The Average-Case Area of Heilbronn-Type Triangles

From among ${n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < \mu_n < C/n^3$ for all large enough values of $n$, where $\mu_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a roll, {\em New Scientist}, November 6, 1999, 44--4

On smallest triangles

Pick n points independently at random in R^2, according to a prescribed probability measure mu, and let D^n_1 <= D^n_2 <= ... be the areas of the binomial n choose 3 triangles thus formed, in non-decreasing order. If mu is absolutely continuous with respect to Lebesgue measure, then, under weak conditions, the set {n^3 D^n_i : i >= 1} converges as n --> infinity to a Poisson process with a constant intensity c(mu). This result, and related conclusions, are proved using standard arguments of Poisson approximation, and may be extended to functionals more general than the area of a triangle. It is proved in addition that, if mu is the uniform probability measure on the region S, then c(mu) <= 2/|S|, where |S| denotes the area of S. Equality holds in that c(mu) = 2/|S| if S is convex, and essentially only then. This work generalizes and extends considerably the conclusions of a recent paper of Jiang, Li, and Vitanyi

A new upper bound for the Heilbronn triangle problem

For sufficiently large $n$, we show that in every configuration of $n$ points chosen inside the unit square there exists a triangle of area less than $n^{-8/7-1/2000}$. This improves upon a result of Koml\'os, Pintz and Szemer\'edi from 1982. Our approach establishes new connections between the Heilbronn triangle problem and various themes in incidence geometry and projection theory which are closely related to the discretized sum-product phenomenon

Distributions of points in the unit square and large k-gons

AbstractWe consider a generalization of Heilbronn’s triangle problem by asking, given any integers n≥k, for the supremum Δk(n) of the minimum area determined by the convex hull of some k of n points in the unit square [0,1]2, where the supremum is taken over all distributions of n points in [0,1]2. Improving the lower bound Δk(n)=Ω(1/n(k−1)/(k−2)) from [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, SIAM Journal on Computing 30 (2000) 383–390] and from [W.M. Schmidt, On a problem of Heilbronn, Journal of the London Mathematical Society (2) 4 (1972) 545–550] for k=4, we show that Δk(n)=Ω((logn)1/(k−2)/n(k−1)/(k−2)) for fixed integers k≥3 as asked for in [C. Bertram-Kretzberg, T. Hofmeister, H. Lefmann, An algorithm for Heilbronn’s problem, SIAM Journal on Computing 30 (2000) 383–390]. Moreover, we provide a deterministic polynomial time algorithm which finds n points in [0,1]2, which achieve this lower bound on Δk(n)

Time-Restricted Feeding Improves Circadian Dysfunction as well as Motor Symptoms in the Q175 Mouse Model of Huntington's Disease.

Huntington's disease (HD) patients suffer from a progressive neurodegeneration that results in cognitive, psychiatric, cardiovascular, and motor dysfunction. Disturbances in sleep/wake cycles are common among HD patients with reports of delayed sleep onset, frequent bedtime awakenings, and fatigue during the day. The heterozygous Q175 mouse model of HD has been shown to phenocopy many HD core symptoms including circadian dysfunctions. Because circadian dysfunction manifests early in the disease in both patients and mouse models, we sought to determine if early intervention that improve circadian rhythmicity can benefit HD and delay disease progression. We determined the effects of time-restricted feeding (TRF) on the Q175 mouse model. At six months of age, the animals were divided into two groups: ad libitum (ad lib) and TRF. The TRF-treated Q175 mice were exposed to a 6-h feeding/18-h fasting regimen that was designed to be aligned with the middle of the time when mice are normally active. After three months of treatment (when mice reached the early disease stage), the TRF-treated Q175 mice showed improvements in their locomotor activity rhythm and sleep awakening time. Furthermore, we found improved heart rate variability (HRV), suggesting that their autonomic nervous system dysfunction was improved. Importantly, treated Q175 mice exhibited improved motor performance compared to untreated Q175 controls, and the motor improvements were correlated with improved circadian output. Finally, we found that the expression of several HD-relevant markers was restored to WT levels in the striatum of the treated mice using NanoString gene expression assays

Knowledge and Civil Society

This open access book focuses on the role of civil society in the creation, dissemination, and interpretation of knowledge in geographical contexts. It offers original, interdisciplinary and counterintuitive perspectives on civil society. The book includes reflections on civil and uncivil society, the role of civil society as a change agent, and on civil society perspectives of undone science. Conceptual approaches go beyond the tripartite division of public, private and civic sectors to propose new frameworks of civic networks and philanthropic fields, which take an inclusive view of the connectivity of civic agency across sectors. This includes relational analyses of epistemic power in civic knowledge networks as well as of regional giving and philanthropy. The original empirical case studies examine traditional forms of civic engagement, such as the German landwomen’s associations, as well as novel types of organizations, such as giving circles and time banks in their geographical context. The book also offers insider reflections on doing civil society, such as the cases of the Umbrella Movement in Hong Kong, epistemic activism in the United States, and the #FeesMustFall movement in South Africa

The Cultural Project : Formal Chronological Modelling of the Early and Middle Neolithic Sequence in Lower Alsace

Starting from questions about the nature of cultural diversity, this paper examines the pace and tempo of change and the relative importance of continuity and discontinuity. To unravel the cultural project of the past, we apply chronological modelling of radiocarbon dates within a Bayesian statistical framework, to interrogate the Neolithic cultural sequence in Lower Alsace, in the upper Rhine valley, in broad terms from the later sixth to the end of the fifth millennium cal BC. Detailed formal estimates are provided for the long succession of cultural groups, from the early Neolithic Linear Pottery culture (LBK) to the Bischheim Occidental du Rhin Supérieur (BORS) groups at the end of the Middle Neolithic, using seriation and typology of pottery as the starting point in modelling. The rate of ceramic change, as well as frequent shifts in the nature, location and density of settlements, are documented in detail, down to lifetime and generational timescales. This reveals a Neolithic world in Lower Alsace busy with comings and goings, tinkerings and adjustments, and relocations and realignments. A significant hiatus is identified between the end of the LBK and the start of the Hinkelstein group, in the early part of the fifth millennium cal BC. On the basis of modelling of existing dates for other parts of the Rhineland, this appears to be a wider phenomenon, and possible explanations are discussed; full reoccupation of the landscape is only seen in the Grossgartach phase. Radical shifts are also proposed at the end of the Middle Neolithic