Generalizations of a theorem of Rolewicz

The aim of this article is to give a unified treatment for the theorems of Rolewicz and Neerven type for uniform exponential stability of evolution families. We obtain necessary and sufficient conditions for uniform exponential stability of evolution families, generalizing a stability theorem due to Rolewicz and we present a new proof for the Rolewicz theorem, based on the theory of Banach function spaces. Finally, we apply our results and we deduce a generalization for a classical stability theorem due to Przyluski and Rolewicz.     

Lifting Property of Direct Sums of Hollow Modules

We give necessary and sufficient conditions for a module M with a decomposition that complements maximal direct summands to be a lifting module. We apply our results to get a necessary and sufficient condition for a direct sum of modules to be a quasi-discrete (discrete) module. We also give some sufficient conditions for a lifting module to be a direct sum of indecomposables.     

Torsion points on families of products of elliptic curves

In a recent paper we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues.     

Acceleration of Convergence of Static and Dynamic Iterations

A new technique for acceleration of convergence of static and dynamic iterations for systems of linear equations and systems of linear differential equations is proposed. This technique is based on splitting the matrix of the system in such a way that the resulting iteration matrix has a minimal spectral radius for linear systems and a minimal spectral radius for some value of a parameter in Laplace transform domain for linear differential systems.This revised version was published online in October 2005 with corrections to the Cover Date.     

Categories of models of R-mingle

We give a new Esakia-style duality for the category of Sugihara monoids based on the Davey-Werner natural duality for lattices with involution, and use this duality to greatly simplify a construction due to Galatos-Raftery of Sugihara monoids from certain enrichments of their negative cones. Our method of obtaining this simplification is to transport the functors of the Galatos-Raftery construction across our duality, obtaining a vastly more transparent presentation on duals. Because our duality extends Dunn's relational semantics for the logic R-mingle to a categorical equivalence, this also explains the Dunn semantics and its relationship with the more usual Routley-Meyer semantics for relevant logics.     

Boundary-value problem for density-velocity model of collective motion of organisms

The collective motion of organisms is observed at almost all levels of biological systems. In this paper the density-velocity model of the collective motion of organisms is analyzed. This model consists of a system of nonlinear parabolic equations, a forced Burgers equation for velocity and a mass conservation equation for density. These equations are supplemented with the Neumann boundary conditions for the density and the Dirichlet boundary conditions for the velocity. The existence, uniqueness and regularity of solution for the density-velocity problem is proved in a bounded 1D domain. Moreover, a priori estimates for the solutions are established, and existence of an attractor is proved. Finally, some numerical approximations for asymptotical behavior of the density-velocity model are presented.     

Estimates of rate of propagation of perturbations in quasilinear divergent degenerate parabolic equations of high order

A method is given for obtaining integral growth lemmas for solutions of boundary problems for a large class of quasilinear evolution equations. As a possible application a sharp estimate is obtained of the dependence of the support of a solution of a mixed problem and a Cauchy problem for a quasilinear divergent parabolic equation on time.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 10, pp. 1451–1456, October, 1992.     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

Toward determination of the eigenvalues of problems on description of the rotation of deep-water drill columns

This paper presents a method for computing the complex eigenvalues for a boundary-value problem describing the rotation of deep-water drill columns for drilling at depths of thousands of meters. A solution is sought for a singularly perturbed boundary-value problem in the form of an asymptotic expansion in a small parameter ε for two parts of the of the drill column: a lower compressed length, and the upper length. The method can be used to investigate the stability of drill columns used for deep-water drilling. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 29–36, 1993.     

Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems

The determination of solutions of the Jacobi partial differential equations (PDEs) for finite-dimensional Poisson systems is considered. In particular, a novel procedure for the construction of solution families is developed. Such a procedure is based on the use of time reparametrizations preserving the existence of a Poisson structure. As a result, a method which is valid for arbitrary values of the dimension and the rank of the Poisson structure under consideration is obtained. In this article two main families of time reparametrizations of this kind are characterized. In addition, these results lead to a novel application which is also developed, namely the global and constructive determination of the Darboux canonical form for Poisson systems of arbitrary dimension and rank two, thus improving the local result provided by Darboux' theorem for such a case.     

Inequalities for Functions of the Sum of the Indicators of Events

We obtain moment inequalities for the sum of the indicators of events and an upper estimate for a convex function of such a sum. Our results generalize inequalities that were obtained earlier for moment characteristics of the sojourn time of a random walk on a half-axis.     

