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.     

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.     

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 Multiplier Space Generated by the Rademacher System

In this paper, we consider the space of multipliers of symmetric spaces with respect to the Rademacher systems. We obtain sufficient conditions under which the space in question coincides with the space L _∞ (with equivalence of norms). These conditions are stated in terms of operator interpolation theory and are essentially weaker than the conditions for the solution of this problem recently obtained by other authors.     

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 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.     

Unbiased estimates of the Neumann series sum by the Monte Carlo method

The conditions of unbiasedness are studied for linear estimates constructed in solving linear integral equations of the second kind by the Monte Carlo method. Earlier results obtained in this field are extended, and the cases are considered in which the well-known sufficient conditions of unbiasedness are found to be also necessary conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 43, pp. 155–168, 1974.     

Properties of the minimum function in the quadratic problem

Perturbations of the quadratic form minimization problem under quadratic constraints of the type of equalities are considered. The minimum function ω in this problem which, to each perturbation of the original problem, assigns a sharp lower bound in the perturbed problem is studied. Sufficient conditions for the upper and lower semicontinuity of the minimum function ω both at zero and in its neighborhood are obtained. Examples showing the importance of these conditions are given.     

Degenerate boundary conditions for the diffusion operator

We describe all degenerate two-point boundary conditions possible in a homogeneous spectral problem for the diffusion operator. We show that the case in which the characteristic determinant is identically zero is impossible for the nonsymmetric diffusion operator and that the only possible degenerate boundary conditions are the Cauchy conditions. For the symmetric diffusion operator, the characteristic determinant is zero if and only if the boundary conditions are falsely periodic boundary conditions; the characteristic determinant is identically a nonzero constant if and only if the boundary conditions are generalized Cauchy conditions.     

Filling the gap between the Gerschgorin and Brualdi theorems

The paper presents new diagonal dominance type nonsingularity conditions for n × n matrices formulated in terms of circuits of length not exceeding a fixed number r ≥ 0 and simple paths of length r in the digraph of the matrix. These conditions are intermediate between the diagonal dominance conditions in terms of all paths of length r and Brualdi’s diagonal dominance conditions, involving all the circuits. For r = 0, the new conditions reduce to the standard row diagonal dominance conditions , i = 1, ..., n, whereas for r = n they coincide with the Brualdi circuit conditions. Thus, they connect the classical Lévy-Desplanques theorem and the Brualdi theorem, yielding a family of sufficient nonsingularity conditions. Further, for irreducible matrices satisfying the new diagonal dominance conditions with nonstrict inequalities, the singularity/nonsingularity problem is solved. Also the new sufficient diagonal dominance conditions are extended to the so-called mixed conditions, simultaneously involving the deleted row and column sums of an arbitrary finite set of matrices diagonally conjugated to a given one, which, in the simplest nontrivial case, reduce to the old-known Ostrowski conditions , i = 1, ..., n, 0 ≤ α ≤ 1. The nonsingularity conditions obtained are used to provide new eigenvalue inclusion sets, depending on r, which, as r varies from 0 to n, serve as a bridge connecting the union of Gerschgorin’s disks with the Brualdi inclusion set. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 128–148.     

On the Itô excursion process

Summary Necessary and sufficient conditions are given, for a process to be the excursion process of some strong Markov process. These are modifications of necessary conditions of Itô, which are here shown by example not to be sufficient.     

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.     

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.     

Spectral sequences for quadratic pencils and the inverse spectral problem for the damped wave equation

We give necessary conditions for a sequence of complex numbers closed under complex conjugation to be the spectrum of the weakly damped wave operator. These restrictions are a consequence of some new results on spectral sequences for Hermitian quadratic pencils which are based on Weyl's and Mirsky's classical eigenvalue inequalities. In the case of finite-dimensional weakly damped pencils our conditions are both necessary and sufficient. We also obtain some conditions for overdamped pencils of degree m, and show that some of the inequalities that have to be satisfied in the weakly damped case are now reversed.     

On the Minimization of Difference Functions

Simple necessary optimality conditions are formulated for a function f of the form f _ g–h, where g and h are nonsmooth functions. Related sufficient conditions are given for local minimization and global minimization.     

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 regularity up to the boundary in the theory of the Navier–Stokes equations with generalized impermeability conditions

In this Note, we extend sufficient conditions for regularity we described in our previous works so that they are valid not only in the interior, but up to the boundary of a flow field. The conditions are based on the integrability properties of either one of the eigenvalues of the rate of deformation tensor or one component of velocity. To cite this article: J. Neustupa, P. Penel, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

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.     

Stochastically minimizing the number of customers in exponential queuing systems

This paper deals with the problem of controlling an exponential queuing system (that is, a system with exponential service times and Poisson arrivals) so as to stochastically minimize the number of customers in the system at any time t > 0. Sufficient (simple) conditions are developed for a policy to be optimal. Similar conditions are sufficient for a policy to stochastically minimize (maximize) any function of the state of the system. Two models are considered to illustrate the results. In both cases, optimal policies are shown to satisfy these conditions by a simple inductive procedure.