Abstract:
We define an equivalence relation on a topological space which is acted by topological monoid as a transformation semigroup. Then, we give some results about the -invariant classes for this relation. We also provide a condition for the existence of relative -invariant classes. 1. Introduction The invariance theory is one of the principal concepts in the topological dynamics system, see [1, 2]. In [3], Colonius and Kliemann introduced the concept of a control set which is relatively invariant with respect to a subset of the phase space of the control system. From a more general point of view, the theory of control sets for semigroup actions was developed by San Martin and Tonelli in [4]. In this paper, we define an equivalence relation on a topological space which is acted by topological monoid as a transformation semigroup. Then, we provide the necessary and sufficient conditions for the equivalence classes to be -invariant classes which correspond with the control sets for control systems. Then, we study the -invariant classes for this relation in , in particular, and we provide the conditions for the existence and uniqueness of invariant classes. Throughout this paper, will denote the closure set of a set , and will denote the interior set of and all topological spaces involved Hausdorff. Definition 1.1 (see [2]). Let be a monoid with the identity element and also a topological space. Then, will be called a topological monoid if the multiplication operation of: is continuous mapping from to . Definition 1.2 (see [4]). Let be a topological monoid and a topological space. We say that acts on as a transformation semigroup if there is a continuous map between the product space and satisfying we further require that for all . The triple is called an flow; will denote . In particular, an flow is called phase flow if is a compact space. The orbit of under is the set . For a subset of , denotes the set . And a subset is called an invariant set if and . A control set for on is a subset of which satisfies(1) ,(2)for all ,(3) is a maximal with these properties. Then, we say that a subset , satisfies the no-return condition if for some and , then . Lemma 1.3 (see [5, Zorn's Lemma]). If each chain in a partially ordered set has an upper bound, then there is a maximal element of the set. 2. Invariant Classes Let be an flow. From the action on , we can define the relation ~ on by It is clear that the relation ~ is an equivalence relation, and will denote the set of all equivalence classes induced by ~ on . We observe that for all , and if , then for all . The

Abstract:
We introduce and study new sequence spaces which arise from the notions of generalized de la Vallée-Poussin means, invariant means, and modulus functions.

Abstract:
We will introduce and study the pairwise weakly regular-Lindelöf bitopological spaces and obtain some results. Furthermore, we study the pairwise weakly regular-Lindelöf subspaces and subsets, and investigate some of their characterizations. We also show that a pairwise weakly regular-Lindelöf property is not a hereditary property. Some counterexamples will be considered in order to establish some of their relations.

Abstract:
We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of threeexamples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

Abstract:
We define the so-called box convolution product and study their properties in order to present the approximate solutions for the general coupled matrix convolution equations by using iterative methods. Furthermore, we prove that these solutions consistently converge to the exact solutions and independent of the initial value.

Abstract:
We consider subclasses of functions with bounded turning for normalized analytic functions in the unit disk. The geometric representation is introduced, some subordination relations are suggested, and the upper bound of the pre-Schwarzian norm for these functions is computed. Moreover, by employing Jack's lemma, we obtain a convex class in the class of functions ofbounded turning and relations with other classes are posed.

Abstract:
We use the concepts of the quasicoincident relation to introduce and investigate some lower separation axioms such as , , , and as well as the regularity axioms and . Further we study some of their properties and the relations among them in the general framework of fuzzy topological spaces. 1. Introduction The fundamental concept of a fuzzy set was introduced by Zadeh in 1965, [1]. Subsequently, in 1968, Chang [2] introduced fuzzy topological spaces (in short, fts). In Chang’s fuzzy topological spaces, each fuzzy set is either open or not. Later on, Chang’s idea was developed by Goguen [3], who replaced the closed interval by a more general lattice L. In 1985, Kubiak [4], and ？ostak [5], in separated works, made topology itself fuzzy besides their dependence on fuzzy sets. In 1991, from a logical point of view, Ying [6] studied Hohles topology and called it fuzzifying topology. This fuzzification opened a rich field for research. As it is well known, the neighborhood structure is not suitable to -topology, and Pu and Liu [7] broke through the classical theory of neighborhood system and established the strong and powerful method of quasicoincident neighborhood system in -topology. Zhang and Xu [8] established the neighborhood structure in fuzzifying topological spaces. Considering the completeness and usefulness of theory of -fuzzy topologies, Fang [9] established -fuzzy quasicoincident neighborhood system in -fuzzy topological spaces and gave a useful tool to study -fuzzy topologies. In ordinary topology, -open sets were introduced and studied by Njastad [10]. Bin Shahna [11], in the same spirit, defined fuzzy -open and fuzzy -closed. Separation is an essential part of fuzzy topology, on which a lot of work has been done. In the framework of fuzzifying topologies, Shen [12], Yue and Fang [13], Li and Shi [14], and Khedr et al. [15] introduced some separation axioms and their separation axioms are discussed on crisp points not on fuzzy points. In 2004, Mahmoud et al. [16] introduced fuzzy semicontinuity and fuzzy semiseparation axioms and examined the validity of some characterization of these concepts. Further, they also defined fuzzy generalized semiopen set and introduced fuzzy separation axioms by using thew semiopen sets concept. In the same paper, the authors also discussed fuzzy semiconnected and fuzzy semicompact spaces and some of their properties. The present paper is organized as follows. It consists of four sections. After this introduction, Section 2 is devoted to some preliminaries. In Section 3, we introduce the notions of some lower

Abstract:
We focus on the equity mutual funds offered by three Malaysian banks, namely Public Bank Berhad, CIMB, and Malayan Banking Berhad. The equity mutual funds or equity trust is grouped into four clusters based on their characteristics and categorized as inferior, stable, good performing, and aggressive funds based on their return rates, variance and treynor index. Based on the cluster analysis, the return rates and variance of clusters are represented as triangular fuzzy numbers in order to reflect the uncertainty of financial market. To find the optimal asset allocation in each cluster we develop a hybrid model of optimization and fuzzy based on return rates, variance. This was done by maximizing the fuzzy return for a tolerable fuzzy risk and minimizing the fuzzy risk for a desirable fuzzy return separately at different confidence levels. 1. Introduction The portfolio optimization is also known as a risk management, and how to obtain the optimal solution of portfolio allocation has atracted many researchers on portfolio decision-making in the recent years. Thus the objective for models, either the return model or the risk model, is to maximize the profit or to minimize the cost for portfolio selection based on mean-variance (MV) theorem that was proposed by Markowitz. Nowadays, mutual funds have become an ideal form of investment for many people since they have the ability to separate risks to the smallest degree. Therefore, it is not surprising at all to have many banks offering equity mutual fund schemes since it is investors’ preference. However, selecting the best equity mutual fund is a difficult process. In the selection process, investors normally analyze the past performance of funds in order to evaluate the future performance. But sometimes, this method cannot be used for prediction of the future performance of funds due to the high volatility of market environment that is there is no assurance past trends will continue. Thus Ammar and Khalifa [1] introduced the formulation of fuzzy portfolio optimization problem as a convex quadratic programming approach and gave an acceptable solution to such problems. They determined how much money should be allocated to each investment so that the total expected return would be greater than or equal to some lowest fuzzy return or the total fuzzy variance less than or equal to some greatest fuzzy variance. In a recent study, by Chen and Huang [2], the uncertainty of future return rates and risk was considered and presented in triangular fuzzy numbers in contrast with the previous research whereby return rates