فهرست مطالب
 Volume:12 Issue: 2, Summer and Autumn 2023
 تاریخ انتشار: 1402/02/11
 تعداد عناوین: 40


Pages 16
Let $(R,m)$ be a commutative Noetherian local ring, $a$ an ideal of $R$. Let $t\in\Bbb N_0$ be an integer and $M$ a finitely generated $R$module such that the $R$module $\mathfrak{F}^i_{\mathfrak{a}}(M)$ is $\mathfrak{a}$cominimax for all $i<t$. We prove that For all minimax submodules $N$ of $\mathfrak{F}^i_{\mathfrak{a}}(M)$, the $R$modules \[ Hom_R(R/\mathfrak{a},\mathfrak{F}^t_{\mathfrak{a}}(M)/N)\hspace{5mm} and \hspace{5mm}Ext^1_R(R/\mathfrak{a},\mathfrak{F}^t_{\mathfrak{a}}(M)/N) \] are minimax. In particular, the set $Ass_R(\mathfrak{F}^t_{\mathfrak{a}}(M)/N)$ is finite.
Keywords: formal local cohomology modules, local cohomology, cominimax modules 
Pages 718
Let $R$ be a prime ring, $\alpha$ an automorphism of $R$ and $b$ an element of $Q$, the maximal right ring of quotients of $R$. The main purpose of this paper is to characterize skew $b$derivations in prime rings which satisfy various differential identities. Further, we provide an example to show that the assumed restrictions cannot be relaxed.
Keywords: Automorphism, Derivation, Skew derivation, Skew derivation, Prime ring 
Pages 1927
Let $R$ be a ring with an endomorphism $\alpha$. A ring $R$ is a skew power series McCoy ring if whenever any nonzero power series $f(x)=\sum_{i=0}^{\infty}a_ix^i,g(x)=\sum_{j=0}^{\infty}b_jx^j\in R[[x;\alpha]]$ satisfy $f(x)g(x)=0$, then there exists a nonzero element $c\in R$ such that $a_ic=0$, for all $i=0,1,\ldots$. We investigate relations between the skew power series ring and the standard ringtheoretic properties. Moreover, we obtain some characterizations for skew power series ring $R[[x;\alpha]]$, to be McCoy, zip, strongly \textit{AB} and has Property (A).
Keywords: Noetherian ring, αcompatible ring, Skew Power series McCoyring, Zip ring, Reversible rin 
Pages 2956
One of the crucial stages in machine learning in highdimensional datasets is feature selection. Unrelated features weaknesses the efficiency of the model. However, merging several feature selection strategies is routine to solve this problem, the way to integrate feature selection methods is problematic. This paper presents a new ensemble of heuristics through fuzzy TypeI based on Ant Colony Optimization (ACO) for ensemble feature selection named AntEHFS. At first, three feature selection methods are run; then, the Euclidean Distance between each pair of features is computed as a heuristic (an M×M matrix is constructed), that M is the total of features. After that, a TypeI fuzzy is used individually to address various feature selections' uncertainty and estimate trustworthiness for each feature, as another heuristic. A complete weighted graph based on combining the two heuristics is then built; finally, ACO is applied to the complete graph for finding features that have the highest relevance together in the features space, which in each ant considers the reliability rate and Euclidean Distance of the destination node together for moving between nodes of the graph. Five and eight robust and wellknown ensemble feature selection methods and primary feature selection methods, respectively, have been compared with AntEHFS on six highdimensional datasets to show the proposed method's performance. The results have shown that the proposed method outperforms five ensemble feature selection methods and eight primary feature selections in Accuracy, Precision, Recall, and F1score metrics.
Keywords: Ant colony optimization, Highdimensional data, Featureselection, Ensemble feature selection, an ensemble of algorithms, Type 
Pages 5775
In this paper, we define the concepts of singlevalued neutrosophic general automaton, complete and deterministic singlevalued neutrosophic general automaton. We present a minimal singlevalued neutrosophic general automaton that preserves the language for a given singlevalued neutrosophic general automaton. Moreover, we present the closure properties such as union and intersection for singlevalued neutrosophic general automata.
