نشریه پژوهشهای ریاضی
سال ششم شماره 2 (پیاپی 13، تابستان 1399)

In this paper, by mixed monotone operators and their fixed points, we investigate the existence and uniqueness of positive solution for the following boundary value problems via nonlinear fractional differential equations.andwhere  is the Riemann–Liouville derivative.  In recent years, fractional differential equations have been studied by many mathematicians both theoretically and practically, for example, in physics, mechanic, chemistry, engineering, biology, economy, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics and etc.First time, mixed monotone operators have been introduced by Guo and Lakshmikantham in 1987. Next, many authors studied them in Banach spaces and obtained some results not only in theory fields but also wide applications in chemistry, engineering, biology, technology and other fields.   In 2009, Xu et. al studied the properties of Greenchr('39')s function for the nonlinear fractional differential equation boundary value problem (2). Here, the existence and uniqueness of its positive solutions are obtained by using the properties of cone and fixed point theorems for mixed monotone operators. As an application, we utilize the obtained results to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems.

The content of this paper is organized as follows. First, we present some definitions, lemmas and basic results that will be used in the proofs of our theorems. Then, we consider the existence and uniqueness of positive solution for the operator equation, also we utilize the results obtained to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems.

In this work, we study the existence and uniqueness of positive solutions for nonlinear fractional differential equation boundary-value problems. Our results guarantee the existence of a unique positive solution, and can be applied for constructing an iterative scheme for obtaining the solution.

The following conclusions were drawn from this research.
Using the properties of cones and the fixed point theorem for mixed monotone operator, the existence and uniqueness of the positive solution are obtained. Our research methods are different from those in the related literature. As an application, we utilize the obtained results to study the existence and uniqueness of positive solution for nonlinear fractional differential equation boundary value problems. Our results extend and improve the related conclusions in the literature.

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The influence of persistence behavior of a dynamical system on tangent bundle of a manifold is always a challenge in dynamical systems. Persistence properties have been studied on whole manifold or on some pieces with independent dynamics. Since shadowing property has an important role in the qualitative theory of dynamical systems, by focusing on various shadowing properties, such as usual shadowing, inverse shadowing, limit shadowing, many interesting results have been  obtained. The notion of limit shadowing property introduced by S. Pilyugin who obtained its relation to other various shadowing. Blank introduced the notion of average-shadowing property. It is known that every Axiom A diffeomorphism restricted to a basic set has the average shadowing property. K. Sakai proved that the -interior of the set of all diffeomorphisms satisfying the average-shadowing property is characterized as the set of all Anosov diffeomorphisms. Asymptotic average shadowing (AASP) defined by R. Gu for continuous maps, combines to the limit shadowing property with the average shadowing property. Here we modify the notion (AASP) and define the limit average shadowing for diffeomorphisms (LASP). R. Gu presented some basic properties of the limit average shadowing for continuous maps. He proved that if a continuous map has the limit average shadowing on a compact metric space, then is chain transitive and that -hyperbolic homeomorphisms with limit average shadowing are topologically transitive. M. Kulczycki et.al., found some relations between LASP and the other notion of topological dynamics. They proved that a surjective map with specification property has the LASP. Also, they found the relation between LASP and shadowing property. They also have been shown that an expansive continuous map with shadowing property is LASP if and only if it is mixing. This paper follows the ideas of R. Gu and M. Kulczycki et.al. Here we define LASP for diffeomorphism with a slight modification of the continuous case. We give an example which shows that shadowing property and LASP are not equivalent. Also, we introduce the notion of - stably limit average shadowing for a closed -invariant subset of  and show that if is -stably limit average shadowing and the minimal points of  are dense there, then  admits a dominated splitting.

