Related papers: On structure theorems and non-saturated examples

Let $d \in \mathbb{N}$, $\delta \in (0, 1/2)$, and $X > 0$. Denote by $N_d(X, \delta)$ the maximum number of points in a subset of the closed Euclidean ball of radius $X$ in $\mathbb{R}^d$ such that every pairwise distance is at least…

An $\infty$-step nilsystem is an inverse limit of minimal nilsystems. In this article is shown that a minimal distal system is an $\infty$-step nilsystem if and only if it has no nontrivial pairs with arbitrarily long finite IP-independence…

For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss' theorem implies that such…

Given a metric space $(X,d_X)$, the earth mover distance between two distributions over $X$ is defined as the minimum cost of a bipartite matching between the two distributions. The doubling dimension of a metric $(X, d_X)$ is the smallest…

We obtain results about fundamental groups of $RCD^{\ast}(K,N)$ spaces previously known under additional conditions such as smoothness or lower sectional curvature bounds. For fixed $K \in \mathbb{R}$, $N \in [1,\infty )$, $D > 0 $, we show…

Let G(d,n) denote the Grassmannian of d-planes in C^n and let T be the torus (C^*)^n/diag(C^*) which acts on G(d,n). Let x be a point of G(d,n) and let \bar{Tx} be the closure of the T-orbit through x. Then the class of the structure sheaf…

Let $S$ be a Dedekind scheme, $X$ a connected $S$-scheme locally of finite type and $x\in X(S)$ a section. The aim of the present paper is to establish the existence of the fundamental group scheme of $X$, when $X$ has reduced fibers or…

Given the action of a group $G$ on a set $X$, an endomorphism of $X$ is a function $f:X \rightarrow X$ which is $G$-equivariant, that is, it commutes with the action, i.e., $f(g\cdot x)= g\cdot f(x)$, for all $x\in X$. The set of…

Let G=(V,E). A set S is independent if no two vertices from S are adjacent. The number d(X)= |X|-|N(X)| is the difference of X, and an independent set A is critical if d(A) = max{d(I):I is an independent set}. Let us recall that ker(G) is…

In this paper we give an answer to Furstenberg's problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable…

Using a cell dynamic system (CDS) simulation scheme, we investigate the phase-ordering dynamics of non-conserved O(n) models without topological defects, i.e. for $n > d+1$ where $d$ is the spatial dimensionality. In particular, we consider…

Recently, G\'{o}rska, Lema\'{n}czyk, and de la Rue characterized the class of automorphisms disjoint from all ergodic automorphisms. Inspired by their work, we provide several characterizations of systems that are disjoint from all minimal…

Recently, interesting empirical phenomena known as Neural Collapse have been observed during the final phase of training deep neural networks for classification tasks. We examine this issue when the feature dimension d is equal to the…

A set of vertices $X\subseteq V(G)$ is a $d$-distance dominating set if for every $u\in V(G)\setminus X$ there exists $x\in X$ such that $d(u,x) \le d$, and $X$ is a $p$-packing if $d(u,v) \ge p+1$ for every different $u,v\in X$. The…

Let $(X,G)$ be a minimal equicontinuous dynamical system, where $X$ is a compact metric space and $G$ some topological group acting on $X$. Under very mild assumptions, we show that the class of regular almost automorphic extensions of…

Given a numerical semigroup $S$ and a positive integer $d$, the fraction $\frac{S}{d}=\{ x \in \mathbb{N} \ | \ dx \in S\}$ is again a numerical semigroup. In this paper we determine a generating set for $\frac{S}{d}$ in terms of the…

Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on…

We review the $N$-Body Problem in arbitrary dimension $d$ at the kinematical level, with modelling Background Independence in mind. In particular, we give a structural analysis of its reduced configuration spaces, decomposing this subject…

A finite set of distinct vectors $\mathcal{X}$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ is called an $s$-distance set if the set of mutual distances between distinct elements of $\mathcal{X}$ has cardinality $s$. In this paper…

For an abelian group $G$, $\vec{g}=(g_1,\ldots,g_d)\in G^d$ and $\epsilon=(\epsilon(1),\ldots,\epsilon(d))\in \{0,1\}^d$, let $\vec{g}\cdot \epsilon=\prod_{i=1}^{d}g_i^{\epsilon(i)}$. In this paper, it is shown that for a minimal system…