Related papers: Arbitrarily Close for Summer 2022 Analysis
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…
Statistical convergence was introduced in connection with problems of series summation. The main idea of the statistical convergence of a sequence l is that the majority of elements from l converge and we do not care what is going on with…
Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension $2$ closed subset in a homogeneous space under a semisimple algebraic group, and for…
Dictionaries are inherently circular in nature. A given word is linked to a set of alternative words (the definition) which in turn point to further descendants. Iterating through definitions in this way, one typically finds that…
A mathematical model is a function taking certain arguments and returning a theoretical prediction of a feature of a physical system. The arguments to the mathematical model can be split into two groups; (a) controllable variables of the…
This short paper discusses continually updated causal abstractions as a potential direction of future research. The key idea is to revise the existing level of causal abstraction to a different level of detail that is both consistent with…
A topological group is (openly) almost-elliptic if it contains a(n open) dense subset of elements generating relatively-compact cyclic subgroups. We classify the (openly) almost-elliptic connected locally compact groups as precisely those…
Any symmetric affinity function $w: V\times V \to \mathbb{R}_+$ defined on a discrete set $V$ induces Euclidean space structure on $V$. In particular, an undirected graph specified by an affinity (or adjacency) matrix can be considered as a…
Discovering patterns in a sequence is an important aspect of data mining. One popular choice of such patterns are episodes, patterns in sequential data describing events that often occur in the vicinity of each other. Episodes also enforce…
This simple note lays out a few observations which are well known in many ways but may not have been said in quite this way before. The basic idea is that when comparing two different Markov chains it is useful to couple them is such a way…
Abstraction is a powerful idea widely used in science, to model, reason and explain the behavior of systems in a more tractable search space, by omitting irrelevant details. While notions of abstraction have matured for deterministic…
Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of…
Covering-based rough set theory is an extension to classical rough set. The main purpose of this paper is to study covering rough sets from a topological point of view. The relationship among upper approximations based on topological spaces…
Clustering is a well-known unsupervised machine learning approach capable of automatically grouping discrete sets of instances with similar characteristics. Constrained clustering is a semi-supervised extension to this process that can be…
We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g.…
We use the notion of topological data analysis to compare metrics on data sets. We provide two different motivating examples for this. The first of these is a point cloud data set that has $\mathbb{R}^2$ as its ambient space, and is…
This paper discusses a more general contractive condition for a class of extended cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same…
Every bounded definable open set is a union of finitely many open strong cells in a weakly o-minimal expansion of a real closed field. We prove this fact and another theorem similar to it.
The octagon abstract domain is a widely used numeric abstract domain expressing relational information between variables whilst being both computationally efficient and simple to implement. Each element of the domain is a system of…
Working over a field ${\mathbb{k}}$ of characteristic $\ne 2$, we study what we call bisector fields, which are arrangements of paired lines in the plane that have the property that each line in the arrangement crosses the paired lines in…