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Introduction A system of integral equations can describe different kind of problems in sciences and engineering. There are many different methods for numerical solution of linear and nonlinear system of integral equations. Material and methods This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods. Results and discussion For showing the accuracy and capability of the proposed method, some numerical examples are proposed that their results compared by results of other methods, and shows the capability and the superiority of this method to other existed methods. Also this paper derived the computational cost and the error analysis of the proposed method. Conclusion The following conclusions were drawn from this research. This paper proposed a numerical method based on modification of Hat functions for solving system of Fredholm-Hammerstein integral equations. The proposed method reduced a system of integral equation to a system of algebraic equations that can be solved easily by known methods. The presented error analysis and solved problems show capability and the superiority of this method to other existed methods../files/site1/files/52/1.pdf

In this paper we study the concept of Arveson spectrum on locally compact hypergroups and for an important class of compact countable hypergroups . In thiis paper we study the concept of Arveson spectrum on locally compact hypergroups and develop its basic properties for an important class of compact countable hypergroups ../files/site1/files/52/6.pdf

Introduction In this paper and by employing some certain techniques and results arisen in the theory of Tanaka, we provide a short proof for the maximum conjecture on the rigidity of Beloshapka's models of the specific CR dimension one and of length . As a consequence, we realize that the Lie group of biholomorphisms (CR automorphisms) associated with each of these models only consists of linear maps. Material and methods Two major approaches employed so far to study Beloshapka's maximum conjecture have been 1) the approach of envelope of holomorphy, applied by Ilya Kossovskii to confirm the mentioned conjecture in the specific lengths three and four and 2) Cartan's approach of solving (biholomorphic) equivalence problems, employed by the present author in CR dimension one. Both of these approaches are geometric. In this paper, we introduce and employ the algebraic approach of applying some techniques in Tanaka's theory of transitive prolongations to study the already mentioned conjecture in the specific CR dimension one. The proofs are based upon some results concerning Tanaka prolongation of rank two fundamental algebras of lengths , achieved by Medori and Nacinovich (1997). Here, we prove first an algebraic parallel version of Beloshapka's conjecture and employ the results to solve this geometric open problem in CR dimension one. Results and discussion As is the main goal of this paper, we confirm the maximum conjecture on the rigidity of Beloshapka's CR models in CR dimension one. As a consequence, we realize that each biholomorphic deformation of these models is linear. It may be worth to notice that the maximum conjecture in CR dimension one has been confirmed before by the present author, using the Cartan approach of solving equivalence problems. But, here we provide a much shorter proof for this result. Conclusion The following conclusions were drawn from this research. We confirm Beloshapka's maximum conjecture in CR dimension one. It is introduced a new approach to consider this conjecture. Comparing the performance of the Tanaka approach, employed in this paper, with two approaches of "envelope of holomorphy" and "Cartan's theory", this algebraic approach seems the best weapon, introduced so far, to attack the maximum conjecture. ./files/site1/files/52/3.pdf

Throughout this paper‎, (R, m) is a‎ commutative Noetherian local ring with the maximal ideal m. ‎The following conjecture proposed by Bass [1]‎, ‎has been‎ proved by Peskin and Szpiro [2] for almost all rings: ‎ (B) If R admits a finitely generated R-module of‎ finite injective dimension‎, ‎then R is Cohen-Macaulay. ‎The problems treated in this paper are closely related to the‎ following generalization of Bass conjecture which is still wide‎ open:‎ (GB) If R admits a finitely generated R-module of‎ finite Gorenstein-injective dimension‎, ‎then R is‎ Cohen-Macaulay. ‎ Our idea goes back to the first steps of the solution of Bass conjecture given by Levin and Vasconcelos in 1968 [3] when R admits a‎ finitely generated R-module of injective dimension 1‎. Levin and Vasconcelos indicate that if‎ is a non-zerodivisor‎, ‎then for‎ every finitely generated R/xR-module M‎, ‎there is‎ ‎. ‎Using this fact‎, ‎they construct a‎ finitely generated R-module of finite injective dimension in‎ the case where R is Cohen-Macaulay (the converse of Conjecture‎ B)‎. ‎ In this paper we study the Gorenstein injective dimension of local cohomology‎. ‎We also show that if R is Cohen-Macaulay‎ with minimal multiplicity‎, ‎then every finitely generated module‎ of finite Gorenstein injective dimension has finite injective‎ dimension.‎ ‎We prove that a Cohen-Macaulay local ring‎ has a finitely generated module of‎ finite Gorenstein injective dimension.‎ ./files/site1/files/52/4.pdf

Paper pages (237-252) Introduction ‎Frames for Hilbert spaces were first introduced by Duffin and‎ ‎Schaeffer in 1952 to study some problems in nonharmonic‎ ‎Fourier series‎, ‎reintroduced in 1986 by Daubechies‎, ‎Grossmann and‎ ‎Meyer‎. ‎Various generalizations of frames have been introduced and many applications of them in different branches have been presented‎. ‎‎Bessel multipliers in Hilbert spaces were introduced by Peter Balazs‎. ‎As we know in frame theory‎, ‎the composition of the synthesis and analysis operators of a frame is called the frame operator‎. ‎A multiplier for two Bessel sequences is an operator that combines the analysis operator‎, ‎a multiplication pattern with a fixed sequence‎, ‎called the symbol‎, ‎and the synthesis operator‎. ‎Bessel multipliers have useful applications‎, ‎for example they are used for solving approximation problems and they have applications as time-variant filters in acoustical signal processing‎‎. We mention that many generalizations of Bessel multipliers have been introduced, also multipliers have been studied for non-Bessel sequences. ‎Approximate duals in frame theory have important applications‎, ‎especially are used for the reconstruction of signals when it is difficult to find alternate duals‎. ‎Approximate duals are useful for wavelets‎, ‎Gabor systems and in sensor modeling‎. ‎Approximate duality of frames in Hilbert spaces was recently investigated by Christensen and Laugesen and some interesting applications of approximate duals were obtained‎. ‎For example‎, ‎it was shown that how approximate duals can be obtained via perturbation theory and some applications of approximate duals to Gabor frames especially Gabor frames generated by the Gaussian were presented‎. Afterwards, many authors studied approximate duals of Bessel sequences and many properties and generalizations of them were presented. In this note, we consider approximate duals for arbitrary sequences. Results and discussion In this paper, we introduce some new kinds of duals and approximate duals in Hilbert spaces using multipliers, invertible operators and symbols. Many papers about approximate duals and their applications have been written so far which in these papers approximate duals have been considered for Bessel sequences. Here, we introduce approximate duals for arbitrary sequences in a Hilbert space, compare them with Bessel approximate duals and we show that they can be useful for the reconstruction of signals though they do not have all of the properties of Bessel approximate duals. Moreover, we obtain some new results for Bessel approximate duals. Conclusion The following conclusions were drawn from this research. New kinds of duals and approximate duals for arbitrary sequences are introduced using multipliers, invertible operators and symbols. Duals and approximate duals of non-Bessel sequences are compared with the Bessel ones and some differences between them are shown by presenting various examples. Some properties and applications of duals and approximate duals of non-Bessel sequences are stated. Some new results about duals and approximate duals of Bessel sequences are obtained especially some important concepts such as closeness of Bessel sequences, nearly Parseval frames and multipliers with constant symbols are related to approximate duals of frames. ./files/site1/files/52/12.pdf

