We present a method for accurately computing the metric entropy (or, equivalently, the Lyapunov exponent) of the absolutely continuous invariant measure for a piecewise analytic expanding Markov map of the interval. We construct atomic measures M supported on periodic orbits up to period M, and prove that h(M) ! h() super-exponentially fast. We illustrate our method with several examples. 0. Th...

We study a new construction of an invariant metric for compact subgroups of the automorphism group of a domain in complex space. Applications are provided.

We show that if the Szlenk index of a Banach space X is larger than the first infinite ordinal ω or if the Szlenk index of its dual is larger than ω, then the tree of all finite sequences of integers equipped with the hyperbolic distance metrically embeds into X. We show that the converse is true when X is assumed to be reflexive. As an application, we exhibit new classes of Banach spaces that ...

Non-rigid shapes are ubiquitous in Nature and are encountered at all levels of life, from macro to nano. The need to model such shapes and understand their behavior arises in many applications in imaging sciences, pattern recognition, computer vision, and computer graphics. Of particular importance is understanding which properties of the shape are attributed to deformations and which are invar...

The possible amplification of gauge invariant metric fluctuations in the infrared sector are very important during reheating stage of inflation. In this stage the inflaton field oscillates around the minimum of the scalar potential. The evolution for the super Hubble scales gauge invariant metric fluctuations can be studied by means of the Bardeen parameter. For a massive scalar field with pote...

We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the invariant volume of the space of two-qubit perfect entanglers. We find that this volume corresponds to more than 84% of the total invariant volume of t...

In this paper we study multidimensional persistence modules [5, 13] via what we call tame functors and noise systems. A noise system leads to a pseudo-metric topology on the category of tame functors. We show how this pseudo-metric can be used to identify persistent features of compact mul-tidimensional persistence modules. To count such features we introduce the feature counting invariant and ...

We introduce a class of metric perceptual distances between textures. The first metric is sensitive to both rotation and scale differences, and provides a basis for two other metrics, one invariant to rotation, and the other invariant to both rotation and scale. Our metrics are based on Earth Mover’s Distance computations on log-polar distributions of spatial frequency computed from Gabor filte...

A Riemannian metric is said to be Einstein if the Ricci curvature is a constant multiple of the metric. Given a manifold M , one can ask whether M carries an Einstein metric, and if so, how many. This fundamental question in Riemannian geometry is for the most part unsolved (cf. [Bes]). As a global PDE or a variational problem, the question is intractible. It becomes more manageable in the homo...

We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when we assume the Galilean invariance, the scale invariant deformation always pres...