Letter to the Editor

The recent paper by Alshabani et al. [Partial size-and-shape distributions, J. Multivariate Anal. (2006), in press] derived the partial size-and-shape distributions motivated by a study in human movement analysis. The paper contained three main results (referred to as Results 1-3), each deriving an expression of the partial size-and-shape distribution. Two of the three results are expressed as infinite sums of terms involving special functions. Here, I would like to point that at least one of these results can be reduced to an explicit and manageable form.     

Statistical theory of shape under elliptical models and singular value decompositions

The size-and-shape and shape distributions based on non-central and non-isotropic elliptical distributions are derived in this paper by using the singular value decomposition (SVD). The general densities require the computation of new integrals involving zonal polynomials. The invariance of the central shape distribution is also proved. Finally, some particular densities are applied in a classical data of Biology, and the inference based on exact distributions is performed after choosing the best model by using a modified BIC^∗ criterion.     

有限深两层流中内孤立波的高阶解*

本文研究有限水深两层流中孤立波的三阶近似理论,并考虑了自由表面对孤立波的影响,运用坐标变形方法得到了三阶内孤立波的发展方程,求得波速的解析表达式。对方程进行了数值计算,得到了几种参数下三阶解曲线,指出自由表面对波型和波速的影响是二阶的。计算表明三阶解对一阶、二阶解有明显的改进,使其更加接近试验结果。     

Partial differential equations and finite-difference methods in image processing,part 1: Image representation

Stochastic representation of discrete images by partial differential equation operators is considered. It is shown that these representations can fit random images, with nonseparable, isotropic covariance functions, better than other common covariance models. Application of these models in image restoration, data compression, edge detection, image synthesis, etc., is possible.Different representations based on classification of partial differential equations are considered. Examples on different images show the advantages of using these representations. The previously introduced notion of fast Karhunen-Loeve transform is extended to images with nonseparable or nearly isotropic covariance functions, or both.     

Singular Stress Behavior in a Bonded Hereditarily-Elastic Aging Wedge. Part I: Problem Statement and Degenerate Case

Stress singularity is investigated in a plane problem for a bonded isotropic hereditarily elastic (viscoelastic) aging infinite wedge. The general solution of the operator Lamé equations, which are partial differential equations in space co-ordinates and integral equations in time, respectively, is represented in terms of one-parametric holomorphic functions (the Kolosov–Muskhelishvili complex potentials depending on time) in weighted Hardy-type classes. After application of the Mellin transform with respect to the radial variable, the problem is reduced to a system of linear Volterra integral equations in time. By using the residue theory for the inverse Mellin transform, the stress asymptotics and strain estimates near the singular point are presented here for non-hereditary Dundurs parameters. The general case of the hereditary Dundurs operators is considered in Part II (see [21]). © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Bayesian size-and-shape regression modelling

Building on Dryden et al. (2021), this note presents the Bayesian estimation of a regression model for size-and-shape response variables with Gaussian landmarks. Our proposal fits into the framework of Bayesian latent variable models and, potentially, allows for a highly flexible modelling framework.     

Studying Dynamical Principles of Human Locomotion using Inverse Optimal Control

Human movement, as for example human gait, can be considered as an optimal realization of some given task. However, the criterion for which the naturally performed human motion is optimal, is generally not known. In this article we formulate an inverse optimal control problem to study the relevance of four different optimization criteria in human locomotion. As a walking model we use an actuated three dimensional spring loaded inverted pendulum (3D-SLIP), which is able to mirror the typical shape of the center of mass trajectory in human gait. Using a direct all-at-once approach, the weighting of the optimization criteria and the position of the footsteps are optimized in such a way, that the center of mass trajectory of the resulting optimal state fits real motion capture data as good as possible. Numerical experiments show, that whereas the so called capture point seems to have a great impact on human walking, minimization of the vertical center of mass movement does not show any relevance at all. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unsupervised classification of children’s bodies using currents

Object classification according to their shape and size is of key importance in many scientific fields. This work focuses on the case where the size and shape of an object is characterized by a current. A current is a mathematical object which has been proved relevant to the modeling of geometrical data, like submanifolds, through integration of vector fields along them. As a consequence of the choice of a vector-valued reproducing kernel Hilbert space (RKHS) as a test space for integrating manifolds, it is possible to consider that shapes are embedded in this Hilbert Space. A vector-valued RKHS is a Hilbert space of vector fields; therefore, it is possible to compute a mean of shapes, or to calculate a distance between two manifolds. This embedding enables us to consider size-and-shape clustering algorithms. These algorithms are applied to a 3D database obtained from an anthropometric survey of the Spanish child population with a potential application to online sales of children’s wear.     

From Damage to Delamination

Delamination models are derived as the limits of models for partial isotropic volume damage via dimension reduction. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Displacement potentials for functionally graded piezoelectric solids

Two new displacement potential functions are introduced for the general solution of a three-dimensional piezoelasticity problem for functionally graded transversely isotropic piezoelectric solids. The material properties vary continuously along the axis of symmetry of the medium. The four coupled equilibrium equations in terms of displacements and electric potential are reduced to two decoupled sixth- and second-order linear partial differential equations for the potential functions. The obtained results are verified with two limiting cases: (i) a functionally graded transversely isotropic medium, and (ii) a homogeneous transversely isotropic piezoelectric solid. The simplified relations corresponding to the special case of similar variation of material properties are also given. Furthermore, the special cases of axisymmetric problems, exponentially graded piezoelectric media and transversely isotropic piezoelectric media with power law variation are discussed in detail.     

