JOURNAL OF NEW RESEARCHES IN MATHEMATICS
Year:2020 | Volume:5 | Issue:22|Papers Count:13

The concept of configuration of groups which is defined in terms of finite partitions and finite strings of elements of the group is presented by Rosenblatt and Willis. To each set of configurations, a finite system of equations known as configuration equations, is associated. Rosenblatt and Willis proved that a discrete group G is amenable if and only if every possible instance of its configuration equations admits a normalized solution. In this paper we compare the existence of such solutions for different systems. We prove that if a system of configuration equations has no normalized solution, then every system related to a refinement of the initial partition, has no normalized solution, as well. The Tarski number of a non-amenable group is the smallest number of the pieces of its paradoxical decompositions. In the present paper we also provide a relation between the Tarski numbers of the subgroups of two configuration equivalent groups.

Equilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm using the resolvent of a maximal monotone operator in Banach space. Under appropriate conditions, we prove the strong convergence of the generated sequence by the algorithm to the zero of the maximal monotone operator. As an application of the main result, and using proved theorems, we can provide a maximal monotone operator for any monotone bifunction so that the zero of the maximal monotone operator is the solution to the equilibrium problem. The results of this paper generalize or improve the obtained results in the various papers.

Bresar in 1993 proved that each biderivation on a noncommutative prime ring is a multiple of a commutatot. A result of it is a characterization of commuting additive mappings, because each commuting additive map give rise to a biderivation. Then in 1995, he investigated biderivations, generalized biderivations and sigma-biderivations on a prime ring and generalized the results of derivations for them. He showed every generalized biderivation G of an ideal in a prime ring R is of the form G(x, y)=xay+ybx, where a, b are the elements of the symmetric Martindale ring of quotionts of R. Ali in 2012, studied *-derivations on a *-ring and showed that *-derivations maps to the center of ring. In this paper, we introduce *-biderivations and *-sigma-biderivations on a *-ring. Then some results obtained by Bresar and Ali generalize for these mapping to a class of *-rings. That is, each *-sigma-biderivation is characterized on a prime *-ring and show that each *-biderivation maps to the center of ring.

A topoframe, denoted by $L_{ tau}$, is a pair $(L, tau)$ consisting of a frame $L$ and a subframe $ tau $ all of whose elements are complementary elements in $L$. $f$-ring $mathcal{R}(L_{ tau})$ is equal to the set $${fin Frm(mathcal{P}(mathbb R), L): f(mathfrak{O}(mathbb R))subseteq tau}. $$ In this paper, for every complemented element $ain L$ with $a, a'in tau$, we introduce an idempotent element $f_{a}$ belong to $mathcal{R}(L_{ tau})$ and we show that an ideal $I$ of $mathcal{R}(L_{ tau})$ is minimal if and only if there exists an atom $a$ of $L$ such that $I$ is generated by $ f_a$ if and only if there exists an atom $a$ of $L$ such that $ I={fin mathcal{R}(L_{ tau}): coz(f)leq a} $. Also, we prove that the socle of $f$-ring $mathcal{R}(L_{ tau})$ consists of those $f$ for which $coz (f)$ is a join of finitely many atoms and finally, we show that the $f$-ring $mathcal{R}(L_{ tau})$ has Property (A) and if $L$ has a finite number of atoms then the residue class ring $ frac{mathcal{R}(L_{ tau})}{Soc (mathcal{R}(L_{ tau}))}$ has Property (A).

In this paper, we define hedge operation on a residuated skew lattice and investigate some its properties. We get relationships between some special sets as dense, nilpotent, idempotent, regular elements sets and their hedges. By examples, we show that hedge of a dense element is not a dense and hedge of a regular element is not a regular. Also hedge of a nilpotent element is a nilpotent and hedge of an idempotent element is not an idempotent and we show that hedge of an idempotent element is idempotent under a condition. We show that the composition of two hedges is a hedge under a condition. By examples, we show that there is no relation between the product of the hedges and the hedge of the products. H(x) is defined and some its properties are proved and is shown that hedge of set H is equal to the H itself. The relationship between H and dense, regular, nilpotent and idempotent elements is investigated too.

Modeling and optimal solving of supply chain management problems lead to efficient decision making in strategic planning and supply chain operations, resulting in a competitive advantage. Today, with the planning of a sustainable supply chain, in addition to achieving economic goals, it is possible to meet social and environmental objectives and considerations. This research deal with sustainable closed-loop supply chain network design and operations planning problem in which is human resource employment and training are considered. First, a three-objective optimization model is developed in which the supply chain network is designed and strategic variables (such as location and capacity determination, technology selection, skilled or semi-skilled employment and training, and etc. ) are obtained. Then, a multi-period model is proposed supply chain operations planning in which the amount of production, inventory, shortage, temporary recruitment of manpower, etc. in each period are determined. In the proposed strategic model, a trade-off between the objectives of minimizing the cost of the supply chain (economic), maximizing employment (social), and minimizing environmental impacts is done by augmented epsilon constraint method. Also, Benders decomposition algorithm is used to solve large-scaled instances. In the final section of the research, some numerical studies are presented to provide numerical results, managerial insights and evaluating the performance of the proposed model and solution approaches.

