Ruin problem for a generalized Poisson process with reflection

We consider a generalized Poisson process with reflection at the level T > 0. Under certain conditions on the distribution of the values of positive jumps of the process, we obtain representations for the characteristic functions of functionals associated with the exit of the indicated process to the negative semiaxis. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1465–1475, November, 2005.     

On an inhomogeneous problem for parabolic-hyperbolic equation

We study a boundary value problem for an inhomogeneous parabolic-hyperbolic equation with a noncharacteristic type change line. Boundary conditions of the first kind are posed on characteristics in the parabolic and hyperbolic parts of the domain where the equation is given, and a condition of the third kind is posed on the noncharacteristic part of the boundary in the parabolic part. First, we study the solvability of an inhomogeneous initial–boundary value problem for a parabolic equation.     

The Mather problem for lower semicontinuous Lagrangians

In this paper we develop the Aubry-Mather theory for Lagrangians in which the potential energy can be discontinuous. Namely we assume that the Lagrangian is lower semicontinuous in the state variable, piecewise smooth with a (smooth) discontinuity surface, as well as coercive and convex in the velocity. We establish existence of Mather measures, various approximation results, partial regularity of viscosity solutions away from the singularity, invariance by the Euler–Lagrange flow away from the singular set, and further jump conditions that correspond to conservation of energy and tangential momentum across the discontinuity.     

On an inhomogeneous Tricomi problem for a parabolic-hyperbolic equation

We consider an inhomogeneous Tricomi problem for a parabolic-hyperbolic equation with noncharacteristic type change line and with Frankl type matching condition for the normal derivatives on the type change line. The auxiliary function method is used to establish an a priori estimate of the solution. The existence of the solution is proved by the spectral method.     

On uniqueness for an inverse problem in inhomogeneous elasticity

We consider the scattering of time-harmonic elastic waves inan isotropic medium which is characterized by constant Lamécoefficients and an inhomogeneous mass density. We prove thata knowledge of the far-field pattern for all incident planewaves and a single frequency provides sufficient informationto determine the mass density uniquely. To this end we constructa periodic Faddeev-type fundamental solution for the Navierequation and derive the existence of complex geometrical opticssolutions to the Navier equation.     

A numerical method for an inhomogeneous nonlinear diffusion problem

A numerical method for the solution of an inhomogeneous nonlinear diffusion problem that arises in a variety of applications is presented. The diffusion coefficient in the underlying diffusion process is concentration- as well as distance- dependent. We wish to determine the concentration of the diffusing substance in a semi-infinite domain at any time, starting with a given initial concentration. The method of solution begins by first mapping the semi-infinite physical domain to a finite computational domain. An implicit finite-difference marching procedure is then used to advance the solution in time. Numerical results are presented for several physical problems. We observe that the present numerical solutions are in good agreement with the analytical solutions obtained previously by other researchers.     

Multiplicity results for an inhomogeneous Neumann problem with critical exponent

In dimension N≥5, we prove the existence of three solutions for the inhomogeneous Neumann problem with critical Sobolev exponent. One of the solutions is obtained with a changing sign property.     

An inverse problem for an inhomogeneous conformal Killing field equation

Let be a Riemannian metric defined on a bounded domain with boundary and let be a vector field on satisfying . We show that if is a gradient field of a solution to the equation on , then both inner products and are uniquely determined by the restriction of the tensor to the gradient field , where is the Lie derivative of the metric tensor under the vector field and . This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.

     

Non-standard asymptotics in an inhomogeneous gamma process

Nonhomogeneous Poisson process (NHPP) is a commonly used stochastic model that is utilized to describe the pattern of repeated occurrence of certain events or conditions. Aninhomogeneous gamma process evolves as a generalization to NHPP, where the observed failure epochs correspond to every successive κ-th event of the underlying Poisson process, κ being an unknown parameter to be estimated from the data. This article focuses on a special class of inhomogeneous gamma process, calledmodulated power law process (MPLP) that assumes the Weibull form of the intensity function. The traditional power law process is a popular stochastic formulation of certain empirical relationships between the time to failure and the cumulative number of failures, often observed in industrial experiments. The MPLP retains this underlying physical basis and provides a more flexible modeling environment potentially leading to a better fit to the failure data at hand. In this paper, we investigate inference issues related to MPLP. The maximum likelihood estimators (MLE’s) of the model parameters are not in closed form and enjoy the curious property that they are asymptotically normal with a singular variance-covariance matrix. Consequently, the derivation of the large-sample results requires non-standard modifications of the usual arguments. We also propose a set of simple closed-form estimators that are asymptotically equivalent to the MLE’s. Extensive simulation results are carried out to supplement the theoretical findings. Finally, we implement our inference results to a failure dataset arising from a repairable system.     