In this paper we give a system which has the limit average shadowing, but not the shadowing property. Also, one can give examples which have the shadowing but not the limit average shadowing property. Thus the limit average shadowing property does not imply the shadowing property. In fact, we can give a class of diffeomorphisms which have LASP, but not the shadowing property. In fact the following proposition gives a large class of diffeomorphisms satisfying the limit average shadowing. Proposition A. Let  be a locally maximal -invariant set. If  is the specific set for then  is limit average shadowable. The main purpose of the paper is to characterize the closed -invariant set via limit average shadowing property in -open condition. So, we consider the notion of limit average shadowing property in geometric differential dynamical systems. First we show that if  has the limit average shadowing property on a closed -invariant set  then  is chain transitive. By using chain transitivity and limit average shadowing property we can prove that  is transitive. Proposition B. If  is -stably limit average shadowable, then there is a neighborhood  of  and a neighborhood  of  such that contains neither almost sinks nor almost sources for any Since we have proved that if  has the limit average shadowing property on a closed f-invariant set  and minimal points of  are dense then  is transitive. It is essentially proved that under assumptions and conclusions of the Proposition B, admits a dominated splitting. Thus we get the main result of this paper. Theorem C. Let  be a closed f-invariant set whose minimal points are dense there. If  is -stably limit average shadowing then  admits a dominated splitting.

Tries (from retrieval) are one of the most practical data structures with a tree construction in computer science. Tries store string data in leaves of tree. They are often used to store such data so that future retrieval can be made efficient. For example, tries are widely used in algorithms for automatically correcting words in texts. The number of nodes of the same type, which are at the same distance from the root of a rooted tree, is called profile. The analysis of the profile of a tree is of great importance. Because many of the parameters of a rooted tree can be expressed in terms of its profile. Although profiles represent one of the most fundamental parameters of tries, they have hardly been studied in the past. A generalized version of trie is called bucket trie where each leaf or bucket has a storage capacity of more than one string. Random trie is obtained by defining a random grow rule for trie. We present a detailed study of the limit behavior of the profiles in a random bucket tries.

In this paper, the methods we apply to derive recurrences satisfied by the expected profiles and to solve them asymptotically for all possible ranges of the distance from the root, are based on the use of Poissonization, Mellin transform, recurrence equations, generating functions, singularity analysis and saddle-point method.

Here, as the number of stored strings in a random bucket trie increases, we investigate the asymptotic expectation, variance and limiting distribution of each of the two internal and bucket profiles (i.e. the number of bucket nodes or leaves, and the number of internal nodes or non-leaves which are at the same distance from the root) in random bucket tries. Both the expectation and variances of the two profiles contain periodic functions, and we show those periodic functions are not zero that this point has not been proven in the paper on the ordinary trie. Also, we examine the amount of asymptotic ratio of the expectations of the bucket and internal profiles.

In this research, we will generalize the most important part of the results for the ordinary tries and also provide a proof for an unproven point in those results. More precisely, the purpose of this article is to study the internal profile (the number of non-leaf nodes at distance  from the root) and bucket profile (the number of buckets at distance  from the root) in an important data structure called random bucket trie (random trie based on  data and maximum capacity of  data per each leaf). By some methods in complex analysis, we have shown that for every  and some fixed constants  and , if  is in the range,then the expectations and variances of both profiles contain non-zero periodic functions (the non-zero property of these periodic functions has not been proved for the ordinary tries up to now). We also provide a graph of those periodic functions for  by MAPLE. Then we study the ratio of the two profiles for  Finally, by finding the asymptotic expansions of  Poisson generating functions for the probability generating functions of profiles and then using the Cauchy integral formula, we obtain the asymptotic expansions for the probability generating functions, which indicate that the limiting distributions of profiles are normal. ./files/site1/files/62/3Abstract.pdf

