Related papers: A structure theorem on doubling measures with diff…

The prolongation structure of a two-by-two problem is formulated very generally in terms of exterior differential forms on a standard representation of Pauli matrices. The differential system is general without making reference to any…

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize…

We prove a common generalization to several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham-sandwich theorem, the necklace splitting theorem for two thieves, a…

In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,\rho,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and…

We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.

This paper presents a set of general strategies for the analysis of structure in amorphous materials and a general approach to assessing the utility of a selected structural description. Measures of structural diversity and utility are…

We obtain a complete description for a probability measure to be doubling on an arbitrarily given uniform Cantor set. The question of which doubling measures on such a Cantor set can be extended to a doubling measure on [0; 1] is also…

In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.

We prove a vector-valued non-homogeneous Tb theorem on certain quasimetric spaces equipped with what we call an upper doubling measure. Essentially, we merge recent techniques from the domain and range side of things, achieving a Tb theorem…

Using the notion of existentially closed structures, we obtain embedding theorems for groups and Lie algebras. We also prove the existence of some groups and Lie algebras with prescribed properties.

Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan…

In the following paper, we prove a dimension bound on the singular set of a Radon measure assuming its doubling ratio converges uniformly on compact sets. More precisely, we prove that if a Radon measure is $n$-Uniformly Asymptotically…

A famous result of Freiman describes the structure of finite sets A of integers with small doubling property. If |A + A| <= K|A| then A is contained within a multidimensional arithmetic progression of dimension d(K) and size f(K)|A|. Here…

We show that when $n$ is a power of two, any $nD$ measures in $\mathbb{R}^n$ can be bisected by an arrangement of $D$ hyperplanes.

This paper is concerned with the lengths of constant length substitutions that generate topologically conjugate systems. We show that if the systems are infinite, then these lengths must be powers of the same integer. This result is a…

We give a constructive proof of a theorem of Naor and Neiman, (to appear, Revista Matematica Iberoamercana), which asserts that if $(E,d)$ is a doubling metric space, there is an integer $N > 0$, that depends only on the metric doubling…

We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent…

This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield…

Given positive measures $\nu,\mu$ on an arbitrary measurable space $(\Omega, \mathcal F)$, we construct a sequence of finite partitions $(\pi_n)_n$ of $(\Omega, \mathcal F)$ s.t. $$ \sum_{A\in \pi_n: \mu(A)>0} 1_{A} \frac{\nu(A)}{\mu(A)}…

Building on recent results regarding symmetric probabilistic constructions of countable structures, we provide a method for constructing probability measures, concentrated on certain classes of countably infinite structures, that are…