Keywords: Neutrosophic set, Automata, Intuitionistic set, Submachine, General fuzzy automat 
Pages 77104
In this article, we aim at obtaining the analytical expression (not previously found and recorded in the literature) for the exact curved surface area of a hyperboloid of two sheets in terms of Appell's double hypergeometric function of second kind and triple hypergeometric function of Srivastava. The derivation is based on MellinBarnes type contour integral representations of generalized hypergeometric function$~_pF_q(z)$, Meijer's $G$function and series manipulation technique. Further, we also obtain the formula for the volume of hyperboloid of two sheets. The closed forms for the exact curved surface area and volume of the hyperboloid of two sheets are also verified numerically by using Mathematica Program.
Keywords: Meijer’s Gfunction, MellinBarnes type contour integrals, Hyperboloid of two sheets, General triple hypergeometric function of Srivastava, Appell’s function of second kind, Mathematica Progra 
Pages 105113
Synchronized systems, has attracted much attention in 1986 by F. Blanchard and G. Hansel, and extension of them has been of interest since that notion was introduced in 1992 by D. Fiebig and U. Fiebig. One was via half synchronized systems; that is, systems having half synchronizing blocks. In fact, if for a left transitive ray such as $\ldots x_{1}x_{0}m$ and $mv$ any block in $X$ one has again $\ldots x_{1}x_{0}mv$ a left ray in $X$, then $m$ is called half synchronizing. A block $m$ is minimal (half)synchronizing, whenever $w \varsubsetneq m$, $w$ is not (half)synchronizing. Examples with $\ell$ minimal (half)synchronizing blocks has been given for $0\leq \ell\leq \infty$. To do this we consider a $\beta$shift and will replace 1 with some blocks $u_i$ to have countable many new systems. Then, we will merge them.
Keywords: Minimal half synchronizing, Synchronizing, Entrop 
Pages 115136
The commutator theory, developed by Fresee and McKenzie in the framework of a congruencemodular variety $\mathcal{V}$, allows us to define the prime congruences of any algebra $A\in \mathcal{V}$ and the prime spectrum $Spec(A)$ of $A$. The first systematic study of this spectrum can be found in a paper by Agliano, published in Universal Algebra (1993).The reticulation of an algebra $A\in \mathcal{V}$ is a bounded distributive algebra $L(A)$, whose prime spectrum (endowed with the Stone topology) is homeomorphic to $Spec(A)$ (endowed with the topology defined by Agliano). In a recent paper, C. Mure\c{s}an and the author defined the reticulation for the algebras $A$ in a semidegenerate congruencemodular variety $\mathcal{V}$, satisfying the hypothesis $(H)$: the set $K(A)$ of compact congruences of $A$ is closed under commutators. This theory does not cover the Belluce reticulation for noncommutative rings. In this paper we shall introduce the quasicommutative algebras in a semidegenerate congruencemodular variety $\mathcal{V}$ as a generalization of the Belluce quasicommutative rings. We define and study a notion of reticulation for the quasicommutative algebras such that the Belluce reticulation for the quasicommutative rings can be obtained as a particular case. We prove a characterization theorem for the quasicommutative algebras and some transfer properties by means of the reticulation.
Keywords: commutator operation, semidegenerate congruence  modular algebras, reticulation, spectral spaces 
Pages 137164
In this paper, closed forms of the sum formulas $ \sum_{k=0}^{n}kx^{k}W_{k} $ and $ \sum_{k=1}^{n} kx^{k}W_{ k} $ for generalized Hexanacci numbers are presented. As special cases, we give summation formulas of Hexanacci, HexanacciLucas, and other sixthorder recurrence sequences.
Keywords: Hexanacci numbers, HexanacciLucas numbers, sum formulas, summing formula 
Pages 165177
Let $G$ be a group. An automorphism $\alpha$ of a group $G$ is called a central automorphism, if $x^{1}x^{\alpha}\in Z(G)$ for all $x\in G$. Let $L_c(G)$ be the central kernel of $G$, that is the set of elements of $G$ fixed by all central automorphisms of $G$ and $Aut_{L_c}(G)$ denote the group of all central automorphisms of $G$ fixing $G/L_c(G)$ elementwise. In the present paper, we investigate the properties of such automorphisms. Moreover, a full classification of $p$groups $G$ of order at most $p^5$ where $Aut_{L_c}(G)=Inn(G)$ is also given.
