Solutions For Different Numbers Research Articles

Optimal Placement of UAVs to Provide Surveillance Coverage for a Ground Vehicle in a Collaborative Search-and-Rescue Operation

A drone-truck combined search-and-rescue operation involves a ground vehicle and a swarm of unmanned aerial vehicles (UAVs), where the UAVs provide surveillance coverage to guide the ground vehicle to navigate through the environment and carry out the search and rescue, and the ground vehicle functions as a service hub for carrying and recharging the UAVs. An effective strategy for providing persistent UAV surveillance coverage around the ground vehicle consists of initially forming the UAV swarm coverage and then controlling the UAV formation to follow the ground vehicle. This paper focuses on the formation of coverage and presents a method for planning an optimal placement of the UAVs to form seamless surveillance coverage around the ground vehicle. The optimization problem is formulated to determine the number and positions of UAVs that minimize the energy consumption in deploying and collecting those UAVs, subject to a set of constraints in UAV positioning, communication, and coverage, specifically the available number of UAVs, allowable range of UAV altitude, allowable energy consumption for deploying and collecting each UAV, communication ranges of UAVs and ground vehicle, safety distance between UAVs for collision and interference avoidance, and seamless coverage. A bi-layer optimization procedure is developed, with an outer layer searching through the allowable numbers of UAVs and an inner layer searching for the optimal positions for each specific number of UAVs. The optimal number and positions of UAVs are chosen by comparing among the solutions for different numbers of UAVs. A simulation study is carried out to validate the proposed optimization formulation and solution approach, where the simulation settings of UAVs, particularly the critical parameters including the UAV energy constants, visibility angle, altitude, and communication range, use the representative values presented in the cited literature. The simulation results show that the proposed approach is effective in planning the optimal number and positions of UAVs to provide seamless surveillance coverage for a ground vehicle. The next step of research will set priorities on comprehending the complexity of the solution space and enhancing the global optimality of the solution.

Modular differential equations with movable poles and admissible RCFT characters

Studies of modular linear differential equations (MLDE) for the classification of rational CFT characters have been limited to the case where the coefficient functions (in monic form) have no poles, or poles at special points of moduli space. Here we initiate an exploration of the vast territory of MLDEs with two characters and any number of poles at arbitrary points of moduli space. We show how to parametrise the most general equation precisely and count its parameters. Eliminating logarithmic singularities at all the poles provides constraint equations for the accessory parameters. By taking suitable limits, we find recursion relations between solutions for different numbers of poles. The cases of one and two movable poles are examined in detail and compared with predictions based on quasi-characters to find complete agreement. We also comment on the limit of coincident poles. Finally we show that there exist genuine CFT corresponding to many of the newly-studied cases. We emphasise that the modular data is an output, rather than an input, of our approach.

An immersed boundary neural network for solving elliptic equations with singular forces on arbitrary domains.

In this paper, we present a deep learning framework for solving two-dimensional elliptic equations with singular forces on arbitrary domains. This work follows the ideas of the physical-inform neural networks to approximate the solutions and the immersed boundary method to deal with the singularity on an interface. Numerical simulations of elliptic equations with regular solutions are initially analyzed in order to deeply investigate the performance of such methods on rectangular and irregular domains. We study the deep neural network solutions for different number of training and collocation points as well as different neural network architectures. The accuracy is also compared with standard schemes based on finite differences. In the case of singular forces, the analytical solution is continuous but the normal derivative on the interface has a discontinuity. This discontinuity is incorporated into the equations as a source term with a delta function which is approximated using a Peskin's approach. The performance of the proposed method is analyzed for different interface shapes and domains. Results demonstrate that the immersed boundary neural network can approximate accurately the analytical solution for elliptic problems with and without singularity.

