This paper introduces a novel mathematical and computational framework, namely Log-Hilbert-Schmidt metric between positive definite operators on a Hilbert space. This is a generalization of the Log-Euclidean metric on the Riemannian manifold of positive definite matrices to the infinite-dimensional setting. The general framework is applied in particular to compute distances between covariance o...

In a recent study, a new concept, namely elucidative fusion systems, was proposed and demonstrated using case based reasoning as the fusion tool. Elucidative fusion systems are designed to offer not only optimally fused decisions but also elucidate the relative contributions made by the different data sources (sensors) to the fused decisions. In the earlier study, the concept was illustrated us...

In this article we define and study a notion of asymptotic rank for metric spaces and show in our main theorem that for a large class of spaces, the asymptotic rank is characterized by the growth of the higher filling functions. For a proper, cocompact, simplyconnected geodesic metric space of non-curvature in the sense of Alexandrov the asymptotic rank equals its Euclidean rank.

Outlier Detection is a critical and cardinal research task due its array of applications in variety of domains ranging from data mining, clustering, statistical analysis, fraud detection, network intrusion detection and diagnosis of diseases etc. Over the last few decades, distance-based outlier detection algorithms have gained significant reputation as a viable alternative to the more traditio...

To begin with, let us agree to some basic conventions. We employ the symbols ∆ and ∇ to denote the Laplace operator ∑nk=1 ∂/∂xk and the gradient vector (∂/∂x1, ..., ∂/∂xn) over the Euclidean space R , n ≥ 2. For notational convenience we use X . Y as X ≤ CY for a constant C > 0. We always assume that u is a smooth real-valued function on R, written u ∈ C∞(Rn), and then it generates a conformal ...

It is a well-known fact that every Riemann surface with negative Euler characteristic admits a hyperbolic metric. But this metric is by no means unique – indeed, there are uncountably many such metrics. In this paper, we study the space of all such hyperbolic structures on a Riemann surface, called the Teichmüller space of the surface. We will show that it is a complete metric space, and that i...

There is a general method, applicable in many situations, whereby a pseudo–Riemannian metric, invariant under the action of some Lie group, can be deformed to obtain a new metric whose geodesics can be expressed in terms of the geodesics of the old metric and the action of the Lie group. This method applied to Euclidean space and the unit sphere produces new examples of complete Riemannian metr...

3 Work done so far 4 3.1 Planning with Homotopy class constraints . . . . . . . . . . . . . . . . . . . . . . 4 3.1.1 The problem in 2 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1.2 The problem in 3 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.3 Generalization and extension to higher dimensions . . . . . . . . . . . . . 6 3.2 Metric information using ...

We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a polynomial map f : C → C is expansive on a neighborhood of the Julia set, J . We show construction of such a metric proves the map is hyperbolic. We also examine the question of whether the algorithm can be improved, and the related question of how to build a metric close to euclide...

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metri...