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We obtain an exact formula for the average order of elements of regular wreath product of two finite groups. Then focussing our attention on $p$-groups for primes $p$, we give an estimate for the verage order of a wreath product $A\wr B$ in terms of maximum order of elements of $A$ and average order of $B$ and an exact formula for the distribution of orders of elements of $A\wr B.$ Finally, we show how wreath products can be used to find several rational numbers which are limits of average orders of a sequence of $p$-groups with cardinalities going to infinity.

A finite switchboard state machine is a specialized finite state machine. It is built by binding the concepts of switching state machines and commutative state machines. The main purpose of this paper is to give a specific algorithm for fuzzy finite switchboard state machine and also, investigates the concepts of switching relation, covering, restricted cascade products and wreath products of fuzzy finite switchboard state machines. More precisely, we study that the direct products/Cartesian compositions of two such fuzzy finite switchboard state machines is again a fuzzy finite switchboard state machine. In addition, we introduce the perfect switchboard machine and establish its Cartesian composition. The relations among the products also been examined. Finally, we introduce asynchronous fuzzy finite switchboard state machine and study the switching homomorphic image of asynchronous fuzzy finite switchboard state machine. We illustrate the definition of a restricted product of fuzzy finite switchboard state machine with the single pattern example.

Background and Aim: What constitutes adequate dialysis has been debated in the nephrology literature over the past years. We believed that short and infrequent dialysis session contributed to poor outcome. This result provides additional evidence that (KT/V) in a flawed concept upon which to based the dose dialysis in general. Base on published evidence from many source we proposed a new index of adequacy of hemodialysis to be called hemodialysis product (HDP).Materials and Methods: The prospective study was performed on 100 dialysis patient in the Imam hospital dialysis ward in Tehran during 2002. delivered (KT/V) and (HDP) achieved for any patient separated.Results: 100 patients including 44 female and 56 male were studied. Result has shown (KT/V) d≥l in 43% and (KT/V) d<1 in 57%. HDP in patient with (KT/V) d> 1 were ≥36. HDP in patient with (KT/V) d < 1 were < 36. Correlation between (KT/V) d and albumin was significant (p< 0.03, Albumin ≥4mg/dl). Correlation between (KT/V) d and cholesterol, hemoglobin, creatinine were not significant. Correlation between (KT/V) d and HDP was significant (p< 0.001).Conclusion: Finding above has shown that higher weekly delivered (KTN) correlate with dialysis result increase hemodialysis product (HDP). More frequent and session dialysis results increase in HDP, provide better weekly (KT/V) d with normal albumin (show good caloric and protein intake).

For any graph, Weisfeiler and  Leman assigned the smallest  matrix algebra which  contains the adjacency matrix of the graph. The coherent configuration underlying this  algebra for a graph $\Gamma$ is called the coherent configuration of $\Gamma$, denoted by $\mathcal{X}(\Gamma)$. In this paper, we study the coherent configuration of circular-arc graphs. We give a characterization of the circular-arc graphs $\Gamma$, where $\mathcal{X}(\Gamma)$  is a homogeneous coherent configuration. Moreover, all homogeneous coherent configurations which are obtained in this way are characterized as a subclass of Schurian coherent configurations.

Let $m>1$ be an integer and $\Omega$ be an $m$-set‎. ‎The Hamming graph $H(n,m)$ has $\Omega ^{n}$ as its vertex-set‎, ‎with two vertices are adjacent if and only if they differ in exactly one coordinate‎. ‎In this paper‎, ‎we provide a new proof on the automorphism group of the Hamming graph $H(n,m)$‎. ‎Although our result is not new (CE Praeger‎, ‎C Schneider‎, ‎Permutation groups and Cartesian decompositions‎, ‎Cambridge University Press‎, ‎2018)‎, ‎we believe that our proof is shorter and more elementary than the known proofs for determining the automorphism group of Hamming graph‎.