Keywords: Automorphism group, central kernel, central autocommutator 
Pages 179186
The numerical tools that have outshinning many others in the history of Geometric Function Theory (GFT) are the Chebyshev and Gegenbauer polynomials in the present time. Recently, Gegenbauer polynomials have been used to define several subclasses of an analytic functions and their yielded results are in the public domain. In this work, analytic univalent functions defined by Gegenbauer polynomials is considered using closetoconvex approach of starlike function. Some early few coefficient bounds obtained are used to establish the famous FeketeSzego inequalities.
Keywords: Starlike function, Convex function, Analytic Univalent function, Coefficient Bounds, Gegenbauer polynomials 
Pages 187200
A hypersurface $ M^n $ in the LorentzMinkowski space $\mathbb{L}^{n+1} $ is called $ L_k $biharmonic if the position vector $ \psi $ satisfies the condition $ L_k^2\psi =0$, where $ L_k$ is the linearized operator of the $(k+1)$th mean curvature of $ M $ for a fixed $k=0,1,\ldots,n1$. This definition is a natural generalization of the concept of a biharmonic hypersurface. We prove that any $ L_k $biharmonic surface in $ \mathbb{L}^3 $ is $k$maximal. We also prove that any $ L_k $biharmonic hypersurface in $ \mathbb{L}^4 $ with constant $ k$th mean curvature is $ k $maximal. These results give a partial answer to the Chen's conjecture for $L_k$operator that $L_k$biharmonicity implies $L_k$maximality.
Keywords: Linearized operator Lk, Lkbiharmonic hypersurface, kmaximalhypersurface, kth mean curvatur 
Pages 201216
Nowadays, analyzing the losses data of the insurance and asset portfolios has special importance in risk analysis and economic problems. Therefore, having suitable distributions that are able to fit such data, is important. In this paper, a new distribution with decreasing failure rate function is introduced. Then, some important and applicable statistical indices in insurance and economics like the moments and moment generating function, value at risk, tail value at risk, tail variance, and Shannon and R\'enyi entropies are obtained. One of the advantages of this distribution is that it has fewer parameters compared to other distributions that have been introduced so far. Finally, this distribution is utilized as a proper distribution to fit on a real data set.
Keywords: Exponential distribution, Mean residual life, Tail ValueatRisk, Tail variance, ValueatRis 
On timelike hypersurfaces of the Minkowski 4space with 1proper second mean curvature vector / Weak convex , Lorentz hypersurface ,Biharmonic , harmonicPages 217233
The mean curvature vector field of a submanifold in the Euclidean $n$space is said to be $proper$ if it is an eigenvector of the Laplace operator $\Delta$. It is proven that every hypersurface with proper mean curvature vector field in the Euclidean 4space ${\Bbb E}^4$ has constant mean curvature. In this paper, we study an extended version of the mentioned subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski 4space ${\Bbb E}^4_1$. Let ${\textbf x}:M_1^3\rightarrow{\Bbb E}_1^4$ be the isometric immersion of a timelike hypersurface $M^3_1$ in ${\Bbb E}_1^4$. The second mean curvature vector field ${\textbf H}_2$ of $M_1^3$ is called {\it 1proper} if it is an eigenvector of the ChengYau operator $\mathcal{C}$ (which is the natural extension of $\Delta$). We show that each $M^3_1$ with 1proper ${\textbf H}_2$ has constant scalar curvature. By a classification theorem, we show that such a hypersurface is $\mathcal{C}$biharmonic, $\mathcal{C}$1type or null$\mathcal{C}$2type. Since the shape operator of $M^3_1$ has four possible matrix forms, the results will be considered in four different cases.

Pages 235246
In the present paper, we aim to generalize the notion of complextype Fibonacci sequences to complextype cyclic Fibonacci sequences. Firstly, we define the complextype cyclicFibonacci sequence and then we give miscellaneous properties of this sequence by using the matrix method. Also, we study the complextype cyclicFibonacci sequence modulo $m$. In addition, we describe the complextype cyclicFibonacci sequence in a $2$generator group and investigate that in finite groups in details. Then, as our last result, we obtain the lengths of the periods of the complextype cyclicFibonacci sequences in dihedral groups $D_{2}$, $D_{3}$, $D_{4}$, $D_{5}$, $D_{6}$ and $D_{8}$ with respect to their generating sets.
