On the Dirichlet–Riquier problem for biharmonic equations

In the paper, it is proved that the distribution of a measurable polynomial on an infinite-dimensional space with log-concave measure is absolutely continuous if the polynomial is not equal to a constant almost everywhere. A similar assertion is proved for analytic functions and for some other classes of functions. Properties of distributions of norms of polynomial mappings are also studied. For the space of measurable polynomial mappings of a chosen degree, it is proved that the L ¹-norm with respect to a log-concave measure is equivalent to the L ¹-norm with respect to the restriction of the measure to an arbitrarily chosen set of positive measure.     

Inverse dynamic problem for a Krein–Stieltjes string

We consider a dynamic inverse problem for a dynamical system describing the propagation of waves in a Krein–Stieltjes string. We offer three methods of recovering the string, that is point masses and distances between them, from the dynamic Dirichlet-to-Neumann operator.     

Cauchy–Gelfand problem and the inverse problem for a first-order quasilinear equation

Gelfand’s problem on the large time asymptotics of the solution of the Cauchy problem for a first-order quasilinear equation with initial conditions of the Riemann type is considered. Exact asymptotics in the Cauchy–Gelfand problem are obtained and the initial data parameters responsible for the localization of shock waves are described on the basis of the vanishing viscosity method with uniform estimates without the a priori monotonicity assumption for the initial data.     

On a Riemann–Hilbert problem for the Fokas–Lenells equation

The solution of the initial value problem (IVP) for the Fokas–Lenells equation (FLE) was constructed in terms of the solution of a 2  2 matrix Riemann–Hilbert problem (RHP) as , and the one-soliton solution of the FLE was derived based on this Riemann–Hilbert problem, in Lenells and Fokas (2009). However, in fact, the derivative with respect to of the solution of the FLE () was recovered from the RHP as . In this paper, we construct the solution of the FLE in terms of the RHP as , because the Lax pair of the FLE contains the negative order of the spectral variable . We show that the one-soliton solution of the FLE obtained in this paper is the same as Lenells and Fokas (2009), but avoiding a complex integral.     

Inference for a change-point problem under a generalised Ornstein–Uhlenbeck setting

Determining accurately when regime and structural changes occur in various time-series data is critical in many social and natural sciences. We develop and show further the equivalence of two consistent estimation techniques in locating the change point under the framework of a generalised version of the one-dimensional Ornstein–Uhlenbeck process. Our methods are based on the least sum of squared error and the maximum log-likelihood approaches. The case where both the existence and the location of the change point are unknown is investigated and an informational methodology is employed to address these issues. Numerical illustrations are presented to assess the methods’ performance.     

The Boussinesq–Mindlin problem for a non-homogeneous elastic halfspace

Boussinesq’s problem for the indentation of an isotropic, homogeneous elastic halfspace by a rigid circular punch constitutes a seminal problem in the theory of contact mechanics as does Mindlin’s problem for the action of a concentrated force at the interior of an isotropic, homogeneous elastic halfspace. The combined action of the surface indentation in the presence of the interior loading is referred to as the Boussinesq–Mindlin problem, which has important applications in the area of geomechanics. The Boussinesq–Mindlin problem, which represents a self-stressing loading configuration, serves as a useful model for interpreting the mechanics of indentation of geologic media for purposes of estimating their bulk elasticity properties. In this paper, the analysis of the problem is extended to include an exponential variation in the linear elastic shear modulus of the halfspace region.     

Infinitely many solutions for a double Sturm–Liouville problem

In this paper, we prove the existence of infinitely many solutions to differential problems where both the equation and the conditions are Sturm?CLiouville type. The approach is based on critical point theory.     

On the Cauchy problem for a two-component Degasperis–Procesi system

This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented.     

Tabu list management methods for a discrete–continuous scheduling problem

A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. Processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation that minimizes the makespan. The tabu search (TS) metaheuristic is presented to attack the problem. Three different tabu list management methods: the tabu navigation method (TNM), the cancellation sequence method (CSM) and the reverse elimination method (REM) are discussed and examined. A computational experiment is described and the results obtained for the methods tested are compared to optimal solutions. A few conclusions and final remarks are presented.     

Multiple solutions for a nonlocal problem in Orlicz–Sobolev spaces

Using the mountain pass theorem combined with the minimum principle, we obtain a multiplicity result for a nonlocal problem in Orlicz–Sobolev spaces. To our knowledge, this is the first contribution to the study of nonlocal problems in this class of spaces.     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

On a problem for Wardʼs equation with a Mittag–Leffler potential

We solve a problem for a type of non-linear partial differential equation (“Ward?s equation”). This is an equation arising naturally in the study of Coulomb gases and random normal matrix ensembles [4]. In this paper, we consider a problem for Ward?s equation whose solutions are precisely the well-known Mittag–Leffler functions. Our solution to this problem generalizes certain results obtained in [4].     

A location–allocation problem for a web services provider in a competitive market

Web Services have become a viable component technology in distributed e-commerce platforms. Due to the move to high-speed Internet communication and tremendous increases in computing power, network latency has begun to play a more important role in determining service response time. Hence, the locations of a Web Services provider’s facilities, customer allocation, and the number of servers at each facility have a significant impact on its performance and customer satisfaction. In this paper we introduce a location–allocation model for a Web Services provider in a duopoly competitive market. Demands for services of these servers are available at each node of a network, and a subset of nodes is to be chosen to locate one or more servers in each. The objective is to maximize the provider’s profit. The problem is formulated and analyzed. An exact solution approach is developed and the results of its efficiency are reported.     

A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.