نشریه پژوهشهای ریاضی
سال ششم شماره 1 (پیاپی 12، بهار 1399)

In this paper, we investigate contact CR submanifolds of contact CR dimension in Sasakian space form and introduce the general structure of these submanifolds and then studying structures of this submanifols with the condition   h(FX,Y)+h(X,FY)=g(FX,Y)zeta,  for the normal vector field zeta, which is nonzero, and we classify these submanifolds../files/site1/files/61/0Abstract.pdf

The Camina triple condition is a generalization of the Camina condition in the theory of finite groups. The irreducible characters of Camina triples have been verified in the some special cases. In this paper, we consider a Camina triple (G,M,N)  and determine the irreducible characters of G in terms of the irreducible characters of M and G/N../files/site1/files/61/0Abstract(1).pdf

   One of ‎the ‎intersecting aspects of the reliability theory and economics is investigating non-negative data generally skewed‎ ‎(lifetimes and income)‎, ‎extrapolation of a suitable model for the data and finding their characteristics‎.
The concepts of inequality measures play an important role in economic, social sciences and the other areas. In recent years, various inequality curves have been developed or investigated as the descriptors of income inequality. The Lorenz curve is a strong tool extensively used in economics to consider the inequality of income distributions and wealth. It is the graphical display of wealth distribution created by the American economist, Max Lorenz, in 1905. It plots the percentage of the total income obtained by various portions of the population when the population is ordered by the size of its income. Several alternative inequality curves have been proposed in the literature. The alternative inequality curves are considered as competitors of the classical Lorenz curve as descriptors of income inequality. The Bonferroni curve and the Zenga-2007 curve appear to be essentially the function of the Lorenz curve.
Reliability is a broad concept. It is applied whenever we expect something to behave in a certain way. It is one of the metrics that are used to measure quality. The notion of reliability, in the statistical sense, is the probability that an equipment or unit will perform the required function under the conditions specified for its operations for a given period of time. The primary concern in reliability theory is to understand the patterns in which failures occur, for different mechanisms and under varying operating environments, as a function of its age. This is accomplished by identifying the probability distribution of the lifetime represented by a non-negative random variable. Accordingly, several concepts have been developed that help in evaluating the effect of age, based on the distribution function of the lifetime random variable and the residual life X. Concepts of aging describe how a component or a system improves or deteriorates with age; and they are very serious in the reliability analysis. In reliability, several aging classes of life distributions have been presented to explain the various forms of aging.
Different order relations have been developed using measures in connection with many fields such as reliability, economics, queuing theory, survival analysis, insurance, operations research, etc.
 

     In this paper, we provide a brief review of the widely used income inequality measures, their inter relationships and their properties and we discuss on basic reliability concepts such as hazard rate, mean residual life, reversed hazard rate and reversed mean residual life in distribution function frame work.
The main aim of this paper is finding the relationship between the indices of economic inequality and reliability theory‎. For example, Zenga curve shall be interpreted as the difference in average age of components which has survived beyond age X from those which has failed before attaining age X, expressed in terms of average age of components exceeding age X. Therefore, we present some of the important results about the relationship between aging concepts and several important theorems have been proved in this subject that actually are novelty of the paper. We also present some characterizations of the reliability concept by using the Lorenz curve. ‎‎In this article‎, ‎first‎, ‎the relationship between inequality indices and reliability indices which have a very close relationship are discussed‎.‎ In fact, proofing of several propositions to express these connections is the main novelty of this article. In addition,‎ some of the ageing concepts can be expressed through inequality indices‎. ‎Finally‎, ‎to gain better understanding of the basic idea the Iranian income and expenditure data between the years 1388 to 1393 are numerically studied.