Keywords: The complextype cyclicFibonacci sequence, Matrix, Group, Perio 
Pages 247254
We investigate on some subclasses of analytic fuctions defined by subordination. Also, we give estimates of $\sup_{z<1}\big(1z^{2}\big)\big\dfrac{f^{''}(z)}{f^{'}(z)}\big$, for functions belonging to extended class of starlike functions. For a locally univalent analytic function $f$ defined on $\Delta =\{z\in \mathbb{C}: Z<1\}$, we consider the preSchwarzian norm by $\Vert T\Vert=\sup _{z<1}\big(1z^{2}\big)\big\dfrac{f^{''}(z)}{f^{'}(z)}\big$. In this work, we find the sharp norm estimate for the functions $f$ in the extended classes of starlike functions.
Keywords: Analytic Functions, Starlike functions, Preschwarzian derivatives, Subordination 
Pages 255266
In this paper, we investigate the pair difference cordial labeling behaviour of some star related graphs.
Keywords: Banana tree, Lilly graph, Shrub, Star 
Pages 267274
Suppose $X$ is a locally compact Hausdorff space and $\Omega \in \bigtriangleup$. If $ F $ is an interval valued function defined in $ \Omega $ with $F:\bar \Omega\rightarrow I_{\mathbb{R}}$. Suppose $F$ is Topological Henstock integrable, is $ F $ Sequential Henstock integrable? Therefore, the purpose of this paper is to provide a positive response to this query.
Keywords: Sequential Henstock integral, Interval valued functions, Topological Henstock, guages, right, left endpoints 
Fixed point results of fuzzy $(\theta, \mathcal {L})$ weak contraction in $\mathbb{G}$metric spacePages 275288
In this paper, the notion of fuzzy $(\theta, \mathcal {L})$weak contraction in $\mathbb{G}$metric space is introduced, and sufficient conditions for the existence of fuzzy fixed points for such mappings are investigated. Relevant illustrative examples are constructed to support the assumptions of our established theorems. It is observed that the principal ideas obtained herein extend and subsume some wellknown results in the corresponding literature. A few of these special cases of our results are noted and discussed as corollaries
Keywords: Fixed point, Fuzzy set, Fuzzy mapping, Contractive type mapping, (θ, L)−weak contraction, G−metric spac 
Pages 289324
The security flaws in cyber security have always put the users and organizations at risk, which as a result created catastrophic conditions in the network that could be either irreversible or sometimes too costly to recover. In order to detect these attacks, Intrusion Detection Systems (IDSs) were born to alert the network in case of any intrusions. Machine Learning (ML) and more prominently deep learning methods can be able to improve the performance of IDSs. This article focuses on IDS approaches whose functionalities rely on deep learning models to deal with the security issue in Internet of Things (IoT), wireless networks, Software Defined Networks (SDNs), and Industrial Control Systems (ICSs). To this, we examine each approach and provide a comprehensive comparison and discuss the main features and evaluation methods as well as IDS techniques that are applied along with deep learning models. Finally, we will provide a conclusion of what future studies are possibly going to focus on in regards to IDS, particularly when using deep learning models.
Keywords: Intrusion Detection System (IDS), Cyber Security, DeepLearning, Internet of Things (IoT), Wireless Network, Software DefinedNetwork (SDN), Industrial Control System (ICS 
Pages 325337
This article discusses the replicating kernel interpolation collocation method related to Jacobi polynomials to solve a class of fractional system of equations. The reproducing kernel function that is executed as an (RKM) was first created in the form of Jacobi polynomials. To prevent Schmidt orthogonalization, researchers compare the numerical solutions achieved by varying the parameter value. Through various numerical examples, it is demonstrated that this technique is practical and precise.
Keywords: Reproducing, Shifted Jacobi Polynomials, Twodimensionalfractional integral equation, Fractional derivative 
Pages 339347
Let $\textbf{M}_{n,m}$ be the set of all $n$by$m$ real matrices, and let $\mathbb{R}^{n}$ be the set of all $n$by$1$ real vectors. An $n$by$m$ matrix $R=[r_{ij}]$ is called grow substochastic if $\sum_{k=1}^{m} r_{ik}\leq 1$ for all $i\ (1\leq i \leq n)$. For $x$, $y \in \mathbb{R}^{n}$, it is said that $x$ is $\textit{sgutmajorized}$ by $y$, and we write $ x \prec_{sgut}y$ if there exists an $n$by$n$ upper triangular grow substochastic matrix $R$ such that $x=Ry$. Define the relation $\sim_{sgut}$ as follows. $x\sim_{sgut}y$ if and only if $x$ is sgutmajorized by $y$ and $y$ is sgutmajorized by $x$. This paper characterizes all (strong) linear preservers of $\sim_{sgut}$ on $\mathbb{R}^{n}$.