Multiscale Model Reduction of the Unsaturated Flow Problem in Heterogeneous Porous Media with Rough Surface Topography

In this paper, we consider unsaturated filtration in heterogeneous porous media with rough surface topography. The surface topography plays an important role in determining the flow process and includes multiscale features. The mathematical model is based on the Richards’ equation with three different types of boundary conditions on the surface: Dirichlet, Neumann, and Robin boundary conditions. For coarse-grid discretization, the Generalized Multiscale Finite Element Method (GMsFEM) is used. Multiscale basis functions that incorporate small scale heterogeneities into the basis functions are constructed. To treat rough boundaries, we construct additional basis functions to take into account the influence of boundary conditions on rough surfaces. We present numerical results for two-dimensional and three-dimensional model problems. To verify the obtained results, we calculate relative errors between the multiscale and reference (fine-grid) solutions for different numbers of multiscale basis functions. We obtain a good agreement between fine-grid and coarse-grid solutions.

Differential protection scheme for power transformers using matched wavelets

A new approach for differential protection based on the mother wavelets which match in shape with inrush and fault current waveforms is proposed in this work. It is demonstrated that if the matched wavelet is used for processing inrush and fault waveforms of transformer then its detection capability improves. In case of false tripping of transformer, the equipment is to be opened for inspection for faults. This leads to equipment outage because there is no mechanism to ascertain that there was no fault. It is needed that inrush and fault be detected independently not only for the purpose of faster and better detection but also for studying mal-operations and post-analysis. For a faster and better detection, a new protection scheme is also proposed making use of independent detection of fault and inrush waveforms. To find matched inrush and fault wavelets, differential evolution algorithm is used. Out of various possible solutions for different number of wavelet coefficients, a choice is made by further testing on various switching angles. The effectiveness of the proposed method is tested through PSCAD simulations on transformers of different ratings.

ЧИСЕЛЬНИЙ АНАЛІЗ РОЗВ’ЯЗКІВ ДВОВИМІРНИХ ЗАДАЧ ТЕПЛОПРОВІДНОСТІ ЗА БЕЗСІТКОВОЮ СХЕМОЮ З ВИКОРИСТАННЯМ ФУНДАМЕНТАЛЬНИХ І ЗАГАЛЬНИХ РОЗВ’ЯЗКІВ

У статті наводиться аналіз чисельних розв’язків двовимірних задач теплопровідності за безсітковою схемою, які отримані з використанням фундаментальних і загальних розв’язків модифікованого рівняння Гельмгольца. У роботі використовувався безсітковий метод, який ґрунтується на комбінації методу подвійного заміщення з використанням радіальних базисних функцій і методу частинних розв’язків. Порівняльний аналіз розв’язків крайових задач продемонстрований на прикладі двох тестових задач. Були отримані чисельні розв’язки двовимірних нестаціонарних задач теплопровідності з використанням фундаментального і загального розв’язків для різного числа інтерполяційних вузлів. Були визначені середньоквадратичні похибки розв’язків розглянутих задач, а також побудовані порівняльні графіки залежності середньоквадратичної похибки від числа інтерполяційних вузлів.

