scholarly journals A geometric formula for multiplicities of $K$-types of tempered representations

2019 ◽  
Vol 372 (12) ◽  
pp. 8553-8586 ◽  
Author(s):  
Peter Hochs ◽  
Yanli Song ◽  
Shilin Yu
Keyword(s):  
Geometric Formula

scholarly journals The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

2019 ◽  
Author(s):  
Kai Xu
Keyword(s):  
Two Dimensions ◽  
Empirical Formulas ◽  
Von Neumann ◽  
Upper Limit ◽  
Edge Distribution ◽  
Cell Topology ◽  
Neighbor Relationship ◽  
New Formula ◽  
Geometric Formula ◽  
Ellipse Packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the new formula of the Aboav-Weaire’s law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.

Second-order stereology for planar fibre processes

1994 ◽  
Vol 26 (04) ◽  
pp. 906-918 ◽  
Author(s):  
V. Weiss ◽  
W. Nagel
Keyword(s):  
Second Order ◽  
Unbiased Estimators ◽  
Order Quantities ◽  
Geometric Formula

Three different stereological methods for the determination of second-order quantities of planar fibre processes which have been suggested in the literature are considered. Proofs of the formulae are given (also by using a new integral geometric formula), relations between the methods are derived and the prerequisites are discussed. Furthermore, edge-corrected unbiased estimators for the second-order quantities are given.

scholarly journals Stable s-minimal cones in ℝ3 are flat for s ~ 1

2020 ◽  
Vol 2020 (764) ◽  
pp. 157-180 ◽  
Author(s):  
Xavier Cabré ◽  
Eleonora Cinti ◽  
Joaquim Serra
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Second Variation ◽  
Classification Result ◽  
Minimal Sets ◽  
Minimal Cones ◽  
The Second Variation ◽  
Geometric Formula

AbstractWe prove that half spaces are the only stable nonlocal s-minimal cones in {\mathbb{R}^{3}}, for {s\in(0,1)} sufficiently close to 1. This is the first classification result of stable s-minimal cones in dimension higher than two. Its proof cannot rely on a compactness argument perturbing from {s=1}. In fact, our proof gives a quantifiable value for the required closeness of s to 1. We use the geometric formula for the second variation of the fractional s-perimeter, which involves a squared nonlocal second fundamental form, as well as the recent BV estimates for stable nonlocal minimal sets.

A New Integral Geometric Formula of the Blaschke-Petkantschin Type

1992 ◽  
Vol 156 (1) ◽  
pp. 57-74 ◽  
Author(s):  
E. B. Vedel Jensen ◽  
K. Kiê
Keyword(s):  
Geometric Formula

On a Geometric Formula for the Fundamental Solution of Subelliptic Laplacians

2009 ◽  
Vol 181 (1) ◽  
pp. 81-163 ◽  
Author(s):  
Richard Beals ◽  
Bernard Gaveau ◽  
Peter Greiner
Keyword(s):  
Fundamental Solution ◽  
Geometric Formula

scholarly journals How tall is an elephant? Two methods for estimating elephant height

Web Ecology ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 1-10 ◽  
Author(s):  
F. Della Rocca
Keyword(s):  
Field Studies ◽  
Height Distribution ◽  
African Elephants ◽  
Linear Distance ◽  
Elephant Population ◽  
Photo Camera ◽  
In The Wild ◽  
Absolute Height ◽  
Free Ranging ◽  
Geometric Formula

Abstract. Shoulder height is a reliable indicator of age for African elephants (Loxodonta africana), and is therefore an important parameter to be recorded in field studies of population ecology of these pachyderms. However, it can be somewhat difficult to estimate with precision the shoulder height of free-ranging elephants because of several reasons, including the presence of drops and vegetation cover and the potential dangerousness of approaching them in the wild. Here I test two alternative models for estimating shoulder height of elephants. In both models, the equipment needed to generate the height estimates is minimal, and include a telemeter and a digital photo-camera furnished with an ×16 zoom. The models are based respectively on a linear regression approach and on a geometric formula approach, and put into a relationship the linear distance between the observer and the animal, the number of pixels of an elephant silhouette as taken from digital photos, and the absolute height of the animal. Both methods proved to have a very small measurement error, and were thus reliable for field estimates of elephant shoulder heights. The model based on a geometric formula was used to estimate the shoulder height distribution of an elephant population in a savannah region of West Africa (Zakouma National Park, Chad). I demonstrated that Zakouma elephants were among the tallest populations in Africa, with growth rates being highest throughout the first five years of life.

