Connes-Chern character for manifolds with boundary and eta cochains

We express the Connes-Chern character of the Dirac operator associated to a b-metric on a manifold with boundary in terms of a retracted cocycle in relative cyclic cohomology, whose expression depends on a scaling/cut-off pa- rameter. Blowing-up the metric one recovers the pair of characteristic currents that represent the corresponding de Rham relative homology class, while the blow-down yields a relative cocycle whose expression involves higher eta cochains and their b-analogues. The corresponding pairing formulae with relative K-theory classes capture information about the boundary and allow to derive geometric consequences. As a by-product, we show that the generalized Atiyah-Patodi-Singer pairing introduced by Getzler and Wu is necessarily restricted to almost flat bundles.Comment: 98 pages, 6 figures; major revision, accepted for publication in the Memoirs of the AM

Relative pairing in cyclic cohomology and divisor flows

We construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to all the essential features of the divisor flows, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.Comment: 43 pages; revision 5.22; expanded by a factor of 1.5, in particular even case adde

Statistics of opinion domains of the majority-vote model on a square lattice

The existence of juxtaposed regions of distinct cultures in spite of the fact that people's beliefs have a tendency to become more similar to each other's as the individuals interact repeatedly is a puzzling phenomenon in the social sciences. Here we study an extreme version of the frequency-dependent bias model of social influence in which an individual adopts the opinion shared by the majority of the members of its extended neighborhood, which includes the individual itself. This is a variant of the majority-vote model in which the individual retains its opinion in case there is a tie among the neighbors' opinions. We assume that the individuals are fixed in the sites of a square lattice of linear size $L$ and that they interact with their nearest neighbors only. Within a mean-field framework, we derive the equations of motion for the density of individuals adopting a particular opinion in the single-site and pair approximations. Although the single-site approximation predicts a single opinion domain that takes over the entire lattice, the pair approximation yields a qualitatively correct picture with the coexistence of different opinion domains and a strong dependence on the initial conditions. Extensive Monte Carlo simulations indicate the existence of a rich distribution of opinion domains or clusters, the number of which grows with $L^2$ whereas the size of the largest cluster grows with $\ln L^2$. The analysis of the sizes of the opinion domains shows that they obey a power-law distribution for not too large sizes but that they are exponentially distributed in the limit of very large clusters. In addition, similarly to other well-known social influence model -- Axelrod's model -- we found that these opinion domains are unstable to the effect of a thermal-like noise

Propagation of social representations

Based on a minimal formalism of social representations as a set of associated cognems, a simple model of propagation of representations is presented. Assuming that subjects share the constitutive cognems, the model proposes that mere focused attention on the set of cognems in the field of common conscience may replicate the pattern of representation from context into subjects, or, from subject to subject, through actualization by language, where cognems are represented by verbal signs. Limits of the model are discussed, and evolutionist perspectives are presented with the support of field data

On the geometry of Siegel-Jacobi domains

We study the holomorphic unitary representations of the Jacobi group based on Siegel-Jacobi domains. Explicit polynomial orthonormal bases of the Fock spaces based on the Siegel-Jacobi disk are obtained. The scalar holomorphic discrete series of the Jacobi group for the Siegel-Jacobi disk is constructed and polynomial orthonormal bases of the representation spaces are given.Comment: 15 pages, Latex, AMS fonts, paper presented at the the International Conference "Differential Geometry and Dynamical Systems", August 25-28, 2010, University Politehnica of Bucharest, Romani

Mass media destabilizes the cultural homogeneous regime in Axelrod's model

An important feature of Axelrod's model for culture dissemination or social influence is the emergence of many multicultural absorbing states, despite the fact that the local rules that specify the agents interactions are explicitly designed to decrease the cultural differences between agents. Here we re-examine the problem of introducing an external, global interaction -- the mass media -- in the rules of Axelrod's model: in addition to their nearest-neighbors, each agent has a certain probability $p$ to interact with a virtual neighbor whose cultural features are fixed from the outset. Most surprisingly, this apparently homogenizing effect actually increases the cultural diversity of the population. We show that, contrary to previous claims in the literature, even a vanishingly small value of $p$ is sufficient to destabilize the homogeneous regime for very large lattice sizes

Extremism propagation in social networks with hubs

One aspect of opinion change that has been of academic interest is the impact of people with extreme opinions (extremists) on opinion dynamics. An agent-based model has been used to study the role of small-world social network topologies on general opinion change in the presence of extremists. It has been found that opinion convergence to a single extreme occurs only when the average number of network connections for each individual is extremely high. Here, we extend the model to examine the effect of positively skewed degree distributions, in addition to small-world structures, on the types of opinion convergence that occur in the presence of extremists. We also examine what happens when extremist opinions are located on the well-connected nodes (hubs) created by the positively skewed distribution. We find that a positively skewed network topology encourages opinion convergence on a single extreme under a wider range of conditions than topologies whose degree distributions were not skewed. The importance of social position for social influence is highlighted by the result that, when positive extremists are placed on hubs, all population convergence is to the positive extreme even when there are twice as many negative extremists. Thus, our results have shown the importance of considering a positively skewed degree distribution, and in particular network hubs and social position, when examining extremist transmission

Social representations of HIV/AIDS in five Central European and Eastern European countries: A multidimensional analysis

Cognitive processing models of risky sexual behaviour have proliferated in the two decades since the first reporting of HIV/AIDS, but far less attention has been paid to individual and group representations of the epidemic and the relationship between these representations and reported sexual behaviours. In this study, 494 business people and medics from Estonia, Georgia, Hungary, Poland and Russia sorted free associations around HIV/AIDS in a matrix completion task. Exploratory factor and multidimensional scaling analyses revealed two main dimensions (labelled ‘Sex’ and ‘Deadly disease’), with significant cultural and gender variations along both dimension scores. Possible explanations for these results are discussed in the light of growing concerns over the spread of the epidemic in this region