General shape distributions in a plane

1991 ◽  
Vol 23 (2) ◽  
pp. 259-276 ◽  
Author(s):  
I. L. Dryden ◽  
K. V. Mardia
Keyword(s):  
Closed Form ◽  
Normal Approximation ◽  
Asymptotic Properties ◽  
Two Dimensions ◽  
General Shape ◽  
Shape Distribution ◽  
Inference Problems ◽  
Point Configurations ◽  
Exact Shape

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

General shape distributions in a plane

1991 ◽  
Vol 23 (02) ◽  
pp. 259-276 ◽  
Author(s):  
I. L. Dryden ◽  
K. V. Mardia
Keyword(s):  
Closed Form ◽  
Normal Approximation ◽  
Asymptotic Properties ◽  
Two Dimensions ◽  
General Shape ◽  
Shape Distribution ◽  
Inference Problems ◽  
Point Configurations ◽  
Exact Shape

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.

Adhesive Contact of Bodies Having an Elliptical Contact Area

2005 ◽  
Author(s):  
K. L. Johnson ◽  
J. A. Greenwood
Keyword(s):  
Closed Form ◽  
Contact Area ◽  
Adhesive Contact ◽  
Contact Radius ◽  
General Shape ◽  
Elliptical Contact ◽  
Jkr Theory ◽  
Two Bodies

The so-called JKR theory of adhesion between elastic spheres in contact (Johnson, Kendall & Roberts 1971, Sperling 1964) has been widely used in micro-tribology. In this paper the theory is extended to solids of general shape and curvature. It is assumed that the area of contact is elliptical which turns out to be approximately true, though the eccentricity is different from that for non-adhesive contact. Closed form expressions are found for the variation with load of contact radius and displacement, as a function of the ratio of principal relative curvatures of the two bodies in contact. The pull-off force is found to decrease with increasing eccentricity from its value of 3πΔγR/2 in the case of contact of spheres of radius R.

Asymptotic properties of the equilibrium probability of identity in a geographically structured population

1977 ◽  
Vol 9 (2) ◽  
pp. 268-282 ◽  
Author(s):  
Stanley Sawyer
Keyword(s):  
Nearest Neighbor ◽  
Asymptotic Properties ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
Three Dimensions ◽  
One Dimension ◽  
Order Zero ◽  
Structured Population ◽  
Equilibrium Probability ◽  
Migration Law

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ2 is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C0 is a constant depending on the migration law, K0(y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ2. For symmetric nearest-neighbor migrations, in one dimension and log mi in two. For is known in one dimension and C0 does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.

scholarly journals Polarization tensor vanishing structure of general shape: Existence for small perturbations of balls

2020 ◽  
pp. 1-32
Author(s):  
Hyeonbae Kang ◽  
Xiaofei Li ◽  
Shigeru Sakaguchi
Keyword(s):  
Small Volume ◽  
Expansion Method ◽  
Two Dimensions ◽  
Degree Zero ◽  
Polarization Tensor ◽  
General Shape ◽  
Small Perturbations ◽  
Geometric Quantity ◽  
The Core ◽  
Boundary Measurements

The polarization tensor is a geometric quantity associated with a domain. It is a signature of the small inclusion’s existence inside a domain and used in the small volume expansion method to reconstruct small inclusions by boundary measurements. In this paper, we consider the question of the polarization tensor vanishing structure of general shape. The only known examples of the polarization tensor vanishing structure are concentric disks and balls. We prove, by the implicit function theorem on Banach spaces, that a small perturbation of a ball can be enclosed by a domain so that the resulting inclusion of the core-shell structure becomes polarization tensor vanishing. The boundary of the enclosing domain is given by a sphere perturbed by spherical harmonics of degree zero and two. This is a continuation of the earlier work (Kang, Li, Sakaguchi) for two dimensions.

The diffraction of SH waves by an arbitrary shaped crack in two dimensions

1992 ◽  
Vol 340 (1659) ◽  
pp. 503-529 ◽  
Keyword(s):  
Function Method ◽  
Scattering Problem ◽  
Elastic Solid ◽  
Two Dimensions ◽  
General Shape ◽  
Sh Waves ◽  
Green Function Method ◽  
Rigid Barrier ◽  
Shaped Crack ◽  
Scalar Scattering

In this paper we consider the two-dimensional scalar scattering problem for Helmholtz’s equation exterior to a smooth open arc of general shape. The problem has a number of physical applications including the diffraction of sound by a rigid barrier immersed in a compressible fluid and by a crack in an elastic solid which supports a state of anti-plane strain (SH-motion). The mathematical method used here is the crack Green function method introduced by G. R. Wickham. This enables the scattering problem to be reduced to the solution of a Fredholm integral equation of the second kind with a continuous kernel. The numerical solution of this equation is discussed and a number of examples are computed.

Existence of weakly neutral coated inclusions of general shape in two dimensions

2020 ◽  
pp. 1-24 ◽  
Author(s):  
Hyeonbae Kang ◽  
Xiaofei Li ◽  
Shigeru Sakaguchi
Keyword(s):  
Two Dimensions ◽  
General Shape
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