Complex analysis is a comparatively active branch in mathematics which has grown significantly. A deep look at the implications of continuity, derivative and integral in complex analysis and their relation with partial differential equations determines the importance of establishing the relation between complex analysis and the theory of partial differential equations.There are three aspects of the aims of complex analysis:First it is possible to interpret the peculiarities of holomorphic functions as properties of solutions of specials systems of partial differential equations. Secondly, complex analysis becomes applicable to general classes of differential equations, not only to special ones. And thirdly, the new general complex analysis is able to construct solutions and to describe the properties of given solutions with the help of solutions of corresponding problems for holomorphic functions. The third aspect is significant, since in the case of nonlinear equations it essentially means their reduction to linear problems. In addition this third aspect shows that the general complex analysis is able to make use of results of classical function theory and of such results which originally have not been connected with partial differential equations.One of the main goals of complex analysis is its systematic application in the branch of the theory of differential equations. By using the contraction function, the Gauss-Oatrogradski integral formula and the Banach fixed point theorem of the complex space, the classical and interesting method of existence and uniqueness of the solution of differential equations with nonlinear partial derivatives are smooth, which shows the importance of complex analysis in the branch of the theory of differential equations. In recent years, complex analysis and their relationship with partial differential equations have attracted a number of researchers.In this paper, we determine the existence and uniqueness of a solution of a nonlinear partial differential equation in the form where

Firstly, we transform the nonlinear partial differential equation in a complex space to a linear differential equation of the first order and show the existence of the solution. Then, we consider the equivalence of the solution of the nonlinear partial differential equation with a singular integral equation system by using weak and strong singular integral operators and the property of the holomorphic function. Also, we consider the solution of the nonlinear partial differential equation on a bounded domain with finite area in the Holder space and Sobolev space.

We obtain the uniqueness of the solution of the nonlinear partial differential equation by applying the Lipshitz condition in the Holder space and Sobolev space based on the contraction function and the Banach fixed point theorem.

In this paper, we discuss on the existence and uniqueness of the solution of the nonlinear partial differential equation in the continuous functions of Holder and Sobolev spaces in general case in complex space. We deduce the proposed method can be extended in other spaces.

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Hom-algebraic ‎structures ‎appeared ‎first ‎as a‎ ‎generalization ‎of ‎Lie ‎algebras ‎in [1,3],  ‎where ‎the ‎authors ‎studied ‎‎q-deformations ‎of ‎Witt ‎and ‎Virasoro ‎algebras. A‎ ‎general ‎study ‎and ‎construction ‎of ‎Hom-Lie ‎algebras ‎were ‎considered ‎in [7, 8]. ‎Since ‎then, ‎other ‎interesting ‎Hom- type ‎algebraic ‎structures ‎of ‎many ‎classical ‎structures ‎were ‎studied ‎Hom-associative ‎algebras, ‎Hom-Lie ‎admissible ‎algebras ‎and ‎Hom-Jordan ‎algebras. ‎Hom-algebraic ‎structures ‎were ‎extended ‎to ‎Hom-Lie ‎superalgebras ‎in .‎As a‎ ‎generalization ‎of ‎Lie ‎superalgebras ‎and ‎Jordan ‎Lie ‎algebras, ‎the ‎notion ‎of ‎‎ δ-Jordan ‎Lie ‎superalgebra ‎was ‎introduced ‎in [6, 12] which is intimately related to both Jordan-super and atiassociative algebras. The case of δ=1 ‎yields ‎the ‎Lie ‎superalgebra, ‎and ‎we ‎call ‎the ‎other ‎case ‎of δ=1 a‎ ‎Jordan ‎Lie ‎superalgebra,   ‎because ‎it ‎turns ‎out ‎to ‎be a‎ ‎Jordan ‎superalgebra. ‎It ‎is ‎often ‎convenient ‎to ‎consider ‎both ‎cases ‎of δ= 1, ‎and ‎call δ-Jordan ‎Lie ‎superalgebras.‎ ‎The ‎motivations ‎to ‎characterize ‎Hom-Lie ‎structurers ‎are ‎related ‎to ‎physics ‎and ‎to ‎deformations ‎of ‎Lie ‎algebras, ‎in ‎particular ‎Lie ‎algebras ‎of ‎vector ‎fields. ‎Hom-Lie superalgebras are a generalization of Hom-Lie algebras, where the classical super Jacobi  identity is twisted by a linear map. If the skew-super symmetric bracket of a Hom-Lie superalgebra is replaced by δ-Jordan-super ‎symmetric‎, it is called a   δ-Jordan-Hom-Lie ‎superalgebra ‎(see [11]).‎There are several notions of differential operators and differential calculus on‎ non-associative algebras (see [4, 5])‎. A ‎ ‎comprehensive definition of differential operators on non-associative algebras fails to be formulated. But many authors was studied a notion of differential operators and differential calculus on ‎Lie ‎algebras ‎and ‎Hom-Lie ‎algebras [9, 10]. ‎ According  ‎to ‎various ‎applications ‎in ‎both ‎mathematics ‎and ‎physics,‎‎‎‎‎ we will investigate a notion of differential operators and differential calculus on‎ ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebras.