Keywords: Generalized row substochastic matrix, (strong) Linear preserver, Twosided sgutmajorizatio 
Pages 349362
Two interesting extensions of Banach contraction principle to mappings that don't to be continuous, are Kannan and Chatterjea's theorems. Before this, in the cyclical form, extensions of these two theorems and Banach contraction principle were produced. But so far, these theorems have not been studied in the noncyclical form. In this paper, we answer the question whether there are versions of these theorems for noncyclic mappings, also we give generalizations of existing results. For this purpose, in the setting of metric spaces we introduce the notions of cyclic and noncyclic contraction of Fisher type. We establish the existence of fixed points for these mappings and iterative algorithms are furnished to determine such fixed points. As a result of our results we give new Theorems for cyclic orbital contractions.
Keywords: Fixed point, Cyclic, noncyclic contractions of Fishertype, Kannan, Chatterjea mappings, Cyclic orbital contractio 
Pages 363374
In this paper, we first study $fs$modules, i.e., modules with finitely many small submodules. We show that every $fs$module with finite hollow dimension is Noetherian. Also, we prove that an $R$module $M$ with finite Goldie dimension, is an $fs$module if, and only if, $M = M_1 \oplus M_2$, where $M_1$ is semisimple and $M_2$ is an $fs$module with $Soc(M_2) \ll M$. Then, we investigate multiplication $fs$modules over commutative rings and we prove that the lattices of $R$submodules of $M$ and $S$submodules of $M$ are coincide, where $S=End_R(M)$. Consequently, $M_R$ and $_SM$ have the same Krull (Noetherian, Goldie and hollow) dimension. Further, we prove that for any selfgenerator multiplication module $M$, to be an $fs$module as a right $R$module and as a left $S$module are equivalent.
Keywords: Small submodules, fsmodules, Multiplication modules, Dimension symmetr 
Pages 375389
Let $R$ be an associative ring and let $n$ be a nonnegative integer. In this paper, we consider $n$super finitely presented, Gorenstein $n$weak injective and Gorenstein $n$weak flat modules under change of rings. For an excellent extension $S\geq R$, we show that the Gorenstein $n$super finitely presented dimensions (resp. weak Gorenstein $n$super finitely presented dimensions) of rings $R$ and $S$ coincide.
Keywords: Almost excellent extension, Excellent extension, Gorensteinnweak injective module, Gorenstein nweak flat module, nsuper finitelypresented modul 
Pages 391410
In this paper, we discuss the twostage and the modified twostage procedures for the estimation of the threshold autoregressive parameter in a firstorder threshold autoregressive model (${\rm TAR(1)}$). This is motivated by the problem of finding a final sample size when the sample size is unknown in advance. For this purpose, a twostage stopping variable and a class of modified twostage stopping variables are proposed. Afterward, we {prove} the significant properties of the procedures, including asymptotic efficiency and asymptotic risk efficiency for the point estimation based on leastsquares estimators. To illustrate this theory, comprehensive Monte Carlo simulation studies is conducted to observe the significant properties of the procedures. Furthermore, the performance of procedures based on YuleWalker estimators is investigated and the results are compared in practice that confirm our theoretical results. Finally, realtimeseries data is studied to demonstrate the application of the procedures.
Keywords: Twostage procedure, Modified twostage procedure, Threshold autoregressive process, Point estimation, Monte Carlo simulatio 
Pages 411423
The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by the trivial automorphism. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V (G)}$ of lists of labels to the vertices of $G$. A distinguishing $L$labeling of $G$ is a distinguishing labeling of $G$ where the label of each vertex $v$ comes from $L(v)$. The list distinguishing number of $G$, $D_l(G)$ is the minimum $k$ such that every list assignment to $G$ in which $L(v) = k$ for all $v \in V (G)$ yields a distinguishing $L$labeling of $G$. In this paper, we determine the listdistinguishing number for two families of graphs. We also characterize graphs with the distinguishing number equal the list distinguishing number. Finally, we show that this characterization works for other list numbers of a graph.