In the present work, we have examined the connection between the other existing inequality measures, the relationship of the concepts of inequality indices with certain reliability concepts are exploited to obtain characterization results for probability distributions. Further some results on a stochastic order using inequality curves are also established. Finally some numerical results were given in order to indicate the usefulness. There are many reasons to study the relationship between the reliability concepts and the inequality measures. For example, the study of the relationship between inequality indices and measurement standards of reliability makes it possible to use each of these two concepts for studying the other one‎. We are able to set some other criteria for exponentiation based on the Lorenz curve and Gini index. Also, we can obtain some new properties for the Lorenz curve, the Gini index, and the other inequality indices. Moreover, we are able to indicate that the Lorenz curve can be expended to create a variant explanation of lifetime data and vice versa and, as well, to determine the bound of the class of lifetime distributions in terms of its Lorenz curve and the other index inequalities.

The following conclusions were drawn from this research.

The bound of the class of lifetime distributions are determined in terms of its Lorenz curve and the other index inequalities.
Some interesting relationships that exist between commonly used notions in reliability theory and economic theory and reliability.
Some new properties for income inequality are obtained‎./files/site1/files/61/3.pdf

The Voronoi game is a simple geometric model for competitive facility location problem which is played by two players, White and Black, in a continuous space (one-dimensional or two-dimensional). In the one-round game, White places all his n points. After that, Black places the same number of points on the game space. The players Cannot change or reuse a point that was placed before. At the end of the game, the Voronoi diagram of all 2n points is constructed and a player who obtains the larger total area is the winner.
Ahn et al. considered the Voronoi game on a unit circle and a line. They presented a winning strategy for the second player and showed that the first player can preserve the winning margin as small as possible. Cheong et al. presented a winning strategy for the second player when
 on a square. Fekete et al. considered the two-dimensional version of Voronoi game on a rectangular area of aspect ratio ρ. They showed that there is a winning strategy for the second player where  with and for  with  and the first player wins in the other situations. Rashid et al. introduced a new version of Voronoi game called neighbor Voronoi and presented the winning strategies based on the neighborhood of points. Bandyapadhyay et al. studied Voronoi game on a weighted graph and showed that the problem of finding a winning strategy for the second player is NP-Complete.
In the previous studies, the researchers considered the Voronoi game in which the facilities are the same. In the real world, the customers usually consider both the preferences and the distances for shopping which is modeled by the multiplicity weighted Voronoi diagram. Accordingly, we introduce a new version of the Voronoi game for modeling this problem, called multiplicity weighted Voronoi diagram.

In this paper, the one-round weighted Voronoi game is studied in both one-dimensional and two-dimensional cases. In the multiplicity weighted Voronoi game, the customer can consider not only the distance of the facility but also its. Therefore, in the one-dimensional, two different models of the facility are considered and it is showed that the second player has a winning strategy in the one-round Voronoi game.
In the weighted Voronoi diagram, the points have the assigned weights according to their performance.
Firstly, we consider the topology of the cells of weighted Voronoi diagram whose edges are the parts of Apollonius circles and it is too complicated to calculate the areas of the weighted Voronoi cells. So Black just tries to earn a little more than White without calculating the exact amount of the winning area. Also, we assume that the sum of all weights of black points equals with the sum of the weights of all whites points, i.e., .

In the one-dimensional model in which the area is a line segment, we study both the same-weight points and the different-weight points separately. We show that Black always has a winning strategy when the points have the same weight. In the other case, Black has a winning strategy when the variance of the distribution of the black points is less than or equal to the variance of the distribution of the white points.
In the two-dimensional model, we study the same-weight points and show that Black has a winning strategy.

