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The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of two-dimensional scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system coupling (1) a standard finite element discretization localized around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established for complex wavenumbers only. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering. [ABSTRACT FROM AUTHOR]

We investigate existence and exact multiplicity of phase-locked solutions of an integro-differential equation derived from a Kuramoto system of coupled oscillators. Under general assumptions on the form of frequency distribution, we derive new, easily verified criteria which guarantee that either (i) exactly one solution exists, or (ii) exactly two solutions coexist over an entire interval of values of the key parameter γ. We illustrate our results with an example in which each of these possibilities occurs. Problems for future research are suggested. [ABSTRACT FROM AUTHOR]

This paper is concerned with the direct and inverse acoustic or electromagnetic scattering problems by a locally perturbed, perfectly reflecting, infinite plane (which is called a locally rough surface in this paper). We propose a novel integral equation formulation for the direct scattering problem which is defined on a bounded curve (consisting of a bounded part of the infinite plane containing the local perturbation and the lower part of a circle) with two corners. This novel integral equation can be solved efficiently by using the Nyström method with a graded mesh introduced previously by Kress and is capable of dealing with large wave number cases. For the inverse problem, we propose a Newton iteration method to reconstruct the local perturbation of the plane from multiple-frequency far-field data, based on the novel integral equation formulation. Numerical examples are carried out to demonstrate that our reconstruction method is stable and accurate even for the case of multiple-scale profiles. [ABSTRACT FROM AUTHOR]

A class of generalized analytic functions arising from the Oseen equations in the axially symmetric case has been identified. For this class of functions, the generalized Cauchy integral formula has been obtained, and a series representation for the region exterior to a sphere has been constructed. The velocity field of the axially symmetric Oseen flow has been represented in terms of two generalized analytic functions, and it has been shown that for an exterior Oseen flow problem, those functions are uniquely determined, provided that they both vanish at infinity. Also the pressure and vorticity have been determined as the real and imaginary parts of the two functions representing the velocity field, and the drag exerted on a solid body of revolution in the axially symmetric Oseen flow has been expressed in terms of one of the involved generalized analytic functions. The problem of the axially symmetric Oseen flow past a solid body of revolution has been reduced to an integral equation based on the generalized Cauchy integral formula. The integral equation has been shown to have computational advantage over an integral equation based on the Oseenlets. The developed framework of generalized analytic functions has been illustrated in solving the problem of the Oseen flow past a solid sphere and solid bispheroids. For different Reynolds numbers, a minimum drag spheroid of fixed volume has been found. [ABSTRACT FROM AUTHOR]

An integral equation constrained optimization approach to finding minimum-drag shapes for solid bodies of revolution in Stokes flows subject to constraints on body's volume and shape has been developed. An axially symmetric Stokes flow problem has been formulated in the form of an integral equation (state equation), and finding an optimal shape has been reduced to an integral equation constrained optimization problem. The total variation of the Lagrangian functional of the problem has been reduced to the variation only with respect to the shape by the adjoint equation-based method, and the steepest descent direction for the coefficients in the function series representing the shape has been found analytically. It has been shown that solutions to the state and adjoint integral equations are related by a simple algebraic formula, which eliminates the need for solving the adjoint equation. The approach has been illustrated in finding the minimum-drag shape for a solid unit volume particle and in finding the optimal shape for the fixed-volume nose of a solid torpedo-shaped body encountering minimum drag along its axis of revolution. In the later problem, the dependence of the nose's optimal shape on the torpedo's length has been investigated. [ABSTRACT FROM AUTHOR]

The framework of generalized analytic functions arising from the related potentials (so-called k-harmonically analytic functions) has been developed in application to three-dimensional (3D) axially symmetric Stokes flow problems. Cauchy's integral formula for the class of k-harmonically analytic functions has been obtained, and series representations for k-harmonically analytic functions for the regions exterior to sphere and prolate and oblate spheroids have been derived. As the central result in the developed framework, a solution form representing the velocity field and pressure for 3D axially symmetric Stokes flows has been constructed in terms of two 0-harmonically analytic functions. It has also been shown that it uniquely determines an external velocity field vanishing at infinity. With the obtained solution form, the problem of 3D Stokes flows due to the axially symmetric translation of a solid body of revolution has been reduced to a boundary-value problem for two 0-harmonically analytic functions, and the resisting force exerted on the body has been expressed in terms of a 0-harmonically analytic function entering the solution form. For regions in which Laplace's equation admits separation of variables, the boundary-value problem can be solved analytically via representations of 0-harmonically analytic functions in corresponding curvilinear coordinates. This approach has been demonstrated for the axially symmetric translation of solid sphere and solid prolate and oblate spheroids. As the second approach, the boundary-value problem has been reduced to an integral equation based on Cauchy's integral formula for k-harmonically analytic functions. As an illustration, the integral equation has been solved for the axially symmetric translation of solid bispheroids and the solid torus of elliptical cross-section for various values of a geometrical parameter. [ABSTRACT FROM AUTHOR]

A skeletal muscle is composed of motor units, each consisting of a motoneuron and the muscle fibers it innervates. The input to the motor units is formed of electrical signals coming from higher motor centers and propagated to the motoneurons along a network of nerve fibers. Because of its complexity, this network still escapes actual direct observations. The present model describes the steady state activation of a muscle, i.e., of its motor units. It incorporates the network as an unknown quantity and, given the latter, predicts the input-force relation (activation curve) of the muscle. Conversely, given a suitable activation curve, our model enables the recovery of the network. This step is performed by using experimental data about the activation curve, and the whole activation process of a muscle can then be theoretically investigated. In this way, this approach provides a link between the macroscopic (activation curve) and microscopic (network) levels. From a mathematical viewpoint, solving the preceding inverse problem is equivalent to solving an integral equation of a new type. [ABSTRACT FROM AUTHOR]

Changes to the relative separation of molecules or other interacting species on account of diffusion accompany their associative or dissociative reaction. The molecules are symbolized, for two distinct types, A,B, by the relations A +B AB, and, if [A], [B], and [AB] denote the corresponding densities, the equation d/dt [AB] = k+[A][B] specifies an associative process with forward rate constant k+. An approximate version of the preceding takes the form of a linear differential equation, which can be employed to obtain significant estimates for both k+ and the flux function d[AB]/dt. Such estimates are presented in different circumstances, including the localization of A,B on a common planar surface or their distribution in space, and also when the domain of A is a half space whereas that of B is a bounding planar surface. It proves advantageous to reformulate the last, a mixed boundary value problem, in terms of a linear integral equation. Biological applications are discussed, including the mechanism for the observed phosphorylation of proteins in resting cells and the incipience of phototransduction in rod photoreceptors. [ABSTRACT FROM AUTHOR]