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 Grünwald-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../files/site1/files/51/%DA%86%D8%B1%D8%A7%D8%BA%DB%8C.pdf

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. ./files/site1/files/51/%D9%85%D9%87%D8%AF%D9%88%DB%8C_%D9%85%D9%86%D8%B4_%D8%A7%DB%8C%D8%B1%D8%A7%D9%86%D9%BE%D9%86%D8%A7%D9%87.pdf

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 ideal‎‎‎‎contains 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 ‎t‏‎he ‎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 T‎he 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. ./files/site1/files/51/%D9%85%D8%A7%D9%81%DB%8C.pdf

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. In this regard obtaining families of simplicial complexes with this property is of great interest. In this paper we present a new family of vertex decomposable simplicial complexes, which is associated to the t-clique ideal of the complement of path graphs. The t-clique ideal is a natural generalization of the concept of the edge ideal of a graph. For a graph G, a complete subgraph of G with t vertices is called a t-clique of G. The ideal generated by the monomials of degree t such that the induced subgraph of G on the set is a complete graph, is called the t-clique ideal of G. We consider the Stanley- Reisner simplicial complex of the ideal , where is a path graph of order n. For such a simplicial complex , we obtain the set of facets of and using this characterization we show that every such simplicial complex is vertex decomposable, whose shedding vertex is an endpoint of the path graph. Indeed, any simplicial complex in this family is Cohen-Macaulay, since it is pure. Since edge ideals of graphs are in fact 2-clique ideals, this family of simplicial complexes contains the independence complexes of complement of path graphs. Finally, as a consequence it is shown that the t-independence ideal of the complement of a path graph is vertex splittable and its Betti splitting is presented Material and methods To prove the vertex decomposability of , first we characterize the set of facets of . This helps us to find a shedding vertex for this simplicial complex and then by an inductive approach the vertex decomposability has been proved. Results and discussion For positive integers and , we show that a subset F of the vertex set of is a facet of if and only if and every component of the induced subgraph is a path graph of even order. Using this characterization, it is shown that any endpoint of the path graph is a shedding vertex of and is vertex decomposable. Moreover, it is proved that the ideal has a Betti splitting. Conclusion The following conclusions were drawn from this research. A characterization for the set of facets of the simplicial complex is presented. The simplicial complex is vertex decomposable for any positive integers and . The ideal has a Betti splitting for any any positive integers and ../files/site1/files/51/%D9%85%D8%B1%D8%A7%D8%AF%DB%8C.pdf

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../files/site1/files/51/%D9%85%D9%82%DB%8C%D9%85_%D8%A8%DB%8C%DA%AF%DB%8C.pdf

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 Aḡargün, 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. ./files/site1/files/51/%D9%86%DB%8C%DA%A9_%D8%B3%D8%B1%D8%B4%D8%AA.pdf

Introduction 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 O (n5) while the execution order of the best available algorithms is O (n17 poly-logn). Another advantage of this algorithm is the independence of restrictions on simple orthogonal polygons. Material and methods Our processing is based on orthogonal or right-angled polygons, so we only have four types of edges. The northern edge, or N, the southern edge or S, the eastern edge or E, the western edge or W-edge. If the interior of polygon lies below/above/left/right of an edge, respectively, then that edge is called N/S/E/W, respectively. There are also four types of dents, including North Dent (N-dent), Southern Dent (S-dent), Eastern Dent (E-dent) and Western Dent (W-dent); see Figure 1. Figure 1. A simple orthogonal polygon with a variety of edges and dents In our proposed algorithm, after rectangles, new edges and vertices are created in the orthogonal polygon P, which we call the sub-edges and sub-vertices. Sub-vertices are known as horizontal and vertical edges, and are formed from the intersection of the adjacent edges with the main or minor edges. The orthogonal polygons can be classified into class-k, such that 1≤k≤4, according to the type of dents which is defined so that its intersections are at most k different directions. Class-2 polygons can be re-categorized into classes 2a, so that 2 directions are parallel to the dents (i.e., N and S or E and W) and Class 2b so that 2 directions are perpendicular to each other. The field of view of the guards or a rectangular field of view is defined as r-visibility. In each orthogonal polygon P, two opposite-corner situated points p and q are called r-visible to each other if the entire rectangle containing them is within P. Now we consider 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 the r-Star problem, we assume that each guard has a 360-degree field of view. Results and discussion In this paper, a new idea is proposed based on the rectangular orthogonal polygon, which creates new edges. In fact, the original orthogonal polygon P with a rectangular method will be divided into a number of rectangles, so that their geometric communities are exactly the same as P. Then the edges are placed in the two groups of main and minor edges. Obviously, the shared edges between the two rectangles are minor and will not prevent the guard from seeing. Therefore, the primary polygon is decomposed into rectangles and then, according to the orthogonal polygon partition method, P is divided into two or more orthogonal polygons, if possible. We make each one of them an r-Star. Then set the deadlock rectangles and set up an incremental guard in each range for each of them. In the next step, we start from the top r-Star with a sweep-line method and check the inserted guards and, if necessary, r-stars and guard rectangles are requested by our Rectangular Guards method. We will continue to work and expand the first r-Star as far as possible, and will do so for the rest of the r-stars, so that the minimum number of guards or let's say the least number of r-stars are obtained. In this article, we are actually looking for the minimum number of r-stars. The steps of our novel algorithm are as follows: 1. Partitioning into rectangles 2. Partitioning the output of Step 1 into orthogonal r-Stars. 3. Determining the guards for the output of Step 2. 4. Scrolling among the r-stars and setting the minimum number of guards 5. Allocating guards for parent, child and minor r-Stars. Conclusion We considered the problem of covering simple orthogonal polygons with no restrictions on the type and number of dents. This algorithm is capable of Partitioning into smaller rectangles, each of which is an r-star polygon. Updating the range of rectangular guards into the registered guards (in order of their priority). Reaching the minimum number of guards. In this paper, the idea was presented on simple orthogonal polygons that do not have a hole. This paper also paves the way for working on other types of polygons that have cavities. In fact, holes can be seen as columns in the gallery, which can also be considered as simple orthogonal rectangles or polygons. The problem here is that these rectangles are different with the rectangles in the polygon that are presented in this article and prevent the guards from seeing../files/site1/files/42/1Abstract.pdf