Stability analysis of set trajectories for families of impulsive equations

In this paper, for a family of impulsive equations, a heterogeneous matrix-valued Lyapunov-like function is considered, the comparison principle is formulated, and stability conditions for the set of stationary solutions are established. In addition, for a class of impulsive equations with uncertain parameters the monotone iterative technique for constructing a set of solutions is adapted.     

Superconvergence of mixed finite element semi-discretizations of two time-dependent problems

We will show that some of the superconvergence properties for the mixed finite element method for elliptic problems are preserved in the mixed semi-discretizations for a diffusion equation and for a Maxwell equation in two space dimensions. With the help of mixed elliptic projection we will present estimates global and pointwise in time. The results for the Maxwell equations form an extension of existing results. For both problems, our results imply that post-processing and a posteriori error estimation for the error in the space discretization can be performed in the same way as for the underlying elliptic problem.     

The impact of cost uncertainty on the location of a distribution center

The location of a distribution center (DC) is a key consideration for the design of supply chain networks. When deciding on it, firms usually allow for transportation costs, but not supplier prices. We consider simultaneously the location of a DC and the choice of suppliers offering different, possibly random, prices for a single product. A buying firm attempts to minimize the sum of the price charged by a chosen supplier, and inbound and outbound transportation costs. No costs are incurred for switching suppliers. We first derive a closed-form optimal location for the case of a demand-populated unit line between two suppliers offering deterministic prices. We then let one of the two suppliers offer a random price. If the price follows a symmetric and unimodal distribution, the optimal location is closer to the supplier with a lower mean price. We also show the dominance of high variability: the buyer can decrease the total cost more for higher price variability for any location. The dominance result holds for normal, uniform, and gamma distributions. We propose an extended model with more than two suppliers on a plane and show that the dominance result still holds. From numerical examples for a line and a plane, we observe that an optimal location gets closer to the center of gravity of demands as the variability of any supplier’s price increases.     

Connection between sets of uniqueness for multiplicative systems of functions and sets of uniqueness for multiplicative transforms

A problem of uniqueness for series over multiplicative systems of functions and for multiplicative transforms is considered. It is shown that each set of uniqueness for a multiplicative transform is specified by a countable collection of sets of uniqueness for series over the corresponding multiplicative system of functions. Each set of uniqueness for a series over a multiplicative system of functions is a portion on [0, 1) of some set of uniqueness for the corresponding multiplicative transform.     

Properties of uniform consistency of the kernel estimators of density and regression functions under dependence assumptions

The paper contains exponential inequalities for dependent random variables. As a measure of dependence we use φand ρ-mixing coefficients, the last one being based on the maximal coefficient of correlation. These results allow us to study the problem of uniform strong convergence for the kernel estimators of a density and for a kernel predictor for stochastic processes. Our uniform consistency theorems extend some known results     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Matrix correction of a dual pair of improper linear programming problems

A matrix is sought that solves a given dual pair of systems of linear algebraic equations. Necessary and sufficient conditions for the existence of solutions to this problem are obtained, and the form of the solutions is found. The form of the solution with the minimal Euclidean norm is indicated. Conditions for this solution to be a rank one matrix are examined. On the basis of these results, an analysis is performed for the following two problems: modifying the coefficient matrix for a dual pair of linear programs (which can be improper) to ensure the existence of given solutions for these programs, and modifying the coefficient matrix for a dual pair of improper linear programs to minimize its Euclidean norm. Necessary and sufficient conditions for the solvability of the first problem are given, and the form of its solutions is described. For the second problem, a method for the reduction to a nonlinear constrained minimization problem is indicated, necessary conditions for the existence of solutions are found, and the form of solutions is described. Numerical results are presented.     

Removable singularities of solutions of degenerate nonlinear elliptic equations on the boundary of a domain

The removability of singularities of solutions for the Dirichlet problem for degenerate nonlinear elliptic equations on the boundary of a domain is studied. A method based on a priori energetic estimates of solutions to elliptic boundary value problems is used. The growth in the vicinity of a boundary point (finite or at infinity) for generalized solutions is studied.     

Adjoints of composition operators on Hilbert spaces of analytic functions

We observe that a formula for the adjoint of a composition operator, known only for special symbols in some spaces of analytic functions, actually holds for every admissible symbol and in any Hilbert space of analytic functions with reproducing kernels. Along with some new results, all known formulas for the adjoint obtained so far follow easily as a consequence, some in an improved form.