Career: hybrid surfaces to control cell adhesion and function

Issued as final reportNational Science Foundation (U.S.

Lifting Hamiltonian loops to isotopies in fibrations

Let $G$ be a Lie group, $H$ a closed subgroup and $M$ the homogeneous space $G/H$. Each representation $\Psi$ of $H$ determines a $G$-equivariant principal bundle ${\mathcal P}$ on $M$ endowed with a $G$-invariant connection. We consider subgroups ${\mathcal G}$ of the diffeomorphism group ${\rm Diff}(M)$, such that, each vector field $Z\in{\rm Lie}({\mathcal G})$ admits a lift to a preserving connection vector field on ${\mathcal P}$. We prove that #\,\pi_1({\mathcal G})\geq #\,\Psi(Z(G)). This relation is applicable to subgroups ${\mathcal G}$ of the Hamiltonian groups of the flag varieties of a semisimple group $G$. Let $M_{\Delta}$ be the toric manifold determined by the Delzant polytope $\Delta$. We put $\varphi_{\bf b}$ for the the loop in the Hamiltonian group of $M_{\Delta}$ defined by the lattice vector ${\bf b}$. We give a sufficient condition, in terms of the mass center of $\Delta$, for the loops $\varphi_{\bf b}$ and $\varphi_{\bf\tilde b}$ to be homotopically inequivalent.Comment: 23 pages, 1 figure. To be published in Int. J. Geom. Methods Mod. Physic

Homogeneous Free Cooling State in Binary Granular Fluids of Inelastic Rough Hard Spheres

In a recent paper [A. Santos, G. M. Kremer, and V. Garz\'o, \emph{Prog. Theor. Phys. Suppl.} \textbf{184}, 31-48 (2010)] the collisional energy production rates associated with the translational and rotational granular temperatures in a granular fluid mixture of inelastic rough hard spheres have been derived. In the present paper the energy production rates are explicitly decomposed into equipartition rates (tending to make all the temperatures equal) plus genuine cooling rates (reflecting the collisional dissipation of energy). Next the homogeneous free cooling state of a binary mixture is analyzed, with special emphasis on the quasi-smooth limit. A previously reported singular behavior (according to which a vanishingly small amount of roughness has a finite effect, with respect to the perfectly smooth case, on the asymptotic long-time translational/translational temperature ratio) is further elaborated. Moreover, the study of the time evolution of the temperature ratios shows that this dramatic influence of roughness already appears in the transient regime for times comparable to the relaxation time of perfectly smooth spheres.Comment: 6 pages; 4 figures; contributed talk at the 27th International Symposium on Rarefied Gas Dynamics (Asilomar Conference Grounds, Pacific Grove, California July 10-15, 2010

Characteristic number associated to mass linear pairs

Let $\Delta$ be a Delzant polytope in ${\mathbb R}^n$ and ${\mathbf b}\in{\mathbb Z}^n$. Let $E$ denote the symplectic fibration over $S^2$ determined by the pair $(\Delta,\,{\mathbf b})$. Under certain hypotheses, we prove the equivalence between the fact that $(\Delta,\,{\mathbf b})$ is a mass linear pair (D. McDuff, S. Tolman, {\em Polytopes with mass linear functions. I.} Int. Math. Res. Not. IMRN 8 (2010) 1506-1574.) and the vanishing of a characteristic number of $E$. Denoting by ${\rm Ham}(M_{\Delta})$ the Hamiltonian group of the symplectic manifold defined by $\Delta$, we determine loops in ${\rm Ham}(M_{\Delta})$ that define infinite cyclic subgroups in $\pi_1({\rm Ham}(M_{\Delta}))$, when $\Delta$ satisfies any of the following conditions: (i) it is the trapezium associated with a Hirzebruch surface, (ii) it is a $\Delta_p$ bundle over $\Delta_1$, (iii) $\Delta$ is the truncated simplex associated with the one point blow up of ${\mathbb C}P^n$.Comment: Revised version which will appear in ISRN Geometr