Double coverings of hyperelliptic real algebraic curves

We consider double and (possibly) branched coverings π:X→X^′ between real algebraic curves where X is hyperelliptic. We are interested in the topology of such coverings and also in describing them in terms of algebraic equations. In this article we completely solve these two problems. We first analyse the topological features and ramification data of such coverings. Second, for each isomorphism class of these coverings we then describe a representative, with defining polynomial equations for X and for X^′, a formula for the involution that generates the covering transformation group, and a rational formula for the covering projection π:X→X^′.     

Stochastic optimal control and algorithm of the trajectory of horizontal wells

This paper presents a nonlinear, multi-phase and stochastic dynamical system according to engineering background. We show that the stochastic dynamical system exists a unique solution for every initial state. A stochastic optimal control model is constructed and the sufficient and necessary conditions for optimality are proved via dynamic programming principle. This model can be converted into a parametric nonlinear stochastic programming by integrating the state equation. It is discussed here that the local optimal solution depends in a continuous way on the parameters. A revised Hooke–Jeeves algorithm based on this property has been developed. Computer simulation is used for this paper, and the numerical results illustrate the validity and efficiency of the algorithm.     

Multiplicity of periodic solutions for second-order discrete Hamiltonian systems with a small forcing term

Some sufficient conditions for the existence of three periodic solutions are obtained for the second-order discrete Hamiltonian systems with a small forcing term by using a three-critical-point theorem.     

Numerical study of the flow in a three-dimensional thermally driven cavity

Solutions for the fully compressible Navier–Stokes equations are presented for the flow and temperature fields in a cubic cavity with large horizontal temperature differences. The ideal-gas approximation for air is assumed and viscosity is computed using Sutherland's law. The three-dimensional case forms an extension of previous studies performed on a two-dimensional square cavity. The influence of imposed boundary conditions in the third dimension is investigated as a numerical experiment. Comparison is made between convergence rates in case of periodic and free-slip boundary conditions. Results with no-slip boundary conditions are presented as well. The effect of the Rayleigh number is studied.     

Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

We present numerical results concerning the solution of the time-harmonic Maxwell equations discretized by discontinuous Galerkin methods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwell equations, is performed in the two-dimensional case.     

Topological decoupling, linearization and perturbation on inhomogeneous time scales

We derive a linearization theorem in the framework of dynamic equations on time scales. This extends a recent result from [Y. Xia, J. Cao, M. Han, A new analytical method for the linearization of dynamic equation on measure chains, J. Differential Equations 235 (2007) 527-543] in various directions: Firstly, in our setting the linear part need not to be hyperbolic and due to the existence of a center manifold this leads to a generalized global Hartman-Grobman theorem for nonautonomous problems. Secondly, we investigate the behavior of the topological conjugacy under parameter variation.These perturbation results are tailor-made for future applications in analytical discretization theory, i.e., to study the relationship between ODEs and numerical schemes applied to them.     

Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.Received: 30 March 2004     

osp(N, 2) AKNS equations

The osp(N, 2) extension of the AKNS scheme is reconsidered. It leads to a general class of integrable nonlinear evolution equations for 2+N(N–1)/2 commuting and 2N anticommuting fields. by reduction, various osp(N, 2) versions of the Korteweg-de Vries equation can be obtained. One of these is shown to be bi-Hamiltonian and its second Hamiltonian structure corresponds to the classical limit of the so(N) superconformal algebra. The nonlinear Schrödinger and modified Korteweg-de Vries reductions are also briefly discussed.Work supported by NSerc and FCAR.     

Hyperfine interactions and lattice distortion of the F center in KCl, NaCl and LiCl crystals

Detailed embedded-cluster quantum-chemistry (ROHF, UHF and UMP2) calculations of hyperfine interactions for the F center in KCl, NaCl and LiCl are presented. Our values for the isotropic hyperfine interaction show a good general agreement with experiments. We show that a careful treatment of the lattice environment, local lattice distortion, spin polarization and correlation are important for this good agreement.     

Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem

Summary It is shown that a boundary-value problem based on a holonomic elastic-plastic constitutive law may be formulated equivalently as a variational inequality of the second kind. A regularised form of the problem is analysed, and finite element approximations are considered. It is shown that solutions based on finite element approximation of the regularised problem converge.