Population-level inference for home-range areas

· Home-range estimates are a common product of animal tracking data, as each range informs on the area needed by a given individual. Population-level inference on home-range areas—where multiple individual home-ranges are considered to be sampled from a population—is also important to evaluate changes over time, space, or covariates, such as habitat quality or fragmentation, and for comparative analyses of species averages. Population-level home-range parameters have traditionally been estimated by first assuming that the input tracking data were sampled independently when calculating home ranges via conventional kernel density estimation (KDE) or minimal convex polygon (MCP) methods, and then assuming that those individual home ranges were measured exactly when calculating the population-level estimates. This conventional approach does not account for the temporal autocorrelation that is inherent in modern tracking data, nor for the uncertainties of each individual home-range estimate, which are often large and heterogeneous. · Here, we introduce a statistically and computationally efficient framework for the population-level analysis of home-range areas, based on autocorrelated kernel density estimation (AKDE), that can account for variable temporal autocorrelation and estimation uncertainty. · We apply our method to empirical examples on lowland tapir (Tapirus terrestris), kinkajou (Potos flavus), white‐nosed coati (Nasua narica), white-faced capuchin monkey (Cebus capucinus), and spider monkey (Ateles geoffroyi), and quantify differences between species, environments, and sexes. · Our approach allows researchers to more accurately compare different populations with different movement behaviors or sampling schedules, while retaining statistical precision and power when individual home-range uncertainties vary. Finally, we emphasize the estimation of effect sizes when comparing populations, rather than mere significance tests.

Convex Polygonal Hull for a Pair of Irregular Objects

Introduction. Optimization placement problems are NP-hard. In most cases related to cutting and packing problems, heuristic approaches are used. The development of analytical methods for mathematical modeling of the problems is of paramount important for expanding the class of placement problems that can be solved optimally using state of the art NLP-solvers. The problem of placing two irregular two-dimensional objects in a convex polygonal region of the minimum size, which is a convex polygonal hull of the minimum area or perimeter, is considered. Continuous rotations and translations of non-overlapping objects are allowed. To solve the problem of optimal compaction of a pair of objects, two algorithms are proposed. The first is a sequentially search for local extrema on all feasible subdomains using a solution tree. The second algorithm searches for a locally optimal extremum on a single subdomain using a "good" feasible starting point. Purpose of the paper. Show how to construct a minimal convex polygonal hull for two continuously moving irregular objects bounded by circular arcs and line segments. Results. A mathematical model is constructed in the form of a nonlinear programming problem using the phi-function technique. Two algorithms are proposed for solving the problem of placing a pair of objects in order to minimize the area and perimeter of the enclosing polygonal area. The results of computational experiments are presented. Conclusions. The construction of a minimal convex polygonal hull for a pair of two-dimensional objects having an arbitrary spatial shape and allowing continuous rotations and translations makes it possible to speed up the process of finding feasible solutions for the problem of placing a large number of objects with complex geometry. Keywords: convex polygonal hull, irregular objects, phi-function technique, nonlinear optimization.

Near optimal minimal convex hulls of disks

The minimal convex hulls of disks problem is to find such arrangements of circular disks in the plane that minimize the length of the convex hull boundary. The mixed-integer non-linear programming model, named [17], works only for small to moderate-sized problems. Here we propose a polylithic framework of the problem for big problem instances by combining the following algorithms and models: (i) A fast disk-packing algorithm based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model taking the output of as its input. We present complete analytic solutions for small problems up to four disks and a semi-analytic mixed-integer linear programming model which yields exact solutions for strip packing problems with up to one thousand congruent disks. It turns out that the proposed polylithic approach works fine for large problem instances containing up to 1,000 disks. Monolithic and polylithic solutions using usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

Gorilla Abundance Estimations within North-East Moukalaba-Doudou National Park, Gabon

Accurate measures of animal population densities are essential to evaluate conservation status and implement action plans to ensure species survival. Gorilla numbers were assessed using the recce survey method within Moukalaba-Doudou National Park (MDNP) in Gabon using fresh nest counts of up to 1 week old. We walked 3,592 km within a 23.01-km² study site totalling a sampling effort of 297 days. Encounter rate was 0.12 fresh nests per kilometre, and gorilla density estimates generated by home range sizes (by minimal convex polygon) ranged between 1.14 and 1.48 gorillas/km². Gorillas preferred mixed forest for nesting over other habitats (Cheason index value 1.31). Results showed that gorilla density values within the study area concurred with previous studies that used line transect methodologies. We conclude that the choice of sampling design is dependent on environmental conditions characterised by each habitat type and target species.

