Manuel Ruiz Marín

On Symmetric Elements and Symmetric Units in Group Rings

ABSTRACT Let R be a commutative ring, G a group, and RG its group ring. Let ϕ: RG → RG denote the R-linear extension of an involution ϕ defined on G. An element x in RG is said to be symmetric if ϕ (x) = x. A characterization is given of when the symmetric elements (RG)ϕ of RG form a ring. For many domains R it is also shown that (RG)ϕ is a ring if and only if the symmetric units form a group. The results obtained extend earlier work of Bovdi (2001), Bovdi et al. (1996), Bovdi and Parmenter (1997), Broche Cristo (2003, to appear), Giambruno and Sehgal (1993), and Lee (1999), who dealt with the case that ϕ is the involution * mapping g ∈ G onto g−1.

ANTISYMMETRIC ELEMENTS IN GROUP RINGS II

Research supported by Onderzoeksraad of Vrije Universiteit Brussel, Fonds voor Wetenschappelijk Onderzoek (Vlaanderen), Flemish-Polish bilateral agreement BIL 01/31, FAPEMIG and CNPq. Proc. 300243/79-0(RN) of Brazil, D.G.I. of Spain and Fundacion Seneca of Region de Murcia.

Free Groups and Subgroups of Finite Index in the Unit Group of an Integral Group Ring

In this article we construct free groups and subgroups of finite index in the unit group of the integral group ring of a finite non-Abelian group G for which every nonlinear irreducible complex representation is of degree 2 and with commutator subgroup G′ a central elementary Abelian 2-group.

A Symbolic Test for Testing Independence between Time Series

In this article we introduce a test for independence between two processes {X} and {Y}. To this end we rely on symbolic dynamics and permutation entropy as a measure of dependence. As a result, a nonparametric (model-free) test for either linear or nonlinear processes is presented. The test is consistent for a broad range of dependent alternatives. Empirical simulations indicate and highlight the general utility of the test for time-series analysts.

Symbolic dynamics of animal interaction

Since its introduction nearly two decades ago, transfer entropy has contributed to an improved understanding of cause-and-effect relationships in coupled dynamical systems from raw time series. In the context of animal behavior, transfer entropy might help explain the determinants of leadership in social groups and elucidate escape response to predator attacks. Despite its promise, the potential of transfer entropy in animal behavior is yet to be fully tested, and a number of technical challenges in information theory and statistics remain open. Here, we examine an alternative approach to the computation of transfer entropy based on symbolic dynamics. In this context, a symbol is associated with a specific locomotory bout across two or more consecutive time instants, such as reversing the swimming direction. Symbols encapsulate salient locomotory patterns and the associated permutation transfer entropy quantifies the ability to predict the patterns of an individual given those of another individual. We demonstrate this framework on an existing dataset on fish, for which we have knowledge of the underlying cause-and-effect relationship between the focal subject and the stimulus. Symbolic dynamics offers an intuitive and robust approach to study animal behavior, which could enable the inference of causal relationship from noisy experimental observations of limited duration.

Symbolic correlation integral

ABSTRACT This paper aims to introduce the concept of symbolic correlation integral SC that is extensively used in many scientific fields. The new correlation integral SC avoids the noisy parameter 𝜀 of the classical correlation integral, defined by Grassberger and Procaccia (1983) and extensively used for constructing correlation-integral-based statistics, as in the BDS test. Once the free parameter 𝜀 disappears, it is possible to construct a nonparametric powerful test for independence that can also be used as a diagnostic tool for model selection. The symbolic correlation integral is also extended to deal with multivariate models, and a test for causality is proposed as an example of the theoretical power of the new concept. With extensive Monte Carlo simulations, the paper shows the good size and power performance of symbolic correlation-integral-based tests.

Detecting the order of spatial dependence via symbolic analysis

In this article, we have proposed a simple diagnostic statistical procedure for testing the order of dependence of a spatial process. The proposed test is of nonparametric nature, and it is able to deal with potential nonlinear spatial dependencies. An added value is that from a methodological point of view, the new test is based on symbolic dynamics and hence on information theory. We characterized the behavior of a symbolic entropy measure in the presence of spatial dependencies of order higher than 1. The good power performance of the new method in detecting higher order spatial lags is notable and gives rise to an expectation that it may form a suitable basis for constructive specification searches.

Information Flow in a Model of Policy Diffusion: An Analytical Study

Networks are pervasive across science and engineering, but seldom do we precisely know their topology. The information-theoretic notion of transfer entropy has been recently proposed as a potent means to unveil connectivity patterns underlying collective dynamics of complex systems. By pairwise comparing time series of units in the network, transfer entropy promises to determine whether the units are connected or not. Despite considerable progress, our understanding of transfer entropy-based network reconstruction largely relies on computer simulations, which hamper the precise and systematic assessment of the accuracy of the approach. In this paper, we present an analytical study of the information flow in a network model of policy diffusion, thereby establishing closed-form expressions for the transfer entropy between any pair of nodes. The model consists of a finite-state ergodic Markov chain, for which we compute the joint probability distribution in the stationary limit. Our analytical results offer a compelling evidence for the potential of transfer entropy to assist in the process of network reconstruction, clarifying the role and extent of tenable confounds associated with spurious connections between nodes.

Plasticity in leader-follower roles in human teams

In humans, emergence of leaders and followers is key to group performance, but little is known about the whys and hows of leadership. A particularly elusive question entails behavioral plasticity in leadership across social contexts. Addressing this question requires to eliminate social feedback between focal individuals and their partners in experiments that could illuminate the spontaneous emergence of social roles. We investigated plasticity in leader–follower roles in cooperation, where members choose the task toward a shared goal, and coordination, where members adjust their actions in real time based on social responsiveness. Through a computer-programmed virtual partner, we demonstrate adaptive plasticity in leader–follower roles. Humans increased their followership to cooperate when the partner led more in the choice of the task, whereas they showed only weak leadership when the partner followed more. We leveraged the information-theoretic notion of transfer entropy to quantify leadership and followership in coordination from their movements. When exhibiting stronger followership in task cooperation, humans coordinated more with the partner’s movement, with greater information being transferred from the partner to humans. The evidence of behavioral plasticity suggests that humans are capable of adapting their leader–follower roles to their social environments, in both cooperation and coordination.

Symbolic recurrence plots to analyze dynamical systems

This paper, based on the concept of symbolic correlation integral, introduces a set of symbolic recurrence plots and associated invariant measures, which are independent of the distance parameter, serving as a tool for quantifying the dynamic structure. These new measures allow the study of transient behavior, coexistence of attractors, bifurcations, and structural change. The final user does not have to choose the free distance parameter. An empirical application to electrocardiography data illustrates the use of symbolic recurrence measures.