Keywords: Distinguishing number, listdistinguishing labeling, list distinguishing chromatic numbe 
Pages 425429
Denote by $ G $ a finite group, by $ {\rm hsn}(G) $ the harmonic mean Sylow number (eliminating the Sylow numbers that are one) in $G$ and by $ {\rm gsn}(G) $ the geometric mean Sylow number (eliminating the Sylow numbers that are one) in $G$. In this paper, we prove that if either $ {\rm hsn}(G)<45/7 $ or $ {\rm gsn}(G)< \sqrt[3]{300} $, then $G$ is solvable. Also, we show that if either $ {\rm hsn}(G)<24/7 $ or $ {\rm gsn}(G)<\sqrt{12} $, then $G$ is supersolvable.
Keywords: Finite group, Sylow subgroup, solvable groups 
Pages 431441
In this paper, we introduce a newly defined subclass $\mathcal{S}_{\Sigma}(\vartheta,\gamma,\eta;\varphi) $ of biunivalent functions by using the Tremblay differential operator satisfying subordinate conditions in the unit disk. Moreover, we use the Faber polynomial expansion to derive bounds for the FeketeSzego problem and first two \emph{TaylorMaclaurin coefficients} $a_2$ and $a_3$ for functions of this class.
Keywords: Analytic function, Biunivalent function, Coefficient estimates, Faber polynomial expansion, Tremblay fractional derivative operato 
Pages 443458
This paper deals with a generalization of the model describing the evolution of a linear viscoelastic body studied by Kirane M. and B.S. Houari in 2011. We prove the existence and uniqueness of the solution of the model using a $C_0$semigroup contraction method with a linear operator parameter. Moreover the strong stability of the solution is shown in a particular case.
Keywords: Viscoelasticity, Homogeneous evolution equation, Monotonemaximal operato 
Pages 459469
In recent years, many researches have been done on the tgsconvex functions and their applications. In this article, we present some properties of the tgsconvex functions by interesting examples. Then we investigate the nonpositive property of the tgsconvex functions. Also, we derive types of the Jensen’s inequality for the tgsconvex functions and obtain several inequalities with respect to the Jensen’s inequality. Finally, we give some applications of these inequalities.
Keywords: Jensen’s inequality, tgsconvex function, Global bounds 
Pages 471479
Let $(C,D)$ be a nonempty pair of disjoint subsets of a metric space. Main purpose of this paper is to present a range of a convergence sequence to $u\in C\cup D$ such that $d(Tu,fu)=dist(C,D)$, for mappings $T,f:C\cup D\to C\cup D$. In fact, we give a generalization of best proximity point results for cyclic contractive mappings. To this end, we consider an example is presented to support the main result.
Keywords: Common best proximity point, coincidence point, metric space, fixed point 
Pages 481496
Krasner $F^{(m,n)}$hyperrings were introduced and investigated by Farshi and Davvaz. In this paper, our purpose is to define and characterize three particular classes of $F$hyperideals in a Krasner $F^{(m,n)}$hyperring, namely prime $F$hyperideals, maximal $F$hyperideals and primary $F$hyperideals, which extend similar concepts of ring context. Furthermore, we examine the relations between these structures. Then a number of major conclusions are given to explain the general framework of these structures.
Keywords: Prime hyperideal, Maximal, hyperideal, Primary, Krasner, hyperring 
Pages 497502
This paper deals with the automatic continuity of multiplicative polynomial operators on a class of topological algebras. Several results are derived in this direction. We also support our results by some examples.
Keywords: Topological algebra, Multiplicative polynomial functional, automatic continuity, Spectral radius 
Pages 503512
The EM algorithm is a powerful tool and generic useful device in a variety of problems for maximum likelihood estimation with incomplete data which usually appears in practice. Here, the term ``incomplete" means a general state and in different situations it can mean different meanings, such as lost data, open source data, censored observations, etc. This paper introduces an application of the EM algorithm in which the meaning of ``incomplete" data is nonprecise or fuzzy observations. The proposed approach in this paper for estimating an unknown parameter in the parametric statistical model by maximizing the likelihood function based on fuzzy observations. Meanwhile, this article presents a case study in the electronics industry, which is an extension of a wellknown example used in introductions to the EM algorithm and focuses on the applicability of the algorithm in a fuzzy environment. This paper can be useful for graduate students to understand the subject in fuzzy environment and moreover to use the EM algorithm in more complex examples.
