Summary From Euler’s reflection formula, the product of gamma functions Γ ( 1 + 1 n ) Γ ( 1 − 1 n ) is shown to have a geometric interpretation as the arc to chord ratio for a sector of a regular convex polygon with n sides. It also applies to any sector of a circle.

Largest similar copies of convex polygons in polygonal domains

Species distribution patterns at range edges can reveal effects of landscape and climate changes which can drive population expansions or contractions. Southern flying squirrel (Glaucomys volans Linnaeus, 1758) populations are common in eastern United States and southeastern Canada, and extend westwardly into the Great Plains. Space-use patterns within their range interior are well documented, though it is unclear if patterns are similar along western range edges. We tracked 10 southern flying squirrels in southeastern Kansas, USA to estimate home-range sizes and patterns in nest-site and tree selection. Forest structure and composition at western range edges differ compared to forest habitat at range interiors, and we predicted differences in home-range sizes and nest-site and tree selection patterns in Kansas compared to previous research. Mean home-range size was 1.55 ha (95% minimum convex polygon) and 5.44 ha (95% kernel density), similar to some populations in the interior of their range. Greater tree genera richness and average tree diameter (DBH) increased probability of nest-site use. Non-hard mast species and greater tree DBH increased nest-tree selection probabilities. Maintaining forest diversity while allowing all tree species to reach maturity could increase available nesting habitat for southern flying squirrels on their western range periphery.

The Thomas langur (Presbytis thomasi) is a primate endemic to Aceh, threatened with extinction due to hunting and habitat destruction. They are found not only in conservation areas but also in other areas and secondary forests. Mount Paroy is one of the important Thomas langur habitats that has been damaged due to encroachment by local communities. In this area, Thomas langur are often hunted because they are considered pests by farmers. The conservation status of Thomas langur is listed as Vulnerable by the IUCN and Appendix II by CITES. This habitat change has resulted in changes in the food supply pattern of Thomas langur. Food trees are replaced by plantation crops. Therefore, Thomas langur must be able to adapt to food sources and sleeping trees of non-forest species. This study aimed to determine the feeding behavior and adaptation of Thomas langur in disturbed secondary forests using exploratory survey methods and data collection techniques using line transect. Recorded data included the horizontal coordinates of the Thomas langur ' feeding points and the height of the trees used for feeding and resting. Data were analyzed using the Maximum Convex Polygon (MCV) method. Results obtained in the Mount Paroy habitat showed horizontal movement of 29 ha, and vertical space was used at altitudes below 10 m. For food, Thomas langur on Mount Paroy consumes more leaves.

We investigated the ecology of the puma (Puma concolor) in the high-altitude plateaus of Tarapacá, northern Chile, with a focus on population density, daily activity patterns, and habitat selection. We hypothesized that prey scarcity and human presence would result in low population density, predominantly nocturnal activity, and a preference for shrubland (“tolar”) habitat. Systematic camera-trap surveys were conducted during two seasonal periods in 2012, totaling 1,602 trap-nights. Puma density was estimated using capture-recapture methods based on Maximum Mean Distance Moved (MMDM) and Minimum Convex Polygon (MCP). Activity patterns were analyzed by hour, and habitat selection was assessed using the Ivlev Selection Index and logistic regression. A total of 72 independent detection events were recorded, corresponding to four identified individuals. Density estimates ranged from 0.2 to 0.6 individuals per 100 km²—the lowest reported for Puma concolor in Chile. Eighty percent of detections occurred between 20:00 and 04:59 h, indicating a bimodal nocturnal activity pattern. Shrublands were preferred (Ivlev Index = +0.15), wetlands were avoided (–0.35), and detection probability was higher in areas with over 40% tolar cover. The low population density likely reflects limited wild prey availability and indirect competition with livestock. Nocturnality appears to be a behavioral adaptation to avoid human activity. The broad spatial distribution of detections suggests wide-ranging movements. Although fieldwork was conducted in 2012, ecological conditions have remained relatively stable, supporting the relevance of our findings, despite emerging threats such as infrastructure development and wildlife decline. This study provides essential baseline data on puma ecology in the Tarapacá highlands and supports evidence-based management. We recommend implementing long-term monitoring with telemetry, improving nighttime livestock protection, and conserving tolar corridors to promote puma persistence in high Andean environments.

