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Abstract: The Riemann problem for a forward–backward parabolic equation of interest in physical and biological models is studied. A complete classification of suitably defined entropy solutions is provided. Thereafter, the existence and uniqueness of a solution is proven, and its type is identified. [Copyright &y& Elsevier]

The paper proposes a general theory for the treatment of a class of optimization problems that arise as a consequence of a search for the optimal replenishment schedule for finite time horizon inventory models. The decision variables in these optimization problems consist of the number of replenishment periods and the lengths of the periods. When the number of replenishment periods is fixed, it is shown that the optimization problems have a unique solution under some partial differential inequalities. Furthermore, the minimal value is convex as a function of the number of replenishment periods. This leads to a simple procedure for determining the optimal control policy. [ABSTRACT FROM AUTHOR]

With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained. [Copyright &y& Elsevier]

Objective Miniature magnets are used in dentistry, principally for the retention of prosthetic devices. The relationship between force and separation of a magnet and its keeper, or, equivalently, two such magnets, has been neither defined theoretically nor described practically in any detail suitable for these applications. The present paper addresses this lacuna. Methods A magnet is considered as a conglomeration of magnetic poles distributed over a surface or a solid in three-dimensional space, with the interaction of poles governed by the Coulomb law. This leads to a suite of mathematical models. These models are analysed for their description of the relationship between the force and the separation of two magnets. Results It is shown that at a large distance of separation, an inverse power law must apply. The power is necessarily integer and at least two. All possibilities are exhausted. Complementarily, under reasonable assumptions, it is shown that at a small distance of separation, the force remains finite. Significance The outcome is in accordance with practical experience, and at odds with the use of simple conceptual models. Consequences relevant to the usage of magnets in dentistry are discussed. [ABSTRACT FROM AUTHOR]

Abstract: Objective: It had previously been shown that the inverse-square law does not apply to the force–distance relationship in the case of a long, thin magnet with one end in close proximity to its image in a permeable plane when simple point-like poles are assumed. Treating the system instead as having a ‘polar disc’, arising from an assumed bundle of dipoles, led to a double integral that could only be evaluated numerically, and a relationship that still did not match observed behavior. Using an elaborate ‘stretched’ exponential polynomial to represent the position of an ‘elastic’ polar disc resulted in a fair representation of the physical response, but this was essentially merely the fitting of an arbitrary function. The present purpose was therefore to find an explicit formula for the force–distance relationship in the polar-disc problem and assess its fit to the previously obtained experimental data. Methods: Starting from Coulomb''s law a corrected integral formula for the force–distance relationship was derived. The integral in this formula was evaluated explicitly using rescaling, changes of order of integration, reduction by symmetry, and change of variables. The resulting formula was then fitted to data that had been obtained for the force exerted by eighty-five rod-shaped magnets (Alnico V, 3mm diameter, 170mm long) perpendicular to a large steel plate, as a function of distance, at small separations (<5mm). Subsequently, the fit of alternative functions was explored. Results: An explicit formula in terms of elliptic integrals was obtained for the polar-disc problem. Despite the greater fidelity, this too was found not to fit the observed physical behavior. Given that failure, nevertheless a simple formula that conforms closely and parsimoniously to the actual magnet data was found. A key feature remains the marked departure from inverse-square behavior. Significance: The failure of the explicit formula to fit the data indicates an inadequate model of the physical system. Nonetheless it constitutes a useful tool for quantifying the force–distance relationship on the premise of polar discs. Given these insights, it may now be possible to address the original motivating problem of the behavior of real dental magnets. [Copyright &y& Elsevier]