Introduction Longitudinal study designs are common in a lot of scientific researches, especially in medical, social and economic sciences. The reason is that longitudinal studies allow researchers to measure changes of each individual over time and often have higher statistical power than cross-sectional studies. Choosing an appropriate sample size is a crucial step in a successful study. A study with insufficient sample size may have a small statistical power to detect significant effects and may lead to incorrect answers to many important research questions. On the other hand, a study with more than what a sample size should be wastes the resources and budget. In longitudinal studies, because of the complexity of the design of experiment, including the selection of the number of individuals and the number of repeated measurements, the sample size determination is less studied. This paper uses a simulation-based method to determine the sample size from a Bayesian perspective. For this purpose several Bayesian criteria of sample size determination for a longitudinal study using marginal model are used. Most of the methods of determining the sample size are based on creation of one hypothesis. In this paper, in addition to using this method, we also present a method to determine the sample size for multiple hypothesis testing. Using several examples the proposed Bayesian methods are discussed. Material and method Based on the Bayesian sample size determination approach proposed by Wang and Gelfand (2002), which is a simulation-based approach, two sets of priors are considered. The first set is called the “fitting” or “analysis” priors which are used for analyzing data. The other set of priors is called “design” or “sampling” sets which drawing upon expertise, we may speculate upon a variety of informative scenarios regarding the unknown parameters and capture each with a suitable sampling prior. In the first step of the simulation-based approach, one generates the parameters from the design priors then a sample from the generated parameters is obtained. Note that the frequentist/classical viewpoint of the sample size determination approach require a point estimate of the variance and the smallest meaningful difference (Diggle et al., 2002). This information is elicited based on pilot data or the opinion of experts. The use of design priors in the proposed approach of Wang and Gelfand (2002) replaces this part of the classical sample size determination approach. After generating data using the design priors in this stage, they are analyzed by the fitting priors. In this paper, non-informative priors are used as fitting priors, as we assume there is no prior information or any expert opinion to be used to construct a prior distribution. If there is any prior information to be used to elicit an appropriate prior distribution, one may use it in the step of fitting prior and for sure, if the specification of the prior is correct, will achieve a better result. Also, in this paper, four criteria are used for determining sample size in longitudinal studies: Bayesian power criterion (BPC), average length criterion (ALC), average posterior variance criterion (APVC) and average converge criterion (ACC). The simulation-based approach of Wang and Gelfand (2002) can be summarized as follows: Specify the value of the effect of an important regression coefficient which is the interest parameter and specify the design and fitting priors. For each sample size, the following steps are repeated M times: Generate values of the unknown parameters from their design priors. Simulate the values of covariates from continuous or discrete distributions and the response variable from its distribution. Analyze the generated data set of step (ii) using the fitting priors. Calculate BPC. Fit a curve or surface through Bayesian power values and find an adequate sample size for a desired power using interpolation. In this paper, the curve is fitted using a polynomial regression. In step 2(iv), one can calculate ACC, APVC, or ALC and determine the sample size based on these criteria. References Diggle, P., Heagerty, P., Liang, K. and Zeger, S. (2002). Analysis of Longitudinal Data. Oxford:Oxford University Press. Wang, F., and Gelfand, A. E. (2002). A simulation-based approach to Bayesian sample size determination for performance under a given model and for separating models. Statistical Science, 193-208../files/site1/files/41/1Extended_Abstract.pdf

This paper aims to introduce the original ideas of normal form theory and bifurcation analysis and control of small amplitude limit cycles in a non-technical terms so that it would be comprehensible to wide ranges of Persian speaking engineers and physicists. The history of normal form goes back to more than one hundreds ago, that is to the original ideas coming from Henry Poincare. This tool plays an important role in the bifurcation analysis of dynamical systems. Many phenomena in chemistry, physics, and engineering are modeled by parametric nonlinear differential systems .These systems demonstrate a complicated dynamics, when the parameters reach singular values. Normal form theory is an efficient tool for the local bifurcation analysis of singular nonlinear‌ systems. The main idea relies in using a nonlinear change of coordinates to convert a given vector field into a simple form, which is named normal form. In most nonlinear systems, behavior and dynamics is involved with local appearance or disappearance of small amplitude limit cycles. These are called bifurcations of limit cycles and they mostly occur through Hopf or degenerate Hopf bifurcations. Center manifolds and normal form theory are considered as among the most powerful tools for the bifurcations and stability analysis of limit cycles. Indeed, the analysis requires the computation of Lyapunov coefficients. Here we present the first (well-known) and the second Lyapunov coefficients for Hopf and degenerate bifurcations. Lyapunov coefficients In this paper, we discuss normal form of systems whose linear part has a pair of imaginary eigenvalues (usually referred by Hopf singularity) and compute the first and second Lyapunov coefficients. In order to use the bifurcation analysis of a Hopf singularity for bifurcation control purposes, one needs to be able to derive the parametric normal forms of the original system so that the bifurcation analysis for the parametric normal form system would give rise to a bifurcation analysis in terms of the parameters of the original parametric system. We have designed a Maple program for this purpose. Here we merely present the Lyapunov coefficients. The program computes Lyapunov coefficients for systems with parameters and symbolic coefficients. Application Computation of Lyapunov coefficients is not important only in theatrical aspects but also in real world applications. So in last section of the Persian language paper, we apply the method on a well-known system, called Linard system, and a non-linear electrical system in order to illustrate how these tools can be applied in applications. We detect Hopf singularity for these two application models and then compute their normal form up to degree 3. ./files/site1/files/41/6Extended_Abstract.pdf

Historically, various methods were suggested for the estimation of Bernoulli and Binomial distributions parameter. One of the suggested methods is the Bayesian method, which is based on employing prior distribution. Their sound selection on parameter space play a crucial role in reducing posterior Bayesian estimator error. At times, large scale of the parametric changes on parameter space brings about an increase in error rate and enlargement of the comparison criteria .therefore, determining appropriate prior distribution on parameter space plays a key part in reducing comparison criteria. Accordingly, in this paper appropriately modified prior distribution was considered for Binomial distribution parameter and then, while extending certain conditions on prior distribution hyperparameters, an effective estimate under the title Expected Bayesian Estimate (AKA E-Bayes) would be proposed. At last, to evaluate the measurement methodologies utilized in this paper, under MSE criteria , extensive simulation studies were conducted, and the results were analyzed and above mentioned methods will be applied in a real example.

