We develop local elliptic gradient estimates for a basic nonlinear f-heat equation with logarithmic power nonlinearity and establish pointwise upper bounds on the weighted heat kernel, all in context of manifolds, where metric potential evolve under Perelman-Ricci type flow. For use is made entropy monotonicity arguments ultracontractivity expressed terms optimal constant Sobolev inequality. So...

We introduce a pointwise variant of the Assouad dimension for measures on metric spaces, and study its properties in relation to global dimension. show that, general, value may differ from counterpart, but many classical cases, exhibits similar exact dimensionality as local dimension, namely it equals almost everywhere. also prove an explicit formula certain invariant with place-dependent proba...

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T p M) for all p ∈ M , which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i. e. S = HId (and thus S is trivially preserved). First we solve an algebraic problem. We determine the...

In this article, we further consider the above system. In particular, we give a sufficient condition under which the above system is Kato chaotic for $eta=0$ and a necessary condition for the above system to be Kato chaotic for $eta=0$. Moreover, it is deduced that for $eta=0$, if $Theta$ is P-chaotic then so is this system, where a continuous map $Theta$ from a compact metric space $Z$ to itse...

Metric learning aims to embed one metric space into another to benefit tasks like classification and clustering. Although a greatly distorted metric space has a high degree of freedom to fit training data, it is prone to overfitting and numerical inaccuracy. This paper presents bounded-distortion metric learning (BDML), a new metric learning framework which amounts to finding an optimal Mahalan...

The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain YII(r, p;K) is obtained in this paper when the parameter K equals p 2 + 1 p+1 . The estimate of holomorphic sectional curvature under this metric is also given which intervenes between −2K and − 2K p and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalen...

We define a family of pseudo metrics on B3 and study elementary properties of the associated metric spaces. As an application we prove that, for any a > 0 and for any countable-to-one function f from (S2, dE) to [0, a], the set NMf = {x ∈ S2 | ∃y ∈ S2 such that f(x) − f(y) > ndE(x, y)} is uncountable for all n ∈ N, where dE is the standard Euclidean metric on S 2 = {(x, y, z) ∈ R3 | x2 + y2 + z...