Given a set of i (i = 1; 2; : : :; k) orientations (angles) in the plane, one can deene a distance function which induces a metric in the plane, called the orientation metric 3]. In the special case where all the angles are equal, we call the metric a uniform orientation metric 2]. Speciically, if there are orientations, forming angles ii ; 0 i ?1, with the x-axis, where 2 is an integer, we cal...

The idea of probabilistic metric space was introduced by Menger and he showed that probabilistic metric spaces are generalizations of metric spaces. Thus, in this paper, we prove some of the important features and theorems and conclusions that are found in metric spaces. At the beginning of this paper, the distance distribution functions are proposed. These functions are essential in defining p...

Any sphere S admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S, m > 1 are known to have another Sp(m + 1)-homogeneous Einstein metric discovered by Jensen [Jen73]. In addition, S has a third Spin(9)-invariant homogeneous Einstein metric discovered by Bourguignon a...

If a connected metric space S is locally separable, then S is separable. If a connected, locally connected, metric space S is locally peripherally separable, then S is separable. Furthermore if a connected, locally connected, complete metric space S satisfies certain "flatness" conditions, it is known to be separable. These "flatness" conditions are rather strong and involve both im kleinen and...

A family of spherically symmetric solutions in the model with mcomponent multicomponent anisotropic fluid is considered. The metric of the solution depends on parameters qs > 0, s = 1, . . . ,m, relating radial pressures and the densities and contains (n− 1)m parameters corresponding to Ricci-flat “internal space” metrics and obeying certain m(m − 1)/2 (“orthogonality”) relations. For qs = 1 (f...

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the -thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log.gCp /. The same result also holds for the -thick part of the moduli space of metric graphs of rank n equipped with the ...

Providing each simplex in C(S) with the standard euclidean metric of side-length 1 equips the complex of curves with the structure of a geodesic metric space whose isometry group is just M̃g,m (except for the twice punctured torus). However, this metric space is not locally compact. Masur and Minsky [MM1] showed that nevertheless the geometry of C(S) can be understood quite explicitly. Namely, C...

We introduce a new Bayesian network (BN) scoring metric called the Global Uniform (GU) metric. This metric is based on a particular type of default parameter prior. Such priors may be useful when a BN developer is not willing or able to specify domain-specific parameter priors. The GU parameter prior specifies that every prior joint probability distribution P consistent with a BN structure S is...

We introduce a new Bayesian network (BN) scoring metric called the Global Uniform (GU) metric. This metric is based on a particular type of default parameter prior. Such priors may be useful when a BN developer is not willing or able to specify domain-specific parameter priors. The GU parameter prior specifies that every prior joint probability distribution P consistent with a BN structure S is...

The density theorem of Lebesgue [l] may be stated in the following form: If 5 is a measurable linear point set, the metric density of S exists and is equal to 0 or 1 almost everywhere. We prove the converse that for every set Z of measure 0 there is a measurable set 5 whose metric density does not exist at any point of Z. We note, however, that in order for Z to be the set of points for which t...