نشریه پژوهشهای ریاضی
سال چهارم شماره 2 (پیاپی 9، پاییز و زمستان 1397)

This paper presents an algorithm for covering orthogonal polygons with minimal number of guards. This idea examines the minimum number of guards for orthogonal simple polygons (without holes) for all scenarios and can also find a rectangular area for each guards. We consider the problem of covering orthogonal polygons with a minimum number of r-stars. In each orthogonal polygon P, two points p and q are called r-visible if these two points are in opposite rectangular corners, then the entire rectangle is placed in P. Now we consider a polygon P to be an r-Star if there is a point p in it so that every point q is r-visible from p. In this paper, we propose an algorithm that applies to all simple orthogonal polygons and is able to locate at least the number of guards. Using a method called a rectangular (orthogonal polygonal division into a number of rectangles), the algorithm separates a number of r-stars and processes them in order to insert guards in their place to reach the target, which has the minimum number of guards. Our proposed algorithm is able to determine the minimum number of guards with a rectangular shape at time while the execution order of the best available algorithms is . Another advantage of this algorithm is the independence of restrictions on simple orthogonal polygons.

In this paper, we deal with solving systems of linear congruences over commutative CF-rings. More precisely, let R be a CF-ring (every finitely generated direct sum of cyclic R-modules has a canonical form) and let I_1,..., I_n be n ideals of R. We introduce congruence matrices theory techniques and exploit its application to solve the above system. Further, we investigate the application of computer algebra techniques (Gröbner bases) in this context whenever R = Z.

This paper deals with the numerical solutions of the 2D time dependent Schr¨odinger equations by using a local strong form meshless method. The time variable is discretized by a finite difference scheme. Then, in the resultant elliptic type PDEs, special variable is discretized with a local radial basis function (RBF) methods for which the PDE operator is also imposed in the local matrices. Despite the global collocation approaches, dividing the global collocation domain into many local subdomains, the stability of the method increases. Furthermore, because of the use of strong form equation and collocation approach, which does not need integration, and since in the matrix operations the matrices are of small size, computational cost decreases. An iterative approach is proposed to deal with the nonlinear term. Two linear and two nonlinear test problems with known exact solutions are considered and then, the simulation to a nonlinear problem with unknown solution and periodic boundary conditions is also presented and the results reveal that the method is efficient.

By using variational methods and critical point theory for smooth functionals defined on a reflexive Banach space, we establish the existence of infinitely many weak solutions for a Steklov problem involving the p(x)-Laplacian depending on two parameters. We also give some corollaries and applicable examples to illustrate the obtained result.

In this study coupled system of nonlinear time fractional Drinfeld-Sokolov-Wilson equations, which describes the propagation of anomalous shallow water waves is investigated. The Lie symmetry analysis is performed on the model. Employing the suitable similarity transformations, the governing model is similarity reduced to a system of nonlinear ordinary differential equations with Erdelyi-Kober fractional derivatives. Moreover the invariant subspace method is used to calculate the explicit analytical solutions of the problem.

Estimation of statistical distribution parameter is one of the important subject of statistical inference. Due to the applications of Lomax distribution in business, economy, statistical science, queue theory, internet traffic modeling and so on, in this paper, the parameters of Lomax distribution under type II censored samples using maximum likelihood and Bayesian methods are estimated. Whereas, selection of prior distribution and loss function plays an important role in Bayesian estimation, therefore the Bayesian estimations are presented by appropriate prior distribution under loss functions, mean square error, Linex and entropy. Whereas the normal equation obtained from estimation methods are not distinct function of parameters, they are estimated using error methods such as EM algorithm and Lindley approximation. At the end, using mean square error criteria, the estimators are compared and the result shows that the Bayesian estimator is better than the maximum likelihood estimator and the accuracy of estimator improves by increasing the sample number while the number of failure is fixed.

One of a nonparametric procedures used to estimate densities is kernel method. In this paper, in order to reduce bias of kernel density estimation, methods such as usual kernel(UK), geometric extrapolation usual kernel(GEUK), a bias reduction kernel(BRK) and a geometric extrapolation bias reduction kernel(GEBRK) are introduced. Theoretical properties, including the selection of smoothness parameter and the accuracy of resultant estimators are studied. Accordingly, the mean integrated squared error of GEBRK method achieve a faster convergence rate when kernels are symmetric, where n is the sample size. In order to evaluate the performance of these new estimators, we conduct a Monte Carlo simulation study. The obtained results are illustrated by analyzing real data. The results show that the amount of bias in the proposed BRK and GEBRK methods significantly decreases.

In this paper, we consider the numerical solution of a class of delay fractional optimal control problems using modification of hat functions. First, we introduce the fractional calculus and modification of hat functions. Fractional integral is considered in the sense of Riemann-Liouville and fractional derivative is considered in the sense of Caputo. Then, operational matrix of fractional integration, product and delay operational matrix of the basis functions vector are introduced. For solving the optimal control problem, all functions in the problem are approximated using the basis functions. A system of nonlinear algebraic equations are obtained using the properties of modification of hat functions and the introduced operational matrices. By solving the obtained system, the unknown coefficients of the state and control input functions are determined and by substituting these values an approximation of the solution of the problem is given. Finally, some examples of different kinds of delay fractional optimal control problems are considered to confirm the accuracy and applicability of the suggested method.

In this study, we use a newly proposed method based on the software structure of the maple, called the Khaters method, and will be introducing exponential, hyperbolic, and trigonometric solutions for one of the Schrödinger equations, called the nonlinear Schrödinger equation with the dual power law nonlinearity. Given the widespread use of the Schrödinger equation in physics and engineering, solving this equation is very important with the above method, which includes a large variety of solutions. Schrödinger's nonlinear equation is a partial differential equation that plays a significant role in modern physics. Since quantum mechanics is present in the most modern technologies, such as nuclear energy, computers made of semiconductor materials, lasers, and all quantum phenomena, all the empirical observations of the world around us are consistent with the results of these equations. And this is the Schrödinger equation describing the system of atomic particle motion and instrumentation over time. Hence, because of the importance of the solutions of the Schrödinger equation, which describes many phenomena in physics and engineering, solving this equation is a great necessity. In every phenomenon and process in nature, there are various parameters that are in accordance with the rules governing that phenomenon. The expression of this relation in mathematical language is a functional equation, and the functional equation is derived from a phenomenon in which the tracks of a function change relative to one or several independent variables are studied, called the differential equation. Due to the nature of the Schrödinger's equation, which contains different nonlinear sentences, is of great use in modern sciences, including Quantum Fiery. We can say that the widest range of applications of equations is related to the Schrödinger equation, especially in physics and modern chemistry and quantum electronics. Wherever there are tiny particles, the Schrödinger equation solves the analysis of the most complex issues associated with them./files/site1/files/42/10Abstract.pdf