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Analytical and Bioanalytical Chemistry -     

Network search games with immobile hider,without a designated searcher starting point

In the (zero-sum) search game Γ(G, x) proposed by Isaacs, the Hider picks a point H in the network G and the Searcher picks a unit speed path S(t) in G with S(0) = x. The payoff to the maximizing Hider is the time T = T(S, H) = min{t : S(t) = H} required for the Searcher to find the Hider. An extensive theory of such games has been developed in the literature. This paper considers the related games Γ(G), where the requirement S(0) = x is dropped, and the Searcher is allowed to choose his starting point. This game has been solved by Dagan and Gal for the important case where G is a tree, and by Alpern for trees with Eulerian networks attached. Here, we extend those results to a wider class of networks, employing theory initiated by Reijnierse and Potters and completed by Gal, for the fixed-start games Γ(G, x). Our results may be more easily interpreted as determining the best worst-case method of searching a network from an arbitrary starting point.     

Nonparametric inference for extrinsic means on size-and-(reflection)-shape manifolds with applications in medical imaging

For all p>2,k>p, a size-and-reflection-shape space of k-ads in general position in R^p, invariant under translation, rotation and reflection, is shown to be a smooth manifold and is equivariantly embedded in a space of symmetric matrices, allowing a nonparametric statistical analysis based on extrinsic means. Equivariant embeddings are also given for the reflection-shape-manifold , a space of orbits of scaled k-ads in general position under the group of isometries of R^p, providing a methodology for statistical analysis of three-dimensional images and a resolution of the mathematical problems inherent in the use of the Kendall shape spaces in p-dimensions, p>2. The Veronese embedding of the planar Kendall shape manifold is extended to an equivariant embedding of the size-and-shape manifold , which is useful in the analysis of size-and-shape. Four medical imaging applications are provided to illustrate the theory.     

An alternative tactic to deal with the contingency of driver absenteeism

On any given workday, unexpected absence of drivers calls for emergency operational tactics to ensure completion of planned deliveries of merchandise to customers. A typical tactic in the workforce scheduling literature is to rely exclusively on a pool of workers who were not scheduled for work; that is, full-time off-duty drivers and part-time drivers. Concerns about the expense of maintaining a staff of part-timers prompt consideration of alternative tactics. This study proposes one such alternative. It minimizes the part-time pool by reassigning routes among drivers who show for work. In multi-route vehicle routing problems, the prospect of this alternative arises because cost-minimizing patterns of travel distances across routes often allow one driver to do extra work by handling more than one route (subject to time-window and workload constraints). By considering costs such as overtime rates when drivers perform extra (emergency) driving duties and the cost to maintain part-time staff, this study tests the efficacy of the aforementioned alternative. Its efficacy is confirmed by the fact that it is less costly than the typical tactic: while the typical tactic adds just over 22% to driver payroll, the proposed alternative adds a noticeably smaller cost of just under 15%.     

Bifurcation of rotating thick-walled elastic tubes

The deformation of a circular cylindrical elastic tube of finite wall thickness rotating about its axis is examined. A circular cylindrical deformed configuration is considered first, and the angular speed analysed as a function of an azimuthai deformation parameter at fixed axial extension for an arbitrary form of incompressible, isotropic elastic strain-energy function. This extends the analysis given previously (Haughton and Ogden, 1980) for membrane tubes.Bifurcation from a circular cylindrical configuration is then investigated. Prismatic, axisymmetric and asymmetric bifurcation modes are discussed separately. Their relative importance is assessed in relation to the wall thickness and length of the tube, the magnitude of the axial extension, and the angular speed turning-points. Numerical results are given for a specific form of strain-energy function.Amongst other results it is found that (i) for long tubes, asymmetric modes of bifurcation can occur at low values of the angular speed and before any possible axisymmetric or prismatic modes and (ii) for short tubes, there is a range of values of the axial extension (including zero) for which no bifurcation can occur during rotation.     

On the incremental equations in non-linear elasticity — I. Membrane theory

A new formulation of the equations of membrane theory in non-linear elasticity is described. It is based on the consistent use of certain conjugate variables averaged through the (undeformed) thickness of the thin shell which the membrane approximates. The deformation gradient is taken as the basic measure of deformation, and its average value as the membrane measure of deformation. It is shown that the average elastic strain energy can be regarded as a function of the average deformation gradient to within an error which is of the second order in a certain small parameter. Moreover, to the same order, the average strain energy is a potential function for the average nominal stress. This means that the averages of the conjugate variables (nominal stress and deformation gradient) are also conjugate.In terms of the average conjugate variables, the membrane equilibrium equations are obtained by averaging from the equilibrium equations of the full three-dimensional theory. Discussion of the order of magnitude of the errors involved in the membrane approximation is a feature of the analysis.The corresponding incremental equations are also derived as a prelude to their application in certain bifurcation problems. One such problem is examined in the companion paper (Part II) in which results for thick shells and membranes are compared.     

Existence of Exact Solutions for the Eversion of Elastic Cylinders

The existence of eversion deformations of an elastic cylinder into another right circular cylinder is studied. It is found that such deformations are associated with strain-energy functions of separable form W(λ₁,λ₂,λ₃) =φ(λ₁)+φ(λ₂)+φ (λ₃), and thus can serve as a test for materials of this kind. Under certain constitutive assumptions, there always exists a cylindrical eversion deformation that satisfies exact pointwise traction free boundary conditions over the entire surface of the cylinder. For such solutions the cavity must remain open upon eversion. This revised version was published online in August 2006 with corrections to the Cover Date.