We present a derivation of relativistic spacetime largely untethered from specific physical considerations, in constrast to the many physically-based derivations that have appeared in the last few decades. The argument proceeds from the inherent magma (groupoid) existing on the union of spacetime frame components [Formula: see text] and Euclidean [Formula: see text] which is consistent with an ...

In this paper we prove that for a large and physically relevant class of smooth complete metrics on open four-manifolds the set of L-norms or energies of (anti-)self-dual finite-energy G-connections is discrete where G is a simple, connected, simply connected and compact Lie group. The proof is based on the Atiyah–Patodi–Singer Index Theorem. To demonstrate our result we consider the Euclidean ...

Modern data analysis tools have to work on high-dimensional data, whose components are not independently distributed. High-dimensional spaces show surprising, counter-intuitive geometrical properties that have a large influence on the performances of data analysis tools. Among these properties, the concentration of the norm phenomenon results in the fact that Euclidean norms and Gaussian kernel...

We consider convolving a Gaussian of a varying scale ǫ against a Borel measure μ on Euclidean δ-dimensional space. The L norm of the result is differentiable in ǫ. We calculate this derivative and show how the upper order of its growth relates to the lower Rényi dimension. We assume p is strictly between 1 and ∞ and that μ is finite and has bounded support. We consider the abstract problem of c...

The operator norm of an n × n complex matrix A is its norm as a linear operator on the Euclidean space I C; i.e., ‖A‖ = sup{‖Ax‖ : x ∈ I C, ‖x‖ = 1}. The Frobenius norm of A is defined as ‖A‖2 = ( ∑ i,j |aij|) 1 2 = (trA∗A) 1 2 . Both these norms are used frequently in analysis of matrices. They can also be described in terms of the singular values of A − the square roots of the eigenvalues of ...

We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. This is applied in two different directions. The first is the determination of whether or not several classes of linear functionals satisfy the uniform Central Limit Theorem or the uniform Law of Large Numbers. The second direction, which was the main motivation of this study, is the ...

This paper aims to develop simple statistical methods for indexing into an aerial image database using a cartographic model. The images contain severe distortions due to the acquisition process and the data cannot be recovered by applying a simple Euclidean transform to the model. The underlying representation of the infra-red image map within the database is based on histograms of line-segment...

A band-dominant function on the Euclidean sphere embedded in R is the restriction to this sphere of an entire function of q + 1 complex variables having a finite exponential type in each of its variables. We develop a method to represent such a function using finitely many bits, using the values of the function at scattered sites on the sphere. The number of bits required in our representation ...

nThis paper transmits a FORTRAN-IV coding of the fuzzy c-means (FCM) clustering program. The FCM program is applicable to a wide variety of geostatistical data analysis problems. This program generates fuzzy partitions and prototypes for any set of numerical data. These partitions are useful for corroborating known substructures or suggesting substructure in unexplored data. The clustering crit...

A bar-joint framework (G,p) in a (non-Euclidean) real normed plane X is the combination of finite, simple graph G and placement p vertices X. globally rigid if every other (G,q) with same edge lengths as arises from an isometry The weaker property local rigidity planes (where only within neighbourhood are considered) has been studied by several researchers over last 5 years after being introduc...