A well-posed integral equation formulation for three-dimensional rough surface scattering

We consider the problem of scattering of time-harmonic acoustic waves by an unbounded sound-soft rough surface. Recently, a Brakhage Werner type integral equation formulation of this problem has been proposed, based on an ansatz as a combined single- and double-layer potential, but replacing the usual fundamental solution of the Helmholtz equation with an appropriate half-space Green's function. Moreover, it has been shown in the three-dimensional case that this integral equation is uniquely solvable in the space L-2 (Gamma) when the scattering surface G does not differ too much from a plane. In this paper, we show that this integral equation is uniquely solvable with no restriction on the surface elevation or slope. Moreover, we construct explicit bounds on the inverse of the associated boundary integral operator, as a function of the wave number, the parameter coupling the single- and double-layer potentials, and the maximum surface slope. These bounds show that the norm of the inverse operator is bounded uniformly in the wave number, kappa, for kappa > 0, if the coupling parameter h is chosen proportional to the wave number. In the case when G is a plane, we show that the choice eta = kappa/2 is nearly optimal in terms of minimizing the condition number.

Optimal protection of stabilised dry live bacteria from bile toxicity in oral dosage forms by bile acid adsorbent resins

We previously found that dried live bacteria of a vaccine strain can be temporarily sensitive to bile acids and suggested that Bile Adsorbing Resins (BAR) can be used in oral vaccine tablets to protect dried bacteria from intestinal bile. Here, we report a quantitative analysis of the ability of BAR to exclude the dye bromophenol blue from penetrating into matrix tablets and also sections of hard capsule shells. Based on this quantitative analysis, we made a fully optimised formulation, comprising 25% w/w of cholestyramine in Vcaps™ HPMC capsules. This gave effectively 100% protection of viability from 4% bile, with 4200-fold more live bacteria recovered from this formulation compared to unprotected dry bacteria. From the image analysis, we found that the filler material or compaction force used had no measurable effect on dye exclusion but did affect the rate of tablet hydration. Increasing the mass fraction of BAR gave more exclusion of dye up to 25% w/w, after which a plateau was reached and no further dye exclusion was seen. More effective dye exclusion was seen with smaller particle sizes (i.e. cholestyramine) and when the BAR was thoroughly dried and disaggregated. Similar results were found when imaging dye penetration into capsule sections or tablets. The predictions of the dye penetration study were tested using capsules filled with dried attenuated Salmonella vaccine plus different BAR types, and the expected protection from bile was found, validating the imaging study. Surprisingly, depending on the capsule shell material, some protection was given by the capsule alone without adding BAR, with Vcaps™ HPMC capsules providing up to 174-fold protection against 1% bile; faster releasing Vcaps Plus™ HPMC capsules and Coni Snap™ gelatin capsules gave less protection.

Singularities of optimal control problems on some 6-D lie groups

This paper considers the motion planning problem for oriented vehicles travelling at unit speed in a 3-D space. A Lie group formulation arises naturally and the vehicles are modeled as kinematic control systems with drift defined on the orthonormal frame bundles of particular Riemannian manifolds, specifically, the 3-D space forms Euclidean space E-3, the sphere S-3, and the hyperboloid H'. The corresponding frame bundles are equal to the Euclidean group of motions SE(3), the rotation group SO(4), and the Lorentz group SO (1, 3). The maximum principle of optimal control shifts the emphasis for these systems to the associated Hamiltonian formalism. For an integrable case, the extremal curves are explicitly expressed in terms of elliptic functions. In this paper, a study at the singularities of the extremal curves are given, which correspond to critical points of these elliptic functions. The extremal curves are characterized as the intersections of invariant surfaces and are illustrated graphically at the singular points. It. is then shown that the projections, of the extremals onto the base space, called elastica, at these singular points, are curves of constant curvature and torsion, which in turn implies that the oriented vehicles trace helices.

Conditioning and preconditioning of the optimal state estimation problem

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.