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From a broad perspective, we study issues related to implementation, testing, and experimentation in the context of geometric algorithms. Our focus is on the effect of quality of implementation on experimental results. More concisely, we study algorithms that compute convex hulls for a multiset of points in the plane. We introduce several improvements to the implementations of the studied algorithms: plane-sweep, torch, quickhull, and throw-away. With a new set of space-efficient implementations, the experimental results—in the integer-arithmetic setting—are different from those of earlier studies. From this, we conclude that utmost care is needed when doing experiments and when trying to draw solid conclusions upon them.

Let P be a set of n points in the plane. We compute the value of θ∈[0,2π) for which the rectilinear convex hull of P, denoted by RHP(θ), has minimum (or maximum) area in optimal O(nlogn) time and O(n) space, improving the previous O(n2) bound. Let O be a set of k lines through the origin sorted by slope and let αi be the sizes of the 2k angles defined by pairs of two consecutive lines, i=1,…,2k. Let Θi=π-αi and Θ=min{Θi:i=1,…,2k}. We obtain: (1) Given a set O such that Θ≥π2, we provide an algorithm to compute the O-convex hull of P in optimal O(nlogn) time and O(n) space; If Θ<π2, the time and space complexities are O(nΘlogn) and O(nΘ) respectively. (2) Given a set O such that Θ≥π2, we compute and maintain the boundary of the Oθ-convex hull of P for θ∈[0,2π) in O(knlogn) time and O(kn) space, or if Θ<π2, in O(knΘlogn) time and O(knΘ) space. (3) Finally, given a set O such that Θ≥π2, we compute, in O(knlogn) time and O(kn) space, the angle θ∈[0,2π) such that the Oθ-convex hull of P has minimum (or maximum) area over all θ∈[0,2π).

The convex hull peeling of a point set is obtained by taking the convex hull of the set and repeating iteratively the operation on the interior points until no point remains. The boundary of each hull is called a layer. We study the number of k-dimensional faces and the outer defect intrinsic volumes of the first layers of the convex hull peeling of a homogeneous Poisson point process in the unit ball whose intensity goes to infinity. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, the growth rates do not depend on the layer.

The convex hull heuristic is a heuristic for mixed-integer programming problems with a nonlinear objective function and linear constraints. It is a matheuristic in two ways: it is based on the mathematical programming algorithm called simplicial decomposition, or SD (von Hohenbalken in Math Program 13:49–68, 1977), and at each iteration, one solves a mixed-integer programming problem with a linear objective function and the original constraints, and a continuous problem with a nonlinear objective function and a single linear constraint. Its purpose is to produce quickly feasible and often near optimal or optimal solutions for convex and nonconvex problems. It is usually multi-start. We have tested it on a number of hard quadratic 0–1 optimization problems and present numerical results for generalized quadratic assignment problems, cross-dock door assignment problems, quadratic assignment problems and quadratic knapsack problems. We compare solution quality and solution times with results from the literature, when possible.

It is shownX^^^\\vphantom {\\widehat {\\widehat {\\widehat {\\widehat {\\widehat {\\widehat {\\widehat X}}} that there exists a compact set XX in CN\\mathbb {C}^N (N≥2N\\geq 2) such that X^∖X\\widehat X\\setminus X is nonempty and the uniform algebra P(X)P(X) has a dense set of invertible elements, a large Gleason part, and an abundance of nonzero bounded point derivations. The existence of a Swiss cheese XX such that R(X)R(X) has a Gleason part of full planar measure and a nonzero bounded point derivation at almost every point is established. An analogous result in CN\\mathbb {C}^N is presented. The analogue for rational hulls of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is established. The results presented address questions raised by Dales and Feinstein.