A ‎key ‎point ‎is ‎that ‎the ‎multiplications ‎on ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebras are their derivations. Therefore, definition of differential operators on a ‎‎‎multiplicative δ-Jordan-Hom-Lie ‎superalgebra must treat the derivations of this algebra as a first-order differential operators too. By our considerations, we will define higher order differential operators as composition of the first-order differential operators on a ‎multiplicative δ-Jordan-Hom-Lie ‎superalgebra. We also consider a geometric aspect to the concept of differential calculus on ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebra by using the cohomology theory for this algebra.

‎The theory of differential operators on associative algebras is not extended to the non-associative algebras in a straightforward way. But, we provide a notion of differential operators of any order on ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras and their modules. We also study some property of differential operators on ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras, for examples, the brackets and composition of two differential operators of higher order on these algebras. Finally, by using theory of cohomology for ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras, we investigate a notion of differential calculus on these algebras. In other words, for a ‎multiplicative   δ-Jordan-Hom- Lie ‎superalgebra L  ‎with ‎center Z(L) ‎and ‎‎Der(L), ‎the ‎derivation ‎of ‎‎ L, ‎we ‎consider ‎the ‎cochain ‎complex ‎of  L ‎as ‎‎Der(L)-module ‎its ‎subcomplex ‎of ‎‎ Z(L)-multilinear  ‎morphism ‎is said ‎to ‎be a‎ ‎‎ differential calculus based on derivation of ‎ L. ‎Next, ‎we ‎compute ‎the‎ differential calculus based on derivation of Hom-Lie super algebra ‎‎‎osp(1, 2).‎

The following conclusions were drawn from this research.
• Definition of the differential operators of any order on ‎ multiplicative   δ-Jordan-Hom-Lie ‎superalgebras and prove several properties of it.‎
• Definition of the differential operators of any order on δ-modul ‎of‎ ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebras and state some properties of it.‎
• The study of ‎‎ differential calculus based on derivation of a ‎ multiplicative δ-Jordan-Hom-Lie ‎superalgebra.
• Compute the ‎‎ differential calculus based on derivation of Hom-Lie superalgebra ‎ osp (1, 2).‎

In this paper, we deal with the existence of at least  two solutions for an anisotropic discrete non-linear problem involving p(k)-Laplacian with Dirichlet boundary value conditions. The technical approach is based on  a two critical points theorem for differentiable functionals. Two examples are inserted to illustrate the importance of main results.

‎One of the most important classes of mappings is the class of‎ ‎monotone mappings due to its various applications‎. ‎For solving many‎ ‎important problems‎, ‎it is required to solve monotone inclusion‎ ‎problems‎, ‎for instance‎, ‎evolution equations‎, ‎convex optimization‎ ‎problems‎, complementarity problems and variational inequalities‎ ‎problems.The first algorithm for approximating the zero points of the‎ ‎monotone operator introduced by Martinet. ‎In the past decades‎, ‎many authors prepared various algorithms and investigated the existence and convergence of zero points for maximal monotone mappings in Hilbert spaces‎.‎A generalization of finding zero points of nonlinear operator is to find zero points of the sum of an‎ ‎-inverse strongly monotone operator and a maximal monotone operator‎. ‎Passty introduced‎ ‎an iterative methods so called forward-backward method for finding zero points of the sum of two operators‎. ‎There are various applications of the problem of finding zero points of the sum of two operators.Recently‎, ‎some authors introduced and studied some algorithms for‎ ‎finding zero points of the sum of a -inverse strongly‎ ‎monotone operator and a maximal monotone operator under different‎ ‎conditions.In this paper‎, ‎motivated and inspired in above‎, ‎a shrinking projection algorithm is introduced for finding zero points of the sum of an inverse strongly monotone operator and a maximal monotone operator‎. ‎We prove the strong convergence theorem‎ ‎under mild restrictions imposed on the control sequences‎.

