MATHEMATICAL RESEARCHES
Year:2019 | Volume:5 | Issue:1 |Papers Count:10

Introduction: Vertex decomposability of a simplicial complex is a combinatorial topological concept which is related to the algebraic properties of the Stanley-Reisner ring of the simplicial complex. This notion was first defined by Provan and Billera in 1980 for k-decomposable pure complexes which is known as vertex decomposable when. Later Bjorner and Wachs extended this concept to non-pure complexes. Being defined in an inductive way, vertex decomposable simplicial complexes are considered as a well behaved class of complexes and has been studied in many research papers. Because of their interesting algebraic and topological properties, giving a characterization for this class of complexes is of great importance and is one of the main problems in combinatorial commutative algebra.....

Introduction: In this paper, we intend to show that without any certain growth condition on the weight function, we always able to present a weighted sup-norm on the upper half plane in terms of weighted sup-norm on the unit disc and supremum of holomorphic functions on the certain lines in the upper half plane. Material and methods: We use a certain transform between the unit dick and the upper half-plane, a translation operator between weighted spaces of holomorphic functions toghther with Phragmen-Lindelof theorem in order to obtain our main results. Results and discussion: We prove 3 Lemma which enable us to get our main results in Theorem 3. Conclusion: The following conclusions were drawn from this research. We find lower bound and upper bound for weighted sup-norm in terms of supremum of the function on the lines in the upper half-plane. We obtain lower and upper bounds for translation operator in terms of weighted Sup-norm.

Introduction: Selection the appropriate statistical model for the response variable is one of the most important problem in the finite mixture of generalized linear models. One of the distributions which it has a problem in a finite mixture of semi-parametric generalized statistical models, is the Poisson distribution. In this paper, to overcome over dispersion and computational burden, finite mixture of semi-parametric generalized linear models using the negative binomial (GFMMNB) distributions instead of finite mixture of semi-parametric generalized linear models using the Poisson distributions (GFMMP) has been proposed. Efficiency of GFMMNB to GFMMP using weighted generalized mean of square error (WGMSE) for both the simulation data and real data are shown. Material and methods: In this scheme, first we have introduced finite mixture of semi-parametric generalized linear models using the Poisson distributions (GFMMP). Then, we have introduced finite mixture of semi-parametric generalized linear models using the negative binomial (GFMMNB) instead of GFMMP. For estimating the parameters in the proposed model, the EM algorithm in two steps computed. We have used the efficiency method using weighted generalized mean of square error (WGMSE) for comparing between GFMMNB and GFMMP model in both the simulation and real data. Results and discussion: Results of real example and simulation study between GFMMNB and GFMMP model are shown that the proposed method is very competitive in terms of estimation accuracy and speed of computational estimation methods. The reported results demonstrate that there is a good agreement between simulation study and real data in the GFMMNB model. Also, the numerical results reported in the tables indicate that the accuracy improve by increasing the n for GFMMNB model. Therefore, to get more accurate results, the larger n is recommended. Conclusion: The following conclusions were drawn from this research. Computation of estimators for proposed model using the EM algorithm are found very easily and therefore many calculations are reduced. Confidence intervals for parameters in GFMMNB model is more accurate than GFMMP model.

Introduction: Many infectious diseases are endemic in a population. In other words they present for several years. Suppose that the population size is constant and the population is uniform. In the SIR model the population is divided into three disjoint classes which change with time t and let, and be the fractions of the population that susceptible, infectious and removed, respectively. This model formulated as the following system of nonlinear Volterra integral equation. Where, and are unknown functions and other constants and functions are known. The susceptibles are transferred at a rate equal to times the number of infectives, where is a constant. is the nonincreasing probablity function of remaining infectious units after becoming infectious, with and and is dominated by a decaying exponential, such as gamma distributed. Since the population size is constant, the birth rate must be equal to the death rate. The death rate is the same for susceptibles, infectives and removed individuals. The fraction of newborns are immunized so that the flow rate of immunized newborns into the removed class is. The initial susceptible and removed fractions be and and be the fraction of the population that was initially infectious and is still alive and infectious at time. Material and methods: We apply the Richardson extrapolation method for numerical solution of this model, so that the nonlinear system is solvable by an iterative process with a good accuracy. The algorithm of such systems completely described. This algorithm has a kind of nested structure, which cause we use the lag data in the future times, and it is the interesting section of programing of the algorithm. This algorithm is ready for programing with every program language, which we do this process by Mathematica programing software. Convergence and accuracy of the method is illustrated by either theoretical and numerical analysis, and some benchmark sample problems. For this aim by using Laplase transform, we sketch a spectrum of sample problems. These problems have analytical solution and appropriate for comparison with numerical solutions. Results and discussion: We solve some test examples by using present technique to demonstrate the efficiency, high accuracy and the simplicity of the present method. The main advantage of the method is the applicability of method for a large interval of time, as the algorithm shows. Numerical results shows the accuracy of the method for a long time interval. Conclusion: The following conclusions were drawn from this research. The proposed algorithm is very suitable for mathematical programing. Many cancer problems have such structure and the method is applicable for them. This method has two characteristics, solve a nonlinear problem and use of previous solution in new interval. So the method is applicable for various kind of problems with little additional works.

