A cochlear model with transversally inhomogeneous cross-section

The transversally inhomogeneous cross-section of the model is constructed from the data obtained by Retzius, Hardesty, Wever, and Miller. On the basis of a three-dimensional linear mechanical WKB model of a cochlea without negative friction we obtain high frequency selectivity and a phase shift in the domain of the peak of the amplitude function comparable with the measurements taken by Sellick and Robles.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 128–138, 1989.     

A system of two inhomogeneous linear functional equations

Given real nonzero coefficients a, A, b, B and additive functions $\phi,\psi : {\mathbb {R}}\to {\mathbb {R}}$ , necessary and sufficient conditions for existence and uniqueness of additive solutions of the system ξ(ax)?Aξ(x)=?(x), ξ(bx)?Bξ(x)=ψ(x) are presented. According to the various possibilities concerning the arithmetic nature of the four coefficients and the algebraic relationships between them, the additive solution(s) of the system are given explicitly for each case.     

Minkowski's conjecture on a system of linear inhomogeneous forms

The possibility of representing third-order unimodular matrices as a product of diagonal, orthogonal, triangular, and integral matrices is investigated.Translated from Matematicheskii Zametki, Vol. 5, No. 1, pp. 107–116, 1969.     

Saint-Venant decay rates for an inhomogeneous isotropic linear thermoelastic strip

In this paper we consider the state of plane strain in an isotropic and inhomogeneous thermoelastic material occupying a rectangular strip. Such a strip is maintained in equilibrium under self-equilibrated traction applied on one of the heated edges, while the other three edges are thermally insulated and traction-free. Our aim is to derive some explicit spatial estimates describing how certain appropriate measures of the Airy stress function and temperature evolve with respect to the distance from the loaded and heated edge, provided specific assumptions are made upon the derivatives of the thermoelastic coefficients. The results of the present paper prove how the spatial decay rate varies with the inhomogeneous constitutive profile.     

Nonlinear mathematical model of inhomogeneous tidal rivers

A nonlinear mathematical model of a tidal river is developed taking into account the effect of the variation of density. Besides the general procedure of solving the equations of continuity and motion, this model incorporates the salt balance equation together with an equation of state relating the salinity with density. The implicit finite difference method is used to solve the nonlinear system. The computational results are then compared with those observed in the river Hooghly. The close agreement between the computed and observed results proves the efficiency and applicability of the model to a variety of other tidal flows in river systems.     

Soliton dynamics in an inhomogeneous Bose-Einstein condensate driven by linear potential

In this paper we study the propagation of solitons in a Bose-Einstein condensate governed by the time dependent one dimensional Gross-Pitaevskii equation managed by Feshbach resonance in a linear external potential. We give the Lax pair of the Gross-Pitaevskii equation in Bose-Einstein condensates and obtain exact N-soliton solution by employing the simple, straightforward Darboux transformation. As an example, we present exact one and two-soliton solution and discuss their transmission, interaction and dynamic properties. We further calculate the particle number, momentum and energy of the solitons and discuss their conservation laws. Knowledge of soliton dynamics helps us in understanding the physical nature of the condensate and in the calculation of the thermodynamic properties.     

A posteriors error bounds for inhomogeneous linear dirichlet and neumann problems

Second-order inhomogeneous linear Dirichlet and Neumann problems in divergent form on a simply-connected to-dimensional domain with Lipschitz-continious boundary of finite length are considered. Conjugate problems, that is, a pair of one Dirichlet and one Neumann problem the minima of energies of which add to a known constant, are introduced. From the conceppt of conjugate problems, two-sided bounds for the energy of the exact solution of any given Dirichlet or Neumann problem are constructed. These two-sided bounds for the energy at the exact solution are in turn used to obtain easily calculable a posteriori error bounds for the norm of the difference of the approximate and exact solutions of the given problem     

Alternative solution of the inhomogeneous linear differential equation of order two

In this paper, we present an alternative method for solving the general inhomogeneous linear ordinary differential equation (ODE) of order two. The solution appears in the standard form as the sum of the solution of the equivalent homogeneous problem and the particular solution of the inhomogeneous problem at hand. The main advantage of the method exposed herein is that the particular solution is computable from two different integrals. This allows the problem solver to choose the simplest integral with which to work with in order to get the final solution. For illustrative purposes we employ the method presented to aid in the solution of some example problems including the inhomogeneous Klein-Gordon equation.     

A mathematical model of structurally inhomogeneous macroisotropic materials

The proposed mathematical model of microinhomogeneous material statistically distributes mechanical properties and contains mutually uninteresting cracks. The physical interpretation of model parameters is given, and the domain of its applicability is established. Analysis of the stress state in microinhomogeneous material is made and a probable failure criterion is suggested. The obtained results coincide with those known in the literature for partial cases.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Physico-Mechanical Institute, Ukrainian Academy of Sciences, Lviv, Ukraine. Institute for Applied Problems of Mechanics and Mathematics, Ukrainian Academy of Sciences, Lviv, Ukraine. Published in Mekhanika Kompozitmykh Materialov, Vol. 32, No. 4, pp. 480–492, July–August, 1996.     

Discrete nonlinear excitations on an inhomogeneous lattice in a spatially linear potential

Discrete nonlinear excitations on an inhomogeneous lattice in a spatially linear potential are investigated. Discrete breathers possessing multiple frequencies are always spatially localized and oscillatory in time, with oscillatory frequencies depending on an external potential. A number of phenomena associated with the translational symmetry of the system is addressed. The spatially localized, time periodic and quasi-periodic nonlinear coherent excitations comprising multiple-soliton components are presented with exact results for the corresponding energy, momentum and norm. The mean width of a pulse as a measure of localization is also studied. Project supported by the National Natural Science Foundation of China.     

On the identification of inhomogeneous parameters in dynamic linear partial differential equations

The task of identifying inhomogeneous (position-dependent) coefficients of linear dynamic partial differential equations on the basis of a finite collection of points of the solution has practical importance and is the subject of many published analyses, some of which are described herein. The purpose of the present paper is to present new developments on a simple yet appealing method due to the hydrologist B. Sagar. The technique exploits the viewpoint that the coefficient values of the partial differential equation at any point x are uniquely determined by the solution values in a small neighborhood of x. The identification algorithm which results from these considerations is extremely simple, and yet, in view of technical considerations and experimental evidence set forth here, it seems effective. In particular, we have been able to derive error bounds, which the authors believe is a new feature in the literature of identification of partial differential systems.     

The linear model revisited

Following Gerber (1982), the annual gain of an insurance concern, G_t, is modeled as an autoregressive moving average process. The surplus process is defined as U_t = u + Σ^t_k=₁G_k. The distribution of U_t is obtained. It is proved that the traditional limit theorems, i.e., the central limit theorem, the strong law of large numbers and the law of the iterated logarithm, hold for U_t as t → ∞. Under certain conditions, bounds are provided for the probability of non-ruin in a finite time interval. It is proved that, if E[G_t] > 0 as t → ∞, the probability of non-ruin in an infinite time interval is positive.     

Three-chamber cochlear model with basilar and Reissner membranes

Results of computing the response of a three-chamber cochlear model including the Reissner membrane are compared with experimental measurements. An estimate is made in the linear approximation of the stiffness of the Reissner membrane on the basis of Bekesy's experiments, whereby the membrane itself is treated as an anisotropic plate.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 132–136, 1984.In conclusion, the author thanks V. M. Babich for his constant help in the work and for valuable discussions.