In this scheme, first we gather some ‎definitions and lemmas of geometry of Banach spaces and monotone‎ ‎operators‎, ‎which will be needed in the remaining sections‎. ‎In‎ the next section‎, ‎a shrinking projection algorithm is proposed and a‎ ‎strong convergence theorem is established for finding a zero point‎ ‎of the sum of an inverse strongly monotone operator and a maximal‎ ‎monotone operator‎.

‎The generated sequence by  the presented algorithm converges strongly to a zero point of the sum of an -inverse strongly‎ ‎monotone operator and a maximal monotone operator‎ ‎in Hilbert spaces. ‎

In this paper‎, ‎we present an iterative algorithm ‎for approximating a zero point of the sum of an -inverse strongly‎ ‎monotone operator and a maximal monotone operator‎ ‎in Hilbert spaces.Under some mild conditions‎, ‎we show the convergence theorem of the mentioned algorithm‎. ‎Subsequently‎, ‎some corollaries and applications of those main result is  provided‎.This observation may lead to the future works that are to analyze and discuss the rate of convergence of these suggested algorithms‎.We obtain some applications of main theorem for solving variational inequality problems and finding fixed points of strict pseudocontractions‎.

This paper deals with the boundary value problem involving the differential equation  -ychr('39')chr('39')+q(x)y=lambda ysubject to the standard boundary conditions along with the following discontinuity conditions at a point  y(a+0)=a1y(a-0),  ychr('39')(a+0)=a2ychr('39')(a-0)+a3y(a-0).  We develop the Hochestadt-Lieberman’s result for Sturm-Liouville problem when there is a discontinuous condition on the closed interval. We show that the potential function and some coefficients of boundary conditions can be uniquely determined by the value of the potential on some interval and parts of two set of eigenvalues.

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  Let ,  and be Banach spaces and  be a bilinear mapping. In 1951 Arens found two extension for  as  and  from  into .  The mapping  is the unique extension of  such that  from  into  is  continuous for every , but the mapping  is not in general  continuous from  into  unless .  Thus for all  the mapping is  continuous if and only if  is Arens regular. Regarding  as a Banach , the operation  extends to  and  defined on . These extensions are known, respectively, as the first (left) and the second (right) Arens products, and with each of them, the second dual space  becomes a Banach algebra.

   The constructions of the two Arens multiplications in  lead us to definition of topological centers for  with respect to both Arens multiplications. The topological centers of Banach algebras, module actions and applications of them were introduced and discussed in some manuscripts. It is known that the multiplication map of every non-reflexive,  -algebra is Arens regular.  In this paper, we extend some problems from Banach algebras to the general criterion on module actions and bilinear mapping with some applications in group algebras.

We will investigate on the Arens regularity of bounded bilinear mappings and we show that a bounded bilinear mapping  is Arens regular if and only if the linear map  with  is weakly compact, so we prove a theorem that establish the relationships between Arens regularity and weakly compactness properties for any bounded bilinear mappings. We also study on the Arens regularity and weakly compact property of bounded bilinear mapping and we have analogous results to that of Dalse, lger and Arikan. For Banach algebras, we establish the relationships between Arens regularity and reflexivity.

The following conclusions were drawn from this research.if and only if the bilinear mapping   is Arens regular.A bounded bilinear mapping  is Arens regular if and only if the linear map  with  is weakly compact. if and only if the bilinear mapping   is Arens regular.Assume that  has approximate identity. Then   is Arens regular if and only if  is reflexive.

