Well-posedness of the Hydrodynamic Model for Semiconductors

This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasi-linear elliptic–parbolic–hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L² sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L² sense. The well-posedness of the model under the boundary conditions is demonstrated.     

On the solution of the basic integral equation of actuarial mathematics by the method of successive approximations

We study the basic integral equation of actuarial mathematics for the probability of (non)ruin of an insurance company regarded as a function of the initial capital. We establish necessary and sufficient conditions for the existence of a solution of this equation, general sufficient conditions for its existence and uniqueness, and conditions for the uniform convergence of the method of successive approximations for finding the solution. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 12, pp. 1689–1698, December, 2007.     

On the equivalence of two conditions for the existence of transversals

There are two conditions which are known to be necessary for the existence of a transversal in any family of sets, but both are sufficient only if the family is countable. This paper proves that these conditions are always equivalent to each other. The families which are compatible with these conditions are characterised, in the sense that each of their subfamilies possesses a transversal if it satisfies the conditions. Using this, a conjecture of Podewski and Steffens is proved.     

On the stability of the σ-scheme with transparent boundary conditions for parabolic equations

Initial-boundary value problems for self-adjoint parabolic equations on a semiaxis and a semibounded strip are considered. For finite-difference σ-schemes, an alternative method for stating approximate transparent boundary conditions is suggested and conditions ensuring unconditional stability in the energy norm with respect to the initial data and free terms for a weight σ ≥ 1/2 are presented. The validity of these stability conditions in the case of discrete transparent boundary conditions is proved (by several methods), and the derivation of the latter conditions is revisited. Published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 4, pp. 671–692. This article was translated by the author.     

Conditions for the self-adjointness of the Schrodinger operator

Sufficient conditions are derived for the self-adjointness of the Schrödinger operator in the whole of space and in bounded regions, without supplementary boundary conditions and without any requirements concerning the existence of a spherically symmetric minorant of the potential satisfying the Titchmarsh-Sears conditions.Translated from Matematicheskie Zametki, Vol. 8, No. 6, pp. 741–751, December, 1970.     

On the essential self-adjointness of the general second order elliptic operators

In this paper, we give sufficient conditions for the essential self-adjointness of second order elliptic operators. It turns out that these conditions coincide with those for the Schrödinger operator on a manifold whose metric essentially depends on the principal coefficients of a given operator.

     

On the Existence and Stability of Spatially Structured Solutions of the Reaction-Diffusion Equations

The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both     

On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

In this article we obtain the asymptotic formulas for eigenfunctions and eigenvalues of the nonself-adjoint Sturm-Liouville operators with periodic and antiperiodic boundary conditions, when the potential is an arbitrary summable complex-valued function. Then using these asymptotic formulas, we find the conditions on Fourier coefficients of the potential for which the eigenfunctions and associated functions of these operators form a Riesz basis inL ₂(0, 1).     

On the zeros of solutions of beam equations

Summary We consider extensible beam equations, Timoshenko beam equations and the system of coupled beam equations. We show that, under suitable conditions, there are bounded domains in which every solution satisfying certain end conditions has a zero. End conditions to be considered are hinged ends and hinged-sliding ends. The results are based on the conditions for the nonexistence of positive solutions of ordinary differential inequalities.     

On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problem

This paper is mainly concerned with the classical KKT reformulation and the primal KKT reformulation (also known as an optimization problem with generalized equation constraint (OPEC)) of the optimistic bilevel optimization problem. A generalization of the MFCQ to an optimization problem with operator constraint is applied to each of these reformulations, hence leading to new constraint qualifications (CQs) for the bilevel optimization problem. M- and S-type stationarity conditions tailored for the problem are derived as well. Considering the close link between the aforementioned reformulations, similarities and relationships between the corresponding CQs and optimality conditions are highlighted. In this paper, a concept of partial calmness known for the optimal value reformulation is also introduced for the primal KKT reformulation and used to recover the M-stationarity conditions.     

