Related papers: Inequalities and bilipschitz conditions for triang…
We consider the classical obstacle problem on bounded, connected Lipschitz domains $D \subset \mathbb{R}^n$. We derive quantitative bounds on the changes to contact sets under general perturbations to both the right hand side and the…
We provide a class of geometric convex domains on which the Carath\'eodory-Reiffen metric, the Bergman metric, the complete K\"ahler-Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other.…
We introduce Gaussian-type measures on the manifold of all metrics with a fixed volume form on a compact Riemannian manifold of dimension $\geq 3$. For this random model we compute the characteristic function for the $L^2$ (Ebin) distance…
Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the…
We discuss how metric limits and rescalings of K\"ahler-Einstein metrics connect with Algebraic Geometry, mostly in relation to the study of moduli spaces of varieties, and singularities. Along the way, we describe some elementary examples,…
Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite…
We prove a generalization for some $\mathbb C$ domains of Gehring - Palka theorem on Moebius transformations regarding the distance ratio metric. Namely, we show that this theorem is valid for arbitrary holomorphic mappings $f : H \to H$ or…
Given two Ahlfors-David regular sets in metric spaces, we study the question whether one of them has a subset bilipschitz equivalent with the other.
Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…
The Dirichlet-to-Neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying Dirichlet problem is not uniquely solvable. The main objective of this paper is to present a systematic study of…
Basic aspects of differential geometry can be extended to various non-classical settings: Lipschitz manifolds, rectifiable sets, sub-Riemannian manifolds, Banach manifolds, Weiner space, etc. Although the constructions differ, in each of…
Basic properties of Hausdorff content, dimension, and measure of subsets of metric spaces are discussed, especially in connection with Lipschitz mappings and topological dimension.
We present a conceptually simple and intuitive method to calculate and to measure the dissimilarities among 2D shapes. Several methods to interpret and to visualize the resulting dissimilarity matrix are presented and compared.
Some sharp inequalities between the triangular ratio metric and the Cassinian metric are proved in the unit ball.
In this paper we study the symmetries of the dual Taub-NUT metrics. Generic and non-generic symmetries of dual Taub-NUT metrics are investigated. The existence of the Runge-Lenz type symmetry is analyzed for dual Taub-NUT metrics. We find…
In this paper the symmetries of the dual manifold were investigated. We found the conditions when the manifold and its dual admit the same Killing vectors and Killing-Yano tensors. In the case of an Einstein's metric $g_{\mu\nu}$ the…
We give conditions for topological and bi-Lipschitz equivalences within a class of mixed singularities of Pham-Brieskorn type. As a consequence, we construct infinite families that are topologically trivial but have distinct bi-Lipschitz…
In this book chapter we survey known approaches and algorithms to compute discrepancy measures of point sets. After providing an introduction which puts the calculation of discrepancy measures in a more general context, we focus on the…
When studying convergence of measures, an important issue is the choice of probability metric. In this review, we provide a summary and some new results concerning bounds among ten important probability metrics/distances that are used by…
For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of…