Related papers: Unimodular Hausdorff and Minkowski Dimensions

Let $\{X_n= e^{2\pi i \theta_n}\}$ be a sequence of Steinhaus random variables, where $\theta_n$ are independent and uniformly distributed on $[0,1]$. We compute the almost sure Hausdorff dimension of the images and graphs of the random…

Diversities are an extension of the concept of a metric space which assign a non-negative value to every finite set of points, rather than just pairs. A general theory of diversities has been developed which exhibits many deep analogies to…

We apply the theory of unimodular random rooted graphs to study the metric geometry of large, finite, bounded degree graphs whose diameter is proportional to their volume. We prove that for a positive proportion of the vertices of such a…

The almost sure Hausdorff dimension of the limsup set of randomly distributed rectangles in a product of Ahlfors regular metric spaces is computed in terms of the singular value function of the rectangles.

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincar\'e inequality. For foliations of a metric space X defined by a…

Given an Euclidean space, this paper elucidates the topological link between the partial derivatives of the Minkowski functional associated to a set (assumed to be compact, convex, with a differentiable boundary and a non-empty interior)…

Generalising a construction of Falconer, we consider classes of $G_\delta$-subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We…

We are going to widen the scope of the previously defined Hausdorff-integral in two ways. First, in the sense, that we develop the theory of the integral on some naturally generalized measure spaces. Second, we extend it to functions taking…

Diakonov theory of quantum gravity, in which tetrads emerge as the bilinear combinations of the fermionis fields,\cite{Diakonov2011} suggests that in general relativity the metric may have dimension 2, i.e. $[g_{\mu\nu}]=1/[L]^2$. Several…

The Lipschitz extension modulus $e(M)$ of a metric space $M$ is the infimum over $L\ge 1$ such that for any Banach space $Z$ and any $C\subset M$, any 1-Lipschitz function $f:C\to Z$ can be extended to an $L$-Lipschitz function $F:M\to Z$.…

For an integer $m\geq 1$, a combinatorial manifold $\widetilde{M}$ is defined to be a geometrical object $\widetilde{M}$ such that for $\forall p\in\widetilde{M}$, there is a local chart $(U_p,\phi_p)$ enable $\phi_p:U_p\to…

The Minkowski vacuum is often presented in textbooks and reviews as a thermofield double (TFD) state, an entangled state of field modes in the left and right Rindler wedges. This picture is widely used to explain the Unruh effect, motivate…

Given a finite set $\mathcal{A} \subseteq \mathrm{SL}(2,\mathbb{R})$ we study the dimension of the attractor $K_\mathcal{A}$ of the iterated function system induced by the projective action of $\mathcal{A}$. In particular, we generalise a…

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and…

We study the four-dimensional effective theory arising from ten-dimensional heterotic supergravity compactified on manifolds with torsion. In particular, given the heterotic superpotential appropriately corrected at $\mathcal{O}(\alpha')$…

The Unruh effect has been investigated from the point of view of the quantum statistical Zubarev density operator in space with the Minkowski metric. Quantum corrections of the fourth order in acceleration to the energy-momentum tensor of…

This work deals with new classes of spinors of mass dimension one in Minkowski spacetime. In order to accomplish it, the Lounesto classification scheme and the inversion theorem are going to be used. The algebraic framework shall be…

We investigate a quasisymmetrically invariant counterpart of the topological Hausdorff dimension of a metric space. This invariant, called the topological conformal dimension, gives a lower bound on the topological Hausdorff dimension of…

This article describes the theoretical foundation of and explicit algorithms for a novel approach to morphology and anisotropy analysis of complex spatial structure using tensor-valued Minkowski functionals, the so-called Minkowski tensors.…

We define angles from-to and between infinite dimensional subspaces of a Hilbert space, inspired by the work of E. J. Hannan, 1961/1962 for general canonical correlations of stochastic processes. The spectral theory of selfadjoint operators…