Related papers: Extreme Geometric Quantiles Under Minimal Assumpti…
Tukey depth regions are important notions in nonparametric multivariate data analysis. A $\tau$-th Tukey depth region $\mathcal{D}_{\tau}$ is the set of all points that have at least depth $\tau$. While the Tukey depth regions are easily…
The new uncertainty relation is derived in the context of the canonical quantum theory with gravity for the case of the maximally symmetric space. This relation establishes a connection between fluctuations of the quantities which determine…
We investigate Gaussian quantum states in view of their exceptional role within the space of all continuous variables states. A general method for deriving extremality results is provided and applied to entanglement measures, secret key…
We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4+n dimensions are given and their…
Pointwise tangential dimensions are introduced for metric spaces. Under regularity conditions, the upper, resp. lower, tangential dimensions of X at x can be defined as the supremum, resp. infimum, of box dimensions of the tangent sets, a…
The quantum corrections to black hole entropy, variously defined, suffer quadratic divergences reminiscent of the ones found in the renormalization of the gravitational coupling constant (Newton constant). We consider the suggestion, due to…
We consider the set of extremal points of the generalized unit ball induced by gradient total variation seminorms for vector-valued functions on bounded Euclidean domains. These are central to the understanding of sparse solutions and…
This article is an expository introduction to our paper Convexity of the K-energy and Uniqueness of Extremal metrics. We present the main ideas behind the proof that Mabuchi's K-energy functional is convex along weak geodesics in the space…
The axial gauge is applied to the analysis of Euclidean quantum gravity on manifolds with boundary. A set of boundary conditions which are completely invariant under infinitesimal diffeomorphisms require that spatial components of metric…
We compute explicitly the limits of tangents of a quasi-ordinary singularity in terms of its special monomials. We show that the set of limits of tangents of Y is essentially a topological invariant of Y .
We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions…
Seeking a relativistic quantum infrastructure for gauge physics, we analyze spacetime into three levels of quantum aggregation analogous to atoms, bonds and crystals. Quantum spacetime points with no extension make up more complex link…
This work employs variational techniques to revisit and expand the construction and analysis of extreme value processes. These techniques permit a novel study of spatial statistics of the location of minimizing events. We develop integral…
The geometric operators of area, volume, and length, depend on a fundamental length l of quantum geometry which is a priori arbitrary rather than equal to the Planck length l_P. The fundamental length l and the Immirzi parameter $\gamma$…
We present canonical quantiles and depths for directional data following a distribution which is elliptically symmetric about a direction $\mu$ on the sphere $\mathcal{S}^{d-1}$. Our approach extends the concept of Ley et al. [1], which…
Consider the unordered configuration spaces of manifolds. Knudsen, Miller and Tosteson proved that the extremal homology groups of configuration spaces of manifold are eventually quasi polynomials. In this paper, we give the precise degree…
In the Hardy spaces $H^1$ and $H^\infty$, there are neat and well-known characterizations of the extreme points of the unit ball. We obtain counterparts of these classical theorems when $H^1$ (resp., $H^\infty$) gets replaced by the…
We establish a regularity result for the metric on any 4-dimensional extremal K\"ahler manifold, and a weak compactness theorem on the space of such metrics. Specifically, the sectional curvature at a point is bounded when the quantity…
Higher-dimensional theories of the kind which may unify gravitation with particle physics can lead to significant modifications of general relativity. In five dimensions, the vacuum becomes non-standard, and the Weak Equivalence Principle…
The Gaussian curvature $K$ is a fundamental geometric quantity discovered by Gauss in the case of surfaces embedded in $\mathbb{R}^3$. One can naturally extend the definition of the Gaussian curvature to arbitrary submanifolds of…