In this study, we introduced the weighted Voronoi game to consider the performance of the facilities as well as the distance from customers. We showed that the second player has a winning strategy. In this model, customers can choose their service based on performances of the facilities such as parking, price, variety, quality of products, etc. Therefore this model is closer to the real world.
In the one-dimensional model, we showed that the second player can always win the game when the points have the same weight. Also, the second player can win the game with the different-weight points when the variance of the distribution of the black points is less than or equals the variance of the distribution of the white points. In the two-dimensional model, we showed that the black player can always win the one-round game../files/site1/files/61/5.pdf

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group and G' is a p-group, 2) if r=0 or r is not a power of a prime, then G is abelian. In this paper, also, we define some generalized Engel conditions on groups, then we present a result about unit group of group algebras which satisfies this kind of generalized Engel conditions. ./files/site1/files/61/6.pdf

Despite wide applications of constant order fractional derivatives, some systems require the use of derivatives whose order changes with respect to other parameters. Samko and Ross produced an extension of the classical fractional calculus with a continuously varying order for differential and integral operators. Variable-order fractional (V-OF) calculus has applications in optimal control, processing of geographical data, diffusion processes, description of anomalous diffusion, heat-transfer problems, etc. Due to the V-OF operators which are non-local with singular kernels, finding the exact solutions of V-OF problems is difficult. Therefore, efficient numerical techniques are necessary to be developed. The numerical solution of V-OF differential equation has been considered in some papers.
    Recently, discrete orthogonal polynomials have been considered as basis functions instead of continuous orthogonal polynomials. Discrete orthogonal polynomials are orthogonal with respect to a weighted discrete inner product. These polynomials have important applications in chemical engineering, theory of random matrices, queuing theory and image coding. In this paper, we focus on a special class of discrete polynomials, called Hahn polynomials.
    In this work, first, a new operational matrix is obtained for V-OF integral of Hahn polynomials. Then, we use a spectral collocation technique combined with the associated operational matrices of V-OF integral for solving weakly singular fractional integro-differential equations.

In this scheme, the operational matrix of fractional integration of Hahn polynomials is calculated. This method converts the weakly singular fractional integro-differential equations into an algebraic system which can be solved by a technique of linear algebra.

In this paper, some numerical examples are provided to show the accuracy and efficiency of the presented method. By using a small number of Hahn polynomials, significant results are achieved which are compared to other methods. A comparison to the numerical solutions by CAS and Haar wavelets and Adomain decomposition method, shows that this technique is accurate enough to be known as a powerful device.

The following results are obtained from this research.

The operational matrix of fractional integration of Hahn polynomials is presented for the first time.
The main advantage of approximating a continuous function by Hahn polynomials is that they have a spectral accuracy at interval [0,N], where N is the number of bases.
Furthermore, for estimating the coefficients of the expansion of approximate solution, we only have to compute a summation which is calculated exactly.
Using Hahn polynomials, the numerical results achieved only by a small number of bases, are accurate in a larger interval and significant results are achieved../files/site1/files/61/7.pdf

In this paper, we have introduced a new method for solving a class of the partial integro-differential equation with the singular kernel by using the finite difference method. One of the best subjects in the numerical analysis is a finite difference method (FDM). We used (FDM) to solve problems in mathematical physics, integral equations, and  engineering, such as electromagnetic potential, fluid flow,  radiation heats transfer, laminar boundary-layer theory and mass transport, Abel integral equations, and problem of mechanics or physics. Also in some physical problems such as fluid flow and heat transfer problems, the Laplace equations and the Poisson equations are describe by (FDM).  In real life most phenomena are modelled by partial differential equations.

First, we employing an algorithm for solving the problem based on the Crank-Nicholson scheme with given conditions. Furthermore, we discrete the singular integral for solving of the problem. Also, the numerical results obtained here can be compared with the cubic B-spline method.

In addition, solving some examples demonstrates the validity and applicability of the approached method, so that the results are reported in the tables and their figures are shown. The high speed of the calculations, and the assurance of having an approximate solution are obtain by proving the stability of the method.

The following conclusions were drawn from this research.