Introduction: Fractional differential equations (FDEs) have attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or numerical scheme may be a good approach, particularly, the schemes in numerical linear algebra for solving a system, , emerging by discretizing the partial derivatives, with large and sparse dimensions. In the procedure of solving a specified FDE, if the dimension of the corresponding system of linear equations is small, one can use the direct methods or the classical iterative methods for the analysis of these systems. However, if the dimension is large, then the proposed methods are not effective. In this case, we use variants of the Krylov subspace methods that are more robust with respect to the computer memory and time. The GMRES (Generalized Minimal Residual) is a well-known method based on Krylov subspace that is used to solve a system of sparse linear equations with an non-symmetric matrix. A main drawback of iterative methods is the slowness of convergence rate which depends on the condition number of the corresponding coefficient matrix. If the condition number of the coefficient matrix is small, then the rate of convergence will be faster. So, we try to convert the original system to another equivalent system, in which the condition number of its coefficient matrix becomes small. A preconditioner matrix is a matrix that performs this transformation. In this paper, we propose the iterative GMRES method, preconditioned GMRES method and examine capability of these methods by solving the space fractional advection-diffusion equation. Material and methods: We first introduce a space fractional advection-diffusion equation in the sense of the shifted Grü nwald-Letnikov fractional derivative. To improve the introduced numerical scheme, we discretize the partial derivatives of equation using the fractional Crank-Nicholson finite difference method. Then we use a preconditioner matrix and present preconditioned GMRES method for solving the derived linear system of algebraic equations. Results and discussion: In this paper, we use the GMRES and preconditioned GMRES to solve a linear system of equations emerging by discretizing partial derivatives appearing in a Advection-Diffusion equation and then asset the accuracy of these methods. Numerical results indicate that we derive a smaller condition number of the equivalent coefficient matrix for different values of M and N, as dimensions of the corresponding linear equations. Hence the convergence rate increases and consequently the number of iterations and the calculation time decreases. Conclusion: The following conclusions were drawn from this research. The GMRES method is a Kyrlov subspace methods to solve large-dimensions non-symmetric system of linear equations, which will be more effective when is applied with preconditioning techniques. One of the common ways to increase the rate of convergence of iterative methods based on the Kyrlov subspace is the applying the preconditioned techniques. An appropriate preconditioner matrix increases the rate of convergence of the iterative method.

Let L be a frame. We denoted the set of all regular ideals of cozL by rId(cozL). The aim of this paper is to study these ideals. For a frame L, we show that rId(cozL) is a compact completely regular frame and the map jc: rId(cozL)→ L given by jc (I)=⋁ I is a compactification of L which is isomorphism to its Stone– Č ech compactification and is proved that jc have a right adjoint rc: L → rId(cozL), given by rc(a)={x∈ cozL: x≺ ≺ a}. Moreover we identify prime and compact elements of rId(cozL) and we investigate the relation between regular ideals of cozL and P-frames. In addition it is shown that a frame L is a P-frame if and only if any ideal of cozL is regular.

