Cyclic permutable subgroups of finite groups

The authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group

The embedding of a cyclic permutable subgroup in a finite group. II

In two previous papers we established the structure of the normal closure of a cyclic permutable subgroup $A$ of a finite group, first when $A$ has odd order and second when $A$ has even order, but with an extra hypothesis that was unnecessary in the odd case. Here we describe the most general situation without any restrictions on $A$

From calls to communities: a model for time varying social networks

Social interactions vary in time and appear to be driven by intrinsic mechanisms, which in turn shape the emerging structure of the social network. Large-scale empirical observations of social interaction structure have become possible only recently, and modelling their dynamics is an actual challenge. Here we propose a temporal network model which builds on the framework of activity-driven time-varying networks with memory. The model also integrates key mechanisms that drive the formation of social ties - social reinforcement, focal closure and cyclic closure, which have been shown to give rise to community structure and the global connectedness of the network. We compare the proposed model with a real-world time-varying network of mobile phone communication and show that they share several characteristics from heterogeneous degrees and weights to rich community structure. Further, the strong and weak ties that emerge from the model follow similar weight-topology correlations as real-world social networks, including the role of weak ties.Comment: 10 pages, 5 figure

On the topological cyclic homology of the algebraic closure of a local field

The cyclotomic trace provides a comparison of the algebraic K-theory spectrum and a pro-spectrum TR that is built from the cyclic fixed points of topological Hochschild homology. In a previous paper with Ib Madsen, we used this comparison and an approximate evaluation of the structure of the pro-spectrum TR to evaluate the p-adic K-groups of a local field K of mixed characteristic (0,p) with perfect residue field. In this paper we completly determine the structure of the pro-spectrum TR for an algebrac closure of the local field K. This leads us to formulate a conjecture for the structure of the pro-spectrum TR for the field K. We also determine the structure of the absolute de Rham-Witt complex of the valuation ring R in the algebraic closure of K. The group in degree one is a p-divisible group whose Tate module is free module of rank one over the ring of Witt vectors in R. We give an explicit generator of this Tate module

Templated synthesis of cyclic poly(ionic liquid)s

Charged cyclic polymers, e.g. cyclic DNAs and polypeptides, play enabling roles in organisms, but their synthesis was challenging due to the well known polyelectrolyte effect. To tackle the challenge, we developed a templated method to synthesize a library of imidazolium and pyridinium based cyclic poly(ionic liquid)s. Cyclic templates, cyclic polyimidazole and poly(2-pyridine), were synthesized first through ring-closure method by light-induced Diels-Alder click reaction. Through quaternization of cyclic templates followed by anion metathesis, the cyclic poly(ionic liquid)s were synthesized, which paired with varied counter anions.Comment: 23 pages, 15 figures, 2 table