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In this paper, we study nilpotent Q ⁢ [ x ] \mathbb{Q}[x] -powered groups that satisfy the following property: for some set of primes 휔 in Q ⁢ [ x ] \mathbb{Q}[x] , every ω ′ \omega^{\prime} -isolated Q ⁢ [ x ] \mathbb{Q}[x] -subgroup in some family of its Q ⁢ [ x ] \mathbb{Q}[x] -subgroups is finite 휔-type separable. [ABSTRACT FROM AUTHOR]

This is the second article by the authors on extraction of roots in exponential A-groups. We prove results on ω-torsion and ω-isolated subgroups, 𝒰ω-groups, ℰω-groups, and 𝒟ω-groups in the category of exponential A-groups, where A is a unique factorization domain (UFD) and ω is a set of primes in A. In particular, we prove that every ω-torsion-free -group is a 𝒰ω-group. We also prove that if R is a principal ideal domain (PID) and G is a finitely R-generated nilpotent R-powered group, then the R-subgroup of ω'-torsion elements of G equals its maximal ℰω-subgroup (ω' denotes the set of primes in R not in ω). [ABSTRACT FROM PUBLISHER]

In this article, we introduce several classes of exponential A-groups. These classes include IA-groups, [image omitted]-groups, and A-groups of type HP. We also record some known results for nilpotent R-powered groups, a thoroughly studied class of exponential A-groups. We derive a residual property of UTn([x]) and establish the freeness of UT3([x]) in a class of [image omitted]-groups. [ABSTRACT FROM AUTHOR]

In order to understand the transmission of a disease across a population we will have to understand not only the dynamics of contact infection but the transfer of health-care beliefs and resulting health-care behaviors across that population. This paper is a first step in that direction, focusing on the contrasting role of linkage or isolation between sub-networks in (a) contact infection and (b) belief transfer. Using both analytical tools and agent-based simulations we show that it is the structure of a network that is primary for predicting contact infection-whether the networks or sub-networks at issue are distributed ring networks or total networks (hubs, wheels, small world, random, or scale-free for example). Measured in terms of time to total infection, degree of linkage between sub-networks plays a minor role. The case of belief is importantly different. Using a simplified model of belief reinforcement, and measuring belief transfer in terms of time to community consensus, we show that degree of linkage between sub-networks plays a major role in social communication of beliefs. Here, in contrast to the case of contract infection, network type turns out to be of relatively minor importance. What you believe travels differently. In a final section we show that the pattern of belief transfer exhibits a classic power law regardless of the type of network involved.