Keywords: EM algorithm, Exponential distribution, Fuzzy Statistics, Fuzzy data, Maximum likelihood estimation 
Pages 513527
A five coupled KaldorKalecki economic model with one delay appeared in the literature, in which the periodic solution of the model was verified by numerical analysis. The periodic solution is an important characteristic of the mutual interactions of economic systems. Also, different investment functions may have different delays. The present paper extends the five coupled KaldorKalecki economic model with one delay to a multiple delay system and discusses the existence of periodic oscillation of this multiple delay model. By linearizing the investment functions at the positive equilibrium and analyzing the instability of the positive equilibrium together with the boundedness of the solutions, some sufficient conditions to guarantee the existence of periodic oscillatory solutions for this model are established. Computer simulations are given to illustrate the validity of the theoretical results. The present result is new.
Keywords: economic model, delay, instability, periodic oscillation 
Pages 529546
Golchin and Rezaei introduced conditions $(PWP)$ and\linebreak $(PWP)_{w}$ in (Subpullbacks and flatness properties of $S$posets). In this paper, we introduce conditions $(PWP_{E})$ and $(PWP_{E})_{w}$ as generalizations of these conditions, respectively, and show that the relevant implications are strict. In general, we observe that condition $(PWP_{E})_{w}$ follows from condition $(PWP_{E})$, but not conversely. Also, we prove that principal weak poflatness follows from condition $(PWP_{E})_{w}$, but not conversely. Then, we obtain some general properties of conditions $(PWP_{E})$ and $(PWP_{E})_{w}$, and find sufficient and necessary conditions for the $S$poset $A(I)$ to satisfy these conditions. Finally, we find conditions on a pomonoid $S$ under which a cyclic or Rees factor $S$poset satisfies condition $(PWP_{E})$ or condition $(PWP_{E})_{w}$. Thereby, we present some homological classifications of pomonoids over which each of the conditions $(PWP_{E})$ and $(PWP_{E})_{w}$ implies a specific property, and vice versa, for Rees factor $S$posets.
Keywords: pomonoid, posets, Conditions, Rees factor 
Pages 547563
In this paper, we study the dynamics of the diphtheria outbreak among the immunocompromised group of people, the Rohingya ethnic group. Approximately 800,000 Rohingya refugees are living in the Balukhali refugee camp in Cox’s Bazar. The camp is densely populated with the scarcity of proper food, healthcare, and sanitation. Subsequently, in November 2017 a diphtheria epidemic occurred in this camp. To keep up with the pace of the disease spread, medical demands, and disaster planning, we set out to predict diphtheria outbreaks among Bangladeshi Rohingya immigrants. We adopted a modified SusceptibleLatentInfectiousRecovered (SLIR) transmission model to forecast the possible implications of the diphtheria outbreak in the Rohingya camps of Bangladesh. We discussed two distinct situations: the daily confirmed cases and cumulative data with unique consequences of diphtheria. Data for statistical and numerical simulations were obtained from \cite{Matsuyama}. We used the fourthorder RungeKutta method to obtain numerical simulations for varying parameters of the model which would demonstrate conclusive estimates. Daily and cumulative data predictions were explored for alternative values of the parameters i.e., disease transmission rate $(\beta)$ and recovery rate $(\gamma)$. Additionally, the average basic reproduction number for the parameters $\beta$ and $\gamma$ was calculated and displayed graphically. Our analysis demonstrated that the diphtheria outbreak would be under control if the maintenance could perform properly. The results of this research can be utilized by the Bangladeshi government and other humanitarian organizations to forecast disease outbreaks. Furthermore, it might help them to make detailed and practical planning to avoid the worst scenario.
Keywords: SLIR model, diphtheria, stability analysis, model validation, numerical analysis 
Pages 565583
In this study, a generalization of the Pell sequence called biperiodic Pell sequence is carried out to matrix theory. Therefore, we call this matrix sequence the biperiodic Pell matrix sequence whose entries are biperiodic Pell numbers. Then the generating function, Binet formula and some basic properties and sum formulas are examined.
Keywords: Pell Sequence, Generating Function, Binet Formula 
Pages 585593
In this paper, we prove the existance of subspacediskcyclic $C_{0}$semigroups on any infinitedimensional separable Banach space. We state that diskcyclic $C_{0}$semigroups are subspacediskcyclic. Also, we establish some criteria for subspacediskcyclic $C_{0}$semigroups. Most of these criteria are based on nonempty relatively open sets and some of them are based on dense sets.
Keywords: Subspacediskcyclicity, Diskcyclicity, semigroups