Application of the network equivalent concept for external system representation for power system transient analysis is well known. However, the challenge to utilize an equivalent network, approximated by a rational function, is to guarantee the passivity of the corresponding model. In this regard, special techniques are required to enforce the passivity of the equivalent model through a post processing approach that minimizes its impact on the original model characteristics. In this paper, the passivity is enforced by expressing the problem in terms of a CONVEX optimization problem that guarantees the global optimal solution. The CONVEX optimization problem is efficiently solved by recently developed numerical interior–point methods. This passivity enforcement is also global which indicates that the passivity enforcement in one region does not lead to passivity violation in other regions.

A new geometric notion on a nonempty and CONVEX pair of subsets of a CONVEX metric space X, called semi-normal STRUCTURE, is introduced and used to investigate the existence of best proximity pairs for a new class of mappings, called strongly noncyclic relatively C-nonexpansive. We also study the STRUCTURE of minimal sets of strongly noncyclic relatively C-nonexpansive mappings in the setting of CONVEX metric spaces.

The conic STRUCTURE of the CONVEX cone of non-negative operator CONVEX functions on (0,¥) (also on (−1, 1)) is clarified. We completely determine the extreme rays, the closed faces, and the simplicial closed faces of this CONVEX cone.

In this paper, we first characterize the CONVEX L-subgroup of an L-ordered group by means of four kinds of cut sets of an L-subset. Then we consider the homomorphic preimages and the product of CONVEX L-subgroups. After that, we introduce an L-CONVEX STRUCTURE constructed by CONVEX L-subgroups. Furthermore, the notion of the degree to which an L-subset of an L-ordered group is a CONVEX L-subgroup is proposed and characterized. An L-fuzzy CONVEX STRUCTURE which results from CONVEX L-subgroup degree is imported naturally, and its L-fuzzy CONVEXity preserving mappings investigated.

Hermite-Hadamard inequality is one of the fundamental applications of CONVEX functions in Theory of Inequality. In this paper, Hermite-Hadamard inequalities for B-CONVEX and B􀀀 1-CONVEX functions are proven.

Following a recent work of Chademan-Mirzapour (1995&1999). we generalize the classical theorems of Jensen, Bernstein-Doetsch, Ostrowski, Blumberg-Sierpinski and Mehdi on approximately midCONVEX functions in real vactor spaces to approximately midCONVEX functions on topological groups. We define also E -Wright-CONVEXity in topological groups and prove a theorem on it