In 1976, A. Lambert characterized subnormal weighted shifts. Then he studied hyponormal weighted composition operators on  in 1986 and in 1988 subnormal composition operators studied again by him. Recently, A. Lambert, et al., have published an interesting paper: Separation partial normality classes with composition operators (2005). In 1978, R. Whitley showed that a composition operator   is normal if and only if  essentially. Normal and quasinormal weighted composition operators were worked by J.T. Campbell, et al. in 1991. In 1993, J.T. Campbell, et al. worked also seminormal composition operators. Burnap C. and Jung I.B. studied  composition operators with weak hyponormality in 2008.

Let  be a complete  -finite measure space and   be a complete  -finite measure space where  is a subalgebra  of . For any non-negative -measurable functions  as well as for any , by the Radon-Nikodym theorem, there exists a unique -measurable function such that  for all  As an operator,   is a contractive orthogonal projection which is called the conditional expectation operator with respect   For a non-singular transformation  again by the Radon-Nikodym theorem, there exists a non-negative  unique function  such that  The function   is called Radon-Nikodym derivative of   with respect . These are two most useful tools which play important roles in this review. For a non-negative finite-valued  - measurable function  and a non-singular transformation  the weighted composition operator  on  induced by  and is given by , where  is called the composition operator on . is bounded on  for   if and only if

In this paper, we review some known classes of composition operators, weighted composition operators, their adjoints and Aluthge transformations on  such as normal, subnormal, normaloid, hyponormal, -hyponormal, -quasihyponormal, -paranormal, and weakly hyponormal, Furthermore, miscellaneous examples are given to illustrate that weighted composition operators lie between these classes. We discuss from the point of view of measure theory and all results depend strongly to the Radon-Nikodym derivative  and the conditional expectation operator  with their various types. Hence we study their fundamental properties in sections 1 and 2. Then, we review some results by A. Lambert, D.J. Harringston, R. Whitley, J.T. Campbell and W.E. Hornor.

Conclusion

According to the given miscellaneous examples in the final section, we can conclude that composition and  weighted composition operators lie between these classes.

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An -dimensional Riemannian manifold  is said to be flat (or locally Euclidean) if  locally isometric with the Euclidean space, that is,  admits a covering of coordinates neighborhoods each of which is isometric with a Euclidean domain.  A Riemannian manifold  is flat if and only if  admits a covering of coordinates neighborhoods on each of, the function  is independent of . A classical result affirms that a Riemannian manifold is flat if and only if its Riemann curvature vanishes (equivalently, the sectional curvature; This is usually taken as the definition of a flat Riemannian manifold in the contexts. The universal Riemannian covering space of a complete and flat Riemannian manifold is the Euclidean space . Up to local isometry, Bieberbach proved that any compact flat Riemannian manifold, is realized as a quotient space , where  is a discrete, co-compact and torsion free subgroup of the Euclidean group  , cf.  The only 1 dimensional complete, flat and connected manifolds are  and . In 2 dimensions,  the only complete, flat and connected manifolds are cylinder, Mӧbius strip, Torus and Klein bottle. In 3 dimensions, there are only 10 complete, flat and connected manifolds including 6 oriented and 4 non-oriented manifolds, cf.  .Likewise the Riemannian case, a Finslerian manifold   is said to be flat (or locally Minkowskian) if,  admits a covering of coordinates neighborhoods each of which isometric with a single Minkowski normed domain. A Finslerian  manifold  is flat if and only if it admits a covering of coordinates neighborhoods on each of, the function  is independent of .  The flag curvature  of any flat Finsler manifold vanishes identically.

Thanks to the works of Bieberbach and Schoenflies, we apply an group theoretic approach to classify flat Randers manifolds. The key idea is that the isometry group of Randers manifold is a subgroup of the Euclidean group. This fact, may ease our approach to find and count discrete, co-compact and torsion frees subgroups of . First we find the Bieberbach subgroups and then, we count those that could form an isometry subgroup.

