We first prove that there always exists a maximal Rectangularly dualizable graph for a given Rectangularly dualizable graph and present an algorithm for its construction. Further, we show that a maximal Rectangularly dualizable graph can always be transformed to an edge-irreducible Rectangularly dualizable graph and present an algorithm that transforms a maximal Rectangularly dualizable graph to an edge-irreducible Rectangularly dualizable graph.

A generic Rectangular partition   is a partition of a rectangle into a finite number of rectangles provided that no four of them meet at a point.  A graph $\mathcal{H}$ is called  dual of a plane graph $\mathcal{G}$ if there is one$-$to$-$one correspondence between the vertices of $\mathcal{G}$ and the regions of $\mathcal{H}$, and  two vertices of $\mathcal{G}$ are adjacent if and only if the corresponding regions of $\mathcal{H}$ are adjacent. A plane graph is a  Rectangularly dualizable graph  if its dual  can be embedded as a  Rectangular partition.   A Rectangular dual  $\mathcal{R}$ of a plane graph $\mathcal{G}$ is a partition of a  rectangle  into $n-$rectangles such that (i) no four rectangles of $\mathcal{R}$ meet at a point, (ii)  rectangles in  $\mathcal{R}$ are mapped to vertices of $\mathcal{G}$,  and (iii)  two rectangles in $\mathcal{R}$ share a common boundary segment if and only if the corresponding vertices are adjacent in $\mathcal{G}$. In this paper, we derive a necessary and sufficient  for a Rectangularly dualizable graph $\mathcal{G}$ to admit a unique Rectangular dual upto combinatorial  equivalence. Further we show that $\mathcal{G}$ always admits   a slicible as well as an area$-$universal  Rectangular dual.

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In this research, in order to investigate the effect of the piezoelectric patch which is used as a sensor or actuator in rotating flexible structures such as a helicopter blade, the free vibrations of the rotating Rectangular sheet with and without the piezoelectric patch have been presented. First-order shear deformation theory is considered for plate displacement and piezoelectric field. Considering the effect of Coriolis acceleration, centrifugal acceleration and centrifugal in-plane forces, the equations of motion are derived from Hamilton's principle and the electromechanical couple equation is obtained from Maxwell's equation. For piezoelectric, two electrical conditions, open circuit and closed circuit, which are used in sensors and actuators, respectively, have been considered. The equations are discretized with the help of the numerical method of generalized differential squares and the matrices of inertia mass, eccentricity, Coriolis and stiffness matrix are obtained. Natural frequency values for beam and rotating plate have been compared in Abaqus software. Also, the values obtained from the numerical solution in MATLAB have been verified with articles and ABAQUS, which have high accuracy. The effect of parameters such as hub radius, rotation speed, sheet thickness, aspect ratio, piezoelectric patch thickness and applied voltage on the natural frequency of the system has also been investigated.

The purpose of this study is to define the concepts of Tsallis entropy and conditional Tsallis entropy of fuzzy Partitions and to obtain some results concerning this kind entropy. We show that the Tsallis entropy of fuzzy Partitions has the subadditivity and concavity properties. We study this information measure under the refinement and zero mode subset relations. We check the chain rules for this information measure and prove some properties about the entropy of independent fuzzy Partitions. Some results of the relationship between the Tsallis entropy and conditional Tsallis entropy of fuzzy Partitions are obtained and, by using conditional Tsallis entropy of fuzzy Partitions, we show that the subadditivity property for Tsallis entropy of fuzzy Partitions is not established in the case that the parameter of this entropy is smaller than one. In general, the Tsallis entropy of fuzzy Partitions has similar properties to the shannon entropy, where the parameter of this entropy is larger than one, and therefore can be used in addition to the Shannon entropy, to measure the amount of information to be extracted from a fuzzy experiment.

