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.

In this paper, we consider the numerical solution of a class of optimal control problems involving variable time points in their cost functions. The control enhancing transform is first used to convert the optimal control problem with variable time points into an equivalent optimal control problem with fixed multiple characteristic time (MCT). Using the control parametrization technique, the time horizon is partitioned into several subintervals. Let the partition points also be taken as decision variables. The control functions are approximated by piecewise constant or piecewise linear functions in accordance with these variable partition points. We thus obtain a finite dimensional optimization problem. The control parametrization enhancing control transform (CPET) is again used to convert approximate optimal control problems with variable partition points into equivalent standard optimal control problems with MCT, where the control functions are piecewise constant or piecewise linear functions with pre-fixed partition points. The transformed problems are essentially optimal parameter selection problems with MCT. The gradient formulae for the objective function as well as the constraint functions with respect to relevant decision variables are obtained. Numerical examples are solved using the proposed method.

Steady two-dimensional free surface flow past a semi-infinite flat plate is considered. The vorticity in the flow is assumed to be constant. For large values of the Froude number F, an analytical relation between F, the vorticity parameter ω and the steepness s of the waves in the far field is derived. In addition numerical solutions are calculated by a boundary integral equation method.

Boundary value problems where resonance phenomena are studied are most often transformable to parameter dependent Sturm-Liouville (SL) eigenproblems with interior singularities. The parameter dependent Sturm-Liouville eigenproblem with interior poles is examined. Asymptotic approximations to the solutions are obtained using an extended Langer's method to take care of the resulting complex eigenvalues and eigenfunctions.

We consider the effect the competing mechanisms of buoyancy-driven acceleration (arising from heating a surface) and streamline curvature (due to curvature of a surface) have on the stability of boundary-layer flows. We confine our attention to vortex type instabilities (commonly referred to as Görtler vortices) which have been identified as one of the dominant mechanisms of instability in both centrifugally and buoyancy driven boundary layers. The particular model we consider consists of the boundary-layer flow over a heated (or cooled) curved rigid body. In the absence of buoyancy forcing the flow is centrifugally unstable to counter-rotating vortices aligned with the direction of the flow when the curvature is concave (in the fluid domain) and stable otherwise. Heating the rigid plate to a level sufficiently above the fluid's ambient (free-stream) temperature can also serve to render the flow unstable. We determine the level of heating required to render an otherwise centrifugally stable flow unstable and likewise, the level of body cooling that is required to render a centrifugally unstableflow stable.

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 considers a cell population model with a general maturation rate. This model is described by a nonlinear PDE. We use the theory of operator semigroups to stud the problem under simple hypotheses on the growth function and the nonlinear term. By showing that a related operator generates a strongly continuous semigroup, we prove the existence of a classical solution of the nonlinear problem and its positivity. It is also proved that under simple hypotheses, the problem generates a semiflow. The invariance of the semiflow is studied as well.

We refine Shannon's inequality, in its discrete and integral forms, by presenting upper estimates of the difference between its two sides.

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.

A rapid spherical harmonic calculation method is used for the design of Nuclear Magnetic Resonance shim coils. The aim is to design each shim such that it generates a field described purely by a single spherical harmonic. By applying simulated annealing techniques, coil arrangements are produced through the optimal positioning of current-carrying circular arc conductors of rectangular cross-section. This involves minimizing the undesirable harmonics in relation to a target harmonic. The design method is flexible enough to be applied for the production of coil arrangements that generate fields consisting significantly of either zonal or tesseral harmonics. Results are presented for several coil designs which generate tesseral harmonics of degree one.

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].

The interaction of a number of self-heating bodies depositing heat into a common finite heat bath and thereby influencing each other is a problem of great practical importance in many areas including storage and transport of self-heating materials, drums of chemicals, foodstuffs etc. Critical conditions for the complete assembly of interacting heat producers (thermons) are derived under various assumptions and modes of ignition are identified. These include cooperative modes as well as modes which are simply perturbations of ignition for single thermons.

For a three-dimensional gravity capillary wave packet in the presence of a thin thermocline in deep water two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained. Reducing these two equations to a single equation for oblique plane wave perturbation, the stability of a uniform gravity-capillary wave train is investigated. The stability and instability regions are identified. Expressions for the maximum growth rate of instability and the wavenumber at marginal stability are obtained. The results are shown graphically.

The storage of bagasse, which is principally cellulose, presents many problems for the sugar industry, one of which is bagasse loss due to spontaneous combustion. This is an expensive problem for the industry as bagasse is used as a fuel by sugar mills, and for cogeneration of electricity. Self-heating occurs in the pile through an oxidation mechanism as well as a moisture dependent reaction. The latter reaction is now known to exhibit a local maximum, similar to the heat release curves found in cool-flame problems. Bagasse typically contains 45–55% by weight of water when milling is completed and the question of how to reduce the moisture content is important for two reasons. Firstly, wet bagasse does not burn nearly as efficiently as dry bagasse, and secondly, self-heating is greatly enhanced in the presence of water, for temperatures less than 60–70°C.

An existing mathematical model is used, but modified to take into account the newly observed peak in the moisture dependent reaction. Most of the previously reported complex bifurcation behaviour possible in this model is not realized when physically realistic parameter values are used. The bifurcation diagram describing the long-time steady-state solution is the familiar S-shaped hysteresis curve. In the presence of the new form of the moisture dependent reaction, an intermediate state can be found which is not a true steady-state of the system as, in reality, the characteristics of the pile slowly change as water is lost. This state corresponds to observations of an elevated temperature (around 60–70°C) which persists for long periods of time. Approximate equations can then be defined which predict this intermediate state, and hence a different hysteresis curve is found. A simple explanation for the process by which water is lost from the pile is obtained from these equations and an analytical expression is given for the exponential decay of water levels in the pile.

In this paper we consider an initial value problem for systems of impulsive differential-difference equations is considered. Making use of the method of comparison and differential inequalities for piecewise continuous functions, sufficient conditions for practical stability of the solutions of such systems are obtained. Applications to population dynamics are also given.

In this paper we use classical linear stability theory to analyse the onset of steady and oscillatory Bénard-Marangoni convection in a horizontal layer of fluid in the more physically-relevant case when both the non-dimensional Rayleigh and Marangoni numbers are linearly dependent. We present examples of situations in which there is competition between modes at the onset of convection when the layer is heated from below.

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.