Now showing items 1-4 of 4

• #### An equivalence class decomposition of finite metric spaces via Gromov products ﻿

Let (X, d) be a finite metric space with elements P-i, i = 1,..., n and with the distance functions d(ij) The Gromov Product of the "triangle" (P-i, P-j, P-k) with vertices P-t, P-j and P-k at the vertex Pi is defined by Delta(ijk) = 1/2(d(ij) + d(ik) - d(jk)). We show that the collection of Gromov products determines the metric. We call a metric space Delta-generic, if the set of all Gromov products at a fixed vertex P-i has a unique smallest element (for i = 1,., n). We consider the function ...

• #### Gromov product structures, quadrangle structures and split metric spaces ﻿

Let (X,d) be a finite metric space with elements Pi, i=1,…,n and with distances dij≔d(Pi,Pj) for i,j=1,…,n. The “Gromov product” Δijk, is defined as [Formula presented]. (X,d) is called Δ-generic, if, for each fixed i, the set of Gromov products Δijk has a unique smallest element, Δijiki. The Gromov product structure on a Δ-generic finite metric space (X,d) is the map that assigns the edge Ejiki to Pi. A finite metric space is called “quadrangle generic”, if for all 4-point subsets {Pi,Pj,Pk,Pl}, ...

• #### Maximal linear subspaces of strong self-dual 2-forms and the Bonan 4-form ﻿

The notion of self-duality of 2-forms in 4-dimensions plays an eminent role in many areas of mathematics and physics, but although the 2-forms have a genuine meaning related to curvature and gauge-field-strength in higher dimensions also, their "self-duality" is something which is almost avoided above 4-dimensions. We show that self-duality of 2-forms is a very natural notion in higher (even) dimensions also and we prove the equivalence of some scattered and rarely used definitions in the literature. ...

• #### Self-duality in higher dimensions ﻿

Let w be a 2-form on a 2n dimensional manifold. In previous work, we called w "strong self-dual, if the eigenvalues of its matrix with respect to an orthonormal frame are equal in absolute value. In a series of papers, we showed that strong self-duality agrees with previous definitions; in particular if w is strong self-dual, then, in 2n dimensions, w(n) is proportional to its Hodge dual w and in 4n dimensions, w(n) is Hodge self-dual. We also obtained a local expression of the Bonan 4-form on 8 ...