ANALYTICAL SOLUTION OF THE FREQUENCY OF THE LOAD OSCILLATION AT AN ARBITRARY GIRDER NODE IN THE SYSTEM MAPLE

Introduction. We study the oscillations of a massive load on a planar statically definable symmetric truss of a regular type with parallel belts. Truss weight is not included. Free vertical oscillations are considered. The stiffness of the truss rods is assumed to be the same, the deformations are elastic. Lattice of the truss is double with descending braces and racks. New in the formulation and solution of the problem is the analytical form of the solution, which makes it possible in practice to easily evaluate the frequency characteristics of the structure depending on an arbitrary number of truss panels and the location of the load. Materials and methods. The operators and methods of the system of computer mathematics Maple are used. To determine the forces in the rods, the knotting method is used. The common terms of the sequence of coefficients of solutions for different numbers of panels are obtained from solving linear homogeneous recurrent equations of various order, obtained by special operators of the Maple system. Dependence on two arbitrary natural parameters is revealed in two stages. First, solutions for fixed load positions are found, then these solutions are summarized into one final formula for frequency. Results. By a series of individual solutions to the problem of load oscillation using the double induction method, it was possible to find common members of all sequences. The solution is polynomial in both natural parameters. Graphs constructed for particular cases, showed the adequacy of the approach. The discontinuous non-monotonic nature of the intermittent change depending on the number of truss panels and some other features of the solution are noted. Conclusions. It is shown that the induction method, previously applicable mainly to statics problems with one parameter (number of truss panels), is fully operational to the problems of the oscillations of system with two natural parameters. It should be noted that significant labor costs and a significant increase in the time symbolic transformations in such tasks

Investigating Reliability on Fuel Cell Model Identification. Part I: A Design of Experiments Approach

AbstractA model‐based Design of Experiments method was employed for the optimization of measurements on a solid oxide fuel cell (SOFC). Based on a simplified SOFC model, a variation of the D‐optimality was used as optimization criterion for the calculation of optimal experimental designs (determinant of the covariance matrix with weighting factors). Solutions for different numbers of design points were calculated and the behavior of optimization criteria as functions of the number of design points as well as of the number of repetitions of measurements was analyzed. A new type of graph was introduced which depicts the behavior of optimization criteria for constant number of measurements. This approach showed that, for constant numbers of measurements, the precision is higher and therefore the reliability in the cell's model identification is improved when repeated measurements of a small set of optimal design points are effectuated, instead of many different measurements. Finally, a sensitivity analysis was performed showing the influence of the parameter values on the values of the optimization criterion and the optimal measurements.The used methodology and its theoretical conclusions may be used as a basis for development of diagnostics tools or filtering existing data for optimal parameter estimations.

Analysis of ground states of0−πlong Josephson junctions

We investigate analytically a long Josephson $0\ensuremath{-}\ensuremath{\pi}$ junction with several $0$ and $\ensuremath{\pi}$ facets which are comparable to the Josephson penetration length ${\ensuremath{\lambda}}_{J}.$ Such junctions can be fabricated exploiting (a) the d-wave order-parameter symmetry of cuprate superconductors; (b) the spacial oscillations of the order parameter in superconductor-insulator-ferromagnet-superconductor structures with different thicknesses of ferromagnetic layer to produce 0 or $\ensuremath{\pi}$ coupling; or (c) the structure of the corresponding sine-Gordon equations and substituting the phase $\ensuremath{\pi}$ discontinuities by the artificial current injectors. We investigate analytically the possible ground states in such a system and show that there is a critical facet length ${a}_{c},$ which separates the states with half-integer flux quanta (semifluxons) from the trivial ``flat phase state'' without magnetic flux. We analyze different branches of the bifurcation diagram, derive a system of transcendental equations which can be effectively solved to find the crossover distance ${a}_{c}$ (bifurcation point), and present the solutions for different number of facets and the edge facets length.

A multireference coupled-cluster method for special classes of incomplete model spaces

A multireference coupled-cluster (MRCC) method for special but general classes of incomplete model spaces is developed within a Hilbert space framework. The formulation avoids the valence universality requirement and related Fock space considerations that require hierarchical solutions for different number of electrons. Consequently, the Hilbert space approach has fewer amplitudes to determine. It is shown that diagonalization of the effective Hamiltonian leads to purely extensive energies, providing the formal basis for a general MR-CC methods for potential energy surfaces.

The buckling of interbraced columns

The paper presents an analysis of a system of parallel structural columns interlinked by elastic braces, and has the particular aim of determining critical or buckling load levels. The analysis reduces the problem to the solution of a pair of simultaneous difference (recurrence) equations. The nature of this solution is investigated and buckling conditions established. Charts are presented which cover the basic solutions for different numbers and stiffness of braces and for different numbers of columns.