Let $R$ be a commutaive ring and $Zd(R)$ be the set of zero divisors of $R$. Define an equivalence relation $\sim$ on $Zd(R)$ as follows: $x\sim y$ if and only if $ann(x)=ann(y)$. The graph $\Gamma_E(R)$ is a graph associated to R whose vertices are the classes of elements in $Zd(R)^*=Zd(R)\backslash\{0\}$, and two distinct classes $[x]\neq[y]$ are joined by an edge if and only if $xy=0$. We show that if $R$ is a local ring and $\Gamma_E(R)$ is a star graph with at least four elements then $m/Soc(R)$ has a partition of vector spaces where $m$ is the maximal ideal of $R$. Also, We construct from a special partition of vector spaces, a ring whose associated graph is a star graph. 1. IntroductionThe compressed zero divisor graph or the graph of equivalence classes of zero divisors of a ring $R$ is denoted by $\Gamma_E(R)$, and is defined in [20]. Let $Zd(R)$ denotes the set of zero divisors of a ring $R$ and $Zd(R)^*=Zd(R)\backslash\{0\}$. Define a relation $\sim$ on $Zd(R)$ as follows [14]: $x\sim y$ if and only if $ann(x)=ann(y)$. It is easily seen that $\sim$ is an equivalence relation. The graph $\Gamma_E(R)$ is a graph associated to R whose vertices are the classes of elements in $Zd(R)^*$, and two distinct classes $[x]\neq[y]$ are joined by an edge if and only if $xy=0$. Another interpretation of $\Gamma_E(R)$ is as follows: The vertices are the elements of $\mathcal{J}=\{ann(a):a\in Zd(R)^*\}$ and two distinct elements $ann(x)$ and $ann(y)$ are adjacent if and only if $xy=0$. In [20], some necessary conditions are obtained for a ring $R$ such that $\Gamma_E(R)$ is a star graph. For example, If $\Gamma_E(R)$ is a star graph with at least four vertices then $|Ass(R)|=1$ and $Char(R)=2,4,8$. In [9] a method for constructing star compressed zero divisor graph is obtained. They used a quotient of a symmetric algebra of a vector space whose relations come from a special partition of that vector space. But it seems that the authors were not aware of partition of vector spaces. By analyzing the proofs in [20] and [9], we see that the star graphs give a partition of a vector space and conversely some Partitions give a star graph. An interesting problem about star compressed zero divisor graph is the size of them. For example is there any ring whose compressed zero divisor graph be a star graph with $36$ vertices? 2. Main ResultsTheorem 2.1. Let $R$ be a ring such that $\Gamma_E(R)$ is a star graph with at least four vertices. Let $[y]$ be the unique vertex with maximal degree and $K=[y]\bigcup \{0\}$. Then $R$ satisfies the following properties:   (1) $Ass(R)=\{P\}$ where $ann(y)=P$. Also $ann(P)=K$. In particular $K$ is an ideal of $R$ and $Zd(R)=P$.   (2) $P^3=0$.   (3) If $ann(x_0)=K$ then $x_0^3=0$ and $[x_0+y]=[x_0]$.   (4) If $J=\{x\in R:K\subsetneqq ann(x)\}$ then $J$ is an ideal of $R$ and $ann(x)=[x]\bigcup K$ for each $x\in J\backslash K$. Also $[x+y]=[x]$ for each $x\in J\backslash K$.   (5) $Char(R)=2,4,8$.Corollary 2.2. Let $(R,m)$ be a local Artinian ring such that $\Gamma_E(R)$ is a star graph with at least four vertices. Let $[y]$ be the unique vertex with maximal degree and $K=[y]\bigcup \{0\}$. If $J=\{x\in R:K\subsetneqq ann(x)\}$ then $\{ann(x)/K:x\in J\backslash (K)\}$ is a partition of $R/m-$vector space $J/K$. Also $Soc(R)=ann(m)=K$.Theorem 2.3. Let $V=V_1\bigoplus\cdots\bigoplus V_t$ be an $n-$dimensional vector space over $\mathbb{F}_2$ and $dim(V_i)=n_i$. Assume $\{X_{i,k}:1\leq k\leq n_i\}$ is a basis of $V_i$. Let $S=\mathbb{F}_2[X_{i,k}:1\leq i\leq t,1\leq k\leq n_i]$ be a polynomial ring over $n$ indeterminates. Let $I=\langle V_i^2,V^3\rangle$ and $R=\frac{S}{I}$. Then $\Gamma_E(R)$ is a star graph with $2^n-(2^{n_1}+\cdots+2^{n_t})+2t$ vertices. 3. Summary of Proofs/ConclusionsIn this article we show that every star compressed zero divisor graph correspond to a partition of vector spaces. Conversely, we construct from a special vector space partition a star compressed zero divisor graph.