# Journal of Mathematical Extension Volume:15 Issue: 3, Summer 2021

• تاریخ انتشار: 1400/01/24
• تعداد عناوین: 18
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• P. Moatamed*, Mahnaz Barkhordari Page 1

In this research an analytic method is used to solve fuzzy heat equation with fuzzy initial values, which is based on ogeneralized Hukuhara differentiability. At rst, we dene fuzzy laplace transform in t considering x as a parameter on fuzzy functions u(x; t) and their derivatives by using generalized Hukuhara differentiability. The fuzzy laplace transform is used in an analytic method for solving the fuzzy partial differential equation with generalized Hukuhara differantiation.finally, the method is explained by presenting examples.ش

Keywords: Fuzzy-number, Fuzzy-valued function, Generalized differentiability, Fuzzy partial differential equation, Fuzzy Laplace transform
• Ali Zohri* Page 2

Abstract. The concept of 2-normed spaces and 2-Banach spacesare considerd as generalization of normed and Banach spaces. Inthe present paper we have studied the existence of square rootsand quasi square roots of some elements of a 2-Banach algebra.Also relation between nth roots and quasi nth root of elements of2-Banach algebras are considered.

Keywords: 2-normed algebras, square root, quasi square root, 2-normed spaces
• Behnam Azizi, Hossein Doostie* Page 3

In this paper‎, ‎we discuss the periodicity problems in the finitely generated algebraic structures and exhibit their natural‎ ‎sources in the theory of invariants of finite groups and it forms an interesting and relatively self-contained nook in the‎ ‎imposing edifice of group theory‎. ‎One of the deepest and important results of the related theory of finite‎ ‎groups is a complete classification of all periodic groups‎, ‎that is‎, ‎the finite groups with periodic properties‎. ‎If an integer‎ ‎be $k\geq 2$‎, ‎let $S$ will be a finite $k$-generated as well as non-associative algebraic structure $S=$‎, ‎where‎ ‎$A=\lbrace a_{1}‎, ‎a_{2},\dots‎, ‎a_{k}\rbrace$‎, ‎and the sequence‎ ‎$$x_{i}=\left\{‎ ‎\begin{array}{ll}‎ ‎a_{i}‎, ‎& 1\leq i\leq k‎, ‎\\‎ ‎x_{i-k}(x_{i-k+1}(\ldots(x_{i-3}(x_{i-2}x_{i-1}))\ldots))‎, ‎& i>k‎, ‎\end{array}‎ ‎\right‎. ‎$$‎ ‎is called the $k$-nacci sequence of $S$ with respect to the generating set $A$‎, ‎as denoted in $k_{A}(S)$‎. ‎When $k_{A}(S)$ is periodic‎, ‎we will use the length of the period of the periodicity length of $S$ proportional to $A$ in $LEN_{A}(S)$‎ ‎and the minimum of the positive integers of $LEN_A(S)$ will be mentioned as periodicity invariant of $S$‎, ‎denoted in $\lambda_k(S)$‎. ‎However‎, ‎this invariant has‎ ‎been studied for groups and semigroups during the years as well as the associative property of $S$ where above sequence was reduced to‎ ‎$x_i=x_{i-k}x_{i-k+1}\dots x_{i-3}x_{i-2}x_{i-1}$‎, ‎for every $i\geq k+1$‎. ‎Thus‎, ‎we attempt to give explicit upper‎ ‎bounds for the periodicity invariant of two infinite classes of‎ ‎finite non-associative $3$-generated algebraic structures‎. ‎Moreover‎, ‎two classes of non-isomorphic Moufang loops of the same periodicity length were obtained in the study‎.

Keywords: Non-associativity‎, ‎Loops‎, ‎Periodic sequences‎, Finitely generated structures‎
• A. Fakharzadeh, N. Naseri Shams* Page 4

Recently, absolute value equations (AVEs) are lied in theconsideration center of some researchers since they are very suitable al-ternatives for many frequently occurring optimization problems. There-fore, nding a fast solution method for these type of problems is verysignicant. In this paper, based on the mixed-type splitting (MTS) ideafor solving linear system of equations, a new fast algorithm for solvingAVEs is presented. This algorithm has two auxiliary matrices whichare limited to be nonnegative strictly lower triangular and nonnega-tive diagonal matrices. The convergence of the algorithm is discussedvia some theorems. In addition, it is shown that by suitable choice ofthe auxiliary matrices, the convergence rate of this algorithm is fasterthan that of the SOR, AOR, Generalized Newton, Picard and SOR-like methods. Eventually, some numerical results for dierent size ofproblem dimensionality are presented which admit the credibility of theproposed algorithm.

