Convexity

We introduce the concept of λ-hyperconvexity in metric spaces, generalizing the classical notion of a hyperconvex metric space. We show that a bounded metric space which is λ-hyperconvex has the fixed point property for nonexpansive mappings provided λ < 2. Uniformly convex Banach spaces are examples of such λ-hyperconvex spaces for some λ < 2. We furthermore investigate the relationship between Penot's Intersection Property and 2hyperconvexity.

Let G be a finite simple graph. Let S ⊆ V (G), its closed interval I[S] is the set of all vertices lying on a shortest path between any pair of vertices of S. The set S is convex if I[S] = S. In this work we define the concept of convex partition of graphs. If there exists a partition of V (G) into p convex sets we say that G is p-convex. We prove that is NP -complete to decide whether a graph G is p-convex for a fixed integer p ≥ 2. We show that every connected chordal graph is p-convex, for 1 ≤ p ≤ n. We also establish conditions on n and k to decide if a power of cycle is p-convex. Finally, we develop a linear-time algorithm to decide if a cograph is p-convex.

The Wave Based Method (WBM) is an alternative numerical prediction method for both interior and exterior steady-state dynamic problems, which is based on an indirect Trefftz approach. It applies wave functions, which are exact solutions of the governing differential equation, to describe the dynamic field variables. The smaller system of equations and the absence of pollution errors make the WBM very suitable for the treatment of Helmholtz problems in the mid-frequency range, where element-based methods are no longer feasible due to the associated computational costs. A sufficient condition for convergence of the method is the convexity of the considered problem domain. As a result, only problems of moderate geometrical complexity can be considered and some geometrical features cannot be handled at all. In this paper, these limitations are alleviated through the development of a general modelling framework based on existing WBM methodologies which allows for the efficient introduction of inclusion configurations in bounded WBM models for problems governed by one or more Helmholtz equations. The feasibility and efficiency of the method is illustrated by means of numerical verification studies in which the methodology is applied to two types of dynamic problems. On the one hand, a single Helmholtz equation associated with the steady-state dynamic behaviour of acoustic cavities is studied. On the other hand, the framework is applied to the solution of the Navier system of partial differential equations that describe the elastodynamic response of two-dimensional perforated solids.

It is known that, if the time variable in the heat equation is complex and belongs to a sector in C, then the theory of analytic semigroups becomes a powerful tool of study. The same is true for the Laplace equation on an infinite strip in the plane, regarded as an initial value boundary value problem. Also, it is known that if both variables, time and spatial, are complex, then, e.g. the Cauchy problem for the heat equation admits as solution, only a formal power series which, in general, converges nowhere. In a recent paper (C.G. Gal, S.G. Gal, and J.A. Goldstein, Evolution equations with real time variable and complex spatial variables, Complex Var. Elliptic Eqns. 53 (2008), pp. 753-774), a complementary approach was made: the study of the complex versions of the classical heat and Laplace equations, obtained by 'complexifying' the spatial variable only (and keeping the time variable real). The goal of this article is to extend that study to the higher-order heat and Laplace-type equations. This 'complexification' is based on integral representations of the solutions in the case of a real spatial variable, by complexifying the spatial variable in the corresponding semigroups of operators. It is of interest to note that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness.