A humanoid robot system may be viewed as a collection of segments coupled at rotational joints which geometrically represent constrained rotational Lie groups. This allows a study of the dynamics of the motion of a humanoid robot. Several formulations are possible. In this paper, dual invariant topological structures are constructed and analyzed on the finite-dimensional manifolds associated with the humanoid motion. Both cohomology and homology structures are examined on the tangent (Lagrangian) as well as on the cotangent (Hamiltonian) bundles on the manifold of the humanoid motion configuration. represented by the toral Lie group. It is established all four topological structures give in essence the same description of humanoid dynamics. Practically this means that whichever of these approaches we use, ultimately we obtain the same mathematical results.

The wave motion of magnetohydrodynamic (MHD) systems can be quite complicated. In order to study the motion of waves in a perfectly conducting fluid under the influence of an external magnetic field in a stratified medium, we make the simplifying assumption that the pressure is constant (to first order). This is the simplest form of the equations with variable coefficients and is not strongly propagative. Alfven waves are still present. The system is further simplified by assuming that the external field is parallel to the boundary. The Green's function for the operator is constructed and then the spectral family is constructed in terms of generalized eigenfunctions, giving four families of propagating waves, including waves “trapped” in the boundary layer. These trapped waves are interesting, since they are not the relics of surface waves, which do not exist in this context when the boundary layer shrinks to zero thickness no matter what (maximal energy preserving) boundary condition is chosen. We conjecture a similar structure for the full MHD problem.

This paper develops a simple model for the containment of oil behind a boom in water. The flow of water beneath the oil is assumed two dimensional (horizontal and vertical) and perpendicular to the boom. We look for steady solutions and assume the oil is so viscous that the fluid velocity within the oil is zero. We are able to calculate what shape the oilslick will form and under which circumstances the boom will be successful (that is, no oil escapes under the boom) based on the predicted depth of the slick at the boom.

In this paper, the results of Freedman and So [13] on global stability and persistence of simple food chains are extended to general diffusive food chains. For global stability of the unique homogeneous positive steady state, our approach involves an application of the invariance principle of reaction-diffusion equations and the construction of a Liapunov functional. For persistence, we use the dynamical system results of Dunbar et al. [11] and Hutson and Moran [29].

We consider the stability of the zero solution of a system of impulsive functional-differential equations. By means of piecewise continuous functions, which are generalizations of classical Lyapunov functions, and using a technique due to Razumikhin, sufficient conditions are found for stability, uniform stability and asymptotical stability of the zero solution of these equations. Applications to impulsive population dynamics are also discussed.

In this paper, multigrid methods for solving the biharmonic equation using some nonconforming plate elements are considered. An average algorithm is applied to define the transfer operator. A general analysis of convergence is given.

This paper is devoted to the study of the solvability of certain one-and multidimensional Abel-type integral equations involving the Gauss hypergeometric function as their kernels in the space of summable functions. The multidimensional equations are considered over certain pyramidal domains and the results obtained are used to present the multidimensional pyramidal analogues of generalized fractional calculus operators and their properties.