It is well known that all subvarieties of the variety of all semigroups are not absolutely closed. So, it is worth to find subvarieties of the variety of all semigroups that are closed in itself or closed in the containing varieties of semigroups. We have gone through this open problem and able to determine that the varieties of (RIGHT)~[LEFT] normal bands and LEFT~[(RIGHT)] quasinormal bands are closed in the varieties of semigroups defined by the identities $axy = xa^ny~[axy = ay^nx],~axy = x^nay~[axy = ayx^n]$ $(n>1)$; and $axy=ax^nay$~$[axy=ayx^ny]$~ $(n>1)$, $axy=a^nxa^ry$ $[axy=ay^rxy^n]$ $(n,r\in \mathbb{N})$, respectively.

A hidden Markov model (HMM) is one of the important statistical tools used for analyzing and modeling the biological data. An HMM is used in many various fields of bioinformatics, for example it is used in the protein secondary structure prediction. Using some algorithms as Viterbi, Forward, Backward and Baum-Welch, one can address some questions about emission generation in HMM. In standard HMM the "t" th hidden state, given the "t-l"st hidden state, is independent of previous states and in every state, emissions are assumed to be independent. In this work the relations among states and emissions are considered. The dependencies between emissions have already been studied by other researchers. In this paper a new approach for consideration of dependencies among emissions is presented. We start with the use of the information of the LEFT hand side of any emission and introduce a new model. We then take the information of the (RIGHT) hand side of any' emission into account. The mathematical structure of the model is presented and the modifications are discussed.

Let B(X, Y) denote the set of all bounded linear operators from Banach space X to Banach space Y. In this paper, we introduce the concepts of LEFT and (RIGHT) decomposably regular operators, LEFT and (RIGHT) decomposably Fredholm operators in the setting of B(X, Y), and the corresponding holomorphic versions in the setting of B(X). By using Harte's techniques, we obtain various characterizations of these classes of operators. As the applications of these characterizations, we can compute the topological interiors and closures of them.

