This paper addresses the problem of testing simple hypotheses about the mean of a bivariate normal distribution with identity covariance matrix against restricted alternatives. The LRTs and their power functions for such types of hypotheses are derived. Furthermore, through some elementary calculus, it is shown that the power function of the LRT satisfies certain monotonicity and symmetry p...

In this paper we extend the notion of a Lorentz cone. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., monotone) with respect to the order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular a Cartesian product between an Euclidean spa...

Let V be a real linear space. The functor ConvexComb(V ) yielding a set is defined by: (Def. 1) For every set L holds L ∈ ConvexComb(V ) iff L is a convex combination of V . Let V be a real linear space and let M be a non empty subset of V . The functor ConvexComb(M) yielding a set is defined as follows: (Def. 2) For every set L holds L ∈ ConvexComb(M) iff L is a convex combination of M . We no...

Generalized circumcenters have been recently introduced and employed to speed up classical projection-type methods for solving feasibility problems. In this note, are enforced in a new setting; they proven provide inward directions sets given by convex inequalities. particular, we show that circumcentric of finitely generated cones belong the interior their polars. We also derive measure interi...

In the conic formulation of a convex optimization problem the constraints are expressed as linear inequalities with respect to a possibly non-polyhedral convex cone. This makes it possible to formulate elegant extensions of interior-point methods for linear programming to general nonlinear convex optimization. Recent research on cone programming algorithms has particularly focused on three conv...

We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E , then the convex hull is the intersection of E with K (resp., C). The existence o...

Recently, Cho et al. [Y. J. Cho, R. Saadati, S. H. Wang, Common xed point theorems on generalized distance in ordered cone metric spaces, Comput. Math. Appl. 61 (2011) 1254-1260] dened the concept of the c-distance in a cone metric space and proved some xed point theorems on c-distance. In this paper, we prove some new xed point and common xed point theorems by using the distance in ordered con...

The purpose of this paper is to provide a modified open mapping theorem, which also applies to maps which are not open, but whose intersection with a convex cone is relatively open. We give a constructive proof, which for the special maps considered here may be seen as an alternative to the indirect proof of Frankowska’s more general open mapping principle. Open mapping theorems are of utmost i...

Our theorems are on ordered cone metric spaces which are not necessarily normal. In the end, we describe the application of the main results in the integral equation.Although Du in [W‎. ‎S‎. ‎Du‎, ‎A note on cone metric fixed point theory and its equivalence‎, ‎Nonlinear Analysis‎, ‎72(2010) 2259-2261.]‎, ‎showed that the fixed point results in the setting of cone...

In a series of nonsmooth versions of the Pontryagin Maximum Principle, we used generalized differentials of set-valued maps, flows, and abstract variations. Bianchini and Stefani have introduced a notion of possibly high-order variational vector that has the summability property. We consider a slightly more general class of variational vectors than that defined by Bianchini and Stefani, and pro...