Let $M$ be an orientable hypersurface in the Euclidean space $R^{2n}$ with induced metric $g$ and $TM$ be its tangent bundle. It is known that the tangent bundle $TM$ has induced metric $overline{g}$ as submanifold of the Euclidean space $R^{4n}$ which is not a natural metric in the sense that the submersion $pi :(TM,overline{g})rightarrow (M,g)$ is not the Riemannian submersion. In this paper...

We study the posterior contraction behavior of the latent population structure that arises in admixture models as the amount of data increases. An admixture model — alternatively known as a topic model — specifies k populations, each of which is characterized by a ∆-valued vector of frequencies for generating a set of discrete values in {0, 1, . . . , d}. The population polytope is defined as t...

The aim of this paper is to characterize $3$-dimensional $N(k)$-paracontact metric manifolds satisfying certain curvature conditions. We prove that a $3$-dimensional $N(k)$-paracontact metric manifold $M$ admits a Ricci soliton whose potential vector field is the Reeb vector field $xi$ if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discuss...

In this note we show that the general theory of vector valued singular integral operators Calderon-Zygmund defined on metric measure spaces, can be applied to obtain Sobolev type regularity properties for solutions dyadic fractional Laplacian. doing so, define partial derivatives in terms Haar multipliers and homogeneous operators.

Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F -valued functions on Y is said to be biseparating if f and g are disjoint if and only if Tf and Tg are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uni...

The geometric structure of stress theory on differentiable manifolds is considered. Mechanics is assumed to take place on an m-dimensional and no additional metric or parallelism structure is assumed. Two different approaches are described. The first is a generalisation of the traditional Cauchy approach where the resulting stresses are represented mathematically as vector valued (m − 1)forms. ...

In this paper we introduce the notion of an F-metric, as a function valued distance mapping, on a set X and we investigate the theory of F-metric spaces. We show that every metric space may be viewed as an F-metric space and every F-metric space (X, δ) can be regarded as a topological space (X, τδ). In addition, we prove that the category of the so-called extended Fmetric spaces properly contai...

1 The Problem Suppose we have a dataset with N datapoints. Each datapoint consists of a vector of inputs and a real valued-output, so the dataset is x0 ; y0 x1 ; y1 .. xN 1 ; yN 1 The inputs need not be real-valued. All we require of them is a distance metric measuring the similarity of a pair of input vectors Dist : x;x0 ! < (1) and a set of M basis functions 0 : x! <; 1 : x! <; : : : M 1 : x!...

We study the p-fine topology on complete metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. We establish a weak Cartan property, which yields characterizations of the p-thinness and the p-fine continuity, and allows us to show that the p-fine topology is the coarsest topology making all psuperharmonic functions continuous. Our p-harmonic and superharmo...