors of e-commerce services. The study provides empirical evidence on unlawful processing of personal data. It comprises a survey on the methods used to seek and obtain consent to process personal data for direct marketing and advertisement, and a test on the frequency of unsolicited commercial emails (UCE) received by customers as a consequence of their registration and submission of personal information to a website. Part One of the article presents a conceptual and normative account of data protection, with a discussion of the ethical values on which EU data protection law is grounded and an outline of the elements that must be in place to seek and obtain valid consent to process personal data. Part Two discusses the outcomes of the empirical study, which unveils a significant departure between EU legal theory and practice in data protection. Although a wide majority of the websites in the sample (69%) has in place a system to ask separate consent for engaging in marketing activities, it is only 16.2% of them that obtain a consent which is valid under the standards set by EU law. The test with UCE shows that only one out of three websites (30.5%) respects the will of the data subject not to receive commercial communications. It also shows that, when submitting personal data in online transactions, there is a high probability (50%) of incurring in a website that will ignore the refusal of consent and will send UCE. The article concludes that there is severe lack of compliance of UK online service providers with essential requirements of data protection law. In this respect, it suggests that there is inappropriate standard of implementation, information and supervision by the UK authorities, especially in light of the clarifications provided at EU level.

Stephen Martin Leake: a Victorian's view of the Common Law

This article considers the life and work of Stephen Martin Leake and seeks to locate his work within the wider context of the procedural and substantive transformation of the mid-to-late Victorian legal world. In particular, the article attempts to rescue Leake from obscurity and emphasise his importance in this process. It is argued that Leake’s work began the process whereby common lawyers conceived of their law as organised in a principled rather than procedural manner. Later common law jurists built upon this work. Consideration is also given to the philosophical and jurisprudential sources upon which Leake drew in constructing his treatises.

From harmonisation to better, smart and fit food law

As the European Union (EU) approaches its 60th anniversary, it is worth assessing progress towards a key objective – the abolition of barriers to the marketing of food in the EU. Food has always created particular problems for the EU as national differences in diets, culture and geography make standardisation impossible. Early attempts focussed on direct measures to harmonise requirements or, later, to create an ‘internal market’. Subsequently a changed emphasis brought about the need to focus more clearly on the harmonisation of food safety. More widely, the recent recognition that too much legislation can itself create barriers has led legislators to attempt to consider more carefully the impact of their efforts. This paper reflects on the various stages in the creation of harmonised food controls and considers how case law has impacted the process. Today there are still differences and complete barrier-free trade seems some way off.

Mapping power law distributions in digital health social networks: methods, interpretations, and practical implications

BACKGROUND: Social networks are common in digital health. A new stream of research is beginning to investigate the mechanisms of digital health social networks (DHSNs), how they are structured, how they function, and how their growth can be nurtured and managed. DHSNs increase in value when additional content is added, and the structure of networks may resemble the characteristics of power laws. Power laws are contrary to traditional Gaussian averages in that they demonstrate correlated phenomena. OBJECTIVES: The objective of this study is to investigate whether the distribution frequency in four DHSNs can be characterized as following a power law. A second objective is to describe the method used to determine the comparison. METHODS: Data from four DHSNs—Alcohol Help Center (AHC), Depression Center (DC), Panic Center (PC), and Stop Smoking Center (SSC)—were compared to power law distributions. To assist future researchers and managers, the 5-step methodology used to analyze and compare datasets is described. RESULTS: All four DHSNs were found to have right-skewed distributions, indicating the data were not normally distributed. When power trend lines were added to each frequency distribution, R(2) values indicated that, to a very high degree, the variance in post frequencies can be explained by actor rank (AHC .962, DC .975, PC .969, SSC .95). Spearman correlations provided further indication of the strength and statistical significance of the relationship (AHC .987. DC .967, PC .983, SSC .993, P<.001). CONCLUSIONS: This is the first study to investigate power distributions across multiple DHSNs, each addressing a unique condition. Results indicate that despite vast differences in theme, content, and length of existence, DHSNs follow properties of power laws. The structure of DHSNs is important as it gives insight to researchers and managers into the nature and mechanisms of network functionality. The 5-step process undertaken to compare actor contribution patterns can be replicated in networks that are managed by other organizations, and we conjecture that patterns observed in this study could be found in other DHSNs. Future research should analyze network growth over time and examine the characteristics and survival rates of superusers.