Coefficients of the approximate function via Crank-Nicholson scheme are found very easily and therefore many calculations are reduced.
The numerical results obtained here can be compared with the cubic B-spline

The assurance of having an approximate solution are obtain by proving the stability of the method../files/site1/files/61/8.pdf

In this paper, we apply the extended triangular operational matrices of fractional order to solve the fractional voltrra model for population growth of a species in a closed system. The fractional derivative is considered in the Caputo sense. This technique is based on generalized operational matrix of triangular functions. The introduced method reduces the proposed problem for solving a system of algebraic equations. Illustrative examples are included to demonstrate the validity and the applicability of the proposed method../files/site1/files/61/9.pdf

One way to identify outlier observations in regression models, is to measure the difference between the observations and their expected values under fitted model. This identification in circular regression, is possible by using of a circular distance. In this paper, the Difference of Means Circular Error statistic that was introduced by ‎Abuzaid et al. [1] for outlier detection in simple circular regression, is applied in linear-circular regression model and the cut-off points of this statistic are obtained by Monte Carlo simulations. In addition, the performance of this statistic is investigated with some simulation studies. Finally, this statistic is applied to identify outlier observations in speed and direction wind data set recorded at Mehrabad weather station in Tehran with parametric Bootstrap simulation method../files/site1/files/61/10.pdf

The graph is a mathematical model for a discrete set whose members are interlinked in some way. The members of this collection can be the different parts of the earth and the connections between them are bridges that tie them together (like the Konigsberg problem). Graph theory is one of the important issues in discrete mathematics, which studies graphs and modeling issues by them. In 1736, Leonard Euler established the graph theory for solving the Konigsberg Bridge problem. But James Joseph Sylvester was the first to use the word "graph" in 1878 to name these mathematical models.
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. The convention of using colors originates from coloring the countries of a map, where each face is literally colored. Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Graph coloring is still a very active field of research. This paper is concerned with a specific coloring, say the distinguishing coloring which is originated from a classic elementary problem, Frank Rubin's key problem, which Stan Wagon circulated in the Macalester College problem column:Professor X, who is blind, keeps keys on a circular key ring.  Suppose there are a variety of handle shapes available that can be distinguished by touch.  Assume that all keys are symmetrical so that a rotation of the key ring about an axis in its plane is undetectable from an examination of a single key.  How many shapes does Professor X need to use in order to keep n keys on the ring and still be able to select the proper key by feel?
The surprise is that if six or more keys are on the ring, there need only be 2 different handle shapes; but if there are three, four, or five keys on the ring, there must be 3 different handle shapes to distinguish them. The answer to the key problem depends on the shape of the key ring. A labeling of a graph G, ɸ: V(G) → {1,2, …, r}, is said to be r-distinguishing if no automorphism of G preserves all of the vertex labels. The point of the labels on the vertices is to destroy the symmetries of the graph, that is, to make the automorphism group of the labeled graph trivial. Consequently, we define the distinguishing number of a graph G by
D(G)=min {r| G has a labeling that is r-distinguishing}.
Similarly, the distinguishing index D′(G) of a graph G is the least integer d such that G has an edge labeling with d labels that is preserved only by a trivial‎ automorphism‎.
Let G be a connected graph of order n ≥ 3 and let c: E(G) → {1, 2, …, k} be a coloring of the edges of G (where adjacent edges may be colored the same). For each vertex v of G, the color code of v with respect to c is the k-tuple c(v) = (a1, a2, …, ak), where ai is the number of edges incident with v that are colored i (1 ≤ i ≤ k). The coloring c is detectable if distinct vertices have distinct color codes. The detection number det(G) of G is the minimum positive integer k for which G has a detectable k-coloring.

We use cycle rank parameter and first consider the case  and prove that and then using spanning trees we obtain an upper bound for distinguishing number.

In this paper, we consider the relationship between the distinguishing number and index with the detection number of a graph. In particular, we show that the distinguishing index of a connected graph is at most equal with the detection number, i.e., D'(G) ≤ det(G).

