Journal of Mahani Mathematical Research
Volume:12 Issue: 2, Summer and Autumn 2023

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.

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.

Let $R$ be a ring with an endomorphism $\alpha$‎. ‎A ring $R$ is a skew power series McCoy ring if whenever any non-zero 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 non-zero 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 ring-theoretic 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)‎.

One of the crucial stages in machine learning in high-dimensional 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 Type-I based on Ant Colony Optimization (ACO) for ensemble feature selection named Ant-EHFS. 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 Type-I 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 well-known ensemble feature selection methods and primary feature selection methods, respectively, have been compared with Ant-EHFS on six high-dimensional 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 F1-score metrics.

In this paper, we define the concepts of single-valued neutrosophic general automaton, complete and deterministic single-valued neutrosophic general automaton. We present a minimal single-valued neutrosophic general automaton that preserves the language for a given single-valued neutrosophic general automaton. Moreover, we present the closure properties such as union and intersection for single-valued neutrosophic general automata.

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 Mellin-Barnes 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.

‎‎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.‎

The commutator theory, developed by Fresee and McKenzie in the framework of a congruence-modular 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 congruence-modular 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 non-commutative rings. In this paper we shall introduce the quasi-commutative algebras in a semidegenerate congruence-modular variety $\mathcal{V}$ as a generalization of the Belluce quasi-commutative rings. We define and study a notion of reticulation for the quasi-commutative algebras such that the Belluce reticulation for the quasi-commutative rings can be obtained as a particular case. We prove a characterization theorem for the quasi-commutative algebras and some transfer properties by means of the reticulation.

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, Hexanacci-Lucas, and other sixth-order recurrence sequences.

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 $G/L_c(G)$ element-wise. 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.

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 close-to-convex approach of starlike function. Some early few coefficient bounds obtained are used to establish the famous Fekete-Szego inequalities.

A hypersurface $ M^n $ in the Lorentz-Minkowski 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 $k=0,1,\ldots,n-1$. 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.

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.

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 4-space ${\Bbb E}^4$ has constant mean curvature. In this paper, we study an extended version of the mentioned subject on timelike (i.e., Lorentz) hypersurfaces of Minkowski 4-space ${\Bbb E}^4_1$. 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 1-proper} if it is an eigenvector of the Cheng-Yau operator $\mathcal{C}$ (which is the natural extension of $\Delta$). We show that each $M^3_1$ with 1-proper ${\textbf H}_2$ has constant scalar curvature. By a classification theorem, we show that such a hypersurface is $\mathcal{C}$-biharmonic, $\mathcal{C}$-1-type or  null-$\mathcal{C}$-2-type. Since the shape operator of $M^3_1$ has four possible matrix forms, the results will be considered in four different cases.

In the present paper, we aim to generalize the notion of complex-type Fibonacci sequences to complex-type cyclic Fibonacci sequences. Firstly, we define the complex-type cyclic-Fibonacci sequence and then we give miscellaneous  properties of this sequence by using the matrix method. Also, we study the complex-type cyclic-Fibonacci sequence modulo $m$. In addition, we describe the complex-type cyclic-Fibonacci 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 complex-type cyclic-Fibonacci sequences in dihedral groups $D_{2}$, $D_{3}$, $D_{4}$, $D_{5}$, $D_{6}$ and $D_{8}$ with respect to their generating sets.

We investigate on some subclasses of analytic fuctions defined by subordination. Also, we give estimates of $\sup_{|z|<1}\big(1-|z|^{2}\big)\big|\dfrac{f^{''}(z)}{f^{'}(z)}\big|$, 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 pre-Schwarzian norm by $\Vert T\Vert=\sup _{|z|<1}\big(1-|z|^{2}\big)\big|\dfrac{f^{''}(z)}{f^{'}(z)}\big|$. In this work, we find the sharp norm estimate for the functions $f$ in the extended classes of starlike functions.

In this paper, we investigate the pair difference cordial labeling behaviour of some star related graphs.

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.

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 well-known results in the corresponding literature. A few of these special cases of our results are noted and discussed as corollaries

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.

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.

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 g-row 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 $\textit{sgut-majorized}$ by $y$, and we write  $ x    \prec_{sgut}y$  if there exists an $n$-by-$n$ upper triangular g-row substochastic matrix $R$ such that $x=Ry$. Define the relation $\sim_{sgut}$ as follows. $x\sim_{sgut}y$ if and only if $x$ is   sgut-majorized  by $y$ and $y$ is sgut-majorized  by $x$.  This paper characterizes all (strong)  linear preservers   of  $\sim_{sgut}$ on $\mathbb{R}^{n}$.

‎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 non-cyclic 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.

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 self-generator multiplication module $M$, to be an $fs$-module as a right $R$-module and as a left $S$-module are equivalent.

Let $R$ be an associative ring and let $n$ be a non-negative 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.

In this paper, we discuss the two-stage and the modified two-stage procedures for the estimation of the threshold autoregressive parameter in a first-order 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 two-stage stopping variable and a class of modified two-stage 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 least-squares 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 Yule-Walker estimators is investigated and the results are compared in practice that confirm our theoretical results. Finally, real-time-series data is studied to demonstrate the application of the procedures.

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 list-distinguishing 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.

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.

In this paper, we introduce a newly defined  subclass $\mathcal{S}_{\Sigma}(\vartheta,\gamma,\eta;\varphi) $ of bi-univalent 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 Fekete-Szego problem and first two \emph{Taylor-Maclaurin coefficients} $|a_2|$ and $|a_3|$ for functions of this class.

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 $C_0$-semi-group contraction method with a linear operator parameter. Moreover the strong stability of the solution is shown in a particular case.

In recent years, many researches have been done on the tgs-convex functions and their applications. In this article, we present some properties of the tgs-convex functions by interesting examples. Then we investigate the non-positive property of the tgs-convex functions. Also, we derive types of the Jensen’s inequality for the tgs-convex functions and obtain several inequalities with respect to the Jensen’s inequality. Finally, we give some applications of these inequalities.

‎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. ‎‎

‎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.

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.

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 non-precise 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 well-known 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.

A five coupled Kaldor-Kalecki 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 Kaldor-Kalecki 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.

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 po-flatness 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.

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 Susceptible-Latent-Infectious-Recovered (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 fourth-order Runge-Kutta 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.

In this study, a generalization of the Pell sequence called bi-periodic Pell sequence is carried out to matrix theory. Therefore, we call this matrix sequence the bi-periodic Pell matrix sequence whose entries are bi-periodic Pell numbers. Then the generating function, Binet formula and some basic properties and sum formulas are examined.

In this paper,  we prove the existance of subspace-diskcyclic $C_{0}$-semigroups   on any infinite-dimensional   separable Banach space.  We state that diskcyclic $C_{0}$-semigroups are subspace-diskcyclic. Also, we establish some criteria for subspace-diskcyclic $C_{0}$-semigroups. Most of these criteria  are based on non-empty relatively open sets  and some of them are based on dense sets.