Abstract The integer sequence π = π 1 ‧‧‧ πn is said to be an inversion sequence if 0 ≤ πi ≤ i – 1 for all i. Let ℐn denote the set of inversion sequences of length n, represented using positive instead of non-negative integers. We consider here two new statistics defined on the bargraph representation b(π) of an inversion sequence π which record the number of unit squares touching the boundary of b(π) and that are either exterior or interior to b(π). We denote these statistics on ℐn recording the number of outer and inner perimeter squares respectively by oper and iper. In this paper, we study the distribution of oper and iper on ℐn and also on members of ℐn that end in a particular letter. We find explicit formulas for the maximum and minimum values of oper and iper achieved by a member of ℐn as well as for the average value of these parameters. We make use of both algebraic and combinatorial arguments in establishing our results.
A common problem in systems and control theory is to provide an approximation to non-linear systems. We provide a novel approach as a general solution to this problem originally conceived by Gamkrelidze. We consider and solve a general approximation problem which provides the fundamentals for various switching-type systems thus encompassing a wide range of systems theory problems.
Abstract It is well known that convexity theory and its generalization play an important role and have many applications in mathematical programming. In the previous article [2], the authors studied E-convex and semi-E-convex functions. In the present article, we deal with a class of E-quasiconvex functions and give a weak condition for a lower semi-continuous function on ℝ n to be an E-quasiconvex function.
Abstract In this paper we investigate disposal activities in a reverse logistics models with linear costs structure. The two models have different disposal strategy. The first model analyzes a continuous disposal strategy, i.e. disposal can happen in every time in the planning horizon. This type of models was investigated in earlier works. The second possible disposal strategy assumes that disposal is only at the end of the planning horizon. After characterization of the optimal strategies of the two models we compare the costs and the trajectories of the optimal paths. The result of the paper is that the relevant costs are higher in case of disposal at the end of the planning horizon. It is cost efficient to dispose off the not necessary products inside of the planning horizon.
Abstract This paper deals with a generalization of polyominoes called (a, b)-connected discrete figures, where a and b respectively denotes the connectivity of the foreground (i.e. black pixels) and background (i.e. white pixels). Formally, a finite set of pixels P is (a, b)-connected if P is a-connected and P̄ is b-connected. By adapting a combinatorial structure enumeration algorithm due to Martin, we successfully generate (a, b)-connected discrete figures up to size n = 18.