On the necessity of the solvable conditions of the typical boundary value problems for quasilinear hyperbolic systems

Some solvable conditions have been derived to ensure the existence and the uniqueness of the C^∞solution for the typical boundary problem on a local angular region for quasilinear hyperbolic systems in two variables[1]. These solvables conditions mean that, under the formulation of the typical boundary problem, the all order derivatives of the solution can be determined uniquely at the vertex. The main purpose of this paper is to show that these solvable conditions are also necessary. In other words, if these solvable conditions fail to hold, then the boundary value problem will either have no solution or have infinite number of solutions.     

On the D-stability and additive D-stability of matrices and Svicobians

We improve the available necessary conditions and sufficient conditions for the Dstability and additive D-stability of matrices. We define these dynamical properties with respect to a finite parallelepiped and propose an algorithm for checking them. For a particular class of matrices called Svicobians, we obtain some constructive necessary and sufficient conditions for Dstabilizability, D-stability, and additive D-stability.     

On the nonexistence of global solutions of some initial-boundary value problems for the Korteweg-de Vries equation

We consider the problem of finite-time blow-up of solutions of a class of initial-boundary value problems for the Korteweg-de Vries equation. By using the method of optimal test functions corresponding to the boundary conditions, we obtain blow-up conditions for local (with respect to t > 0) solutions and estimate the blow-up time.     

Integrability and linearizability of the Lotka-Volterra systems

Integrability and linearizability of the Lotka-Volterra systems are studied. We prove sufficient conditions for integrable but not linearizable systems for any rational resonance ratio. We give new sufficient conditions for linearizable Lotka-Volterra systems. Sufficient conditions for integrable Lotka-Volterra systems with 3:−q resonance are given. In the particular cases of 3:−5 and 3:−4 resonances, necessary and sufficient conditions for integrable systems are given.     

Tensor conditions for the existence of a common solution to the Lyapunov equation

Necessary and sufficient conditions for the existence of a common solution to the Lyapunov equation for two stable complex matrices are derived. These conditions are applied to the cases when a common weak solution to the Lyapunov equation exists. Conditions for the existence of a common solution to the Lyapunov equation for two complex 2 × 2 and two complex 3 × 3 matrices are derived.     

On the convergence of Wang-Zheng's method

The choice of initial conditions ensuring safe convergence of the implemented iterative method is one of the most important problems in solving polynomial equations. These conditions should depend only on the coefficients of a given polynomial P and initial approximations to the zeros of P. In this paper we state initial conditions with the described properties for the Wang-Zheng method for the simultaneous approximation of all zeros of P. The safe convergence and the fourth-order convergence of this method are proved.     

On the existence of the free-boundary for some problems arising in plasma physics

We study some conditions for the existence of a free-boundary for two different bidimensional models arising in the magnetic confinement of a plasma. We derive estimates on the size and location of the region surrounded by the free-boundary (known as plasma region) for both models. We also derive sufficient conditions for the non-existence of the free-boundary.     

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Consider the homogeneous equation $$u'(t) = l(u)(t){\rm{ for a}}{\rm{.e}}{\rm{. }}t \in [a,b]$$ where ?: C([a, b];?) → L([a, b];?) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.     

On the stability of retrial queues

We consider the following Type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the call (customer) waits in line, and will be served according to the FIFO order. If λ is the arrival rate (average number per time unit) of calls and μ is one over the expected service time in the facility, it is well known that μ > λ is not always sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition. In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent i.i.d. sequences and the retrial times are exponentially distributed; (ii) we establish conditions for strong coupling convergence to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival and retrial times are i.i.d. exponentially distributed; or when inter-arrival times are general stationary ergodic, and service and retrial times are i.i.d. exponentially distributed; (iii) we obtain conditions for the existence of uniform exponential bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions for boundedness in distribution for the case of nonpatient (or non persistent) customers. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the existence of nonnegative solutions to the Dirichlet boundary value problem for the p-Laplace equation in the presence of exterior mass forces

We consider the Dirichlet problem for an inhomogeneous p-Laplace equation with nonlinear source in the presence of exterior mass forces.We obtain new sufficient conditions for the existence of a weak nonnegative bounded solution. The sufficient conditions are written in explicit form through the data of the problem.