Sets and Numbers

This chapter covers set theory. The topics include set algebra, relations, orderings and mappings, countability and sequences, real numbers, sequences and limits, and set classes including monotone classes, rings, fields, and sigma fields. The final section introduces the basic ideas of real analysis including Euclidean distance, sets of the real line, coverings, and compactness.

An Upper Bound for the Size of $s$-Distance Sets in Real Algebraic Sets

In a recent paper, Petrov and Pohoata developed a new algebraic method which combines the Croot-Lev-Pach Lemma from additive combinatorics and Sylvester’s Law of Inertia for real quadratic forms. As an application, they gave a simple proof of the Bannai-Bannai-Stanton bound on the size of $s$-distance sets (subsets $\mathcal{A}\subseteq \mathbb{R}^n$ which determine at most $s$ different distances). In this paper we extend their work and prove upper bounds for the size of $s$-distance sets in various real algebraic sets. This way we obtain a novel and short proof for the bound of Delsarte-Goethals-Seidel on spherical s-distance sets and a generalization of a bound by Bannai-Kawasaki-Nitamizu-Sato on $s$-distance sets on unions of spheres. In our arguments we use the method of Petrov and Pohoata together with some Gröbner basis techniques.

Scale

A crucial consideration for any approach to landscape is that of scale. People in the past operated at a series of scales from the very local to long distance sets of connections, so that often most evidence is generated by life in the local area, structuring the nature of evidence. The balance between local, regional and long-distance action varies between periods, with longer distance connections most obvious in the Roman period, when Britain was connected to the empire. We start with a general consideration of questions of scale, before moving to consider the Roman period more specifically. We focus on the nature of villas as microcosms of the landscapes in which they sit, looking at where building materials come from, using good information from Isle of Wight villas as a case study. We play with the ideas that villas sit in landscapes, but also represent those landscapes in a condensed form.

Biangular Lines Revisited

Line systems passing through the origin of the d-dimensional Euclidean space admitting exactly two distinct angles are called biangular. It is shown that the maximum cardinality of biangular lines is at least $$2(d-1)(d-2)$$ 2 ( d - 1 ) ( d - 2 ) , and this result is sharp for $$d\in \{4,5,6\}$$ d ∈ { 4 , 5 , 6 } . Connections to binary codes, few-distance sets, and association schemes are explored, along with their multiangular generalization.