In this paper, we analyze the eighth grade students performance in Fars province in cognitive and content domains in mathematics by TIMSS 2019 assessment results. Due to the cultural, social and economic proximity of Fars and Isfahan provinces, corresponding results have been obtained for students of Isfahan province and comparisons have been made in each domain by using appropriate tests. Also, gender analysis has been derived in both cognitive and content dimensions and in all of their domains in mathematics for students of both provinces

Suppose $G$ is a finite non-abelian group and $\Gamma(G)$ is a simple graph with the non-central conjugacy classes of $G$ as its vertex set. Two different non-central conjugacy classes $A$ and $B$ are assumed to be adjacent if and only if there are elements $a,b \in G$ such that $a \in A$, $b \in B$ and $ab = ba$. This graph is called the commuting conjugacy class graph of G. In this paper, the structure of the commuting conjugacy class graph of the generalized dihedral group $D_{(m,n)}$ and the generalized dicyclic group $Dic (A, y, x)$ are completely determined.

There are many topics in graph theory that can be called ``moving objects around a graph". For example; In network optimization, shipments are transferred from some vertices (resources) to other vertices (demand) according to the costs allocated to the edges, so that this can be done in the cheapest way. A pebble motion in a graph involves removing two pebbles from the vertex of a graph and then placing a pebble at the adjacent vertex. If a distribution (or configuration) of pebbles allows us to move at least one pebble to each vertex by repeatedly applying pebble movements, then that distribution is called a pebble of the graph. One of the most basic questions is how many pebbles are needed to ensure that any configuration with this number can target a pebble at any particular target. The smallest number of stones that meet this condition is called the graph pebble number. In this paper, after examining the roots of the number theory of the pebble graph model, which in turn is a productive subject, we will study the pebble number for specific graphs and also consider an optimization approach to this subject called weight functions.

The mann fixed point algorithm play an importmant role in the approximation of fixed points of nonexpansive operators. In this paper, by considering new conditions, we prove the weak convergence of mann fixed point algorithm, for finding a common fixed point of two nonexpansive mappings in real Hilbert spaces. This results extend the privious results given by Kanzow and Shehu. Finally, we give an application of our results, by using the John von Neumann's method.

The Aluffi algebra is an algebraic version of characteristic cycles in intersection theory which is an intermediate graded algebra between the symmetric algebra (naive blowup) and the Rees algebra (blowup). Let R be a commutative Noetherian ring and J ⊂I ideals of R. We say that J ⊂I satisfy linearity condition if the Aluffi algebra of I/J is isomorphic with the symmetric algebra. In this paper, we present the necessary and sufficient conditions for satisfying linearity condition. We give classes of ideals J ⊂I which satisfy linearity condition. We prove that if J is a complete intersection ideal contained in I, then J ⊂I satisfy linearity condition if and only if the ideal I is of linear type.

In this paper, we are intend to present a numerical algorithm for computing approximate solution of linear and nonlinear Fredholm, Volterra and Fredholm-Volterra integro-differential equations. The approximated solution is written in terms of fractional Jacobi polynomials. In this way, firstly we define Riemann-Liouville fractional operational matrix of fractional order Jacobi polynomials, then by using this matrix and the least squares method the solution of equation reduce to a system of algebraic equations which is solved through the Newton’s iterative method. In the next step we analyze convergence of the solution, and then to confirm the theoretical issue we examine some numerical examples. The results indicate the accuracy and efficiency of the method. The excellence of this method is its generality, which includes the fractional order Legendre and Chebyshev polynomials. Also it is also easy to use for linear and nonlinear integro-differential equations and provides good results.

In this paper, based on variational methods and critical point theory, we guarantee the existence of infinitely many classical solutions for a two-point boundary value problem with fourth-order Sturm-Liouville equation; Some recent results are improved and by presenting one example, we ensure the applicability of our results.

Invariance principles is one of the ways to summarize sample information and by these principles invariance or equivariance decision rules are used. In this paper, first, the methods for finding the maximal invariant function are introduced. As a new method, maximal invariant statistics are constructed using equivariant functions. Then, using several equivariant functions, the maximal invariant statistic is obtained. If the group acts uniquely transitively, the equivariant functions will be converted into estimators that can be used in statistical branches. Finally, it is shown that in the case that the group action is not uniquely transitive but a normal subgroup has this property, the maximal invariant statistic can also be calculated.

Tries are the most popular data structure on strings. We can construct d-ary tries by using strings over an alphabet leading to d-ary tries. Throughout the paper we assume that strings stored in trie are generated by an appropriate memory less source. In this paper, with a special combinatorial approach we extend their analysis for average profiles to d-ary tries. We use this combinatorial approach for studying of average profile, since its probability distribution is unknown. We obtain the probability distribution of depth and the distribution function of height as n is large. These results follow from the study of certain recurrence equations that we solve by a analytic method.

In this paper, a fractional order model is presented to study the effect of the drug therapy on the control of HIV infection. For this purpose, we calculate the reproduction number of the model with the next generation operator method and by using that, we study the stability of equilibrium point. Reproduction number $R_0$ plays an important role in spreading or non-spreading of HIV. So that, if $R_0 < 1$, the infection-free equilibrium point is locally asymptotically stable and the infection is cleared from the $CD{4^ +}T$-cell population. In the res, we analyze the relationship between the reproduction number and the therapy parameter values, for some different values of the order of model. Also, we evaluate the effect of the therapy parameter on the immune system cells and HIV virus population. Finally, we simulate the variations of one of the parameters that show the amount of virus production by actively infected cells. Adams predictor-corrector method will be used to solve the fractional order model.

