Sets of Metrical Hypotheses for Geometry

The Geometry of Melodic, Harmonic, and Metrical Hierarchy

Music is hierarchically structured in numerous ways, and all of these forms of organization share essential mathematical features. A geometrical construct called the Stasheff polytope or associahedron summarizes these similarities. The Stasheff polytope has a robust mathematical literature behind it demonstrating its wealth of mathematical structure. By recognizing hierarchies that arise in mus...

Three Hypotheses about the Geometry of Mind

What set of concepts and formalizations might one use to make a practically useful, theoretically rigorous theory of generally intelligent systems? We present a novel perspective motivated by the OpenCog AGI architecture, but intended to have a much broader scope. Types of memory are viewed as categories, and mappings between memory types as functors. Memory items are modeled using probability ...

Explicit Salem Sets and Applications to Metrical Diophantine Approximation

Let Q be an infinite subset of Z, let Ψ : Z→ [0,∞), and let θ ∈ R. Define E(Q,Ψ, θ) = {x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q}. We prove a lower bound on the Fourier dimension of E(Q,Ψ, θ). This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimens...

Geometry of antimatroidal point sets

The notion of ”antimatroid with repetition” was conceived by Bjorner, Lovasz and Shor in 1991 as a multiset extension of the notion of antimatroid [2]. When the underlying set consists of only two elements, such two-dimensional antimatroids correspond to point sets in the plane. In this research we concentrate on geometrical properties of antimatroidal point sets in the plane and prove that the...

On the Hypotheses which lie at the Bases of Geometry