# Journal of Mathematical Extension Volume:15 Issue: 2, Spring 2021

• تاریخ انتشار: 1400/01/10
• تعداد عناوین: 18
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The Birnbaum-Saunders (BS) distribution is one of the most con-sidered right-skewed distributions to model failure times for materials subjectto lifetime data. In this paper, a new extension of the BS model is initiallyproposed based on the family of mean-mixtures of normal distributions. Then,we present a new probabilistic mixture model based on the new extended BSdistribution for modeling and clustering right-skewed and heavy-tailed data.The maximum likelihood (ML) parameter estimates of the model in questionare estimated by employing an expectation-maximization (EM) type algorithm.Moreover, the empirical information matrix is derived by using an information-based approach. Simulations and real data analysis illustrate the performanceof the proposed methodology.

Keywords: Birnbaum-Saunders distribution, Mean-mixtures of normal distributions, Finite mixture model, ECM algorithm
• Mandana Moccari, Taher Lotfi Page 2

This ‎paper‎ deals with the study of relaxed conditions for semi-local convergence for a general iterative method, k-step Newton's method, using ‎majorizing‎ sequences. Dynamical behavior of the mentioned method is also analyzed via Julia set and basins of attraction. Numerical examples of nonlinear systems of equations will be examined to clarify the given theory.

Keywords: Majorizing sequence, High order of convergence, Semi-local convergence, Julia set, Basin of attraction
• MohammadReza Heidari Tavani Page 3

In the present paper, by using the variational methods incritical point theory, the existence and multiplicity of periodic solutionsfor a class of p-Hamiltonian systems is established. In fact, using twofundamental theorems that are attributed to Bonanno, we get someimportant results. Are presented the results were extention of someexisting results.

In this paper, we investigate conditions under which the numerical range of a composition operator, acting on a Hilbert space, contains zero as an interior point and we investigate extreme points of the numerical range of an operator acting on an arbitrary Banach space. Also, we give necessary and suﬃcient conditions under which the numerical range of an operator on some Banach spaces, to be closed. Finally, we characterize the structure of the numerical range of an operator acting on Banach weighted Hardy spaces.

Keywords: Banach weighted Hardy spaces, bounded point evaluation, composition operator, numerical range, extreme point
• Habibollah Ansari Toroghy, Faranak Farshadifar, Sepideh ‎Maleki Roudposhti Page 5

‎In this paper‎, ‎we introduce the concepts of strongly 2-absorbing primary ideals (resp.‎, ‎submodules) and strongly 2-absorbing ideals (resp.‎, ‎submodules) as generalizations of strongly prime ideals‎. ‎Furthermore‎, ‎we investigate some basic properties of these‎ ‎classes of ideals‎.

Keywords: Strongly prime ideal‎, ‎strongly 2-absorbing primary ideal‎, ‎strongly 2-absorbing primary submodule‎, ‎strongly 2-absorbing ideal‎, ‎strongly 2-absorbing submodule
• Sheila A. Bishop, E.O. Ayoola Page 6

We study the properties of the existence and uniqueness of solu-tions of a class of evolution quantum stochastic differential equations(QSDEs) dened on a locally convex space whose topology is gen-erated by a family of seminorms dened via the norm of the rangespace of the operator processes. These solutions are called strong solutions in comparison with the solutions of similar equations denedon the space of operator processes where the topology is generated bythe family of seminorms dened via the inner product of the rangespace. The evolution operator generates a bounded semigroup. Weshow that under some more general conditions, the unique solutionis stable. These results extend some existing results in the literatureconcerning strong solutions of quantum stochastic differential equa-tions.

Keywords: Strong solutions, Stability, Bounded semigroup, General Lipschitz condition

In this investigation, we solve the Caputo's fractional parabolic partial integro-differential equations (FPPI-DEs) by Gaussian-radial basis functions (G-RBFs) method. The main idea for solving these equations is based on RBF which also provides approaches to higher dimensional spaces.In the suggested method, FPPI-DEs are reduced to nonlinear algebraic systems. We propose to apply the collocation scheme using G-RBFs to approximate the solutions of FPPI-DEs. Error analysis of the proposed method is investigated. Numerical examples are provided to show the convenience of the numerical schemes based on the G-RBFs. The results reveal that the method is very efficient and convenient for solving such equations.

