In this paper we investigate the problem of searching on d-dimensional arrays with partial order. We generalize Linial and Saks’ search algorithm [2] for 3 dimension to arbitrary dimension d. Our new algorithms require at most d d−1 n + O(n) comparisons. It is optimal for d = 4.

In 1995, Hochberg, McDiarmid, and Saks [4] proved that the vertex-bandwidth of the triangular grid Tn is precisely n+1; more recently Balogh, Mubayi, and Pluhár [1] posed the problem of determining the edge-bandwidth of Tn. We show that the edge-bandwidth of Tn is bounded above by 3n− 1 and below by 3n− o(n).

1 Domains with Convex Closure Saks and Yu (2005) proved that if D is convex then every monotone deterministic allocation rule is implementable. We prove in this appendix the following generalization of their result: Theorem 1 Every domain with a convex closure is a proper monotonicity domain.

In this paper, we have studied the Henstock - Kurzweil integral which is a generalized Riemann means. Hen-stock natural extension of integral. We defined Banach space valued function with respect to bounded variation an real increasing function. investigated elementary properties variation. proved convergence theorems and Saks lemma functions vari-ation. Equi-integrability equi-integrable theor...