Keywords: Absolute value equations, M−splitting, Mixed-type splitting method, Unique solution, Spectral radius

‎We first introduce a novel notion named $C^{*}$-algebra-valued $b_{v}(s)$-metric spaces‎. ‎Then‎, ‎we give proofs of the Banach contraction principle‎, ‎the expansion mapping theorem‎, ‎and Jungck's theorem in $C^{*}$-algebra-valued $b_{v}(s)$-metric spaces‎. ‎As an application of our results‎, ‎we establish a result for an integral equation in a $C^{*}$-algebra-valued $b_{v}(s)$-metric space‎. ‎Finally‎, ‎a numerical method is presented to solve the proposed integral equation‎, ‎and the convergence of this method is also studied‎. ‎Moreover‎, ‎a numerical example is given to show applicability and accuracy of the numerical method and guarantee the theoretical results‎.

Keywords: $C^{*}$-algebra, $b, {v}(s)$-metric space, Fixed point theorem, Integral equation, Contractive mapping
• Fateme Alinaghian, Farhad Khaksar Haghani, Shahram Heidarian* Page 6

In this paper, we generalize the concepts of para and quasi topological MV -algebras, which was first introduced by Najafi et al. in 2017, to BL-algebras as para and quasi topological BL-algebras and elaborate these concepts via some examples. We further derive and prove some theorems by employing pre-filters and a fundamental system of neighborhoods.

Keywords: BL-algebra, Para, Quasi-Topological BL-algebra, Ideal, Pre-filter, Filter
• Amir Peyravi* Page 7

This paper deals with the spatial behavior of solutions for aviscoelastic wave equations with nonlinear dissipative terms in asemi-infinite $n$-dimensional cylindrical domain. An alternativeof Phragm\'{e}n-Lindel\"{o}f type theorems is obtained in theresult. In the case of decay, an upper bound will be derived forthe total energy by means of the boundary data. The main point ofthe contribution is the use of energy method.

Keywords: Spatial estimates, hyprbolic equations, viscoelasticity
• Akram Mohammadpouri, Rana Abbasi, Roya Abbasi, Mitra Narimani* Page 8

In this paper, we give a complete classification of the translation hypersurfaces in the (n+1)-dimensional Lorentz-Minkowski space whose Gauss map G satisfies the condition Ln−1G = AG where Ln−1 is the linearized operator of the first variation of the the Gauss-Kronecker curvature of the hypersurface and A ∈ R (n+1)×(n+1) is a constant matrix.

Keywords: Gauss-Kronecker curvature, Lorentz-Minkowskispace, Linearized operator Ln−1, Translation hypersurface

The theory of algebraic frames for a Hilbert space $H$ is a generalization of the theory of frames and generalized frames. The paper applies the theory of unbounded operators to define the dual of algebraic frames with densely defined unbounded analysis operators. It is shown that every algebraic frame has an algebraic dual frame, and if an algebraic frame has a nonzero redundancy, then it is not Riesz-type. An example of an algebraic frame with finite redundancy is constructed which is not a Riesz-type algebraic frame. Finally, for a lower bounded analytic frame, the discreteness of its indexing measure space and the uniqueness of its algebraic dual are studied and shown to be interrelated.

Keywords: Unbounded operators, algebraic frames, algebraic dual frame, dual frame, generalized frames
• Halil İbrahim Yoldaş* Page 10

In this paper, we study $N(k)-$contact metric manifolds endowed with a torse-forming vector field and give some characterizations for such manifolds. Then, we deal with $N(k)-$contact metric manifolds admitting a Ricci soliton and find that the potential vector field $V$ of the Ricci soliton is a constant multiple of $\xi$. Also, we obtain a necessary condition for a torse-forming vector field to be recurrent and Killing on $M$.

Keywords: N(k)−contact metric manifold, Recurrenttensor field, Ricci Soliton, Sasakian manifold, Torse-forming vector field
• Sedigheh Barootkoob, Hosein Lakzian* Page 11

The concept of a $w$-distance on a‎ ‎metric space has been introduced by Kada et al‎. ‎\cite{Kst}‎. ‎They generalized Caristi‎ ‎fixed point theorem‎, ‎Ekeland variational principle and the‎ ‎nonconvex minimization theorem according to Takahashi‎. ‎In the present paper‎, ‎we first introduce the notion of quasi $w$-distances in quasi-metric spaces and then we will prove some fixed point theorems for $\mathcal{L}$-contractive mappings in the class of quasi-metric spaces with $w$-distances via a control function introduced by Jleli and Samet \cite{JL}‎. ‎These results generalize many fixed point theorems by Kada et al‎. ‎\cite{Kst}‎, ‎Suzuki \cite{S}‎, ‎Ciri\'{c} \cite{ciric}‎, ‎Aydi et al‎. ‎\cite{Aydbarlak}‎, ‎Abbas and Rhoades \cite{Ar}‎, ‎Kannan \cite{Kannan}‎, ‎Hicks and Rhoades \cite{H}‎, ‎Du \cite{D}‎, ‎Lakzian et al‎. ‎\cite{LAR}‎, ‎Lakzian and Rhoades \cite{LR} and others‎. ‎Some examples in support of the given concepts and presented results‎.

