A method for calculating the behavior and formation of smoke particle agglomerates in crossed ultrasonic fields is proposed. The proposed method takes into account: the convergence of agglomerates with allowance for the moments of flow forces from the gas flow, the rotation of agglomerates due to the moments of forces from the flow around the gas flow, and a phase shift between ultrasonic cross fields, leading to the rotation of the resulting vibrational velocity vector and, consequently, to an increase in the effective collision cross-sectional area due to the rotation of the agglomerates. A description of the morphology and position of agglomerates is constructed when representing the agglomerate as a solid body, typical for solid-phase aerosols in the form of smoke. An equation for the dynamics of translational and rotational motion of an agglomerate is obtained taking into account the interaction of particles. The statement about the presence of rotational motion with a limited angular velocity proportional to the sound pressure level is proven, with a phase shift of cross fields equal to 90 deg. By means of numerical experiments a critical value of the phase shift angle is established at which the pseudo-rotational motion (change in the rotation angle in a limited range the width of which is less than 180 deg) transforms into rotational motion (the agglomerate makes a full revolution through 360 deg). The critical phase shift depends (weakly) on the sound pressure level and ranges from 82 to 85 deg. The transition to rotational motion increases the efficiency of forming larger agglomerates due to an increase in the average cross section during particle collisions, caused by the difference in the sizes of agglomerates along different axes. It was established that for the practical implementation of increasing the efficiency of smoke deposition, the most appropriate and feasible is ultrasonic action in cross fields with a frequency difference much lower than the fundamental frequency (from approximaterly 100 to 200 Hz). Such action is much easier to implement compared to maintaining a constant phase shift between the cross fields and increases the collision cross-sectional area up to several times.
This study focuses on investigating an optimal control problem that represents the unilateral frictional contact between a thermo-electro-elastic body and a foundation with electrical and thermal conductivity. The problem is considered in a static setting, and the material behavior is described using linear thermo-piezoelectric constitutive laws. Tresca’s friction law is used to model the contact, along with regularized conditions for electrical and thermal conductivity. The mathematical properties of the problem, including its variational formulation, existence, and uniqueness of weak solutions, are discussed. Additionally, an optimal control problem is formulated, and the existence of at least one solution is established. A regularized version of the unilateral contact and Tresca’s law is then incorporated into the model. The existence of a weak solution to the regularized control problem is proven, and its convergence to the solution of the original optimal control problem is verified. Finally, an optimality condition is presented for this type of problem.
The paper considers the following problem of scheduling theory. Jobs are serviced by a single machine. Each job is characterized by a positive weight and a release date. The service durations are the same. Interruptions are allowed in service. Time is assumed to be discrete. It is necessary to find a schedule for servicing jobs that minimizes the weighted sum of moments of completion of processing jobs. The complexity status of the problem is currently unknown. The paper presents a new Boolean linear programming model for this problem that includes some necessary conditions for the optimality of schedules. These conditions are presented as linear inequalities. As a result of the computational experiment, it was noted that the model produces an integer solution even without imposing integer conditions on the variables.
A nonpendant vertex of a graph is called a support vertex if it is adjacent to any leaf. This paper examines how two classes of extremal trees among all trees with a prescribed degree sequence can relate to each other: the class of trees minimizing the number of support vertices and the class of trees minimizing the domination number. Two specially defined values play important role here: the Slater number, proposed by Slater as a lower bound on domination number, and the lower bound on the number of the support vertices of a tree proposed by Kurnosov. In the paper, the problem of comparing the classes is completely solved in the case where the maximum of two values is the latter.
In this note, we single out some promising classes of differential-algebraic equations (DAEs) with nonlinearity of hysteresis type modeled by a sweeping process. DAEs is a well recognized and extensively studied area of the modern applied mathematics, arisen as a natural generalization of the concept of ordinary differential equations (ODEs). The unsolvability measure with respect to the derivatives for some DAE is an integer that is called the index of the DAE. The analysis is carried out under the assumption of the existence of a structural form with separated “differential” and “algebraic” subsystems. This structural form is equivalent to the initial system in the sense of solution, and the operator that transformes the DAE into the structural form possesses the left inverse operator. Finding the structural form is constructive and does not use a change of variables. In addition, the problem of consistency of initial data is solved automatically. Systems of DAEs are attracting more and more attention due to mathematical modeling problems in many applied domains: automated control theory, optimal control with mixed constraints, mechanics, chemical kinetics, hydrodynamics, thermal engineering, etc. The systems under investigation arise in modeling various physical processes, in particular, in electrical circuits with hysteresis phenomena. For such a DAE, we design an equivalent structural form (in the sense of solutions). Necessary and sufficient conditions for the existence and uniqueness of a solution to an initial value problem and controllability are proved. Illustrative examples are given in the conclusions.