Let $R$ be a commutative ring, Γ be a group, and Γ′ be a subgroup of finite index. In this paper, we study the class of Gorenstein flat (resp. Gorenstein cotorsion) modules over the group ring RΓ. In particular, we discuss the relationship between Gorenstein flat (resp. Gorenstein cotorsion) dimension of RΓ−modules and that of RΓ′−modules. We will prove that, for any RΓ−module $M$ of finite Gorenstein flat dimension (∞>GfdRΓM), there is an inequality GfdRΓM ⩽ GfdRM + hdRΓ, where hdRΓ denotes the homological dimension of Γ over $R$ . Analogously, we prove that if Γ is finite, there is an inequality GctdRΓM ⩽ GctdRM + cdRΓ, where GctdRΓM (GctdRM) is the Gorenstein cotorsion dimension of module $M$ over RΓ (resp. over $R$) and cdRΓ is the cohomological dimension of $Γ over $R$.

Wavelet analysis is one of the useful techniques in mathematics which is used much in statistics science recently. In this paper, in addition to introduce the wavelet transformation, the wavelet threshold estimation of semiparametric regression model with correlated errors with having Gaussian distribution is determined and the convergence ratio of estimator computed. To evaluate the wavelet threshold estimation, the block function and sinusoidal function are used as objective functions and using the simulation method the average of mean square error and standard deviation of this estimator are compared with the average of mean square error and standard deviation of kernel method. Also, the wavelet semiparametric regression model has been fitted to data on the growth rate of the teeth.

The fractional calculus deals with the generalization of integration and differentiation of integer order to those ones of any order. The q-fractional differential equation usually describe the physical process imposed on the time scale set Tq. In this paper, we first propose a difference formula for discretizing the fractional q-derivative of Caputo type with order and scale index . We establish a rigorous truncation error boundness and prove that this difference formula is unconditionally stable. Then, we consider the difference method for solving the initial problem of q-fractional differential equation: . We prove the unique existence and stability of the difference solution and give the convergence analysis. Numerical experiments show the effectiveness and high accuracy of the proposed difference method.

In this paper we, obtain the weight of radial basis finite difference formula for some differential operators. These weights are used to obtain the local truncation error in powers of the inter-node distance and the shape parameter of radial basis functions. We show that for each difference formula, there is a value of the shape parameter for which RBF-FD formulas are more accurate than the corresponding standard FD formulas. We apply these formulas for Poisson equation with irregular domains and show that the proposed method can be used as a fully meshfree methods.

A Fibonacci string of length $n$ is a binary string $b = b_1b_2ldots b_n$ in which for every $1 leq i < n$, $b_icdot b_{i+1} = 0$. In other words, a Fibonacci string is a binary string without 11 as a substring. Similarly, a Lucas string is a Fibonacci string $b_1b_2ldots b_n$ that $b_1cdot b_n = 0$. For a natural number $ngeq1$, a Fibonacci cube of dimension $n$ is denoted by $Gamma_n$ and is defined as a graph whose vertices are Fibonacci strings of length $n$ such that two vertices $b_1b_2ldots b_n$ and $b'_1b'_2ldots b'_n$ are adjacent if $b_ineq b'_i$ holds for exactly one $iin{1,ldots, n}$. A Lucas cube of dimension $n$, $Lambda_n$, is a subgraph of $Gamma_n$ induced by the Lucas strings of length $n$. Let $G=(V,E)$ be a simple undirected graph. A perfect code is a subset $C$ of $V$ in such a way that for every $vin C$, the sets ${uin V | d(u, v) = 1}$ are pairwise disjoint and make a partition for $V$. In other words, each vertex of $G$ is either in $C$ or is adjacent to exactly one of the elements of $C$. It is proved that Fibonacci cube $Gamma_n$, admits a perfect code if and only if $nleq3$. In this paper, we prove the same result for Lucas cubes i.e, $Lambda_n$ admits a perfect code if and only if $nleq3$.

In this paper, subcategories with threshold c are examined from an active general fuzzy automaton and the generator from an active general fuzzy automaton is defined and the connections between them are examined. Then, under the main and maximum automata with threshold c, one of the active general fuzzy automata will be defined, and it will be proved that each active general fuzzy automaton can be written socially from below the distinct primary and maximum automata with threshold c.

In this paper, the robust vertex centdian location problem with uncertain vertex weights on general graphs is studied. The used criterion to solve the problem is the min-max regret criterion. This problem is investigated with objective function contains $lambda$ and a polynomial time algorithm for the problem is presented. It is shown that the vertex centdian problem on general graphs is solved in cubic time.

In this paper, first, the concept of the fuzzy semi maximal filters in BL-algebras has been introduced, then several properties of fuzzy semi maximal filters have been studied and its relationship with other fuzzy filters including fuzzy (positive)implicative fuzzy filters and fuzzy fantastic filters are investigated. In the following, using the level subsets of fuzzy sets in a BL-algebra, fuzzy semi maximal filters have been studied. It has also been proved that a fuzzy filter is a fuzzy semi maximal filters if and only if all level subsets are semi maximal filters filter. In addition, it proves that the preimage of a semi maximal fuzzy filter under the BL-homomorphism and the image of a fuzzy semi maximal filter under the BL-isomorphism , is also a fuzzy semi maximal filter. Also, the Extension property has been proved for fuzzy semi maximal filters in BL-algebra. Finally, the relationships between the fuzzy semi maximal filters and semi simple BL-algebras have been studied.

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

The goal of this paper is to determine the best-fit probability distribution to describe the annual maximum daily rainfall for the Isfahan (period 1951-2010) and Kashan (period 1966-2010) stations located distantly in Isfahan province as an arid geographical region of Iran. The Gamma, Lognormal, Weibull, Pearson type III, inverse Gumbel, inverse Gamma, inverse Normal, and Generalized Extreme Value distributions are fitted for these purposes. Parameters of these distributions were estimated by the method of maximum likelihood. The performances of the distributions are evaluated using the Kolmogorov-Smirnov goodness-of-test. Finally, the inverse Normal distribution and Lognormal distribution were empirically proved to be the most appropriate distribution of the annual maximum daily rainfall for Isfahan and Kasha stations, respectively using the Akaike information criterion

One of the key principles in any scientific-research activities is that of coordination and consistency of theory and related data in the corresponding scientific discipline. This paper demonstrates the power of theory-oriented models (compared to statistical models) in testing and analyzing a scientific hypothesis for the study of social systems from a sociological perspective and shows that (not the methodological aspect but) the theoretical aspect of scientific activities represents the value and the validity of mathematical research in sociology. The evaluation of the existing theoretical models can be considered as an important and key step in providing a unified mathematical approach to the study and analysis of the complex behavior of social systems with common characteristics. As an objective example of social phenomena, we study the process of opinions formation in a social system.