Estimated functional space of centric condyle positions in temporomandibular joints of asymptomatic individuals using MRI

Magnetic resonance imaging (MRI) studies on centric condyle positions lack 3D comparisons of guided and unguided methods, which are used for accomplishing centric relation reference positions. The purpose of this study was to describe the space, in which mandibular condyles are placed in vivo by dental intercuspation, Dawson’s bimanual manipulation, and neuromuscular position. Twenty asymptomatic individuals aged 23 to 37 years underwent separate bite registrations using bimanual manipulation and the unguided neuromuscular technique. Subsequent 3-Tesla MRI scans of both temporomandibular joints yielded 3D data of the most superior condylar points at maximum intercuspation and both centric relation positions. We found concentric condyle positions in maximum intercuspation but considerable variation of condyle position after bimanual manipulation and neuromuscular technique. Their 95% predictive confidence ellipses overlapped substantially and created a space of reference positions. Its smallest volume averaged 2 mm3 for a minimal convex hull (95% confidence interval 1.1–3.2) and 3.5 mm3 for a minimal ellipsoid hull (95% confidence interval 1.8–5.4). Visualized in vivo by MRI, condyle positions at bimanual manipulation and neuromuscular position were not predictable and showed substantial variation in asymptomatic subjects. Clinicians should be aware of the functional space and its effect on dental intercuspation.

Fixed Points Features in N-Point Gravitational Lenses

A set of fixed points in N-point gravitational lenses is studied in the paper. We use complex form of lens mapping to study fixed points. There are some merits of using a complex form over coordinate. In coordinate form gravitational lens is described by a system of two equations and in complex form is described by one equation. We transform complex equation of N-point gravitational lens into polynomial equation. It is convenient to study polynomial equation. Lens mapping presented as a linear combination of two mappings: complex analytical and identity. Analytical mapping is specified by deflection function. Fixed points are roots of deflection function. We show, that all fixed points of lens mapping appertain to the minimal convex polygon. Vertices of the polygon are points into which dimensionless point masses are. Method of construction of fixed points in N-point gravitational lens is shown. There are no fixed points in 1-point gravitational lens. We study properties of fixed points and their relation to the center of mass of the system. We obtained dependence of distribution of fixed points on center of mass. We analyzed different possibilities of distribution in N-point gravitational lens. Some cases, when fixed points merge with the center of mass are shown. We show a linear dependence of fixed point on center of mass in 2-point gravitational lens and we have built a model of this dependence. We obtained dependence of fixed point to center of mass in 3-point lens in case when masses form a triangle or line. In case of triangle, there are examples when fixed points merges. We study conditions, when there are no one-valued dependence of distribution of fixed points in case of 3-points gravitational lens and more complicated lens.

Minimal convex extensions and finite difference discretisation of the quadratic Monge–Kantorovich problem

We present an adaptation of the Monge–Ampère (MA) lattice basis reduction scheme to the MA equation with second boundary value condition, provided the target is a convex set. This yields a fast adaptive method to numerically solve the optimal transport (OT) problem between two absolutely continuous measures, the second of which has convex support. The proposed numerical method actually captures a specific Brenier solution which is minimal in some sense. We prove the convergence of the method as the grid step size vanishes and show with numerical experiments that it is able to reproduce subtle properties of the OT problem.

THE GREEN–OSHER INEQUALITY IN RELATIVE GEOMETRY

In this paper we give a proof of the Green–Osher inequality in relative geometry using the minimal convex annulus, including the necessary and sufficient condition for the case of equality.