Low-order digital controller design based on l2 parametric stability margin.

Catalan numbers are a concept derived from the study of convex polygon dis- sections. Based on this idea, this article aims to discover the importance and similarity of Catalan numbers in solving different problems in enumerative combinatorics, according to the generalization of its recursive formula. With the introduction of Fuss Catalan numbers, which is a generalization of Catalan numbers, the following section will show its relation with j-ary trees and discover solutions of specific problems from different perspective. The arti-cle also expands Catalan numbers in particular Avoiding Permutations and Standard Young Tableaux, which indicates the breadth of applications of Catalan numbers.

ABSTRACT Understanding elephant habitat use and movement is essential for conservation in dynamic, resource‐limited semi‐arid ecosystems. Remote sensing and GPS telemetry provide powerful tools for quantifying elephant ecological patterns across heterogeneous landscapes. This study investigates the habitat preferences, movement patterns and spatial clustering of African savannah elephants ( Loxodonta africana ) in Mana Pools National Park (MPNP), Zimbabwe, using GPS telemetry data and remote sensing–derived environmental variables. Habitat suitability was modelled using the maximum entropy (MaxEnt) modelling technique. Seasonal home ranges and movement dynamics were analysed using minimum convex polygons (MCP), Voronoi polygons and the Time local convex hull (T LoCoH) techniques. Clusters were identified using the K‐means method. The MaxEnt results revealed that proximity to permanent water sources particularly, the Zambezi River was the most significant predictor of elephant distribution, contributing over 96% to the model's performance. Home range size estimates varied across methods and seasons. During the dry season, MCP, Voronoi polygons and time local convex hull (T‐LoCoH) estimated ranges of 406 km 2 , 521 km 2 and 325 km 2 , respectively. In the wet season, home ranges expanded markedly to 975 km 2 (MCP), 713 km 2 (Voronoi) and 527 km 2 (T‐LoCoH). The transition season recorded the largest ranges, with 1065 km 2 (MCP), 1032 km 2 (Voronoi) and 714 km 2 (T‐LoCoH). MCP consistently produced the largest estimates, while T‐LoCoH yielded the smallest, highlighting methodological sensitivity in quantifying elephant ranging behaviour across seasonal landscapes. K‐means clustering identified spatially distinct movement clusters across all seasons, showing non‐random habitat use driven by environmental constraints such as water availability, vegetation productivity and terrain. Findings support data‐driven conservation strategies for sustainable elephant management in protected areas.

A polygon model of the functional base-of-support during standing improves the accuracy of balance analysis.

Convex polygon sandwich panels with gradient continuous controllable core: Design, simulation, and experiment

ABSTRACT Luciobarbus esocinus (mangar), a critically endangered mega‐fish species (≤ 140 kg) endemic to the Euphrates and Tigris river systems, faces severe threats due to habitat fragmentation. To assess its spatial ecology in the Keban Reservoir, 30 individuals were acoustically tagged and monitored for 218–437 days using 18 receivers deployed in the reservoir. The tagged fish exhibited low site fidelity, with station residency indices (RI) ranging from 0.00 to 0.46 and detection periods lasting 59–153 days. Depth and temperature data (115,318 and 112,789 records, respectively) revealed seasonal habitat shifts: individuals occupied deeper waters in winter (mean 10.1 m), shallower depths in spring (4.4 m) and summer (3.8 m), and tolerated temperatures of 7.4°C–28.4°C. Although depth use was weakly correlated with temperature, trends indicated shallower occupancy during warmer periods. Home range analyses showed a maximum linear home range (LHR) of 36.5 km and the areal home range estimated via minimum convex polygon (MCP) was 6422.8 ha. In addition, kernel density (KD) analysis revealed the 95% KD area was 2877.0 ha, the 75% KD area was 67.7 ha and the 50% KD area was 33.8 ha. Seasonal migrations spanned 25–30 km upstream/downstream from release sites, particularly toward tributaries in spring. Around the Keban Reservoir, 14 hydroelectric power plants (HPPs) present and most of these HPP are not designed suitable fish passages. Urgent conservation measures are needed, including Fish Conservation Zones in critical habitats and connectivity restoration via functional fishways, to safeguard this imperilled species in fragmented river‐reservoir ecosystems.

