Coordinate-free Approach Research Articles

Numerical modeling of elastic wave propagation and scattering with EFIT — elastodynamic finite integration technique

The basic equations of EFIT, the Elastodynamic Finite Integration Technique, are formulated for anisotropic inhomogeneous media in 3D. For isotropie inhomogeneous media we discuss the discrete equations on a staggered grid resulting in a unique way to discretize material parameters, and evaluate stability conditions and consistency for isotropic homogeneous unbounded media. For the sake of simplified visualization, numerical results are presented for various two-dimensional problems as they originate from nondestructive testing applications. In particular, EFIT wavefronts are related to group velocity curves for anisotropic media with transverse isotropy and cubic symmetry, the latter one being derived by a coordinate-free approach in the appendix.

Discussion of the Paper by Vardi and Lee

Discussion of the Paper by Vardi and Lee

Euclidean Distance Matrix Analysis (EDMA): Estimation of mean form and mean form difference

Euclidean Distance Matrix Analysis (EDMA) of form is a coordinate free approach to the analysis of form using landmark data. In this paper, the problem of estimation of mean form, variance-covariance matrix, and mean form difference under the Gaussian perturbation model is considered using EDMA. The suggested estimators are based on the method of moments. They are shown to be consistent, that is as the sample size increases these estimators approach the true parameters. They are also shown to be computationally very simple. A method to improve their efficiency is suggested. Estimation in the presence of missing data is studied. In addition, it is shown that the superimposition method of estimation leads to incorrect mean form and variance-covariance structure.

Coordinate-free description of rotations

Isometric transformations of the three-dimensional space are considered in a concise coordinate-free approach. Introducing their symmetrical and antisymmetrical components, we show that they are direct and inverse rotations. The Gibbs representation and other well known properties of rotations are then derived.

On limits of spacetimes-a coordinate-free approach

A coordinate-free approach to limits of spacetimes is developed. The limits of the Schwarzschild metric as the mass parameter tends to 0 or infinity are studied, extending previous results. Besides the known Petrov-type-D and type-O limits, three vacuum plane-wave solutions of Petrov type N are found to be limits of the Schwarzschild spacetime.

The limits of Brans-Dicke spacetimes: a coordinate-free approach

We investigate the limit of Brans-Dicke spacetimes as the scalar field coupling constant omega tends to infinity applying a coordinate-free technique. We obtain the limits of some known exact solutions. It is shown that these limits may not correspond to similar solutions in the general relativity theory.

On integrability of 3×3 semi-Hamiltonian hydrodynamic type systems uit = vij( u) ujx which do not possess Riemann invariants

We give a complete classification of the integrable 3×3 semi-Hamiltonian hydrodynamic type systems which do not possess Riemann invariants. The “integrability” is understood as the existence of at least one higher integral. All these equations are transformable to the system of resonant 3-wave interaction by appropriate differential substitution and hence are actually integrable via inverse scattering transform. Considerations are based on a convenient coordinate-free approach to nondiagonalizable systems.

Functional model for dissipative operators. A coordinate-free approach

The paper has, basically, a methodical character. Dissipative operators are investigated with the aid of the Cayley contraction-transformation on the basis of the “coordinate-free model” of V. L Vasyunin and N. K. Nikol'skii. It is shown that the known forms of the characteristic function and of the selfadjoint dilation can be obtained as various realization of a general scheme. One obtains also new formulas for the dilation and its eigenfunctions, generalizing B. S. Pavlov's formulas for the Schrodinger operator on the semiaxis with a real potential and complex boundary condition. Examples are considered.

Two classical theorems of function model theory in coordinate-free presentation

This paper is devoted to the presentation, within the framework of a coordinate-free model, of two known Sz. Nagy—Foiaş theorems: The first one deals with the correspondence between the invariant subspaces of a contraction T and the regular factorizations of its characteristic function θT, while the second one is the commutant lifting theorem. The proofs are based on a coordinate-free approach to the model. In the first theorem an essential point is the singling out of the role of functional imbeddings and the formulation of a criterion for the existence of an invariant subspace in terms of a functional imbedding of a special form. As far as the commutant lifting theorem is concerned, our approach enables us to give a parametrization of the lifted operators with the aid of one free parameter instead of two dependent ones, as done by Sz.-Nagy and Foiaş.

Light propagation characteristics for arbitrary wavevector directions in biaxial media by a coordinate-free approach

The general case of light propagation in lossless anisotropic media occurs in crystals that are biaxial, either naturally so or by induced means (e.g., by electrooptic effect). In these crystals the optical properties, such as the refractive indices, change with propagation direction and are conveniently described by the two-sheeted wavevector surface. Most published work treats light propagation only in the principal planes of the crystal, where the wavevector surface reduces to a circle and an ellipse and the mathematics is simplified. Commonly, however, a biaxial bulk or waveguide device, especially an active device, will be oriented so that the light propagation is not in a principal plane. A complete and concise coordinate-free approach is presented for isolating each sheet, thereby providing a convenient means for calculating the directional optical properties of the two decoupled waves for arbitrary wavevector directions and birefringence levels. The versatility of this approach coupled with available graphics software is demonstrated by displaying numerous cross sections of the wavevector surfaces.

Some examples of the use of distances as coordinates for Euclidean geometry

Distance geometry provides us with an implicit characterization of the Euclidean metric in terms of a system of polynomial equations and inequalities. With the aid of computer algebra programs, these equations and inequalities in turn provide us with a coordinate-free approach to proving theorems in Euclidean geometry analytically. This paper contains a brief summary of the mathematical results on which this approach is based, together with some examples showing how it is applied. In particular, we show how it can be used to derive the topological structure of a simple linkage mechanism.

Discussion of the Paper by Goodall

Discussion of the Paper by Goodall

A coordinate-free approach to surface kinematics

A thin three-dimensional material system, such as a thin shell or fluid-fluid interface, is often modelled as a bidimensional continuous body which at any instant “occupies” some geometrical surface. The time evolution of such surfaces is usually described in terms of curvilinear coordinates [2], [4], [6], a procedure which can mask the geometry involved. An alternative, coordinate-free, approach has been employed [1], [3] which patently exhibits the fundamental geometric (and algebraic) aspects of the kinematics of deforming surfaces. The foundations of this approach are presented in Section 2, following introductory remarks on notation and calculus in Euclidean point spaces, and hitherto unpublished results are developed in Section 3. Account is taken both of material and non-material surfaces: in the former case (surface) mass is conserved (this will be true for thin solid shells) while in the latter context mass exchange with contiguous phases is possible (as is to be expected in the case of fluid-fluid interfaces). The results are also pertinent to singular surfaces [2], [6], [7] (such as shock waves) which are not endowed with intrinsic material attributes but rather with discontinuities of bulk quantities.

Pseudo-Euclidean Hurwitz pairs in any signature

The author considers the pseudo-Euclidean Hurwitz problem, as formulated in a recent study by Lawrynowicz and Rembielinsky (1987). However, he adopts a more coordinate-free approach, and employs the spinorial formalism developed by Crumeyrolle (1989) to solve this problem completely. His results in part disagree with those of Lawrynowicz and Rembielinsky; in particular he uncovers some extra possibilities. He also gives a link between pseudo-Euclidean Hurwitz pairs and Z2-graded Lie algebras, and discusses the introduction of some fibre bundle with pseudo-Euclidean Hurwitz pairs.

A coordinate free approach to score tests

This paper presents a coordinate free approach to score tests and Cα tests. This approach allows the tests to be understood by analogy to the usual linear model. The invariance propertieswith respect to changes in parametrizations are stressed. Finally, aresult of Pregibon relating score tests to the Pearson statistic is extended.

The coordinate-free estimation in finite population sampling

This article presents the coordinate-free approach to extend a result of Bellhouse (1987) for the estimation of a finite population total under model-based inference to include models with correlated error structure. The result is applied to derive the optimal linear model unbiased estimator of the finite population total when the observations are correlated.

A coordinate-free approach to testable hypotheses

Using the coordinate-free version of the linear model and geometrical considerations, the usual definition of testable hyptheses is extended. The concept of partially testable hypotheses is defined in that context. This new approach provides an alternative to the one given by Peixoto (1986), offering some complementary aspects. Simple and direct techniques for testing the new testable hyptheses by means of the classical testing procedure are given.

Nonorthogonal Analysis of Variance Using Gradient Methods

Abstract Because they require very little storage and can be computationally quite efficient, gradient algorithms are attractive methods for fitting large nonorthogonal analysis of variance (ANOVA) models. A coordinate-free approach is used to provide very simple definitions for a number of well-known gradient algorithms and insights into their similarities and differences. The key to finding a good algorithm is finding an algorithm metric that leads to easily computed gradients and that is as close as possible to the metric defined by the ANOVA problem. This leads to the proposal of a new class of algorithms based on a proportional subclass metric. Several new theoretical results on convergence are derived, and some empirical comparisons are made. A similar, but much briefer, treatment of analysis of covariance is given. On theoretical convergence of the methods it is shown, for example, that the Golub and Nash (1982) algorithm requires at most d + 1 iterations if all but d of the cells in the model have the same cell count, that the proportional subclass algorithm converges in one step for proportional subclass problems, and that it demands at most 2 min(a, b) −1 iterations when fitting the two-way additive ANOVA model of size a by b. This can, for example, lead to large savings for models with many more rows than columns. For empirical comparisons a two-way ANOVA model is fitted to some artificial and nonartificial data. For the problems considered, the proportional subclass algorithm requires the fewest iterations followed by the Golub and Nash, optimized steepest descent, Hemmerle, and Yates algorithms, in that order. Some of the differences are quite substantial, involving factors of 10 or more.

Nonorthogonal Analysis of Variance Using Gradient Methods

Abstract Because they require very little storage and can be computationally quite efficient, gradient algorithms are attractive methods for fitting large nonorthogonal analysis of variance (ANOVA) models. A coordinate-free approach is used to provide very simple definitions for a number of well-known gradient algorithms and insights into their similarities and differences. The key to finding a good algorithm is finding an algorithm metric that leads to easily computed gradients and that is as close as possible to the metric defined by the ANOVA problem. This leads to the proposal of a new class of algorithms based on a proportional subclass metric. Several new theoretical results on convergence are derived, and some empirical comparisons are made. A similar, but much briefer, treatment of analysis of covariance is given. On theoretical convergence of the methods it is shown, for example, that the Golub and Nash (1982) algorithm requires at most d + 1 iterations if all but d of the cells in the model have...

Comments, with reply, on "A coordinate-free approach to wave reflection from a uniaxially anisotropic medium" by H.C. Chen

It is pointed out that parts of a previously published paper (see ibid., vol.31, no.4, p.331-36, 1983) have earlier appeared in Russian technical literature not available in the West. A reply from the original author is also given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>