Keywords: Quasi-metric space, Fixed point, $, mathcal{L}$-contraction
• Ali Khalouta, Abdelouahab Kadem* Page 12

The main objective of this paper is to propose a new analytical method called the inverse fractional Aboodh transform method for solving fractional differential equations. Fractional derivatives are taken in the Riemann-Liouville and Caputo sense. The main advantages of this method it that it is direct and concise. Various examples are given to shows that the proposed method is very efficient and accurate.

Keywords: Fractional differential equations, Riemann-Liouville derivative, Caputo derivative, Aboodh transform method
• Nawal Alsarori, Kirtiwant P. Ghadle* Page 13

Nonlocal functional fractional differential inclusions with impulses effect in Banach spaces are studied. This paper deals with the case when the multivalued function is lower semicontinuous and nonconvex as well as the linear term generates a semigroup which is not,in general, compact. Our results are obtained by using NCHM (noncompactness Hausdorff measure), multivalued properties and theorems of fixed point. We finally present an example to lighten our results.

Keywords: Semilinear impulsive differential inclusions, Nonlocal conditions, Fixed point theorems, Mild solutions
• Nesa Eshagh Nimvari, Seadat Ollah Faramarzi* Page 14

Let (R, m) be a d-dimensional Noetherian local ring and T be a commutative strict algebra with unit element 1T over R such that mT 6= T. We define almost exact sequences of T-modules and characterize almost flat T-modules. Moreover, we define almost (faithfully) flat homomorphisms between R-algebras T and W, where W has similar properties that T has as an R-algebra. By almost (faithfully) flat homomorphisms and almost flat modules, we investigate Cousin complexes of T and W-modules. Finally, for a finite filtration F = (Fi)i≥0 of length less than d of Spec(T) such that it admits a T-module X, we show that IE 2 p,q := TorT p M, H d−q (CT (F, X)) p⇒ Hp+q(Tot(T )) and IIE 2 p,q := Hd−p TorT q (M, CT (F, X)) p⇒ Hp+q(Tot(T )), where M is an any flat T-module and as a result we show that IE 2 p,q and IIE 2 p,q are almost zero, when M is almost flat

Keywords: Almost flat, almost flat homomorphism, Cousin complex, filtration, spectral sequence
• Rasoul Soleimani* Page 15

Let G be a group and AutΦ(G) denote the group of all automorphisms of G centralizing G/Φ(G) elementwise. In this paper, we give a necessary and sufficient condition on a finite p-group G for the group AutΦ(G) to be elementary abelian.

Keywords: Automorphism group, Finite p-group, Frattini subgroup
• Vali Torkashvand, Mohammad Ali Fariborzi Araghi* Page 16

An efficient family of the recursive methods of adaptive is proposed for solving nonlinear equations, is developed such that all previous information are applied. These methods have reached the maximum degree of convergence improvement of 100%, and also have an efficiency index of 2. Three families have been examined from Steffensen-Like single, two, and three-step methods that have used 2, 3 and 4 parameters respectively. Numerical comparisons are made with other existing methods one-, two-, three-, and four-point to show the performance of the convergence speed of the proposed method and confirm theoretical results.

Keywords: Adaptive method with memory, Accelerator parameter, Nonlinear equations, Newton's interpolatory polynomial, Order convergence
• Azizalah Kord, Ali Payan, Sabar Saati* Page 17

The agricultural sector ensures food security in every country.Optimal agricultural practices presuppose the optimal allocationof resources, including water, soil, etc., by official authorities in everycountry because excessive use of natural resources would have harmfulconsequences for posterity despite meeting ad hoc needs. Therefore,sustainable agricultural practices in different regions should be basedon environmental, social, and economic criteria in the decision-makingprocess for the future. This study investigated the agricultural practicesin two stages: environmental stage (planting and maintaining)and economic stage (harvesting), which use shared resources. A networkDEA model was proposed for developing sustainable agriculturalpractices based on the proposed process. The development of sustainableagricultural practices in different regions presupposes the optimal allocation of water and human resources, which is realized by the improvementof irrigation methods and the quality of life of farmers. Innetwork DEA models, weight restrictions are used to determine sustainabledevelopment. The proposed model was analyzed with and withoutweight restrictions to determine the sustainable development of agriculturein Sistan and Baluchestan Province, Iran, between 2013 and 2017.On the other hand, given the important role human resources play inthe development of sustainable agricultural practices, some models weresuggested for determining the maximum required human resources foreach stage according to the obtained performance levels.

Keywords: Sustainability, Resource Allocation, Agricultural Practices, Network DEA
• Mohsen Kian, Mahdi Dehghani, Mostafa Sattari* Page 18

‎The main purpose of this note is to define an analogues of the numerical radius related to the matricial range‎. ‎However‎, ‎we will find relations between the numerical radius and matricial range of an operator‎. ‎The tone of the paper is mostly expository‎.

Keywords: Numerical range‎, ‎matricial range‎, ‎completely positive mapping‎, numerical radius‎