Introduction: Let be a Noetherian ring with unity and be a regular ideal of, that is, contains a nonzerodivisor. Let. Then. The: union: of this family, , is an interesting ideal first studied by Ratliff and Rush in [15]. The Ratliff-Rush closure of is defined by. A regular ideal for which is called Ratliff-Rush ideal. The present paper, reviews some of the known properties, and compares properties of Ratliff-Rush closure of an ideal with its integral closure. We discuss some general properties of Ratliff-Rush ideals, consider the behaviour of the Ratliff-Rush property with respect to certain ideal and ring-theoretic operations, and try to indicate how one might determine whether a given ideal is Ratliff-Rush or not. For a proper regular ideal, we denote by the graded ring (or form ring) . All powers of are Ratliff-Rush ideals if and only if its positively graded ideal contains a nonzerodivisor. An ideal is called a reduction of if for some A reduction is called a minimal reduction of if it does not properly contain a reduction of. The least such is called the reduction number of with respect to , and denoted by. A regular ideal I is always a reduction of its associated Ratliff-Rush ideal The Hilbert-Samuel function of is the numerical function that measures the growth of the length of for all . This function, , is a polynomial in, for all large . Finally, in t he last section, we review some facts on Hilbert function of the Ratliff-Rush closure of an ideal. Ratliff and Rush [15, (2. 4)] prove that every nonzero ideal in a Dedekind domain is concerning a Ratliff-Rush ideal. They also [15, Remark 2. 5] express interest in classifying the Noetherian domains in which every nonzero ideal is a Ratliff-Rush ideal. This interest motivated the next sequence of results. A domain with this property has dimension at most one. Results and discussion: The present paper compares properties of Ratliff-Rush closure of an ideal with its integral closure. Furthermore, ideals in which their associated graded ring has positive depth, are introduced as ideals for which all its powers are Ratliff-Rush ideals. While stating that each regular ideal is always a reduction of its associated Ratliff-Rush ideal, it expresses the command for calculating the Rutliff-Rush closure of an ideal by its reduction. This fact that Hilbert polynomial of an ideal has the same Hilbert polynomial its Ratliff-Rush closure, is from our other results. Conclusion: T he Ratliff-Rush closure of ideals is a good operation with respect to many properties, it carries information about associated primes of powers of ideals, about zerodivisors in the associated graded ring, preserves the Hilbert function of zero-dimensional ideals, etc.

Introduction: Statistical analysis of the data on the Earth's surface was a favorite subject among many researchers. Such data can be related to animal's migration from a region to another position. Then, statistical modeling of their paths helps biological researchers to predict their movements and estimate the areas that are most likely to constitute the presence of the animals. From a geometrical view, spherical data are points that take their values on the surface of a unit sphere. There are many methods to fit a curve, especially regression curves to spherical data. For example, Gould [1] used the corresponding angles of spherical data coordinates to introduce regression models. He considered Fisher distribution as a candidate density for the error in his analysis. A non-parametric version of his model was proposed by Thompson and Clark [2]. Usually, the data that are close to the North or South pole have different behavior. Hence, their proposed model was failing to work there, and so they tried to keep the data somewhat away from the pole via adopting their model. They advocated overcoming this problem by using the tangent plane and suggested the use of splines there [3]. On the other hand, Fisher et al. [4] proposed two families of the spherical spline for spherical data. They introduced two families of curves using differential geometry suitable for fitting the splines. One of the methods to predict statistics is to utilize non-parametric regression models. Another strategy is to consider some forms of smooth models. Both of these procedures, along with other approaches in non-Euclidean statistics context are somewhat an initiative method in analyzing the spherical data. It worths mentioning that the benefits of using the spline path by employing the rotation parameters were of interest in directional statistics in [5], albeit for circular data. One of the interesting techniques to construct the non-parametric regression model was to minimize the Euclidean risk function, first proposed in [6]. We also follow the same procedure in this paper. In particular, the primary objective of this paper is to introduce a non-parametric regression model based on minimizing the mean square errors risk function for spherical data. To apply this idea, we used the suggested method in [6] for data on the circle. We initiate our model by considering two separate models for two common angles on the sphere. Then, we impose a correlation among these angles using an appropriate risk function. The proposed models will be evaluated using simulated and real-life data. Material and methods: In this paper, we presented two methods for modeling spherical data. The first one considers, separately, a regression model for each angle on the sphere. To construct a feasible model, a risk function is then suggested for modeling spherical data using Haversine distance. A non-parametric longitudinal model is derived by minimizing the proposed risk function. Hence, a parametric longitudinal model for spherical data, as the second method, is built. The estimates of the parameters in the latter model are done using the quadratic risk function. Results and discussion: Some of the data sets are intrinsically on the surface of a sphere in many scientific disciplines. For example, the location of quakes on earth can be considered a point with a constant norm on a unit sphere. Many researchers paid attention to construct a proper model to analyze such data. Regression models are among popular forms of treating spherical data, statistically. In this paper, we also attempted to provide an efficient model to analyze spherical data. To aim this, we first adopted a regression model for each angle on the sphere, independently. Our methods included two different approaches; a non-parametric longitudinal regression modeling and minimizing a least square error framework to construct a parametric longitudinal model. In the first method, the Haversine distanced, and its minimization were considered. The validity of this approach was studied using simulated and real-life data. Then, regression modeling was proposed using the least-square error approach with an appropriate link function. Although the efficiency of this latter method in comparison with the former was in doubt, it was able to provide a suitable smooth paths prediction on the sphere. Moreover, the proposed method was more appropriate while using Haversine distance. The idea to increase the efficiency of the current model is using other distances having a secure connection with the least square method suitable for spherical data. Conclusion: The following conclusions were drawn from this research. A non-parametric model inspired by previous models and a generalized version of it from circle to sphere was introduced. A risk function was proposed based on the Haversine distance on sphere. Two separated longitudinal models were suggested for the angles on the sphere and then a correlation was imposed using the least square risk function. Although the non-parametric method was more accurate in analyzing real data, the parametric method predicts more smooth paths.