Solvability of an inhomogeneous boundary value problem for steady MHD equations

In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions under sharp assumption on boundary data for the magnetic field.     

Numerical realization of an algorithm for reconstructing an inhomogeneous medium in an X-ray tomography problem

Under study is the X-ray tomography problem that is an inverse problem for the transport differential equation. We take into account the absorption of particles by the medium and their single scattering. The statement of the problem corresponds to multiple probing. The medium is unknown; while the densities of the outcoming flux averaged over energy are given. The object in question is the discontinuity surfaces of the coefficients of the equation. This corresponds to searching for the boundaries between various substances contained in the medium that we probe. The solution is constructive, and a numerical realization of the obtained algorithm is presented.     

A boundary blow-up elliptic problem with an inhomogeneous term

By a perturbation method and constructing comparison functions, we reveal how the inhomogeneous term h$h$ affects the exact asymptotic behaviour of solutions near the boundary to the problem △u=b(x)g(u)+λh(x)$△u=b\left(x\right)g\left(u\right)+\lambda h\left(x\right)$, u>0$u>0$ in Ω$\Omega$, u_|∂Ω=∞$u{|}_{\partial \Omega }=\infty$, where Ω$\Omega$ is a bounded domain with smooth boundary in R^N${\mathbb{R}}^N$, λ>0$\lambda >0$, g∈C¹[0,∞)$g\in C^1[0,\infty )$ is increasing on [0,∞)$[0,\infty )$, g(0)=0$g\left(0\right)=0$, g^′$g^{\prime }$ is regularly varying at infinity with positive index ρ$\rho$, the weight b$b$, which is non-trivial and non-negative in Ω$\Omega$, may be vanishing on the boundary, and the inhomogeneous term h$h$ is non-negative in Ω$\Omega$ and may be singular on the boundary.     

A mixed problem for an inhomogeneous wave equation with a summable potential

A mixed problem for an inhomogeneous wave equation with fixed ends in the case of a summable potential is studied. Using the Krylov method for acceleration of the convergence of Fourier series, a classical solution under minimal conditions on the initial data and a generalized solution in the case of quadratic summable initial data and perturbing function are obtained.     

经济环境下带扩散扰动古典风险过程的破产概率

对于经济环境下带扩散扰动古典风险过程的重要性质进行了讨论,给出了有限时间的破产概率的上界估计.     

Stable methods for an inverse problem in acoustic scattering by an obstacle and an inhomogeneous medium

We consider (in two-dimensional Euclidean space) the scattering of a plane, time-harmonic acoustic wave by an inhomogeneous medium Ω with compact support and a bounded obstacle D lying completely outside of the inhomogeneous medium. We show that one may determine the shape of D and the local speed of sound in Ω from a knowledge of the asymptotic behavior of the scattered wave (i.e. the far field). This is done by considering a constrained optimization problem and employing integral equation and conformal mapping techniques. By assuming a priori that the functions which determine the shape of D and the local speed of sound in Ω lie in given compact sets, we show that the problem is stable, in the sense that the solution of the inverse scattering problem depends continuously on the far field data.     

Uniqueness criterion in an inverse problem for an abstract differential equation with nonstationary inhomogeneous term

In a Banach space E, we consider the inverse problem du(t)/dt = Au(t) + (t)p , u(0) = u ₀, u(T) = u₁, with an unknown function u(t) and an element p E. The operator A is assumed linear and closed. In this paper, we establish minimal constraints on the function C([0, T]) for which the uniqueness of the solution of the inverse problem is completely described in terms of the eigenvalues of the operator A.Translated from Matematicheskie Zametki, vol. 77, no. 2, 2005, pp. 273–290.Original Russian Text Copyright © 2005 by I. V. Tikhonov, Yu. S. Éidelman.This revised version was published online in April 2005 with a corrected issue number.