The paper is devoted to solving the problem of local control of in-flows in regular resource networks with low resource. For such networks, a set of controlled vertices is specified. The local control problem is to determine such capacities of arcs entering the controlled vertices that the unique limit state Q^* of regular resource network is the closest to a given state Q^' . Conditions for the unreachability of the limit state that coincides with the state Q^' are obtained. Various configurations of resource networks with respect to the distribution of controlled vertices in them are considered. It is shown that if the conditions for the unreachability of the limit state are not satisfied, then there exists a set of capacities of arcs entering the controlled vertices for which the limit state Q^* is equal to the given state Q^' .
The article studies a model of international trade between two countries under monopolistic competition of producers. The utility functions of consumers are additively separable. Transport costs are of the iceberg type. The production cost function is nonlinear: marginal costs are a decreasing function of R D investments. The article considers market equilibrium in autarky situation, when transport costs are so high that international trade ceases. Comparative statics is carried out on transport costs of equilibrium variables (individual consumption, size and mass of firms, and prices), as well as social welfare.
The inverse problem of determining two unknown functions σ _0(x) and σ _1(x) included in the absorption coefficient σ (x,u)=σ _0(x)+σ _1(x)u in the equation of electrodynamics is considered. An a priori estimate of the solution to the direct problem is obtained, and the existence and uniqueness theorems of solutions to the direct and inverse problems are proved.
In an undirected graph, a 3-vertex induced subgraph having exactly 2 edges is called an open triangle (OT). We consider the class of graphs where the difference between the numbers of edges and vertices is a fixed constant c . The complete characterization of graphs on at least c+7 vertices with the maximum number of OTs is obtained for this class.
This paper presents a new approach to the joint construction of topologies of diameter-optimal circulant networks C(N; ± 1, ± s_2) and optimal routing algorithms of complexity O(1) implemented for them. New routing algorithms are based on the use of scalable parameters of L -shaped patterns in a dense packing of graphs on the plane for families of optimal networks. The scalability of the parameters of L -shaped templates for many families of optimal networks C(N; ± 1, ± s_2) has been proven, analytical formulas for the dependence of these parameters on the diameter of the graphs have been obtained, reducing the time for setting up the routing algorithm at the preliminary stage from O (log N) to O(1) . A comparison of the new routing algorithm with the optimal routing algorithm known in the literature showed its greater efficiency by an average of 10 percent in terms of time spent on routing in families of optimal graphs. Due to their good scalability and ease of routing, optimal degree-four circulant networks are of interest as efficient and reliable communication networks for networks-on-chip, multiprocessor supercomputer systems, telecommunications network structures, and neural communication networks.
A line segment (barrier) is specified on the plane, as well as the location of depots. Each sensor is able to travel a limited-length path, starting and ending at its depot. The part of the barrier along which the sensor moves is covered by this sensor. It is necessary to place some number of mobile sensors (drones) in each depot in order to cover the entire barrier with a minimum number of drones (MinNum), or to minimize the total length of paths traveled by drones (MinSum), or to minimize the maximum distance traveled by a drone (MinMax). Previously, the authors investigated a similar problem with an unlimited number of drones and, for its solution, proposed a pseudopolynomial algorithm depending on the length of the barrier L. In this paper, a generalized problem with a limited number of drones is considered and, to construct an optimal solution, we propose an algorithm with the same complexity. However, in the case of an unlimited number of drones, the new algorithm has complexity L times less than the previous one.
The existence of a unique solution to a nonlocal conjugation problem for a third-order partial differential equation of mixed parabolic-hyperbolic type is established. In the upper half-plane, the characteristic equation has a triple root, while in the lower half-plane, it has one simple root and two multiple ones. By applying the order reduction method, Green’s and Riemann’s functions, and the method of integral equations, the problem is equivalently reduced to a nonlocal problem with an integral condition imposed on the trace of the unknown function along the type-change line of the equation. This, in turn, is reduced to solving a Fredholm integral equation of the second kind, the solvability of which is proven using the method of successive approximations. The solution in the parabolic part of the domain is constructed using Green’s function, whereas in the hyperbolic part, the Riemann function method is employed, reducing the problem to a two-dimensional Volterra integral equation of the second kind. Examples are provided.
