An extremum problem for the power moment of a convex polygon contained in a disc

2021 ◽  
Vol 21 (4) ◽  
pp. 599-609
Author(s):  
Irmina Herburt ◽  
Shigehiro Sakata
Keyword(s):  
Convex Polygon ◽  
Extremum Problem ◽  
Classical Theorem ◽  
Power Moment ◽  
Area Functional ◽  
The Mean

Abstract In this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.

scholarly journals UNIQUENESS PROPERTIES OF THE INVARIATOR, LEADING TO SIMPLE COMPUTATIONS

2012 ◽  
Vol 31 (2) ◽  
pp. 89 ◽  
Author(s):  
Luis Manuel Cruz-Orive
Keyword(s):  
Convex Polygon ◽  
Smooth Boundary ◽  
Three Dimensional ◽  
Computational Formula ◽  
Compact Domain ◽  
Particle Volume ◽  
Pivotal Point ◽  
Wedge Volume ◽  
The Mean ◽  
The One

It is shown that, for a three dimensional particle  (namely an arbitrary compact domain with piecewise smooth boundary in R^3) the mean wedge volume defined on a given pivotal section is equal to the average nucleator estimator of the particle volume defined on that section. Further, if the particle is convex and it contains the pivotal point, then the flower area of a given pivotal section equals the average surfactor estimator defined on that section. These results are intended to throw some light on the standing conjecture that the functional defined on a pivotal section according to the invariator has a unique general expression. As a plus, the former result leads to a computational formula for the mean wedge volume of a convex polygon which is much simpler than the one published recently, and it is valid whether the fixed pivotal point is interior or exterior to the particle.

Gap-crossing in fragmented habitats by mahogany gliders (Petaurus gracilis). Do they cross roads and powerline corridors?

2010 ◽  
Vol 32 (1) ◽  
pp. 10 ◽  
Author(s):  
Yumiko Asari ◽  
Christopher N. Johnson ◽  
Mark Parsons ◽  
Johan Larson
Keyword(s):  
Convex Polygon ◽  
Harmonic Mean ◽  
Home Ranges ◽  
Minimum Convex Polygon ◽  
Narrow Strip ◽  
North East ◽  
Large Trees ◽  
Scattered Trees ◽  
The Mean ◽  
Gap Crossing

The mahogany glider (Petaurus gracilis) is one of the most threatened arboreal mammals in Australia. Although its habitat is affected by fragmentation, gap-crossing behaviour of the species has not been studied. A radio-tracking survey was undertaken on six individuals (three males, three females) in a woodland patch bisected by a 35.8-m-wide highway and a 31.5-m-wide powerline corridor, in north-east Queensland. The mean home ranges of males were 20.1 ± 3.3 ha, 21.3 ± 7.9 ha and 20.9 ± 8.2 ha, as measured by the Minimum Convex Polygon, Kernel and Harmonic Mean methods respectively. The mean home ranges of females were 8.9 ± 0.5 ha, 9.0 ± 4.2 ha and 8.8 ± 2.3 ha, as measured by the Minimum Convex Polygon, Kernel and Harmonic Mean methods respectively. Two males and one female were observed crossing linear gaps. However, there was less crossing than expected, and females were less likely to cross than males. One male used a narrow strip of woodland at the opposite side of the highway as supplemental habitat for foraging. This individual also used scattered trees in a grassland matrix for foraging or denning. Another male used a wooden power pole as a launching site to cross the highway. This study emphasises the importance of protecting large trees along linear barriers in open habitat, and suggests that gliding poles may be used to facilitate gap-crossing by mahogany gliders.

Movement Patterns of Feral Goats in a Semi-Arid Woodland in Eastern Australia.

1999 ◽  
Vol 21 (1) ◽  
pp. 71 ◽  
Author(s):  
D Freudenberger ◽  
J Barber
Keyword(s):  
Convex Polygon ◽  
New South ◽  
Movement Patterns ◽  
Shrub Cover ◽  
Local Populations ◽  
South Wales ◽  
Eastern Australia ◽  
Radio Transmitters ◽  
The Mean ◽  
Semi Arid

The movement patterns of ten feral goats fitted with radio transmitters were examined over a 20 month period in a semi-arid woodland of western New South Wales. The mean distance between locations (fixes) was 3.1 km at 42 day intervals. The mean interfix distance for male goats was 1.1 km greater than for females. The mean home range for the five males was 29.4 km2 and 10.9 km2 for the five females (95% convex polygon). The movement patterns of feral goats in this woodland system were predictable. Goats usually moved small distances and remained close to intermittent lakes and creeks with abundant tree and shrub cover. Goats commercially harvested in this area were likely to have come from local populations living in an area of 15-35 km2, an area encompassed by 1-2 paddocks on a single property.