The home range represents the space in which individuals of a determined species carry out their essential activities and is influenced by ecological and behavioural factors. Accurate estimates of a given home range are essential for understanding the spatial requirements of a species and for developing effective conservation strategies. Therefore, this study aimed to estimate and characterise the minimum home range of a group of blonde capuchin monkeys (Sapajus flavius) in a Caatinga dry forest in northeastern Brazil, using tool-use sites as spatial indicators. Between November 2023 and December 2024, 370 stone tool sites with evidence of recurrent use (presence of food remains) were recorded and georeferenced using handheld GPS, posteriorly confirmed by camera traps. Based on these stone tool sites, the home range of the group was delineated using the minimum convex polygon method with 100% of the locations. The size of the minimum home range was 53.7ha; the majority of the home range consisted of Savannah Formations (67.7%, 36.35ha), followed by anthropogenic areas (27.6%, 14.9ha). We conclude that lithic sites can serve as an alternative indicator for spatial estimates of blonde capuchin home ranges in contexts where direct observation is difficult. Furthermore, human-modified areas in the home range may reduce the availability of food resources and consequently, stone tool use sites. Our results provide the first home range estimate for blonde capuchins in the Caatinga dry forest and highlight the need to improve habitat quality.

The Wood Turtle (Glyptemys insculpta) is an endangered species in the state of Iowa and a species of conservation concern across their entire range. The Iowa population is characterized by high levels of adult and egg predation, displays little or no annual recruitment, and harbors an extremely low number of juveniles (7.3%). Home range and habitat usage studies of hatchling and juvenile Wood Turtles are limited to a few studies, and only one study of juveniles exists from the state of Iowa. Over a 10 yr period, we conducted a radiotelemetry study in Iowa on seven juvenile wood turtles for 32–182 weeks, and a 6-week study on six head-started hatchlings to determine home range sizes and habitat usage patterns and to provide comparisons with similar studies on adult Wood Turtles. Mean home range sizes of hatchling Wood Turtles were significantly smaller than the mean home range of older juvenile turtles for 100%, 95%, and 50% minimum convex polygons (MCPs), for 95% and 50% kernel density estimators (KDEs), and for linear home range (LHR) and stream home range (SHR). Habitat usage patterns of hatchlings and juveniles also differed. During periods of terrestrial activity, older juveniles utilized grass and forb clearings significantly more frequently than did hatchlings, and hatchlings used riverbank habitat more frequently than did juvenile turtles. In addition, juveniles were, on average, located significantly farther from the stream than were hatchlings. Our study provides important data on the home range size and habitat usage patterns of two under-represented age classes of this endangered species. These data will inform conservation agencies regarding relevant habitat protection and age-class management strategies of riparian areas that are necessary for the continued survival and protection of this imperiled species.