Some Notes on the Double Geometric Formula

1936 ◽  
Vol 15 (1) ◽  
pp. 229-234 ◽  
Author(s):  
A. E. King ◽  
A. R. Reid
Keyword(s):  
Geometric Formula

An asymptotic formula for the transition density of random genetic drift

1978 ◽  
Vol 15 (01) ◽  
pp. 184-186 ◽  
Author(s):  
Peter L. Antonelli
Keyword(s):  
Population Genetics ◽  
Asymptotic Formula ◽  
Genetic Drift ◽  
Asymptotic Expansions ◽  
Stochastic Models ◽  
Transition Density ◽  
Fundamental Solutions ◽  
Diffusion Equations ◽  
Random Genetic Drift ◽  
Geometric Formula

Stochastic models in population genetics which lead to diffusion equations are considered. A geometric formula for the asymptotic expansions of the fundamental solutions of these equations is presented. Specifically, the random genetic drift process of one-locus theory and the Ohta–Kimura model of two-locus di-allelic systems with linkage are studied. Agreement with the work of Keller and Voronka for the two-allele one-locus case is obtained. For the general n-allele problem, the formulas obtained here are apparently new.

scholarly journals The geometric formulas of the Lewis’s law and Aboav-Weaire’s law in two dimensions based on ellipse packing

2019 ◽  
Author(s):  
Kai Xu
Keyword(s):  
Two Dimensions ◽  
Empirical Formulas ◽  
Von Neumann ◽  
Upper Limit ◽  
Edge Distribution ◽  
Cell Topology ◽  
Neighbor Relationship ◽  
New Formula ◽  
Geometric Formula ◽  
Ellipse Packing

The two-dimensional (2D) Lewis’s law and Aboav-Weaire’s law are two simple formulas derived from empirical observations. Numerous attempts have been made to improve the empirical formulas. In this study, we simulated a series of Voronoi diagrams by randomly disordered the seed locations of a regular hexagonal 2D Voronoi diagram, and analyzed the cell topology based on ellipse packing. Then, we derived and verified the improved formulas for Lewis’s law and Aboav-Weaire’s law. Specifically, we found that the upper limit of the second moment of edge number is 3. In addition, we derived the geometric formula of the von Neumann-Mullins’s law based on the new formula of the Aboav-Weaire’s law. Our results suggested that the cell area, local neighbor relationship, and cell growth rate are closely linked to each other, and mainly shaped by the effect of deformation from circle to ellipse and less influenced by the global edge distribution.

Stereological versions of integral geometric formulae for n-dimensional ellipsoids

1986 ◽  
Vol 23 (04) ◽  
pp. 1031-1037 ◽  
Author(s):  
E. B. Jensen ◽  
J. Møller
Keyword(s):  
Explicit Form ◽  
Particle Volume ◽  
Unbiased Estimators ◽  
Ellipsoidal Shape ◽  
Close Relationship ◽  
Geometric Formula ◽  
Lower Dimensional

Recently, unbiased stereological estimators of moments of particle volume, based on measurements on lower-dimensional sections through the particles, have been developed by Jensen and Gundersen (1985). In this note, we derive an explicit form of these unbiased estimators valid for particles in Rnof ellipsoidal shape, and we establish a close relationship between the estimators and a known integral geometric formula for ellipsoids, due to Furstenberg and Tzkoni (1971). Furthermore, a stereological version of another integral geometric formula forn-dimensional ellipsoids, due to Guggenheimer (1973), is derived.

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