In this paper, the notion of Direct sum of branches in hyper BCK-algebras is introduced and some related properties are investigated. Applying this notion to lower hyper BCK-semi lattice, a necessary condition for a hyper BCK-ideal to be prime is given. Some properties of hyper BCK-chain are studied. It is proved that if H is a hyper BCK-chain and [a) is finite for any aÎH, then |Aut (H)|=1.

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In this paper, we describe all maximal dissipative, maximal accretive and selfadjoint extensions of the minimal symmetric Direct sum differential operators. Further using the equivalence of the Lax-Phillips scattering function and the Sz.-Nagy-Foias characteristic function we show that all eigen and asso- ciated functions of the maximal dissipative extension of the minimal symmetric Direct sum operator are complete in L2w(W), where W=W1ÈW2, W1=(0, C) and W2=(c, ¥).

We study the thoery of best approximation in tensor product space and the Direct sum of some lattice normed spaces Xi: We introduce quasi tensor product space and discuss about the relation between tensor product space and this new space which we denote it by X⊠ Y. We investigate best approximation in Direct sum of lattice normed spaces by elements which are not necessar-ily downward or upward and we call them Im􀀀 quasi downward or Im􀀀 quasi upward. We show that these sets can be interpreted as downward or upward sets. The relation of these sets with down-ward and upward subsets of the Direct sum of lattice normed spaces Xi is discussed. This will be done by homomorphism functions. Fi-nally, we introduce the best approximation of these sets.

For a coloring c of a graph G, the edge-di erence coloring sum and edge-sum coloring sum with respect to the coloring c are respectively Σ c D(G) = Σ jc(a) 􀀀 c(b)j and Σ s S(G) = Σ (c(a) + c(b)), where the summations are taken over all edges ab 2 E(G). The edge-di erence chromatic sum, denoted by Σ D(G), and the edge-sum chromatic sum, denoted by Σ S(G), are respectively the minimum possible values of Σ c D(G) and Σ c S(G), where the minimums are taken over all proper coloring of c. In this work, we study the edge-di erence chromatic sum and the edge-sum chromatic sum of graphs. In this regard, we present some necessary conditions for the existence of homomorphism between two graphs. Moreover, some upper and lower bounds for these parameters in terms of the fractional chromatic number are introduced as well.