Keywords: Fractional parabolic partial integro-differential equations, Radial basis functions, Collocation method, Quadrature methods
• M. E. Samei∗, G. K. Ranjbar, D. Nazari Susahab Page 8

. In the current work, we present some innovative solutions for the attractivity of fractional functional q-differential equations involving Caputo fractional q-derivative in a k-dimensional system, by using some fixed point principle and the standard Schauder’s fixed point theorem. Likewise, we look into the global attractivity of fractional q-differential equations involving classical Riemann-Liouville fractional q-derivative in a k-dimensional system, by employing the famous fixed point theorem of Krasnoselskii. Also, we must note that, this paper is mainly on the analysis of the model, with numerics used only to verify the analysis for checking the attractivity and global attractivity of solutions in the system. Lastly, we give two examples to illustrate our main results.

Keywords: Positive attractivity, fractional q-differentialequations, fractional Caputo type q-derivative, Riemann-Liouville fractional q-derivative
• A. A. Alijani Page 9

Let £ be the category of all locally compact abelian (LCA) groups. Let G ∈ £ and H ⊆ G. The maximal torsion subgroup of G is denoted by tG and the closure of H by H. A proper short exact sequence 0 → A ϕ→ B ψ→ C → 0 in £ is said to be a generalized textension if 0 → tA ϕ→ tB ψ→ tC → 0 is a proper short exact sequence. We show that the set of all generalized t-extensions of a torsion group A ∈ £ by a compact group C ∈ £ is a subgroup of Ext(C, A). We establish conditions under which the generalized t-extensions split.

Keywords: : Generalized t-extensions, t-extensions, locally compact abelian groups
• G. S¸enel Page 10

This paper is about, the detailed analysis of soft ditopological space theory (SDT - space theory) is ameliorated by introducing new soft sets called ˜δ-b-unclosed sets, ˜δ-b-closed sets and ˜δ-b-dense sets, which are needed for the definition of extremally disconnected spaces and submaximal spaces in soft ditopological spaces. Moreover, ˜δ-regular- unclosed, ˜δ-preunclosed, ˜δ-semi unclosed, ˜δ- α-unclosed and ˜δ-β- unclosed sets in SDT-space are determined and studied relations between these sets in detail. A new idea is introduced in order to prove relations, which gives an affirmative answer to understand the structure of SDT-spaces. It is demonstrated that separately these frameworks can in any case be very confounded, a perhaps increasingly tractable errand is to portray their conceivable joint dispersions by using recently characterized ˜δ-sets.

Keywords: ˜δ-b-unclosed set, ˜δ-b-closed set, ˜δ-b-denseset, ˜δ-regular- unclosed set, ˜δ-preunclosed set, ˜δ-semi unclosed set, ˜δα-unclosed set, ˜δ-β- unclosed set, ˜δ-b-extremally disconnected space, ˜δ-b-submaximal space
• N. Ilaghi, M. Maani Shirazi, Sh. Khoshdel ∗ Page 11

In this paper, we introduce the concept of primary submodules over S which is a generalization of the concept of S-prime submodules. Suppose S is a multiplicatively closed subset of a commutative ring R and let M be a unital R-module. A proper submodule Q of M with (Q :R M) ∩ S = ∅ is called primary over S if there is an s ∈ S such that, for all a ∈ R, m ∈ M, am ∈ Q implies that sm ∈ Q or san ∈ (Q :R M), for some positive integer n. We get some new results on primary submodules over S. Furtheremore, we compare the concept of primary submodules over S with primary ones. In particular, we show that a submodule Q is primary over S if and only if (Q :M s) is primary, for some s ∈ S.

Keywords: Multiplicatively closed subset, Multiplication module, Primary module, primary module over S
• C. Chesneau∗, F. Jamal Page 12

In this paper, we introduce a new trigonometric family of continuous distributions called the sine Kumaraswamy-G family of distributions. It can be presented as a natural extension of the wellestablished sine-G family of distributions, with new perspectives in terms of applicability. We investigate the main mathematical properties of the sine Kumaraswamy-G family of distributions, including asymptotes, quantile function, linear representations of the cumulative distribution and probability density functions, moments, skewness, kurtosis, incomplete moments, probability weighted moments and order statistics. Then, we focus our attention on a special member of this family called the sine Kumaraswamy exponential distribution. The statistical inference for the related parametric model is explored by using the maximum likelihood method. Among others, asymptotic confidence intervals and likelihood ratio tests for the parameters are discussed. A simulation study is performed under varying sample sizes to assess the performance of the model. Finally, applications to two practical data sets are presented to illustrate its potentiality and robustness.