Here, flatness of a generic Finsler manifold is aimed to be defined so that it generalizes the flatness for Riemannian manifolds. The following result outcome in dimension 2 and 3:Theorem 1. The only connected and closed -dimensional () closed flat Randers manifolds is the torus  , respectively.To classify the flat Randers manifolds, we find out that the flat Randers manifolds are flat Riemannian manifolds.  Besides, the isometry group of a Randers manifold  is a subgroup of the isometry group . Our discussion also apply the following results:Every dimensional flat Randers manifold is itself a flat Riemannian manifolds.Every dimensional flat Randers manifold is orientable.The non-Riemannian properties for generic Finsler metrics may cause obstructions for a Finsler manifold to be falt.

The following conclusions were drawn from this research.In dimensions 2 and 3, the only connected and closed flat Randers manifolds are the tori  , respectively.Every dimensional flat Randers manifold is itself a flat Riemannian manifolds.

‎Throughout this paper‎,  will denote a commutative ring with‎ ‎identity and  will denote the ring of integers.Let be an -module‎. A submodule  of is said to be pure if for every ideal of .  has the copure sum property if the sum of any two copure submodules is again copure‎.  is said to be a comultiplication module if for every submodule of  there exists an ideal  of such that .  satisfies the double annihilator conditions if for each ideal  of , we have . is said to be a strong comultiplication module if  is a comultiplication R-module which satisfies the double annihilator conditions. A submodule  of  is called fully invariant if for every endomorphism In [5]‎, ‎H‎. ‎Ansari-Toroghy and F‎. ‎Farshadifar introduced the dual notion of pure submodules (that is copure submodules) and investigated the first properties of this class of modules‎. ‎A submodule  of  is said to be copure if  for every ideal of .

We say that an -modulehas the copure intersection property if the intersection of any two copure submodules is again copure‎. In this paper, we investigate the modules with the copure intersection property and obtain some related results.

The following conclusions were drawn from this research.Every distributive -module has the copure intersection property.Every strong comultiplication -module has the copure intersection property.An -module  has the copure intersection property if and only if for each ideal  of and copure submodules  of  we have If  is a , then an -module  has the copure intersection property if and only if  has the copure sum property. Let , where is a submodule of . If  has the copure intersection property, then each  has the has the copure intersection property. The converse is true if each copure submodule of  is fully invariant.

General linear methods(GLM) was developed by Butcher in 1966 as an extension of the traditional Runge-Kutta and linear multistep methods [1]. The classification of GLMs is an important open and active research area. Many authors studied GLMs and developed different GLM classes for solving stiff and non-stiff problems. For example, the Diagonally-implicit multi-stage integration methods known as DIMSIM has four subclasses which is appropriate to stiff and non-stiff problems with capability of implementing in parallel or sequential forms [2].  In this paper we study the GLM representation of the predictor-corrector (PC) schemes in finite mode. The PC schemes are a class of practical schemes for solving initial value problems. The basic scheme (corrector) is an implicit scheme in which the implementation is accomplished by an iteration process with a rather good initial approximation evaluated by predictor method. The built in error estimation with local extrapolation is one of major advantages of PC schemes based on Adams schemes. In new formulation based on GLM we have enforced Milne estimation in the internal or external iterations. We present a closed form stability function for derived schemes. Numerical implementation and comparisons illustrate that in new formulation based on GLM framework we obtain more accurate solutions. This new formulation provide the opportunity of more

The methodology of this paper is based on the constructing matrix-vector formulation of the PC schemes. The matrix dimensions depend on the step numbers and the number of iterations of the PC finite mode. The derived GLM method is a scheme with r internal and s external stages. These stages are specified by the step numbers and a finite number of iterations. 

The general linear methods are a large class of schemes that include the traditional schemes such as Runge-Kutta methods. Among the PC schemes based on Adams linear multistep methods we can easily choice the methods with different step numbers and different orders. It is easy to estimate the local truncation errors. The stability function is available in the closed form. The local extrapolation method provides the application of Milne estimation in the internal and external stages of the PC iterations. In fact, we can control the error and improve the accuracy and locally increase the order of the method.

