相关论文: Measuring sets in infinite groups

Communities are clusters of nodes with a higher than average density of internal connections. Their detection is of great relevance to better understand the structure and hierarchies present in a network. Modularity has become a standard…

In this paper, we define a new structure analogous to group, called partial group. This structure concerns the partial stability by the composition inner law. We generalize the three isomorphism theorems for groups to partial groups.

What is a population? This review considers how a population may be defined in terms of understanding the structure of the underlying genetics of the individuals involved. The main approach is to consider statistically identifiable groups…

We explore the interplay between random and deterministic phenomena using a representation of uncertainty based on the measure-theoretic concept of outer measure. The meaning of the analogues of different probabilistic concepts is…

The question on connection between the structure of a finite group $G$ and the properties of the indices of elements of $G$ has been a popular research topic for many years. The $p$-index $|x^G|_p$ of an element $x$ of a group $G$ is the…

Symmetry is a cornerstone of much of mathematics, and many probability distributions possess symmetries characterized by their invariance to a collection of group actions. Thus, many mathematical and statistical methods rely on such…

We consider the problem of constructing matched groups such that the resulting groups are statistically similar with respect to their average values for multiple covariates. This group-matching problem arises in many cases, including…

Our confidence set quantifies the statistical uncertainty from data-driven group assignments in grouped panel models. It covers the true group memberships jointly for all units with pre-specified probability and is constructed by inverting…

For large classes of group testing problems, we derive lower bounds for the probability that all significant items are uniquely identified using specially constructed random designs. These bounds allow us to optimize parameters of the…

Algorithmic modeling relies on limited information in data to extrapolate outcomes for unseen scenarios, often embedding an element of arbitrariness in its decisions. A perspective on this arbitrariness that has recently gained interest is…

We consider the group testing problem, in the case where the items are defective independently but with non-constant probability. We introduce and analyse an algorithm to solve this problem by grouping items together appropriately. We give…

Known and new results on free Boolean topological groups are collected. An account of properties which these groups share with free or free Abelian topological groups and properties specific of free Boolean groups is given. Special emphasis…

In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain…

Topological models of empirical and formal inquiry are increasingly prevalent. They have emerged in such diverse fields as domain theory [1, 16], formal learning theory [18], epistemology and philosophy of science [10, 15, 8, 9, 2],…

This paper covers two topics: first an introduction to Algorithmic Complexity Theory: how it defines probability, some of its characteristic properties and past successful applications. Second, we apply it to problems in A.I. - where it…

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations…

We introduce a new mathematical framework for the probabilistic description of an experiment upon a system of any type in terms of initial information representing this system. Based on the notions of an information state, an information…

This is a review of group entropy and its application to permutation complexity. Specifically we revisit a new approach to the notion of complexity in time serie analysis, based on both permutation entropy and group entropy. As a result,…

We examine the extent to which random samplings from the values of a random set, determine the distribution of the random set itself. We also comment on how, given the statistics of the sampling, to detect the distribution. Several methods…

The problem of defining a statistical ensemble of random graphs with an arbitrary connectivity distribution is discussed. Introducing such an ensemble is a step towards uderstanding the geometry of wide classes of graphs independently of…