Introduction In recent years, the use of the Internet has been increasing. Attractions and many educational and recreational applications, etc. have caused the emergence of a new phenomenon called Internet addiction. Internet addiction is a new and interesting topic that arises from social changes including the cellularization of society and the family. Poor family support of their members, poor social skills, and easy access to Internet has contributed to this addiction. Today, there are Internet addiction treatment clinics around the world. In addition to the treatment, the person in these clinics receives the necessary training to use the Internet properly. The effects of Internet addiction are similar to other addictions. These effects include mental disorders, euphoria and desire to consume more, etc. Because of the great similarities between Internet addiction and other addictions, in this article we have developed the epidemic model of heroin addiction in White and Comiskey [1]. Material and methods In the proposed model, we have studied the effect of education and prevention. To examine this issue, we have considered five classes: 1- Susceptible individuals, 2- Internet users without training, 3- Educated Internet users and fully aware of the harms of the Internet addiction, 4- Internet addicts 5- Addicts under treatment and education. We examine the dynamic behavior of the model such as backward bifurcation, local and global stability equilibrium points. We investigate the boundary and obtain the basic reproduction number of the system. We study the existence of endemic equilibrium points and, using the Chavez-Sang theorem, show that the backward bifurcation occurs. We obtain sufficient conditions for the global stability, the addiction-free equilibrium point, and the endemic equilibrium point using the Lyapunov function and the geometric stability method. Results and discussion The epidemic model of heroin addiction was first introduced by White and Comiskey. In this article, we have developed this model. First, we have considered the age group of 15 to 65 years old as people who are inclined to use the Internet. Then some of these people become Internet users and a group of these people may become addicted to the Internet due to the excessive use of the Internet. At a certain rate, these addicts are treated and educated. On the one hand, Internet users may not become addicted and receive the necessary training and become professional Internet users. Conclusion In this paper, we have conducted a complete qualitative study of the model including the existence and evaluation of local and global stability of the equilibrium points of the model. We have shown that under certain conditions the addiction-free equilibrium point is local and global asymptotical stable. Using the compound matrix, we have obtained the conditions for the global stability of the endemic equilibrium point. We have shown the occurrence of backward bifurcation. This bifurcation indicates that when, addiction remains in the society. To control addiction, we need to get. Reduce to less than.

Introduction Suppose E is a Banach lattice. A net (xα) in E is said to be unbounded absolute weak convergent (uaw-convergent, for short) to x∈E provided that the net (xα-x˄u) convergences to zero, weakly, whenever u∈E+. In this note, we further investigate unbounded absolute weak convergence in E. We show that this convergence is stable under passing to and from ideals and sublattices. Compatible with un-convergenc, we show that uaw-convergence is topological, which means that E with uaw-topology forms a topological vector space. We consider some closedness properties for this type of convergence. Some examples are given to make the context more understandable. Finally, we introduce the notion of strongly continuous operators between Banach lattices and investigate some properties about them. Specially, we characterize Banach lattices with a strong unit in tems of this type of operators. Material and methods In this paper, we combine the order structure and the norm structure in a Banach lattice to consider the unbounded convergences in the category of all Banach lattices. Results and discussion We shall show the following main results. 1. The uaw-convergence in a Banach lattice is topological. 2. In an order continuous Banach lattice, uaw-convergence is stable under passing to and from sublattices and ideals. 3. Introduce strongly continuous operators between Banach lattices and investigate some properties of them. Conclusion The following main conclusions were drawn from this research. Theorem 2. Theorem 4. Proposition 10. Theorem 11.

Introduction Regression analysis is a common method for modeling relationships between variables. Usually Ordinary Least Squares method is applied to estimate regression model parameters. These estimators are unbiased and appropriate when design matrix is nonsingular. In presence of multicollinearity, design matrix is singular and Ordinary Least Squares estimates cannot be obtained. In this situation, other methods‎, ‎such as Lasso‎, ‎Ridge‎ and ‎Lars may be considered. Other hand, in many fields such as medicine, number of variables is greater than the number of observations‎ and usual methods such as Ordinary Least Squares are not proper and shrinkage methods‎, ‎such as Lasso‎, ‎Ridge‎ ‎and‎ ... ‎have a better performance to estimate regression model coefficients‎. ‎In the shrinkage methods‎, ‎tuning parameter plays an essential role in selecting variables and estimating parameters‎. ‎Bridge shrinkage estimators is an estimator that can be obtained by changing its tuning parameter‎. In other words, Bridge method is the extension of Ridge and Lasso regression methods. Selecting the appropriate amount of tuning parameter is important. There are many studies ‎on each of these methods under the assumed conditions. In this paper‎, performance of Bridge shrinkage estimators‎, ‎such as Lasso and Ridge‎ ‎are compared with Lars and Ordinary Least Squares estimators in a simulation study. Material and Methods A simulation study is applied to compare performance of the regression methods Ridge, Lasso, Lars and Ordinary Least Squares. MSE criterion is applied to evaluate the method performance. Statistical software R is applied for simulation and comparing the regression methods. Results and discussion In the presence of collinearity, Bridge regression estimators will result in appropriate estimators. These estimators are biased but their performance is better than unbiased estimators such as Ordinary Least Squares. Indeed, Bridge estimators have the best performance in the class of biased estimators. Conclusion In this article, Ridge and Lasso estimators as special cases of Bridge estimators are compared with Lasso and Ordinary Least Squares in a simulation study. This study shows that under the supposed conditions, Ridge, Lasso and Lars have better action than Ordinary Least Squares method. Lars method has the best performance and Ridge estimators is better than Lasso Regression.

