An Efficient Class of Modulus-Based Matrix Splitting Methods for Nonlinear Complementarity Problems

In this paper, we present an efficient method for finding a numerical solution for nonlinear complementarity problems (NCPs). We first reformulate an NCP as an equivalent system of fixed-point equations and then present a modulus-based matrix splitting iteration method. We prove the convergence of the proposed method with theorems with the relevant conditions. Our preliminary numerical results show that the method is feasible and effective.

Classical equations of an electron from the special form of the Dirac equation

The equivalent system of equations corresponding to the Dirac equation is derived and the WKB approximation of this system is found. Similarly, the WKB approximation for the equivalent system of equation corresponding to the squared Dirac equation is found and it is proved that the Lorentz equation and the Bargmann-Michel-Telegdi iquations follow from the new Dirac-Pardy system. The new tensor equation with sigma matrix is derived for the verification by adequate laboratories.

Lockdown: a non-pharmaceutical policy to prevent the spread of COVID-19. Mathematical modeling and computation

In this paper, we derive and analyze an extended SIRS-model which includes lockdown policies at the early stages of the pandemic. The latter play a salient role for flattening the curve of infectious diseases such as COVID-19, and is introduced as a model compartment. An error function is reported, which serves as a bridge between the outcomes of the model and available databases; we estimate the values of the model parameters by minimizing the error function. The intervention function, obtained from the equivalent system of the proposed model, and effective reproduction function are also derived to understand the underline scenario of the coronavirus outbreak. We then estimate the epidemiological variables such as susceptible, recovered, lockdown etc. for Canada and three of its provinces, Ontario, Qu\’ebec and British Columbia, significantly affected by the coronavirus. Some improvements, such as spatial dependence or “at risk’‘ vs “healthy” population, will finally be proposed in order to increase the accuracy of the modeling.

An efficient convergent method for the delay reaction-diffusion equations

In this manuscript, we consider the delay reaction-diffusion equation and implement an efficient spectral collocation method to approximate the solution of this equation. We first replace the delay function in the delay reaction-diffusion equation and achieve an equivalent system of equations. We then utilize the Legendre-Gauss-Lobatto and two-dimensional interpolating polynomial to approximate the solution of obtained system. Moreover, we prove the convergent of method under some mild conditions. Finally, the capability and efficiency of the method is illustrated by providing several numerical examples and comparing them with others

Analysis of the Dynamical Behaviour of a Two-Dimensional Coupled Ecosystem with Stochastic Parameters

Because the two-dimensional coupled ecosystem has perfect symmetry, the dynamical behavior of symmetric dynamical system is discussed. The analysis of the dynamical behavior of a two-dimensional coupled ecosystem with stochastic parameters is explored in this paper. Firstly, a two-dimensional coupled ecosystem with stochastic parameters is established, it is transformed into a deterministic equivalent system by orthogonal polynomial approximation. Then, analysis of the dynamical behaviour of equivalently deterministic coupled ecosystems is performed using stability theory. At last, we analyzed the dynamical behaviour of non-trivial points by means of the mathematics analysis method and found the influence of random parameters on asymptotic stability in coupled ecosystem is prominent. The dynamical behaviour analysis results were verified by numerical simulation.

A Recursive Least-Squares Approach with Memorizing Factor for Deriving Dynamic Equivalents of Power Systems

In this research, a two-stage identification-based approach is proposed to obtain a two-machine equivalent (TME) system of an interconnected power system for transient stability studies. To estimate the parameters of the equivalent system, a three-phase fault is applied near and/or at the bus of a local machine in the original multimachine system. The electrical parameters of the equivalent system are calculated in the first stage by equating the active and reactive powers of the local machine in both the original and the predefined equivalent systems. The mechanical parameters are estimated in the second stage by using a recursive least-squares estimation (RLSE) technique with a factor called “memorizing factor”. The approach is demonstrated on New England 10-machine 39-bus system, and its accuracy and efficiency are verified by computer simulation in MATLAB software. The results obtained from the TME system agree well with those obtained from the original multimachine system.

Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

The multidimensional parabolic integro-differential equation with the time-convolution in- tegral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. In this paper it is studied the inverse problem consisting in finding of a time and spatial dependent kernel of the integrated member on known in a hyperplane xn = 0 for t > 0 to the solution of direct problem. With use of the resolvent of kernel this problem is reduced to the investigation of more convenient inverse problem. The last problem is replaced with the equivalent system of the integral equations with respect to unknown functions and on the bases of contractive mapping principle it is proved the unique solvability to the direct and inverse problems

Extension of Equivalent System Mass for Human Exploration Missions on Mars

NASA mission systems proposals are often compared using an equivalent system mass (ESM) framework wherein all elements of technology to deliver an effect- its components, operations and logistics of delivery- are converted to effective masses since this has a known cost scale in space operations. To date, ESM methods and the tools for system comparison have not considered complexities wherein systems that serve a mission span multiple transit and operations stages, such as would be required to support a crewed mission to Mars, and thus do not account for the different mass equivalency factors operational during each period and the inter-dependencies of the costs across the mission. Further, ESM does not account well for the differential reliabilities of the underlying technologies. Less reliability should incur an equivalent mass cost for technologies that might otherwise provide a mass advantage. We introduce an extensions to ESM to address these limitations and show that it provides a direct method for analyzing, optimizing and comparing different mission systems. We demonstrate our extended ESM (xESM) calculation with crop production technologies -- an aspect of the developing offworld biomanufacturing suite -- since it represents a case with strong coupling among stages of the mission and a relatively high-risk profile.