Suppose that $G$ is a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. A subset $S=\{s_1, s_2,\ldots , s_l \}$ of vertices of graph $G$ is called a doubly resolving set of $G$, if for any distinct vertices $u$ and $v$ in $G$ there are elements $x$ and $y$ in the set $S$ such that $d(u, x)-d(u, y)\ne d(v, x)-d(v, y)$. The minimum size of a doubly resolving set of the vertices of graph $G$ is denoted by ${\psi} (G)$. In this paper, we calculate the resolving sets of vertices with the minimum size for the line graph $L(C_n\circ{\overline{K}}_m)$ and graph $\LEFT((C_n\circ{\overline{K}}_m)\square P_k\(RIGHT))$, in which the symbols $\circ$ and $\square$ denote the Corona product and Cartesian product between two graphs, respectively. In particular, we show that if $n\geq 3$ and $m, k\geq 2$ are integers, then ${\psi}((C_n\circ{\overline{K}}_m)\square P_k )={\psi}(C_n\circ{\overline{K}}_m)+{\psi}(P_k )-1$, which gives a partial answer to the problem of characterizing graphs $G$ and $H$ satisfying the equality ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$, which is recently posed in [K. Nie and K. Xu, The doubly metric dimension of cylinder graphs and torus graphs, Bull. Malays. Math. Sci. Soc., 46 (2023) 19 pp]. 1. IntroductionLet $G$ be a simple and connected graph with the set of vertices $V(G)$ and the set of edges $E(G)$. We denote the length of the shortest path between two vertices $u$ and $v$ in the graph $G$ by $d(u, v)$. We use $C_n$, $\overline{K_m}$ and $P_k$ to denote the cycle graph of order $n$, complement of the complete graph on $m$ vertices and the path graph of order $k$, respectively. Also, the line graph $G$ is denoted by $L(G)$, that the set of vertices of $L(G)$ are the same as the edges of the graph $G$, and two vertices are adjacent in the graph $L(G)$, if their corresponding edges in graph $G$ have a common vertex [6]. Our goal is to calculate some resolving sets depending on the line graph of the Corona product $C_{n}\circ \overline{K_{m}} $ and the Cartesian product $(C_{n}\circ \overline{K_{m}} )\square P_{k}$, so we give first some explanations about the Corona product and Cartesian product of graphs. Suppose $G$ and $H$ are two graphs with $n$ and $m$ vertices, respectively. If we consider $n$ copies of $H$ and for $i=1, 2,\cdots ,n$, all the vertices of the $i^{th}$ copy of $H$ are adjacent to the vertex $i$ of $G$, then the desired graph is called the Corona product of two graphs $G$ and $H$ and we denote it by $G\circ H$. Also, if $G$ and $H$ are two graphs, we denote the Cartesian product of these graphs by $G \square H$ or $G \times H$ and define in this way $V\LEFT(G\square H\(RIGHT))=V(G)\times V(H)$ and two vertices $(g, h)$ and $(g',h')$ in $G\square H$ are adjacent if and only if $g=g'$ and $hh'\in E(H)$ or $h=h'$ and $gg'\in E(G)$. For any ordered subset $S=\{ s_{1}, s_{2},\cdots,s_{k}\}$ of the vertices of graph $G$ and the vertex $v$ of $G$, representation the vertex $v$ with respect to the ordered set $S$ denoted by $r(v{|S)}$ and so it is $ r(v{|S)}=\LEFT({d}\LEFT(v, s_1\(RIGHT)),{d}\LEFT(v, s_2\(RIGHT)),\ldots,{d}\LEFT(v, s_k\(RIGHT))\(RIGHT))$. If all the vertices of the graph $G$ have distinct metric representations with respect to the ordered set $S$, then $S$ is called a resolving set of vertices of $G$. A resolving set of vertices of $G$ with the minimum size is called the metric dimension of the graph $G$ and it is represented by ${\beta }\LEFT(G\(RIGHT))$. The study of resolving set of vertices in graph theory dates back to the 1970s, and such concepts were first introduced in the articles [7,18]. The metric dimension of complete graphs, trees, paths, and the Cartesian product have been taken into consideration, also in the article [5] all graphs of order $n$ that have the metric dimension greater than or equal to $n-2$ are fully characterised. The subset $ S=\{s_{1},s_{2},\cdots,s_{l}\}$ of the vertices of graph $G$ is called a doubly resolving set for $G$, if for any distinct pair vertices $u$ and $v$ of $G$, there are the elements $x$ and $y$ of $S$, in which $d(u, x)-d(u, y)\ne d(v, x)-d(v, y)$. We denote the size of the minimum doubly resolving set in graph $G$ by ${\psi}\LEFT(G\(RIGHT)) $. Concepts related to resolving sets and doubly resolving sets of graphs have been studied in the articles [4,5]. The graph $(C_5\circ \overline{K}_{3})$ and graph $L(C_{5}\circ \overline{K}_{3})$ are drawn in Figure 1. Also, the graph $(C_3\circ \overline{K}_{3})$ and graph $(C_{3}\circ \overline{K}_{3})\square P_{2}$ are drawn in Figure 2. 2. Main ResultsTheorem 2.1. If $n$ and $m$ are fixed positive integers, in which $n\geq 3$, $m\geq 2$, then the cardinality of minimum doubly resolving set of the line graph of graph $C_n\circ{\overline{K}}_m$ is $nm-n$. Theorem 2.2. If $n$, $m$ and $k$ are fixed positive integers, in which $n\geq 3$ and $m, k\geq 2$, then ${\beta }((C_n\circ{\overline{K}}_m)\square P_k )=nm-n+1.$ Theorem 2.3. If $n$, $m$ and $k$ are fixed positive integers, in which $n\geq 3$ and $m, k\geq 2$, then ${\psi }\LEFT((C_n\circ \overline{K}_{m})\square P_{k}\(RIGHT))={\psi } (C_n\circ \overline{K}_{m})+{\psi }(P_{k})-1.$ 3. ConclusionsConsidering the importance of the minimum resolving sets in graphs, in this paper, we first calculated some resolving sets of vertices with the minimum size for the line graph of graph $C_n\circ{\overline{K}}_m$ and graph $\LEFT((C_n\circ{\overline{K}}_m)\square P_k\(RIGHT))$. In particular, we show that if $n\geq 3$ and $m, k\geq 2$ are integers, then ${\psi}((C_n\circ{\overline{K}}_m)\square P_k )={\psi}(C_n\circ{\overline{K}}_m)+{\psi}(P_k )-1$, which gives a partial answer to the problem of characterizing graphs $G$ and $H$ satisfying the equality ${\psi}(G\square H)={\psi}(G)+{\psi}(H)-1$, which is recently posed in [15].