Introduction Estimating the spatial hazard, or in other words, the probability of exceeding a certain boundary is one of the important issues in environmental studies that are used to control the level of pollution and prevent damage from natural disasters. Risk zoning provides useful information to decision-makers; For example, in areas where spatial hazards are high, zoning is used to design preventive policies to avoid adverse effects on the environment or harm to humans. Generally, the common spatial risk estimating methods are for stationary random fields. In addition, a parametric form is usually considered for the distribution and variogram of the random field. Whereas in practice, sometimes these assumptions are not realistic. For an example of these methods, we can point to the Indicator kriging, Disjunctive kriging, Geostatistical Markov Chain, and simple kriging. In practice utilize the parametric spatial models caused unreliable results. In this paper, we use a nonparametric spatial model to estimate the unconditional probability or spatial risk: rcs0=PZs0⩾c. (1) Because the conditional distribution at points close to the observations has less variability than the unconditional probability, nonparametric spatial methods will be used to estimate the unconditional probability. Material and methods Let Z=Zs1,…,ZsnT be an observation vector from the random field {Zs;s∈D⊆Rd} which is decomposed as follows Zs=μs+εs, (2) where μ(s) is the trend and ε(s) is the error term, that is a second-order stationary random field with zero mean and covariogram Ch=Covεs,εs+h. The local linear model for the trend is given by μHs= e1TSsTWsSs-1 SsTWsZ≡ ϕTsZ, where e1 is a vector with 1 in the first entry and all other entries 0, Ss is a matrix with ith row equal to (1, (si-s)T), Ws = diag {KHs1 – s,…,KH(sn-s)}, KHu=H-1K(H-1u), K is a triple multiplicative multivariate kernel function and H is a nonsingular symmetric d×d bandwidth matrix. In this model, the bandwidth matrix obtained from a bias corrected and estimated generalized cross-validation (CGCV). From nonparametric residuals ε(s) = Z(s) -μ(s) a local linear estimate of the variogram 2 γ(⋅)is obtained as the solution of the following least-squares problem minα.βinεi-εj2-α-βT si-sj-u2 KGsi-sj-u, where G is the corresponding bandwidth matrix, that obtained from minimizing cross-validation relative squared error of semi-variogram estimate. Algorithm1: Semiparametric Bootstrap Obtain estimates of the error covariance and nonparametric residuals covariance. Generate bootstrap samples with the estimated spatial trend μHs and adding bootstrap errors generated as a spatially correlated set of errors. Compute the kriging prediction Z*s0 at each unsampled location s0 from the bootstrap sample Z*s1,…,Z*sn. Repeat steps 2 and 3 a larger number times B. Therefore, for each un-sampled location s0, B bootstrap replications Z*(1)s0.…. Z*(B)(s0) are obtained. Calculate (1) at position s_0 by calculating the relative frequency of Bootstrap repetition as follows to estimate the unconditional probability of excess of boundary c. rcs0= 1B j=1BIZ*js0≥ c Results and discussion To analysis the practical behavior of the proposed methods a simulation study is conducted under different scenarios. For N=150 samples and n=16×16 were generated on a regular grid in the unit square following model (2), with mean function μs=2.5 + sin( 2π x1) + 4x2 - 0.5 2, and random errors normally distributed with zero mean and isotropic exponential covariogram Ch= 0.04 + 2.01 1- exp-3 ∥ h∥0.5, h∈ R2. For comparing nonparametric spatial methods for estimate unconditional risk, conditional risk, and Indicator kriging, we considered 7 missing observations in certain situations. Empirical spatial risk and its estimates are presented in Table 1. The Indicator kriging is overestimating and estimate spatial risk larger than 1. Generally, an estimated risk with unconditional and conditional methods is near value to empirical value. Table 1. Empirical spatial risk and its estimates Locations (0.13, 0) (0.87, 0.87) (0.80, 0.20) (0.94, 0.27) (0, 0.47) (074, .60) (0.34, 0.60) Methods 0.999 0.300 0.069 0.317 0.504 0.011 0.989 Empirical 0.998 0.351 0.054 0.347 0.494 0.057 0.954 Conditional 1.002 0.230 0.091 0.091 0.652 0.006 0.996 Indicator 1.000 0.388 0.418 0.481 0.602 0.024 0.994 Unconditional The spatial risk mapping for the maximum temperature means of Iran in 364 stations in March 2018 is obtained. By applying Algorithm 1 final trend and semi-variogram estimates are smoother than the pilot version. The conditional and unconditional spatial risk with 150 bootstrap replicates for two values of threshold 25 and 31 on a 75×75 grid are estimated. The unconditional risk estimate is smoother than the conditional risk estimate. Because of this in the unconditional version, biased residual unused directly in the spatial prediction but in the conditional risk estimating, original residuals and simple kriging used. Conclusion The spatial risk estimated with the nonparametric spatial method. For the trend and variability of the random field, modeling applied a local linear nonparametric model. In the simulation study, this method better results than Indicator kriging. Because the flexibility of the nonparametric spatial method could apply for the construction of confidence or prediction intervals and hypothesis testing.

Introduction Since linear and multilinear algebra has many applications in different branches of sciences, the attention of many mathematicians has been attracted to it in recent decades. The determinant and the permanent are the most important functions in linear algebra and so a generalized matrix function, which is a generalization of the determinant and the permanent, becomes significant. Generalized matrix functions connect some branches of mathematics such as theory of finite groups, representation theory of groups, graph theory and combinatorics, and linear and multilinear algebra. Let be the symmetric group of degree , be a subgroup of , and be a function. The function given by is called the generalized matrix function associated to and . Note that if and , the principal character of , then is the permanent and if and , the alternating character of , then is the determinant. Results In this paper, using permutation matrices or symmetric matrices, necessary and sufficient conditions are given for a generalized matrix function to be the determinant or the permanent. We prove that a generalized matrix function is the determinant or the permanent if and only if it preserves the product of symmetric permutation matrices. Also we show that a generalized matrix function is the determinant if and only if it preserves the product of symmetric matrices. To be precise, we show that: If and is a nonzero function, then the following are equivalent: 1) or ; 2) ; 3) ; where is the permutation matrix induced by and . Also if and is a nonzero function, then if and only if for all symmetric matrices .