The problem of vibrations of objects with moving boundaries is presented as a differential equation with boundary and initial conditions and is a nonclassical generalization of a hyperbolic problem. In the present paper, equivalent integro-differential equations with symmetric and time-dependent kernels and time-varying integration limits are constructed. An expansion of the integro-differential equation of motion of variable-length objects into an infinite system of ordinary differential equations with variable coefficients is given. The concept of eigenfunctions and eigenvalues is defined for a boundary value problem in a domain bounded by time-varying integration limits. Solutions to homogeneous integro-differential equations describing vibrations of variable-length objects and systems of ordinary differential equations with changing parameters are constructed using asymptotic methods. Expressions for the amplitudes and phases of vibrations are obtained. This approach is especially useful in studying complex dynamical systems with lumped masses that oscillate under the influence of moving loads.
The paper considers the transverse vibrations of two strings connected to each other at a certain point. A mathematical model of this process, based on the law of conservation of momentum, expressed in the form of an integro-differential equation is constructed. This equation connects the deviations of the strings during vibrations with their characteristics such as density, tension, and sources of external forces. This approach can be considered as a development of the method proposed by A.N. Tikhonov for the equation of string vibrations with nonsmooth data. For the case where the densities of the strings are constants, we set a nonclassical problem for the integro-differential equation. In addition to the Cauchy data, the problem includes necessary matching conditions. A theorem on the existence and uniqueness of a solution to the problem is proved, and explicit formulas are obtained for its solution.
We analyze the complexity of one extremal problem of choosing a subset of p points in a given finite set in a metric space. The chosen subset of points is required to describe given clusters in the best way from the point of view of some geometric criterion. This problem is a formalization of one applied problem from data mining that consists in finding a subset of typical representatives of a dataset based on the rival similarity function. We prove that the problem under consideration is NP-hard by polynomially reducing the well-known NP-hard 3D-Matching problem to this one.
We consider a stadium antenna deployment problem. The stadium is divided into sectors. Several antennas are assigned to each sector. Users should receive a signal of a certain quality from antennas assigned to their sector. The problem is to choose locations of antennas, their types, angles, and assignments to sectors to maximize three quality criteria: the mean signal to interference ratio (SIR), the number of clients with good signal quality, and the assignment consistency. We use a simulation to compute the signal quality. We present a three-stage heuristic approach to the problem. It uses a constructive heuristic, a local improvement procedure, and a decomposition-based MIP heuristic. We carry out numerical experiments on test instances with 94 antennas of 7 types, 19 sectors, and 4426 clients. It is possible to improve the provided baseline solutions in 2 h and obtain solutions comparable to running a metaheuristic package for 24 h.
Rounding errors inevitably occur when performing calculations on a computer. In this paper, we study the impact of these errors on the accuracy of calculating the correction in the Krylov subspace. An estimate of the accuracy of the calculated correction, expressed via perturbations of the original data, is obtained.
A coalition in a graph G is a pair of disjoint nondominating subsets of its vertices V_1, V_2 ⊂ V(G) such that V_1∪ V_2 is a dominating set. In the coalition partition π (G)={ V_1,V_2,… ,V_k } , every nondominating set V_i is included in some coalition and if V_i is dominating, then it is a single-vertex set. A coalition partition of vertices of a graph G generates a coalition graph CG(G,π ) whose vertices correspond to the partition sets, while two vertices are adjacent if the corresponding sets form a coalition. It is well known that all simple cycles of order greater than three generate in total 26 coalition graphs of order at most six. A universal cycle generates all such graphs. It is shown that only the cycles C_3k , k ≥ 5 , are universal.
The two-dimensional inverse problem of determining the kernel of an elasticity equation of memory type is studied. It is assumed that the coefficients of the equations depend on only one spatial variable. Applying the linearization principle, the inverse problem is reduced to an equivalent linear system of integral equations. The generalized principle of compressed maps is applied to the latter in the space of continuous functions. The theorem of unique solvability is proved and an estimate of the stability of the solution to the inverse problem is obtained.
As a model of shear rupture in the Earth’s crust at the depths of seismic activity, which grows with a velocity exceeding the velocity of longitudinal waves, we consider a Volterra edge dislocation moving in an infinite isotropic elastic medium under the action of preliminary tangential stresses. In the plane strain approximation, the equations of stationary motion of the medium around the dislocation are reduced to a hyperbolic system of equations for velocities and stresses, which is integrated by the method of characteristics. Using the invariant J –integral, an estimate of the energy released during the motion of dislocation is obtained, depending on the velocity, the value of tangential stress at infinity, the length of the fan adjacent to the vertex of dislocation, and on the nature of the distribution of the Burgers vector in the fan.
