Coupled-mode equations and gap solitons in a two-dimensional nonlinear elliptic problem with a separable periodic potential

We address a two-dimensional nonlinear elliptic problem with a finite-amplitude periodic potential. For a class of separable symmetric potentials, we study the bifurcation of the first band gap in the spectrum of the linear Schr\"{o}dinger operator and the relevant coupled-mode equations to describe this bifurcation. The coupled-mode equations are derived by the rigorous analysis based on the Fourier--Bloch decomposition and the Implicit Function Theorem in the space of bounded continuous functions vanishing at infinity. Persistence of reversible localized solutions, called gap solitons, beyond the coupled-mode equations is proved under a non-degeneracy assumption on the kernel of the linearization operator. Various branches of reversible localized solutions are classified numerically in the framework of the coupled-mode equations and convergence of the approximation error is verified. Error estimates on the time-dependent solutions of the Gross--Pitaevskii equation and the coupled-mode equations are obtained for a finite-time interval.Comment: 32 pages, 16 figure

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.Comment: 26 pages, 3 Tables, 2 Figures; final versio

Understanding preventive behaviors among mid-Western African-American men: a pilot qualitative study of prostate screening

http://dx.doi.org/10.1016/j.jomh.2011.03.00

Expression of AMPA/kainate receptors during development of chick embryo retina cells: in vitro versus in vivo studies

The activity and the subunit expression of [alpha]-amino-3-hydroxy-5-methylisoxazole-4-propionate (AMPA)/kainate ionotropic glutamate receptors were studied in retina cells developing in chick embryos and in retina cells cultured as retinospheroids, at the same stages of development. In the retinospheroids, the activity of the AMPA/kainate receptors was monitored by following the changes in the intracellular free calcium concentration ([Ca2+]i), in response to AMPA, kainate or to -glutamate, and the expression of the receptor subunits GluR1, GluR2/3, GluR4 and GluR6/7 was determined in the retinospheroids and in chick retinas by immunodetection using polyclonal antibodies. The changes in [Ca2+]i in response to 400 [mu]M kainate increased from 5 h in vitro to 3 days, and remained constant until day 14, whereas the [Ca2+]i in response to 500 [mu]M -glutamate or 400 [mu]M AMPA increased from 5 h in vitro to 3 days, and thereafter decreased slightly until day 14. The [Ca2+]i responses to kainate are mainly due to AMPA receptor stimulation, since the signals were abolished by LY303070, the AMPA receptor antagonist, and were not affected by MK-801, the NMDA receptor antagonist. In retinospheroids, the levels of expression of GluR1 subunit increased from 5 h in vitro until day 7, then decreased until day 14. The levels of expression of GluR2/3 and GluR4 subunits increased from 5 h in vitro until day 10, and remained constant until day 14. The levels of kainate receptor subunits GluR6/7 increased from 5 h in vitro until day 3, and thereafter decreased slightly until day 14. In the retinas, the expression of GluR1 and GluR6/7 subunits increased from day 8 until day 15, and then decreased until day 22 (post-natal 1). The subunits GluR2/3 and GluR4 increased from day 8 until day 18, and remained constant until day 22. The results suggest that AMPA/kainate receptors are expressed at early embryonic stages, although at low levels and before synapse formation (E12). However, the AMPA receptors are not completely functional at the first stage studied since they do not respond to the agonist AMPA. Also, the patterns of AMPA/kainate receptor subunit expression in retinospheroids of chick embryo retina cells cultured in vitro and in retina cells developing in the embryo (in vivo) were similar, indicating that the AMPA/kainate receptor subunits expression in these primary cultures mimics their expression in the developing chick retina.http://www.sciencedirect.com/science/article/B6T01-45CW0B3-1/1/a35858f1d7808b1301de428de93fa9e