We have studied a reformulation of a class of predictor-corrector schemes in the new framework of general linear methods. The new schemes have the following properties:The Adams linear multistep methods are a large class of schemes including implicit and explicit schemes. We can choose Adams choice methods with different step numbers and orders.There is cheap error estimation for PC schemes known as Milne estimation and it is possible to use this error estimation in internal and external stages. The stability function of the resulting GLM is available for further developments and study of the stability properties.The new GLM formulation provides accurate results comparing with the original formulation in PC form.

An important class of Finsler metric is named Kropina metrics which is defined by Riemannian metric α and 1-form β  which have many applications in physic, magnetic field and dynamic systems. In this paper, conformal transformations of χ-curvature and H-curvature of Kropina metrics are studied and the conditions that preserve this quantities are investigated. Also it is shown that in the special cases the  conformal transformations reduced to homothetic transformations.  

Hepatitis C virus (HCV) was first identified in the year 1989. Globally, hepatitis has infected an estimated 200 million people, most of whom are chronically infected. Mathematical modeling of the spread of infectious diseases continues to provide important insights into diseases behavior and control. Over the years, it has also become an important tool in understanding the dynamics of diseases and in decision making processes regarding intervention programs for controlling these diseases in many countries. The impact of a chronic stage on the disease transmission and behavior in an exponentially growing or decaying population, is the focus of this paper. In this article a new approach for controlling hepatitis C disease for a homogeneous population based on optimal control theory and next generation matrix is proposed. In the first, we consider the Hepatitis C model using predefined parameters and variables by Yuan and Young paper [20]. The model is a SEIR epidemiological model, that the factors are Corresponds to special blocks of possible states for individuals and next calculate reproduction number ([21], [22]). In fact, we divide the population in researched area into four classes: S—susceptible, E—exposed, I—infected with acute hepatitis C, V—infected with chronic hepatitis C. The total number in time t is . In this paper, it is assumed that, after the initial infection, a host stays in a latent period before becoming infectious. We obtain the basic reproduction number of this model, which completely decides the dynamics of this model. If  the disease cannot be prevalent and If  the disease can be prevalent. In the following by the optimal control theory, we check the infectious states to control the outbreak, and then solve the control problem. So we get the numerical solution and we will check witch of them (variables) is the best control and compare the results.

In this scheme, first we Consider the disease model and then check it. This model is a four-dimensional SEIR model. There are a few chronic diseases that have been analyzed in a chronic fashion. We consider the population in four Blocks. (infected people (i), exposed population(e) and Chronic infected population)The costs associated with each of these strategies are also investigated by formulating the costs function problem as an optimal control problem, and we then use the Pontryaginchr('39')s Maximum Principle to solve the optimal control problems.Then we create the optimal control problem, next with using Pontryaginchr('39')s maximum principle we will solve the problem.

After the modelling the hepatitis C Disease, a nonlinear ordinary differential equations system is obtained, and with simplifying the system by applying the appropriate variable change, we can optimize some of the variables that can be controlled (for reducing the prevalence) until converted to zero.
Also, the numerical results reported in the tables that controlling on infected people(i), exposed population(e) and Chronic infected population. The reported results demonstrate that the disease can be have the maximum reduction by controlling on the Chronic infected population (v).These populations can be controlled by medication, using different therapies, Quarantine them to reduce the spread of the disease or with Other medical procedures.

The following conclusions were drawn from this research.The epidemic is transmitted through peoplechr('39')s direct contacts, so It is important to reduce the incidence of the disease and we can use the Optimal control theory for some of the variables that can be controlled by reducing the prevalence until converted to zero.The results show that control over the variable sensitive(s) is weak and studying it does not lead to a desirable result.The reported results demonstrate that the disease can be the maximum reduction by controlling on the Chronic infected population (v).