In this note we continue the study of asymptotic invariants of Riemannian tori (e.g. see [BuI] and references there). By asymptotic invariants we mean invariants which do not change under passing to finite covers. In [BuI] we show that the asymptotic volume growth of a Riemannian torus is at least as fast as that of a flat one. One may ask what are the possible values of “asymptotic isoperimetr...

Locally flat Finsler metrics arise from information geometry. Some speciel locally dually flat Finsler metrics had been studied in Cheng et al. [3] and Xia [4] respectively. As we konw, a new class of Finsler metrics called general (α, β)-metrics are introduced, which are defined by a Riemannian metrics α and 1-form β. These metrics generalize (α, β)-metrics naturally. In this paper, we give a ...

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Section 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis. Many natural applications arise in the settin...

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q = 1 case these are the normal distributions. Some methods to endow this parameter space with Riemannian metric is presented: the second derivative of the q-Rényi entropy, Tsallis-entropy and the relative entropy gi...

Introduction Having applications to Form recognition in mind, we want to be able to compare shapes of surfaces in R3 in a way that does not depend on parameterizations. To accomplish such so-called gauge invariance, we defined a metric on the space of parameterized surfaces that is degenerate in the direction of reparameterization. 1 What are the surfaces under consideration? The surfaces we wi...

U. Introduction In Doyle and Bricksen [1956. p. 77] it is observed that the Cauchy stress tensor 0 can be derived by varying the internal energy e with respect to the Riemannian metric on space: oab =Zpae/agab. Their formula has gone virtually unnoticed in the elasticity literature. In this lecture we shall explain some of the reasons why this formula is, in fact, of fundamental significance. S...

Let $(M,g)$ be a Riemannian manifold and $(TM,\tilde{g})$ its tangent bundle with the $g-$natural metric. In this paper, family of metallic structures $J$ is constructed on $TM,$ found conditions under which these are integrable. It proved that $(TM,\tilde{g},J)$ decomposable if only flat.

In this paper, as an application of the inverse problem calculus variations, we investigate two compatibility conditions on spherically symmetric Finsler metrics. By making use these conditions, focus our attention Landsberg We classify all manifolds or Berwald types. For higher dimensions $n\geq 3$, prove that: are either Riemannian their geodesic sprays have a specific formula; regular metric...

We show that for generic Riemannian metrics on a closed spin manifold of dimension three the Dirac operator has only simple eigenvalues.

Nearest-neighbor search dominates the asymptotic complexity of sampling-based motion planning algorithms and is often addressed with k-d tree data structures. While it is generally believed that the expected complexity of nearest-neighbor queries is O(log(N)) in the size of the tree, this paper reveals that when a classic k-d tree approach is used with sub-Riemannian metrics, the expected query...