Tarjomeh is an Arabic term whose correct pronunciation is tarjama, which is commonly pronounced tarjomeh. The term is said to stand for: 1. expressing discourse form a language to another; 2. interpretation, hermenutic study, and explanation; 3. biography, to state characteristics, morality, and predecessors of an individual. The method to be adopted in Arabic-Persian TRANSLATION is just ordinary TRANSLATION, proved that the translator has a good command of Arabic grammar. The good method of TRANSLATION is to consider Arabic morphology and syntax as well as other things such as correct pronunciation, rhetoric, and finding the (RIGHT) lexical equivalents. In another part of the article translating strategies and methods are practically presented by using the blessed verses of the holy Quran, Sonnat and so on.

Let A be a Banach algebra with unity 1, and θ: A → A be an continuous automorphism. In this paper we characterize a continuous linear map T: A → A which satisfies one of the following conditions: a, b ∈ A, ab = w =⇒ θ(a)T(b) = T(w), a, b ∈ A, ab = w =⇒ T(a)θ(b) = T(w), or a, b ∈ A, ab = w =⇒ θ(a)T(b) = T(a)θ(b) = T(w), where w 6= 0 is a LEFT ((RIGHT)) separating point of A.

We report a rare case of anomalous origin of (RIGHT) coronary artery (RCA) from the LEFT coronary sinus with a peculiar medusa head pattern of LEFT coronary system in an octogenarian presenting with inferior wall myocardial infarction. Although anomalous origin of RCA from the LEFT coronary sinus is the most common anomaly to be reported, this rare association of extensive arborization of LEFT coronary system appearing like a medusa head is not reported in literature so far. The anomalous RCA harbored critical mid-RCA lesion, but the crux of coronary intervention was that we engaged the anomalous RCA with extra back up guide catheter in place of conventionally used Judkins (RIGHT) guide catheter to engage the anomalous origin and accomplished the coronary intervention with buddy wire technique. Our case is unique and the first to describe the association of medusa head appearance of LEFT coronary circulation in a case of anomalous origin of RCA from the LEFT coronary sinus.

Introduction: A limited number of studies demonstrated that the developmental potential of human oocytes was affected by the side of ovary. In addition, the competence of an oocyte to resume and complete meiosis and, after fertilization, to develop into a blastocyst is markedly increased during preovulatory development through a process called oocyte capacitation. To our knowledge, there are no reports regarding in oocyte development in LEFT and (RIGHT) ovaries. The objective of the present study was to evaluate oocyte development in LEFT and (RIGHT) ovaries.Materials and Methods: Bovine LEFT and (RIGHT) ovaries were obtained from an abattoir and transported to the laboratory. The follicles were aspirated at 33 ºC and divided according to their diameter into three groups: small follicles:<3 mm, medium follicles: 3-5 mm and large follicles:>5 mm. The means of oocyte diameter (by ocular micrometric method) and percentage of fully grown oocytes (by brilliant cresyl blue test) was recorded for each follicle categories.Results: Oocyte diameter and percentage of fully grown BCB+oocytes increases in parallel with follicular size in LEFT and (RIGHT) side ovaries. The results set out in the current study, indicate that the oocyte diameter, cytoplasm volume and percentage of fully grown oocytes in (RIGHT)-side ovary was higher than that LEFT-side ovary.Conclusion: We can deduce a relationship between side of ovary and oocyte development. In addition, we can conclude a positive relationship between follicle diameter, fully grown oocytes, and oocyte diameter in LEFT and (RIGHT) side ovaries.

‎‎‎In this paper first we introduce (RIGHT)(resp‎. ‎LEFT) induced hyperlattices and investigate some of their properties‎. ‎Especially‎ ‎a characterization of the smallest strongly regular relation for the class of distributive (RIGHT)/LEFT induced hyperlattice is investigated‎. ‎Next we propose and study the generated hyperlattices from hyperlattices‎. ‎Finally‎, ‎the (RIGHT) induced hyperlattices of two Boolean lattices of four DNA bases and physico-chemical properties of amino acids of four DNA bases are investigated‎.

‎‎‎In this paper first we introduce (RIGHT)(resp‎. ‎LEFT) induced hyperlattices and investigate some of their properties‎. ‎Especially‎ ‎a characterization of the smallest strongly regular relation for the class of distributive (RIGHT)/LEFT induced hyperlattice is investigated‎. ‎Next we propose and study the generated hyperlattices from hyperlattices‎. ‎Finally‎, ‎the (RIGHT) induced hyperlattices of two Boolean lattices of four DNA bases and physico-chemical properties of amino acids of four DNA bases are investigated‎.