Introduction Throughout this paper, we consider monomial ideals of the polynomial ring over a filed. We try to give some properties of the polymatroidal ideals, which are the special class of monomial ideals. Herzog and Takayama constructed explicit resolutions for all ideals with linear quotients which admit regular decomposition functions. They also shaw that this class contains all matroidal ideals. We generalize their result to the polymatroidal ideals. Therefore, we can give an explicit linear resolution for any polymatroidal ideal. We also characterize generic polymatroidal ideals. The author and Jafari [1] characterized generalized Cohen-Macaulay polymatroidal ideals. Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals. Material and methods A monomial ideal is said to be polymatroidal, if it is single degree and for any two elements such that there exists an index with such that. In the case that the polymatroidal ideal is squarefree, it is called matroidal. We know that the powers of a polymatroidal ideal are again polymatroidal and polymatroidal ideals have linear quotients. Therefore all powers of polymatroidal ideal have linear resolutions. Let has linear quotients with the order of elements of. We can associate a unique decomposition function, that is a function which maps a monomial to, if is the smallest index such that , where . The decomposition function is called regular, if for all and. We show that any polymatroidal ideal has a regular decomposition function. Therefore we can give an explicit linear resolution for any polymatroidal ideal. By an example, we show that our result can not be extended to the weakly polymatroidal ideals even if they are generated in a single degree. Recall that, a monomial ideal is called generic if two distinct minimal generators and have the same positive degree in some variable , there is a third generator which and , where is the least common multiple of and . In the next result, we characterize generic polymatroidal ideals. A monomial ideal is called generalized Cohen-Macaulay, whenever is equidimensional and monomial localization is Cohen-Macaulay for all monomial prime ideals, where is unique homogenous maximal ideal of . Finally, we characterize monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals. Results and discussion For the first result, we show that any polymatroidal ideal has a regular decomposition function. So we have an explicit linear resolution of any polymatrodal ideal. In the next, we show that ifis a fully supported polymatroidal ideal generated in degree. Then is generic if and only if is either a complete intersection or. Finally, we prove that if is a fully supported monomial ideal in and generated in degree. Then is a generalized Cohen-Macaulay polymatroidal ideal for all if and only if where and for some integers and one of the following statements holds true: is a principal ideal. is a Veronese ideal. is equidimensional and for all . is an unmixed matroidal ideal of degree 2. Conclusion The following conclusions were drawn from this research: Any polymatroidal ideal has a regular decomposition function. characterization of generic ideals. characterization of monomial ideals which all their powers are generalized Cohen-Macaulay polymatroidal ideals.

The inherent characteristics of real-world data is uncertainty. If data is generated in valid experiments or collected standard, probability theory or fuzzy theory is a powerful tool for analysis it in the uncertainty conditions. But data is not always reliable; especially when it is not possible to perform multiple tests or reliable data collection. In this context, referring to the beliefs of experts in the field in question is an alternative approach and uncertainty theory is a tool by which the beliefs of experts can be mathematically incorporated into the problem-solving structure. A stable set has a wide range of applications in many fields, while in most cases its problems are without reliable data. In this paper, we investigate the finding of stable weighted sets with uncertain weights. These weights have an uncertain distribution based on the degree of belief of the field expert. For this purpose, we offer two methods. In the first method, by introducing the concept of chance constraint, we come to an integer linear programming model with definite coefficients. The second method is based on the concept of uncertain expected value. Finally, a numerical example for these two methods is presented.

Translator's abstractThis paper deals with one of the important and interesting topics in Number Theory under the title of prime number races, which is less discussed in Persian texts. All prime numbers (except 2) can be written in the form 4n+1 or 4n+3. The question that naturally arises is, which category contains more prime numbers? Certainly, the infinity of prime numbers makes it difficult for the spectator to predict the competition. The present article, which was written by two prominent mathematicians in this field, has comprehensively discussed the history of this problem, the types of prime number races, and the research work done in this field

In this paper, by introducing well-established epidemiological models, compartmental models and Ising models, examples of their applications in describing epidemics, including some aspects of Covid-19, and several perspectives are presented. The importance of social distancing in preventing the spread of Covid-19 will also be demonstrated. We make a connection between the standard mathematical models employed in epidemics and well-established concepts in condensed matter Physics, such as the Fermi gas and the Fermi-liquid picture. Given the impossibility of making a precise forecast of the disease spread, the importance of taking into account additional factors, such as climate changes and urbanization, in the mathematical description of epidemics are considered.

In this paper, we first consider $4D$ conformally flat homogeneous pseudo-Riemannian space with trivial isotropy, then, we investigate some geometrical properties such as being Ricci solitons and Walker on the spaces under consideration. Indeed, we give a complete classification of the cases giving rise to conformally flat homogeneous Ricci solitons or Walker spaces with trivial isotropy.

Auslander‐Reiten Conjecture is one of the most important and long‐standing conjectures in representation theory of finite dimensional algebras. It is known to be in close relationship with a string of homological conjectures on top of which lies the well‐known Finitistic Dimension Conjecture. Recently, a dual version of Auslander‐Reiten conjecture has been posed. This dual statement deserves further study from the point of view that its validity implies the validity of the conjecture itself. This paper is devoted to deal with this Dual Auslander‐Reiten Conjecture. We discuss it for Gorenstein rings of Krull dimension at least 2. To this end, we firstly show that it suffices to focus only on the case where the Krull dimension is exactly 2. Then, switching to such rings, we show that this conjecture holds for modules of finite length over Gorenstein rings of Krull dimension at least 2.

In this paper, random vectors following the multivariate generalized hyperbolic (GH) distribution are compared using the hessian stochastic order. This family includes the classes of symmetric and asymmetric distributions by which different behaviors of kurtosis in skewed and heavy tail data can be captured. By considering some closed convex cones and their duals, we derive some necessary and sufficient conditions for some important applied stochastic orderings. The linear convex orderings are shown to be equivalent with a certain kind of hessian orderings. Based on copulas generated by the GH distributions, it is revealed that the ordering of the GH distributions in terms of their dependence structures corresponds to some hessian stochastic orderings being satisfied. The results are shown to be relevant to some insurance and economic applications.

We investigate the existence of a weak nontrivial solution for the following problem. Our analysis is generally bathed on discussions of variational based on the Mountain Pass theorem and some recent theories one the generalized Lebesgue-Sobolev space. This paper guarantees the existence of at least one weak nontrivial solution for our problem. More precisely, by applying Ambrosetti and Rabinowitz’s mountain pass theorem and under appropriate conditions, we show that there exists a positive number such that our problem has at least one nontrivial weak solution.

A well-known result of Green [4] shows for any finite p-group G of order p^n, there is an integer t(G) , say corank(G), such that |M(G)|=p^(1/2n(n-1)-t(G)) . Classifying all finite p-groups in terms of their corank, is still an open problem. In this paper we classify all finite abelian p-groups by their coranks.