Home range of stoats (Mustela erminea) in podocarp forest, south Westland, New Zealand: implications for a control strategy

2001 ◽  
Vol 28 (2) ◽  
pp. 165 ◽  
Author(s):  
Craig Miller ◽  
Mike Elliot ◽  
Nic Alterio
Keyword(s):  
New Zealand ◽  
Home Range ◽  
Breeding Season ◽  
Convex Polygon ◽  
Home Ranges ◽  
Convex Polygons ◽  
Minimum Convex Polygon ◽  
Mustela Erminea ◽  
The Mean ◽  
Breeding Seasons

The home range of stoats (Mustela erminea) was determined as part of a programme to protect Okarito brown kiwi chicks (Apteryx australis) ‘Okarito’, from predation. Twenty-seven stoats were fitted with radio-transmitters and tracked in two podocarp (Podocarpaceae) forests, in south Westland, New Zealand, from July 1997 to May 1998. Home-range area was determined for 19 animals by minimum convex polygons and restricted-edge polygons, and core areas were determined by hierarchical cluster analysis. The mean home ranges of males across all seasons calculated by minimum convex polygon (210 28 ha ( s.e.)) and restricted-edge polygon (176 29 ha) were significantly larger than those of females across all seasons (89 14 ha and 82 12 ha). The mean home range of males calculated by minimum convex polygon during the breeding season (256 38 ha) was significantly larger than the mean home range pooled across the non-breeding seasons (149 16 ha), whereas that calculated by restricted-edge polygon was not significantly different. The mean home range of females during the breeding season was not significantly different from that in the non-breeding seasons when estimated by either method. Overlap of home ranges was observed within and between sexes in all seasons, with the greatest proportion of home range overlap being male–female. The mean home range of females in spring and summer is used to guide the spacing of control stations.

ON THE FOURTH POWER MOMENT OF Δx AND E(x) IN SHORT INTERVALS

2009 ◽  
Vol 05 (02) ◽  
pp. 355-382 ◽  
Author(s):  
YOSHIO TANIGAWA ◽  
WENGUANG ZHAI
Keyword(s):  
Fourth Power ◽  
Mean Square ◽  
Divisor Function ◽  
Power Moment ◽  
Short Intervals ◽  
The Riemann Zeta Function ◽  
Error Terms ◽  
Riemann Zeta ◽  
The Mean ◽  
Power Moments

Let Δ(x) and E(x) be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper, their fourth power moments for short intervals of Jutila's type are considered. We get an asymptotic formula for U in some range.

Photo-electric Hβ and uvby photometry

1966 ◽  
Vol 24 ◽  
pp. 170-180
Author(s):  
D. L. Crawford
Keyword(s):  
Narrow Band ◽  
Line Strength ◽  
Interference Filter ◽  
B Stars ◽  
Relative Line ◽  
The Mean ◽  
F Stars ◽  
Two Parameters ◽  
Filter Photometry ◽  
The Continuum

Early in the 1950's Strömgren (1, 2, 3, 4, 5) introduced medium to narrow-band interference filter photometry at the McDonald Observatory. He used six interference filters to obtain two parameters of astrophysical interest. These parameters he calledlandc, for line and continuum hydrogen absorption. The first measured empirically the absorption line strength of Hβby means of a filter of half width 35Å centered on Hβand compared to the mean of two filters situated in the continuum near Hβ. The second index measured empirically the Balmer discontinuity by means of a filter situated below the Balmer discontinuity and two above it. He showed that these two indices could accurately predict the spectral type and luminosity of both B stars and A and F stars. He later derived (6) an indexmfrom the same filters. This index was a measure of the relative line blanketing near 4100Å compared to two filters above 4500Å. These three indices confirmed earlier work by many people, including Lindblad and Becker. References to this earlier work and to the systems discussed today can be found in Strömgren's article inBasic Astronomical Data(7).

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

The motion of a lunar orbiter

1966 ◽  
Vol 25 ◽  
pp. 373
Author(s):  
Y. Kozai
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Artificial Satellite ◽  
Equations Of Motion ◽  
Triaxial Ellipsoid ◽  
The Moon ◽  
Secular Motion ◽  
The Earth ◽  
Close Satellite ◽  
The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

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