The deterministic border collision normal form describes the bifurcations of a discrete time dynamical system as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, but within some bounded region, we give conditions which imply that the attractor close to the bifurcation point is the attractor of an Iterated Functio...

Invented in the 1960’s, permutation codes have reemerged in recent years as a topic of great interest because of properties making them attractive for certain modern technological applications. In 2011 a decoding method called LP (linear programming) decoding was introduced for a class of permutation codes with a Euclidean distance induced metric. In this paper we comparatively analyze the Eucl...

We give a sufficient condition for a projective metric on a subset of a Euclidean space to admit a bi-Lipschitz embedding into Euclidean space of the same dimension.

The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from the eigenvalues and eigenfunctions of the Laplace-Beltrami operator determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete heat kernel and the discrete Riemannian metric (u...

Unsupervised phoneme segmentation aims at dividing a speech stream into phonemes without using any prior knowledge of linguistic contents and acoustic models. In [1], we formulated this problem into an optimization framework, and developed an objective function, summation of squared error (SSE) based on the Euclidean distance of cepstral features. However, it is unknown whether or not Euclidean...

We attach the degenerate signature (n,0,1) to the dual Grassmann algebra of projective space to obtain a real Clifford algebra which provides a powerful, efficient model for euclidean geometry. We avoid problems with the degenerate metric by constructing an algebra isomorphism between the Grassmann algebra and its dual that yields non-metric meet and join operators. We focus on the cases of n=2...

Introduction. This paper is concerned with several disconnected developments in distance geometry. §1 deals with the congruent imbedding of metric spaces in euclidean or Hubert spaces. By showing that the validity of the Pythagorean theorem insures the essentially euclidean character of the metric, the basic role this theorem plays in euclidean geometry is seen to be fully justified. In §2 the ...

Quantifying the dissimilarity (or distance) between two sequences is essential to the study of action potential (spike) trains in neuroscience and genetic sequences in molecular biology. In neuroscience, traditional methods for sequence comparisons rely on techniques appropriate for multivariate data, which typically assume that the space of sequences is intrinsically Euclidean. More recently, ...

We show that every n-point metric of negative type (in particular, every n-point subset of L1) admits a Fréchet embedding into Euclidean space with distortion O (√ log n · log log n), a result which is tight up to the O(log log n) factor, even for Euclidean metrics. This strengthens our recent work on the Euclidean distortion of metrics of negative into Euclidean space.