Narrow Results

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

Chapter 7 covers the connections between the Mereon Matrix and the work of R. Buckminster Fuller and Ron Resch are explored in detail, deepening the knowledge found in the Context. The Context’s dynamics defines five “Jitterbugs’, a process described by Fuller. The concept of scalability is expanded to include horizontal and vertical, inside and outside, up and down; it is shown how these characterisations are essential for application in natural and social systems. Fuller’s impact on the world is realised today through his ongoing influence on engineering, design, architecture, science and the arts; the foundation he laid is used to those committed to creating sustainable systems. Authors suggest that in many ways this work rests on Fuller’s foundation and evolves it, that applying the Mereon Matrix will catalyse further positive change, stable and long-lasting benefits. Evidence is also presented to demonstrate comparisons between the Context and the 120 Cell Dodecaplex.

Creating direct and dramatic visual use of the Mereon Trefoil as time and a knotted fold in space-time, the dynamics are imagined as geometric lines of force that turn the Mereon Matrix inside out. It is shown that Mereon’s toroidal dynamics are reflected in nature’s patterns and human social patterning. This chapter presents how the Matrix’s 33 planar ‘landshapes’, similar to Russian dolls, imprint the knot with particular symmetries that define the Mereon Trefoil as a knotted ‘timeshape’, all aspects time and space connected to and embedded with prior ‘time-slices’. Chirality is shown to be implicit in the knot, and us revealed to have a fundamental relationship with the smallest 3D polyhedra, two types of tetrahedron. The Mereon Trefoil is demonstrated equal to a minimal Enneper surface; all polyhedra are shown to ‘tie’ the knot, and like the geometry, the topology scales to infinity.