We develop in this paper efficient and robust numerical methods for solving anisotropic Cahn-Hilliard systems. We construct energy stable schemes for the time discretization of the highly nonlinear anisotropic Cahn-Hilliard systems by using a stabilization technique. At each time step, these schemes lead to a sequence of linear coupled elliptic equations with constant coefficients that can be efficiently solved by using a spectral-Galerkin method. We present numerical results that are consistent with earlier work on this topic, and also carry out various simulations, such as the linear bi-Laplacian regularization and the nonlinear Willmore regularization, to demonstrate the efficiency and robustness of the new schemes.

The induced magnetic fields generated by a line mechanical singularity in a magnetized anisotropic half plane are considered in this paper. The linear theory for a soft ferromagnetic elastic with multidomain structure, which has been developed by Pao and Yeh [1] is adopted to investigate this problem. By applying the Fourier transform technique, the exact solutions for the generated magnetic inductions due to various mechanical singularities such as single force, a dipole, single couple and dislocation are obtained in a closed form. The distributions of the generated inductions on the surface are shown graphically.

A two-dimensional (2D) multigroup radiation transfer hydrodynamics code LARED-R-1 is used to simulate a supersonic wave experiment performed earlier by the Livermore group. The main result is that, contrary to the conclusion of Back et al. (2000a), the average-atom opacity model is sufficient to explain the obtained experimental results, provided that an adequate description of the radiation transport was used. The simulation results from LARED-R-1 show the spectrum of radiation in foam with radius and length of several optical depths is not in Planckian distribution and the angular intensity distribution is anisotropic.

We consider the pseudo-p-Laplacian, an anisotropicversion of the p-Laplacian operator for $p\not=2$ . We studyrelevant properties of its first eigenfunction for finite p andthe limit problem as p → ∞.

With the aid of a six-dimensional special eigenvector q, T.C.T.Ting finds five new invariants of anisotropic elasticity constants. The purpose of this paper is to consider some character of the eigenvector q. It is pointed that the six-dimensional special eigenvector q is unique, if it is independent of the coordinate transformation, and the general form of a three-rank orthogonal matrix is given if it has a three-dimensional special eigenvector like q. In addition, the concept of the special eigenvector q is extended and 20 invariants of anisotropic elasticity constants are obtained under rotation about x3-axis.

The general approximate solutions for the two-dimensional thermoelastic problems with a nearly circular hole are provided in this study. Based on Stroh formalism and the method of conformal mapping, the boundary perturbation analysis is applied to solve the problems of a hole with arbitrary shape. The radius of the hole considered here is represented as a sum of a reference constant and a perturbation magnitude that is expanded into a Fourier series. In order to illustrate the applicability and efficiency of the present approach, special examples associated with polygonal hole problems are solved explicitly and discussed in detail. Since the general solutions have not been found in the literature, comparison is made with some special cases for which the analytical solutions exist, which shows that our proposed method is effective and general.

The backbone dynamics and overall tumbling of protein G have been investigated using 15N relaxation. Comparison of measured R2/R1 relaxation rate ratios with known three-dimensional coordinates of the protein show that the rotational diffusion tensor is significantly asymmetric, exhibiting a prolate axial symmetry. Extensive Monte Carlo simulations have been used to estimate the uncertainty due to experimental error in the relaxation rates to be D∥/D⊥ = 1.68 ± 0.08, while the dispersion in the NMR ensemble leads to a variation of D∥/D⊥ = 1.65 ± 0.03. Incorporation of this tensorial description into a Lipari–Szabo type analysis of internal motion has allowed us to accurately describe the local dynamics of the molecule. This analysis differs from an earlier study where the overall rotational diffusion was described by a spherical top. In this previous analysis, exchange parameters were fitted to many of the residues in the alpha helix. This was interpreted as reflecting a small motion of the alpha helix with respect to the beta sheet. We propose that the differential relaxation properties of this helix compared to the beta sheet are due to the near-orthogonality of the NH vectors in the two structural motifs with respect to the unique axis of the diffusion tensor. Our analysis shows that when anisotropic rotational diffusion is taken into account NH vectors in these structural motifs appear to be equally rigid. This study underlines the importance of a correct description of the rotational diffusion tensor if internal motion is to be accurately investigated.

The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfacesin natural norms. Furthermore we prove a global existence result for thecontinuous problem and present some examples of computed surfaces.

The concept of isotropically random mosaic introduced by Pielou, is recalled. The notion of anisotropically random mosaic with respect to a point is defined. An example is given.

Examples of non-stationary random set processes are presented. The reconstruction of a pattern, generated by such a process, is studied; a particular arrangement of the sample-points is considered.