Recent issues that have emerged in regard to raccoon dog (Nyctereutes procyonoides) include interaction with humans and disease transmission. Therefore, understanding their habitat characteristics and preferences is crucial in the effort to limit conflicts with humans. A total of thirteen raccoon dogs were captured from three regions in South Korea, each with distinct habitat characteristics. GPS trackers were attached for tracking the raccoon dogs’ movements. Utilizing GPS tracking data, Kernel Density Estimation (KDE), Minimum Convex Polygon (MCP), and Jacobs Index were applied to learn more about the habitat preferences of the raccoon dogs. According to the results, the habitat composition ratios for KDE and MCP showed that forests had the largest proportion. However, a habitat composition ratio similar to the land proportion of the area that they inhabit indicated that raccoon dogs had the ability to adapt to various habitats. Jacobs Index analysis revealed different habitat selection patterns compared to KDE and MCP, with forests showing neutral to negative selection despite comprising large proportions of home ranges. Our results highlight important methodological considerations when inferring habitat preferences from spatial data, suggesting that multiple analytical approaches provide complementary insights into animal space use.

Futaki invariants of the classical moduli space of 4dN=1 supersymmetric gauge theories are conjectured to determine whether they have a conformal fixed point in the IR. We systematically compute the Futaki invariants for a large family of 4dN=1 supersymmetric gauge theories coming from D3-branes probing Calabi-Yau 3-fold singularities whose bases are Gorenstein Fano surfaces. In particular, we focus on the toric case where the Fano surfaces are given by the 16 reflexive convex polygons and the moduli spaces are given by the corresponding toric Calabi-Yau 3-folds. We study the distribution of and conjecture new bounds on the Futaki invariants with respect to various topological and geometric quantities. These include the minimum volume of the Sasaki-Einstein base manifolds as well as the Chern and Euler numbers of the toric Fano surfaces. Even though the moduli spaces for the family of theories studied are known to be K-stable, our work sheds new light on how the topological and geometric quantities restrict the Futaki invariants for a plethora of moduli spaces.

ABSTRACTOrnate box turtle (Terrapene ornata) and eastern box turtle (Terrapene carolina carolina) are long‐lived, relatively small, and slow‐moving ectotherms experiencing range‐wide declines primarily due to habitat loss. Understanding home range and movement patterns of box turtles is crucial for conservation efforts in fragmented landscapes. Very High Frequency (VHF) radio transmitters are commonly used to locate box turtles, but sample size and availability of personnel can limit how often each turtle is tracked. In this study, we evaluated whether the rate of tracking events affects the estimated home range and average daily movement of T. ornata in a short‐grass prairie located in Roosevelt County, New Mexico, and T. c. carolina in an urban forest located in Wake County, North Carolina, USA. We tracked three T. ornata and four T. c. carolina four to 7 days per week during their active season. Using the full dataset, we first calculated 100% Minimum Convex Polygon for each turtle. Then, we randomly sampled the full dataset, representing scenarios where turtles were tracked from one to four times per week, and recalculated home ranges. We also calculated distances between consecutive locations in the full dataset to evaluate how rates of tracking events affect estimates of mean daily movement. Mixed effect models revealed that home range size estimates significantly decreased with less frequent tracking events. Furthermore, we found that turtles occasionally moved longer distances within a single day. Our findings suggest that these rare bursts of movement are ecologically relevant but may be overlooked, and consequently home range sizes underestimated, if turtles are not tracked frequently.

