Articles in "Mathematics"

Summary: Parcel delivery networks have grown rapidly during the last few years due to the intensive evolution of online marketplaces. We address the issue of managing the operation of a network’s pick-up point, including the selection of the warehouse’s capacity and the policy for accepting orders for delivery. The existence of the time lag between order placing and delivery to the pick-up point is accounted for via modeling the order’s processing as the service in the dual tandem queueing system. Distinguishing features of this tandem queue are the account of possible irregularity in order generation via consideration of the versatile Markov arrival process and the possibilities of batch transfer of the orders to the pick-up point, group withdrawal of orders there, and client no-show. To reduce the probability of an order rejection at the pick-up point due to the overflow of the warehouse, a threshold strategy of order admission at the first stage on a tandem is proposed. Under the fixed value of the threshold, tandem operation is described by the continuous-time multidimensional Markov chain with a block lower Hessenberg structure for the generator. Stationary performance measures of the tandem system are calculated. Numerical results highlight the dependence of these measures on the capacity of the warehouse and the admission threshold. The possibility of the use of the results for managerial goals is demonstrated. In particular, the results can be used for the optimal selection of the capacity of a warehouse and the policy of suspending order admission.

Summary: This work is dedicated to the impulsive Sturm - Liouville operator on the whole axis with complex almost periodic potentials and the discontinuous coefficient on the right – hand side. We investigated the main characteristics of the fundamental solutions of the Sturm – Liouville equation. From the impulsive condition we found the transfer matrix. Using the impulsive condition and transfer matrix, we constructed Green’s function and obtained the resolvent of the impulsive Sturm – Liouville operator. In future works, eigenvalues of the impulsive Sturm - Liouville operator will be investigated. The inverse problem will be formulated, a constructive procedure for the solution of the inverse problem will be provided.

Summary: The paper considers a boundary value problem generated by a differential diffusion equation and nonseparated boundary conditions. One of the boundary conditions contains a quadratic function of the spectral parameter. The multiplicity of eigenvalues of the boundary value problem under consideration is investigated. A criterion for the multiplicity of eigenvalues and zeros of the characteristic function of a boundary value problem is obtained.The found necessary and sufficient conditions are expressed through the values of the fundamental solutions of the diffusion equation and the coefficients of the boundary conditions. Note that the results obtained can be used in the study of direct and inverse problems of spectral analysis for various differential operators. These results also play an important role in studying the structure of the spectrum, in establishing the order of arrangement of eigenvalues of boundary value problems, and in finding sufficient conditions for the reconstruction of the corresponding problems.

Summary: In this article, the issue of controlling the movement of a hexacopter-type unman-ned aerial vehicle (UAV) along a route is investigated. The movement of the hexacopter is modeled as the movement of a rigid body, and in this process, the forces of gravity and aerodynamic resistance are taken into account. The spatial orientation of the hexacopter is expressed using quaternions. The movement route is considered as a broken line consisting of straight segments, and the parameters controlling its flight are determined when one of the hexacopter's motors is not working. Mathematical justification is provided for how the operational motors are controlled to continue the hexacopter's movement in its previous manner when one motor fails.

Summary: The paper examines different types of hypersingular integrals with the Cauchy kernel on a segment and a unit circle and defines them using specific methods. It presents more general definitions for one-dimensional hypersingular integrals with the Cauchy kernel based on Hadamard's integral in the sense of a finite part. The paper also establishes the existence theorems of these hypersingular integrals and formulas, which demonstrates the accuracy of the resulting integrals that are applied in various applications and engineering problem-solving. The proposed formulas are straightforward to calculate, making the new approximate method reliable and easy to apply and the obtained numerical results demonstrate the stability and efficiency of the approach.

Summary: The study investigates the inverse scattering problem for the Schrodinger operator with complex potentials, considering indefinite discontinuous coefficients on the axis. Using the integral representation of the Jost solutions on the real and imaginary axes, solved the direct scattering problem. An additional study of the operator's spectrum was conducted, scattering data was introduced, and the eigenfunction expansion was obtained. Integral equations derived play a crucial role in solving the inverse problem and finally prove the uniqueness theorem for the solution. © 2024

Summary: Green function of a 2n-th order differential equation with normal coefficients on the half-axis is studied. We first consider the Green function of our equation with “frozen” coefficients. Using Levi’s method, we obtain a Fredholm-type integral equation for the Green function of our problem, whose kernel is a Green function of a problem with constant coefficients. We prove an existence and uniqueness theorem for this integral equation in some Banach spaces of operator-valued functions. The main result of this paper is a theorem stating that the solution of the obtained integral equation is a Green function of our problem.