Parameter-uniform numerical method for singularly perturbed convection-diffusion problem on a circular domain

A linear singularly perturbed elliptic problem, of convection-diffusion type, posed on a circular domain is examined. Regularity constraints are imposed on the data in the vicinity of the two characteristic points. The solution is decomposed into a regular and a singular component. A priori parameter-explicit pointwise bounds on the partial derivatives of these components are established. By transforming to polar co-ordinates, a monotone finite difference method is constructed on a piecewise-uniform layer-adapted mesh of Shishkin type. Numerical analysis is presented for this monotone numerical method. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established.     

The inverse scattering problem for cavities with impedance boundary condition

We consider the inverse scattering problem of determining the shape of a cavity with impedance boundary condition from sources and measurements placed on a curve inside the cavity. It is shown that both the shape ?D\partial D of the cavity and the surface impedance λ are uniquely determined by the measured data and numerical methods are given for determining both ?D\partial D and λ where neither one is known a priori. Numerical examples are given showing the viability of our method.     

基于经验似然的部分线性模型的统计诊断

经验似然方法己经被广泛应用于许多模型的统计推断.本文基于经验似然对部分线性模型进行统计诊断.首先给出模型的估计方程,进而得到模型参数的极大经验似然估计;其次,基于经验似然研究了三种不同的影响曲率;最后通过随机模拟和实例分析,说明了统计诊断方法的有效性.     

New approach for describing transient pressure response generated by horizontal wells of arbitrary geometry

This paper is aimed at developing a methodology for studying the transient pressure behavior of horizontal wells with any curvilinear trajectory in an isotropic/anisotropic arbitrarily shaped reservoir. This methodology employs generalized functions to represent the tortuous horizontal well. A particular way of removing the singularities involved in the partial differential equation is based on reducing the original problem to the conventional solution of the homogeneous diffusivity equation under any given initial and boundary conditions. The Green function method and any standard numerical technique are combined in a single computational strategy to obtain the transient pressure response generated by a curved and twisted horizontal well in reservoirs with irregular boundaries. Analytical methods can be also used, whenever possible, to solve the reduced problem. This proposal can be easily broadened to analyze the performance of the pressure transient of any kind of reservoir sources or sinks that can be modeled using generalized functions. Some models are presented.     

Asymptotic Behavior of Solutions to the Cahn-Hilliard Equation in Spherically Symmetric Domains

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. The asymptotic behavior of solutions to the Cahn-Hilliard equation with Dirichlet boundary conditions and the associated stationary problem have been studied. In particular, it is proved that the only possible stable equilibrium solutions in spherically symmetric domains are spherically symmetric and monotone in the radial direction.     

Spatio-temporal analysis of kinematic signals in classical ballet

Motions of markers arranged on a dancer's body can be approximated by the sum of a minimal set of linear trajectories with given accuracy. The composition of approximating linear trajectories features the movement traits and discloses the level of movement expertise in the dancers. We suggest the computationally simple methods for the analysis of trajectories and body shape changes attested directly from the motion capture data. We have tested our approach for 6 figures from the classical ballet repertoire performed by 24 dancers varying in expertise. The methods allow to estimate the level of movement expertise, to draw the detailed structure of movements, and to classify movements into a given repertoire automatically.     

Numerical Modelling of PIN Diodes

The coupled non-linear partial differential equations describingthe behaviour of PIN diodes are solved using a conservativefinite-difference scheme. For such problems the conservationof certain quantities is particularly important and the consistencyof the numerical scheme is demonstrated. Problems are consideredin cylindrical polar co-ordinates, and particular attentionis paid to the correct discretization of the boundary conditions.The modelling of large thin diodes is also considered. Numericalresults illustrating various features are presented.     

Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging

For all p>2,k>p, a size-and-reflection-shape space of k-ads in general position in R^p, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold , a space of orbits of scaled k-ads in general position under the group of isometries of R^p, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p>2. The Veronese embedding of the planar Kendall shape manifold is extended to an equivariant embedding of the size-and-shape manifold , which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.     

Curvilinear Co-ordinate Systems Associated with a Class of Boundary-Value Problems

A system of non-linear integrability equations is derived whichis associated with the differential form of a transformationfrom Cartesian co-ordinates to non-orthogonal curvilinear co-ordinates.A solution for this system is established when the curvilinearco-ordinate system contains two identical scaling factors andone right angle provided the unit normal to a smooth, finiteor infinite tube is prescribed. The general form of the transformationis obtained. It is shown that the transformation can also beobtained when the unit normal to the tube boundary section bythe plane normal to the given curve defining the orientationof the tube is prescribed. Moreover conditions are establishedunder which a completely orthogonal co-ordinate system can befound. An example is treated for a tube with circular cross-sectionand both the non-orthogonal and orthogonal co-ordinate systemsassociated with this are discussed.     

Comparative study of some constitutive equations characterising non-Newtonian fluids

This paper compares, in a general way, the predictions of the constitutive equations given by Rivlin and Ericksen, Oldroyd, and Walters. Whether we consider the rotational problems in cylindrical co-ordinates or in spherical polar co-ordinates, the effect of the non-Newtonicity on the secondary flows is collected in a single parameterα which can be explicitly expressed in terms of the non-Newtonian parameters that occur in each of the above-mentioned constitutive equations. Thus, for a given value ofα, all the three fluids will have identical secondary flows. It is only through the study of appropriate normal stresses that a Rivlin-Ericksen fluid can be distinguished from the other two fluids which are indistinguishable as long as this non-Newtonian parameter has the same value.