ABSTRACTThe African savannah elephant (Loxodonta africana) is listed as endangered on the IUCN Red List. The Ruaha–Rungwa ecosystem supports Tanzania's largest elephant population and remains a critical yet understudied stronghold for elephant conservation in East Africa. Effective conservation in this ecosystem requires understanding elephant movements across large landscapes and varying levels of disturbance. This study addressed the gap by utilizing GPS‐satellite tracking of 28 elephants over a 4‐year period in Tanzania's Ruaha–Rungwa ecosystem. It examined their home ranges and movement patterns inside and outside protected areas and identified key elephant hotspots. The findings aim to improve conservation strategies and mitigate human–elephant conflicts through better habitat management and protection efforts. The study found no significant difference between home range estimates from Minimum Convex Polygon and Adaptive Kernel Density Estimation, with 28 collared elephants averaging 2536 km2 at the 99% isopleth. Home range size varied by age and season but not by sex. Elephants moved faster at night in less protected areas during both wet and dry seasons. Population‐level habitat use was similar across protection levels. Village lands in western Rungwa Game Reserve and its surrounding areas, as well as the eastern regions outside Ruaha National Park, served as key refuge sites during the wet season. Findings suggest elephants in the Ruaha–Rungwa ecosystem require vast areas, with adjacent lands playing a crucial role in their survival. High elephant activity in village lands within the western Rungwa Game Reserve and Lunda‐Mkwambi Game Controlled Area highlights the potential for human–elephant conflict, emphasizing the need to reopen the wildlife corridor for linking Ruaha–Rungwa and Rukwa–Katavi ecosystems. This study offered crucial insights into elephant movement and behavior in a landscape increasingly shaped by human–elephant conflict and habitat fragmentation, informing strategies for connectivity, land‐use planning, and coexistence.

The gelada (Theropithecus gelada), Ethiopia's only endemic primate and the last surviving graminivorous cercopithecid, was studied in Susgen Natural Forest, South Wollo, to examine seasonal variations in activity budgets and ranging ecology. From February to August 2023, encompassing both dry and wet seasons, 3519 behavioral scans were collected from 1680 group observations using instantaneous scan sampling at 15-min intervals (07:00–17:00 h). Data were analyzed with descriptive statistics and nonparametric tests (Kruskal–Wallis H and Mann–Whitney U), while home ranges were mapped via minimum convex polygon (MCP) and kernel density estimation (KDE). Results revealed that geladas allocated 43.2% of their time to feeding, 15% to movement, 15.5% to social activities, 13.1% to resting, and 13.2% to other behaviors. Dry seasons elicited significantly greater feeding effort (46.1% vs. 40.4%; p < 0.05) and daily travel distances (3658.4 ± 0.902 m vs. 3132.1 ± 2.367 m in wet season; Mann–Whitney U, p ≤ 0.05), with home ranges analyzed through the MCP method expanding to 190.1 ha in dry season as compared with 118.18 ha in wet season. KDE analysis identified the intensive use of core areas (54 ha) within broader ranges (164.95 ha). These findings underscore how geladas in human-modified landscapes face chronic nutritional stress, adapting through extended foraging and ranging patterns. We recommend immediate conservation measures, including habitat restoration and buffer zone establishment, to mitigate anthropogenic pressures on this threatened endemic species.

We give a simple and complete description of those convex lattice polygons in the plane that can be dissected into lattice triangles of integer area. A new version of Sperner’s lemma plays a central role.

Land-use change has resulted in natural habitats becoming fragmented and disjunct. Wetland ecosystemsin the United States are one example that has impacted wetland-adapted species. One species, the Spotted Turtle(Clemmys guttata), lives and relies on freshwater wetland habitats. Unfortunately, relatively little is known about thespatial and habitat requirements for C. guttata, especially in fragmented habitats. Using ground-based radio-telemetryand thermal ecology, we calculated seasonal home ranges and examined overwintering ecology for 30 adult turtlesfrom April 2022 to January 2023 at two sites in northern Indiana. We then calculated annual and seasonal (seasonalmovement areas, SMA) home ranges using 100% Minimum Convex Polygons (MCP) and annual 95%, 90%, and50% kernel density estimates (KDs). Our results show MCPs were not significantly different between sexes or sitesbut differed significantly across seasons (highest in spring). Annual KDs (95%, 90%, and 50%) did not differ betweensites. Daily mean air temperatures prior to overwintering were 12.01 °C and no turtle was recorded to have a carapacialtemperature below 0.5 °C. Results of this study show that even in these highly fragmented areas Spotted Turtles canoperate similar to other populations in more natural habitats. These data can be used to help develop managementplans for Spotted Turtle populations in disjunct and/or urban areas.