We study the canonical metric on a compact Riemann surface of genus at least two. This natural metric is the pullback, via the period map, from the Euclidean metric on the Jacobian variety of the surface. While it is known that the canonical metric is of nonpositive curvature, we show that its Gaussian curvatures are not bounded away from zero nor negative infinity when the surface is close to ...

We describe a new proof of the complete integrability of the two related dynamical systems: the billiard inside the ellipsoid and the geodesic flow on the ellipsoid (in Euclidean, spherical or hyperbolic space). The proof is based on the construction of a metric on the ellipsoid whose nonparameterized geodesics coincide with those of the standard metric. This new metric is induced by the hyperb...

In any subspace of the real line R with the usual Euclidean metric d(x, y) = |x− y|, every triangle is degenerate. In R or R with the usual Euclidean metrics, a triangle is degenerate if and only if its vertices are collinear. With our intuition of a degenerate triangle having “collinear vertices” extended to arbitrary metric spaces, we might expect that a metric space in which every triangle i...

Abstract: Motivated by problems related to quasi-local mass in general relativity, we study the static metric extension conjecture proposed by R. Bartnik [4]. We show that, for any metric on B̄1 that is close enough to the Euclidean metric and has reflection invariant boundary data, there always exists an asymptotically flat and scalar flat static metric extension in M = R \B1 such that it satis...

We prove, with a few minor exceptions, that if P1 and P2 are probability distributions on the countable set S for which the fixed events E and F are independent, then, both for the standard Euclidean metric and for any metric inducing a topology coarser than the Euclidean topology, there exists a third probability distribution P3 on S that preserves this independence and is equidistant from P1 ...

We investigate the performance of a stack algorithm to decode binary Reed-Muller codes. As metric to evaluate the different elements in the stack we use a recently derived lower bound on the squared Euclidean distance. Similar to the A -algorithm the new metric comprises a lower bound on the increase of the squared Euclidean distance in future decoding steps. Simulation results show that in cas...

Conventional physical dogma, justified by the local success of Newtonian dynamics for particles, assigns a Euclidean metric with signature (plus, plus, plus) to the three spatial dimensions. Minimal surfaces are of zero mean curvature and negative Gauss curvature in a Euclidean space, which supports affine evolutionary processes. However, experimental evidence now indicates that the non-affine ...

The grid cells discovered in the rodent medial entorhinal cortex have been proposed to provide a metric for Euclidean space, possibly even hardwired in the embryo. Yet, one class of models describing the formation of grid unit selectivity is entirely based on developmental self-organization, and as such it predicts that the metric it expresses should reflect the environment to which the animal ...

This paper presents an improved version of anchor model applied to solve the two-class classification tasks of the INTERSPEECH 2012 speaker trait Challenge. To build the anchor model space of each task, we include the class models of all tasks. The introduction of within-class covariance normalization (WCCN) applied to the log-likelihood scores of the anchor space not only improves the results ...

INTRODUCTION Unsupervised clustering methods such as Self-Organizing Map (SOM) or Hierarchical Clustering (HC) are data-driven techniques, which have been used successfully in fMRI data analyses [Ngan, Peltier]. In conventional SOM or HC methods, the Euclidean distance is used as the similarity metric to compare signals [Kohonen]. However, Euclidean distance does not accurately delineate the in...