Introduction: In some biological, environmental or ecological studies, there are situations in which obtaining exact measurements of sample units are much harder than ranking them in a set of small size without referring to their precise values. In these situations, ranked set sampling (RSS), proposed by McIntyre (1952), can be regarded as an alternative to the usual simple random sampling (SRS) to draw a more representative sample from the population of interest than what is possible in SRS. To draw a ranked set sample, one first draws n simple random samples, each of size n, from the population of interest and ranks them in an increasing magnitude. The ranking process is done without measuring sample units and therefore it need not to be accurate. One then identifies the ith sample unit from the ith sample for actual quantification (for i=1, … , n). Finally, he repeats this process m times (cycle) if he/she is required to obtain a sample of size mn. Since a ranked set sample contains information from both measured sample units and their corresponding ranks, one intuitively expects that statistical inference based on RSS to be more accurate than what is possible to obtain based on SRS. This paper is concerned with problem of estimating variance of the normal distribution in RSS. Several methods of estimation of variance of the normal distribution are described and compared via a Monte Carlo simulation study. Material and methods: All simulation studies in this paper have been done using R statistical software version R-3. 3. 1 Results and discussion: In this paper, we consider estimation of the normal variance based on a ranked set sample with single (multiple) cycle(s) and propose different unbiased estimators for each case. Our simulation results indicate that the mean square error (MSE) of each estimator is decreased as the values of n or m increases while the other parameters are kept fixed. It is also found that the estimator based on combining variance estimators of within and between ranking classes has typically better performance than the others. Conclusion: The following results can be obtained based on our simulation study: If there is a single cycle in RSS, then the proposed estimator in the case of single cycle beats Stokes-modified unbiased estimator. In the multiple cycle case in RSS, the estimator based on combining variance estimators of within and between ranking classes is the best one.

Introduction: Suppose that is a commutative ring with identity, is a unitary-module and is a multiplicatively closed subset of. Factorization theory in commutative rings, which has a long history, still gets the attention of many researchers. Although at first, the focus of this theory was factorization properties of elements in integral domains, in the late nineties the theory was generalized to commutative rings with zero-divisors and to modules. Also recently, the factorization properties of an element of a module with respect to a multiplicatively closed subset of the ring has been investigated. It has been shown that using these general views, one can derive new results and insights on the classic case of factorization theory in integral domains. An important and attractive question in this theory is understanding how factorization properties of a ring or a module behave under localization. In particular, Anderson, et al in 1992 showed that if is an integral domain and every principal ideal of contracts to a principal ideal of, then there are strong relations between factorization properties of and. In the same paper and also in another paper by Aḡ argü n, et al in 2001 the concepts of inert and weakly inert extensions of rings were introduced and the relation of factorization properties of and, under the assumption that is (weakly) inert, is studied. In this paper, we generalize the above concepts to modules and with respect to a multiplicatively closed subset. Then we utilize them to relate the factorization properties of and. Material and methods: We first recall the concepts of factorization theory in modules with respect to a multiplicatively closed subset of the ring. Then, we define multiplicatively closed subsets conserving cyclic submodules of and say that conserves cyclic submodules of, when the contraction of every cyclic submodule of to is a cyclic submodule. We present conditions on equivalent to conserving cyclic submodules of and study how factorization properties of is related to those of, when coserves cyclic submodules of Finally we present generalizations of inert and weakly inert extensions of rings to modules and investigate how factorization properties behave under localization with respect to, when is inert or weakly inert. Results and discussion: We show that if is an integral domain, is torsion-free and conserves cyclic submodules of, then splits (as defined by Nikseresht in 2018) and hence factorization properties of and those of are strongly related. Also we show that under certain conditions, the converse is also true, that is, if splits, then conserves cyclic submodules of. Suppose that is a multiplicatively closed subset of containing and. We show that if is a-weakly inert extension, then there is a strong relationship between-factorization properties of and-factorization properties of. For example, under the above assumptions, if is also torsion-free and has unique (or finite or bounded) factorization with respect to, then has the same property with respect to. Conclusion: In this paper, the concepts of a multiplicatively closed subset conserving cyclic submodules and inert and weakly inert extensions of modules are introduced and utilized to derive relations between factorization properties of a module and those of its localization. It is seen that many properties can be delivered from one to another when conserves cyclic submodules or when is a weakly inert extension, especially when is an integral domain and is torsion-free.