Non-destructive testing in reinforced concrete (RC) for damage detection is still limited to date. In monitoring the damage in RC, 18 beam specimens with varying water cement ratios and reinforcements were casted and tested using a four-point bending test. Repeated step loads were designed and at each step load acoustic emission (AE) signals were recorded and processed to obtain the acoustic emission source location (AESL). Computational geometry using a convex hull algorithm was used to determine the maximum volume formed by the AESL inside the concrete beam in relation to the load applied. The convex hull volume (CHV) showed good relation to the damage encountered until 60% of the ultimate load at the midspan was reached, where compression in the concrete occurred. The changes in CHV from 20 to 40% and 20 to 60% load were five and 13 times from CHV of 20% load for all beams, respectively. This indicated that the analysis in three dimensions using CHV was sensitive to damage. In addition, a high water-cement ratio exhibited higher CHV formation compared to a lower water-cement ratio due to its ductility where the movement of AESL becomes wider.

Home range estimation is routine practice in ecological research. While advances in animal tracking technology have increased our capacity to collect data to support home range analysis, these same advances have also resulted in increasingly autocorrelated data. Consequently, the question of which home range estimator to use on modern, highly autocorrelated tracking data remains open. This question is particularly relevant given that most estimators assume independently sampled data. Here, we provide a comprehensive evaluation of the effects of autocorrelation on home range estimation. We base our study on an extensive data set of GPS locations from 369 individuals representing 27 species distributed across five continents. We first assemble a broad array of home range estimators, including Kernel Density Estimation (KDE) with four bandwidth optimizers (Gaussian reference function, autocorrelated-Gaussian reference function [AKDE], Silverman's rule of thumb, and least squares cross-validation), Minimum Convex Polygon, and Local Convex Hull methods. Notably, all of these estimators except AKDE assume independent and identically distributed (IID) data. We then employ half-sample cross-validation to objectively quantify estimator performance, and the recently introduced effective sample size for home range area estimation (N̂area) to quantify the information content of each data set. We found that AKDE 95% area estimates were larger than conventional IID-based estimates by a mean factor of 2. The median number of cross-validated locations included in the hold-out sets by AKDE 95% (or 50%) estimates was 95.3% (or 50.1%), confirming the larger AKDE ranges were appropriately selective at the specified quantile. Conversely, conventional estimates exhibited negative bias that increased with decreasing N̂area. To contextualize our empirical results, we performed a detailed simulation study to tease apart how sampling frequency, sampling duration, and the focal animal's movement conspire to affect range estimates. Paralleling our empirical results, the simulation study demonstrated that AKDE was generally more accurate than conventional methods, particularly for small N̂area. While 72% of the 369 empirical data sets had >1,000 total observations, only 4% had an N̂area >1,000, where 30% had an N̂area <30. In this frequently encountered scenario of small N̂area, AKDE was the only estimator capable of producing an accurate home range estimate on autocorrelated data.

Given a dataset of two-dimensional points in the plane with integer coordinates, the method proposed reduces a set of n points down to a set of s points s ≤ n, such that the convex hull on the set of s points is the same as the convex hull of the original set of n points. The method is O(n). It helps any convex hull algorithm run faster. The empirical analysis of a practical case shows a percentage reduction in points of over 98%, that is reflected as a faster computation with a speedup factor of at least 4.

AIM: One of the main gaps in the assessment of biodiversity is the lack of a unified framework for measuring its taxonomic and functional facets and for unveiling the underlying patterns. LOCATION: Europe, 25 large river basins. METHODS: Here, we develop a decomposition of functional β‐diversity, i.e. the dissimilarity in functional composition between communities, into a functional turnover and a functional nestedness‐resultant component. RESULTS: We found that functional β‐diversity was lower than taxonomic β‐diversity. This difference was driven by a lower functional turnover compared with taxonomic turnover while the nestedness‐resultant component was similar for taxonomic and functional β‐diversity. MAIN CONCLUSIONS: Fish faunas with different species tend to share the same functional attributes. The framework presented in this paper will help to analyse biogeographical patterns as well as to measure the impact of human activities on the functional facets of biodiversity.