Keywords: Sine-G family of distributions, Kumaraswamydistribution, moments, practical data sets
• Wajih Ashraf, Shabir Ahmad Ahangar, Abdus Salam, Noor Mohammad Khan Page 13

Dominions have been studied from different perspectives however their major application lies to study the closure property for monoids. The most useful characterization of semigroup domonions is provided by the famous Isbell’s Zigzag Theorem. In this paper, we introduce the dominion of a $\Gamma$-semigroups and give the analogue of Isbell's zigzag theorem in $\Gamma$-semigroups.

Keywords: Dominions, Zigzag equations, $, Gamma$-semigroups
• M. R. Foroutan∗, A. Ebadian, M. R. Yasamian Page 14

In this work, we introduce and investigate an interesting operator Q ν λ based on fractional derivative which is introduced by Owa and Srivastava in [10]. We consider a new technique to prove our results and then, we introduce two subclasses of analytic functions in the open unit disk U concerning with this operator. Some results such as inclusion relations, subordination properties, integral preserving properties and argument estimate are investigated.

Keywords: Analytic function, subordination, integraloperator, fractional derivative, argument
• M. Shah Hosseini∗, H. R. Moradi, B. Moosavi Page 15

This paper is mainly devoted to studying operator Jensen inequality. More precisely, a new generalization of Jensen inequality and its reverse version for convex (not necessary operator convex) functions have been proved. Several special cases are discussed as well.

Keywords: Operator order, Jensen’s inequality, convexfunctions, self-adjoint operators, positive operators
• G. Tohidi, S. Razavyan∗, S. Tohidnia Page 16

In traditional data envelopment analysis (DEA), the efficiency and productivity changes computations are based on an optimistic perspective and an efficient unit may perform rather poorly when the realist weights are assigned to inputs and outputs. Hence, the results of productivity change of decision making units (DMUs) between two time periods may change from progress to regress or vice versa when the weights are modified. Because of the cross efficiency merits, we use it to obtain a common set of weights so called common set of cross weights. On the other hand, we need a base for comparing the productivity change of DMUs. To this end, the common set of cross weights are used to approximate the cross efficient frontier as a base for determining cross Malmquist (CM) index for evaluating the productivity change. This leads to introduce a new efficiency, weight efficiency, and the decomposition of the cross efficiency. Some DEA and cross efficiency models are modified to find the value of the proposed CM index and its components. An empirical example is used to compare the proposed method and the technical Malmquist index.

Keywords: Data envelopment analysis (DEA), Cross efficiency, Malmquist index, Productivity, Common set of weights (CSW)
• Z. Izadi, R. Soltani∗ Page 17

Let H be a separable infinite dimensional complex Hilbert space and SA(H) be the real Jordan algebra of all bounded self-adjoint operators acting on H. In this paper, we study the general form of surjective non-linear maps ξ : SA(H) → SA(H), that preserve the difference of minimum and surjectivity moduli of self-adjoint operators in both directions. It turns out that ξ(PLet H be a separable infinite dimensional complex Hilbert space and SA(H) be the real Jordan algebra of all bounded self-adjoint operators acting on H. In this paper, we study the general form of surjective non-linear maps ξ : SA(H) → SA(H), that preserve the difference of minimum and surjectivity moduli of self-adjoint operators in both directions. It turns out that ξ(P) = EP E∗ + R, (P, R ∈ SLet H be a separable infinite dimensional complex Hilbert space and SA(H) be the real Jordan algebra of all bounded self-adjoint operators acting on H. In this paper, we study the general form of surjective non-linear maps ξ : SA(H) → SA(H), that preserve the difference of minimum and surjectivity moduli of self-adjoint operators in both directions. It turns out that ξ(P) = EP E∗ + R, (P, R ∈ SA(H)) where E : H → H, is either a bounded unitary or an anti-unitary operator.A(H)) where E : H → H, is either a bounded unitary or an anti-unitary operator.) = EP E∗ + R, (P, R ∈ SA(H)) where E : H → H, is either a bounded unitary or an anti-unitary operator.

Keywords: Non-linear preserver problems, algebraicoperators, algebraic singularity
• Masoumeh Neghabi, Abasalt Bodaghi, Abbas Zivari Kazempour Page 18

In this paper, we show that every $(n,m)$-Jordan homomorphism between two commutative algebras is an $(n,m)$-homomorphism. For the non-commutative case, we prove that every surjective $(2,m)$-Jordan homomorphism from an algebra $\mathcal{A}$ to a semiprime commutative algebra $\mathcal{B}$ is $(2,m)$-homomorphism.

Keywords: $n$-homomorphism, $n$-Jordan homomorphism, mixed $n$-Jordan homomorphism, automatic continuity, semisimple