The following conclusions were drawn from this research.
An upper bound for D(G) by the detection number of its spanning trees.
The upper and lower bounds for the distinguishing number by the detection number of a graph.
Every detectable coloring is a distinguishing labeling of the edges of a graph.
The upper bounds for the distinguishing index by the detection number of a graph../files/site1/files/61/11.pdf

Optimal control problems (OCPs) appear in a wide class of applications. In the classical control problems, the state-space equations are expressed as differential equations. Many physical systems, technology, biology, viscoelastic, electrochemical, economic, and generally the systems that have a memory effect cannot properly be described as ordinary differential equations. Hence, the equation of these systems expresses as integral equations, integro-differential equations, fractional differential equations and fractional integro-differential equations. Almost every system of controlled ordinary differential equations or controlled integro-differential equations can be modeled by a class of systems of controlled Volterra integral equations. There are many methods for solving optimal control problems with the state space of the system in the form of ordinary, fractional, and integral equations; can be mentioned the Euler-Lagrange method, the method of using Pontryagin’s maximum principle, the numerical methods based on finite difference, finite element methods, conjugate gradient method, spectral methods, the methods of continuous orthogonal functions, the operational matrices of integrals and embedding method. The method which we used in this paper is based on using the operational matrix of cubic B-spline scaling functions and wavelets with collocation method to reduce the optimal control problem governed by the nonlinear integral equation and integro-differential equation system with quadratic performance index to a nonlinear programming. The semi-orthogonal B-spline scaling functions and wavelets and their dual functions used in this paper have compact support, vanishing moments. These properties make many of the operational matrix elements be very small compared with the largest ones. These scaling functions and wavelets can be represented in a closed form so working with them is easy. The convergence of control and state functions and the performance index of the optimal approximation of the proposed method and also the upper bound of the error are given.
 

In this paper, a numerical method based on cubic B-spline scaling functions and wavelets for solving optimal control problems with the dynamical system of the integral equation or the differential-integral equation is discussed. The Operational matrices of derivative and integration of the product of two cubic B-spline wavelet vectors, collocation method and Gauss-Legendre integration rule for the discretization of the continuous optimal control problem and its transformation into a problem of non-linear programming is used.

We solve two examples of optimal control with the dynamical system of integral equation and two examples with the dynamical system of the integro-differential equation by using present method to demonstrate validity, applicability and the simplicity of the new technique, then compare the present method with hybrid pseudo-spectral and Legendre wavelets method. These results illustrate that the accuracy of our numerical solutions are a few better than the numerical solutions obtained in the other method and there is a good agreement between the approximate solution and exact solution. Also, the numerical results reported in the tables and convergence analysis demonstrate that the accuracy improve by increasing the. Therefore, to get more accurate results, using the larger  is recommended.

The following conclusions were drawn from this research.
The operation matrix can be simply obtained for any basis of the approximation space and it is always available, therefore it can be applied to obtain the numerical solution of various kind of optimal control problems.
Numerical results and convergence analysis indicates that the approximation solution fairly matches with the exact solution and the upper bound of error exponentially decreases by growing of approximation space.
Due to the characteristics of the B-spline wavelet and dual of them, a nonlinear objective function can be obtained without calculating the integral.
The semi-orthogonal B-spline scaling functions and wavelets used in the present paper have the properties of compact support, vanishing moments, smoothness function and the representation by a closed-form expression. With these assumptions, time is reduced, computer memory is less occupied and the operation matrix is always available../files/site1/files/61/12.pdf

One of the important concepts in topos theory is the concept of (weak) Lawvere-Tierney topology. A class of (weak) Lawvere topology on the  topos  Act-S of right acts over a  fixed monoid S is ideal topology which has been introduced by the authors in [13]. In this paper we first give some characterizations of sheaves with respect to this kind of topologies. Then, using this topology, we construct a Hoehnke radical on this topos. Finally, we investigate the relationship between the corresponding sheaf to the closure operator obtained from this radical and the corresponding sheaf with respect to the ideal topology../files/site1/files/61/13Abstract.pdf