Landscapes of data sets and functoriality of persistent homology

The aim of this article is to describe a new perspective on functoriality of persistent homology and explain its intrinsic symmetry that is often overlooked. A data set for us is a finite collection of functions, called measurements, with a finite domain. Such a data set might contain internal symmetries which are effectively captured by the action of a set of the domain endomorphisms. Different choices of the set of endomorphisms encode different symmetries of the data set. We describe various category structures on such enriched data sets and prove some of their properties such as decompositions and morphism formations. We also describe a data structure, based on coloured directed graphs, which is convenient to encode the mentioned enrichment. We show that persistent homology preserves only some aspects of these landscapes of enriched data sets however not all. In other words persistent homology is not a functor on the entire category of enriched data sets. Nevertheless we show that persistent homology is functorial locally. We use the concept of equivariant operators to capture some of the information missed by persistent homology

Vibration Control of an existing building through the Vibrating Barrier

Vibration control of structures is normally addressed through devices such as isolators, dampers and tuned mass dampers. Although those devices are technically sound their use might become unpractical in existing buildings such as heritage structures, whereas the alteration of part of the structure is forbidden for various socio-economic issues. In this context, a novel passive control device called Vibrating Barrier (ViBa) has been recently proposed. The Vibrating Barrier is a massive structure, hosted in the soil and detached from the existing building, calibrated for absorbing portion of the ground motion input energy. The working principle is based on the generally know structure-soil-structure interaction between two vibrating structures and the soil. In this paper the Vibrating Barrier is designed to control the vibration of an existing masonry structure forced by ground motion acceleration. The structure, the soil and the ViBa are assumed to be linear behaving and modelled through a pertinent Finite Element approach. The design is pursued through a simplified discrete model of the structure and the ViBa in which the soil is represented by linear elastic springs. Significant reduction of the dynamic response has been achieved manifesting the potential of the Vibrating Barrier to be a valid alternative whereas the traditional vibration control techniques cannot be applied