相关论文: Configurations of infinitely near points
The aim of this paper is to revise the theory of clusters of infinitely near points for arbitrary fields. We describe in particular the intersection matrix of such a cluster, we introduce the notion of curvette over an arbitrary field and…
To any finite ordered subset and any finite partition of a group a set of tuples of positive integers, named as configurations, is associated that describes the group's behavior. The present paper provides an exposition of this notion and…
I review recent progress in the study of the large-scale structure of the Universe through the distribution of clusters of galaxies, concentrating on new results using X-ray selected samples. After discussing the importance of understanding…
We study the Orchard relation for generic configurations of points in the plane (also called order types). We introduce infinitesimally-close points and analyse the relation of this notion with the Orchard relation. The second part of the…
The paper develops a general framework for constrained clustering which is based on the close connection of geometric clustering and diagrams. Various new structural and algorithmic results are proved (and known results generalized and…
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
Groups of almost upper triangular infinite matrices with entries indexed by integers are studied. It is shown that, when the matrices are over a finite field, these groups admit a nondiscrete totally disconnected, locally compact group…
We show that the locations where finite- and infinite-time clustering occurs for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time…
We show how to use topological ideas, such as compactness, to establish orderability properties of infinite groups. A new application is to provide a left-ordering for the group of PL homeomorphisms of a connected surface with boundary…
In this paper I survey some recent results on finite determination, convergence, and approximation of formal mappings between real submanifolds in complex spaces. A number of conjectures are also given.
We consider and characterize classes of finite and countably categorical structures and their theories preserved under $E$-operators and $P$-operators. We describe $e$-spectra and families of finite cardinalities for structures belonging to…
We consider the problem of enumerating planar constellations with two points at a prescribed distance. Our approach relies on a combinatorial correspondence between this family of constellations and the simpler family of rooted…
This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as…
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…
We develop a general theory for estimating the probability that a galaxy cluster of a given shape exists. The theory is based on the observed result that the distribution of galaxies is very close to quasi-equilibrium, in both its linear…
We start by introducing the basics of configurations of points and lines, and then move into discussing symmetry groups of these configurations. Specifically, we explore how we might classify the symmetries of $(9_3)$ and $(10_3)$ geometric…
Some aspects of a mathematical theory of rigidity and flexibility are developed for general infinite frameworks and two main results are obtained. In the first sufficient conditions, of a uniform local nature, are obtained for the existence…
We consider the system of $N$ points on the segment of the real line with the nearest-neighbor Coulomb repulsive interaction and external force $F$. For the fixed points of such systems (fixed configurations) we study the asymptotics (in…
We introduce ordered and unordered configuration spaces of 'clusters' of points in an Euclidean space $\mathbb{R}^d$, where points in each cluster satisfy a 'verticality' condition, depending on a decomposition $d=p+q$. We compute the…
We derive, in order of magnitude, the observed astrophysical and cosmological scales in the Universe, from neutron stars to superclusters of galaxies, up to, asymptotically, the observed radius of the Universe. This result is obtained by…