The non-abelian tensor product of groups which has its origins in algebraic K-theory as well as in homotopy theory, was introduced by Brown and Loday in 1987. Group theoretical aspects of non-abelian tensor products have been studied extensively. In particular, some studies focused on the relationship between the exponent of a group and exponent of its tensor square. On the other hand, computation of c-fold tensor products is generally a difficult problem. Several authors have given upper bounds for the order of G, when G is a finite p-group. In this paper we determine a bound for the exponent triple tensor product which sharpens a bound of G. Ellis. Let G be a nilpotent group of nilpotency class k≥ 3 and prime power exponent P^e (where p is a prime and is not equal 3). In this paper, we show that the exponent of triple tensor product of G, that is, (G⨂ G)⨂ G, divides P^([k/2]-1)e where [k/2] denotes the smallest integer n such that n≥ k/2. In this way, the exponent provided by Ellis is improved.

In this paper we defined the concept of module amenability of Banach algebras and module connes amenability of module dual Banach algebras. Also we assert the concept of module Arens regularity that is different with [1] and investigate the relation between module amenability of Banach algebras and connes module amenability of module second dual Banach algebras. In the following we study the relation between module amenability, weak module amenability and module approximate amenability of Banach algebra. The notation of amenability of Banach algebras was introduced by B. Johnson in [7]. A Banach algebra A is amenable if every bounded derivation from A into any dual Banach A-bimodule is inner, equivalently if H(A; X) = 0 for any Banach A-bimodule X, where H(A; X) is the first Hochschild co-homology group of A with coefficient in X. Also, a Banach algebra A is weakly amenable if H(A; A) = 0. Bade, Curtis and Dales introduced the notion of weak amenability on Banach algebras in [4]. They considered this concept only for commutative Banach algebras. After that Johnson defined the weak amenability for arbitrary Banach algebras.

The existence of infinitely many solutions for an anisotropic discrete non-linear problem with variable exponent according to p(k)– Laplacian operator with Dirichlet boundary value condition, under appropriate behaviors of the non-linear term, is investigated. The technical approach is based on a local minimum theorem for differentiable functionals due to Ricceri. We point out a theorem as a special case of our main results. We define two differentiable functionals and set up the variational framework and present an applied lemma which $lambda$ lying in a well-defined interval. Bearing in mind this fundamental lemma and the local minimum theorem due to Ricceri, we obtain our result which is the existence of a sequence of infinitely many solutions which converges to zero depending on the nonlinear term has suitable behavior at zero. We ensure exact interval of the parameter $lambda$, in which the anisotropic discrete non-linear problem admits infinitely solutions such that their norm converges to zero. Some remarks and corollaries and the proof of especial case theorem are provided. Some examples are inserted to illustrate the importance of the main results.

The most exchange volume in a country will be obtained through bank system whose correct function will have a determinant role in improving economic activities. Nowadays, the customer’ s rating and accreditation subject has been considered more than before by the banks due to increase the volume of overdue claims and banks’ past over dues. One of the most important tools for controlling the banks’ credit risk is to implement the credit rating system and to recognize effective factors on withholding payment of the bank advances. The goal of this article is to investigate the relation between efficiency and risk in banking system. Based on this, with the help of data envelopment analysis method and the information related to evaluation, rating and interpretation of credit risk which has been suggested by three main rating institutes like “ Fitch, Moody’ s and Standard and Poor’ s” , the credit risk of 24 legal customers of Arak Melli Bank was evaluated and the credit rate of each legal companies was determined and their quality analysis was expressed.

Considering profiles as functional variables, two control charts are proposed for their monitoring in phase II. Due to its conformity with the nature of real-world profiles, applying functional model leads to proposed control charts obtained through functional data analysis techniques with desired features. These include simplicity in calculation and possibility of using them for different profiles such as linear and non-linear (even in the presence of complex within-profile Autocorrelation forms). These features distinguish the functional model from the regression models common in profile monitoring. Simulated computer simulations show that, in different states, the proposed control charts have a lower average run length than other methods, which indicates the desired performance of the proposed functional approach. Morevere, in some non-linear cases with complex autocorrelation, other methods completely fail, and only the proposed control charts are able to detect the occurring deviation, and even in these cases, the average run length of these control charts is highly desirable.

Weighted composition operators appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of weighted composition operators, are generalized weighted composition operators which in special cases of their inducing functions give different types of well-known operators like: weighted composition operators, composition operators, multiplication operators and composition operators followed by differentiation operators. In this paper we study generalized weighted composition operators and give estimates for the essential norm of such operators on certain Banach spaces of analytic functions into weighted type spaces. The underlying Banach spaces of analytic functions include Bloch spaces, Zygmund spaces and weighted type spaces. Our estimates for the essential norms of generalized weighted composition operators imply necessary and sufficient conditions for the compactness of such operators. As another application of our results, we obtain essential norm estimates of certain well-known operators which are special cases of generalized weighted composition operators.