Twistings, crossed coproducts and Hopf-Galois coextensions

Let $H$ be a Hopf algebra. Ju and Cai introduced the notion of twisting of an $H$-module coalgebra. In this note, we study the relationship between twistings, crossed coproducts and Hopf-Galois coextensions. In particular, we show that a twisting of an $H$-Galois coextension remains $H$-Galois if the twisting is invertible.Comment: 20 page

An extension of heat hierarchy

We propose a formally completely integrable extension of heat hierarchy based on the space of symmetries isomorphic to the Weyl algebra $\mathcal{A}_1$. The extended heat hierarchy will be the basic model for the analysis of the extension of KP hierarchy, and other integrable equations.Comment: This note is incorporated into arXiv:1409.7024, arXiv:1408.324

On the convergence of autonomous agent communities

This is the post-print version of the final published paper that is available from the link below. Copyright @ 2010 IOS Press and the authors.Community is a common phenomenon in natural ecosystems, human societies as well as artificial multi-agent systems such as those in web and Internet based applications. In many self-organizing systems, communities are formed evolutionarily in a decentralized way through agents' autonomous behavior. This paper systematically investigates the properties of a variety of the self-organizing agent community systems by a formal qualitative approach and a quantitative experimental approach. The qualitative formal study by applying formal specification in SLABS and Scenario Calculus has proven that mature and optimal communities always form and become stable when agents behave based on the collective knowledge of the communities, whereas community formation does not always reach maturity and optimality if agents behave solely based on individual knowledge, and the communities are not always stable even if such a formation is achieved. The quantitative experimental study by simulation has shown that the convergence time of agent communities depends on several parameters of the system in certain complicated patterns, including the number of agents, the number of community organizers, the number of knowledge categories, and the size of the knowledge in each category

Interlaminar crack growth in fiber reinforced composites during fatigue, part 3

Interlaminar crack growth behavior in fiber-reinforced composites subjected to fatigue loading was investigated experimentally and theoretically. In the experimental phase, inter-laminar crack propagation rates and mechanisms were determined for the cases of various geometries, laminate parameters and cyclic stress levels. A singular hybrid-stress finite element method was used in conjuction with the experimental results to examine the local crack-tip behavior and to characterize the crack propagation during fatigue. Results elucidate the basic nature of the cyclic delamination damage, and relate the interlaminar crack growth rate to the range of mixed-mode crack-tip stress intensity factors. The results show that crack growth rates are directly related to the range of the mixed-mode cyclic stress intensity factors by a power law relationship

Projectively deformable Legendrian surfaces

Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{$\,\Psi$-deformation}, and give a differential geometric characterization of surfaces admitting maximum three parameter family of such deformations. Two explicit examples of maximally $\, \Psi$-deformable surfaces are constructed; the first one is given by a Legendrian map from \, \PP^2 blown up at three distinct collinear points, which is an embedding away from the -2-curve and degenerates to a point along the -2-curve. The second one is a Legendrian embedding of the degree 6 del Pezzo surface, \, \PP^2 blown up at three non-collinear points. In both cases, the Legendrian map is given by a system of cubics through the three points, which is a subsystem of the anti-canonical system.Comment: 33 page