Magnetic flows on Sol-manifolds: dynamical and symplectic aspects

We consider magnetic flows on compact quotients of the 3-dimensional solvable geometry Sol determined by the usual left-invariant metric and the distinguished monopole. We show that these flows have positive Liouville entropy and therefore are never completely integrable. This should be compared with the known fact that the underlying geodesic flow is completely integrable in spite of having positive topological entropy. We also show that for a large class of twisted cotangent bundles of solvable manifolds every compact set is displaceable.Comment: Final version to appear in CMP. Two new remarks have been added as well as some numerical calculations for metric entrop

Laminin γ1 chain peptide, C-16 (KAFDITYVRLKF), promotes migration, MMP-9 secretion, and pulmonary metastasis of B16–F10 mouse melanoma cells

Laminin-1, a heterotrimer of α1, β1, and γ1 chains specific to basement membrane, promotes cell adhesion and migration, proteinase secretion and metastases of tumour cells. Several active sites on the α1 chain have been found to promote B16–F10 melanoma lung colonisation and here we have determined whether additional tumour promoting sites exist on the β1 and γ1 chains. Recently, we have identified novel cell adhesive peptides derived from laminin β1 and γ1 chains by systematic screening of synthetic peptides. Nine β1 peptides and seven γ1 peptides active for cell adhesion were tested for their effects on experimental pulmonary metastases of B16–F10 mouse melanoma cells in vivo. The most active adhesive peptide derived from the γ1 chain globular domain, C-16 (KAFDITYVRLKF), significantly enhanced pulmonary metastases of B16–F10 cells, whereas no other peptides showed enhancement. C-16 also stimulated migration of B16–F10 cells in the Boyden chamber assay in vitro. Furthermore, C-16 significantly induced the production of MMP-9 from B16–F10 cells. These results suggest that this specific laminin γ1 chain peptide has a metastasis-promoting activity and might be a new molecular target of anti-cancer treatment

Landau-De Gennes theory of nematic liquid\ud crystals: the Oseen-Frank limit and beyond

We study global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals, in three-dimensional domains, subject to uniaxial boundary conditions. We analyze the physically relevant limit of small elastic constant and show that global minimizers converge strongly, in W 1,2 , to a global minimizer predicted by the Oseen-Frank theory for uniaxial nematic liquid crystals with constant order parameter. Moreover, the convergence is uniform in the interior of the domain, away from the singularities of the limiting Oseen-Frank global minimizer. We obtain results on the rate of convergence of the eigenvalues and the regularity of the eigenvectors of the Landau-De Gennes global minimizer.\ud \ud \ud We also study the interplay between biaxiality and uniaxiality in Landau-De Gennes global energy minimizers and obtain estimates for various related quantities such as the biaxiality parameter and the size of admissible strongly biaxial regions

Quantum deformations of associative algebras and integrable systems

Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton equations too. In particular, such deformations for the three-dimensional and infinite-dimensional algebras are described by the Boussinesq equation and KP hierarchy, respectively.Comment: Numeration of the formulas is correcte

Bundles over Nearly-Kahler Homogeneous Spaces in Heterotic String Theory

We construct heterotic vacua based on six-dimensional nearly-Kahler homogeneous manifolds and non-trivial vector bundles thereon. Our examples are based on three specific group coset spaces. It is shown how to construct line bundles over these spaces, compute their properties and build up vector bundles consistent with supersymmetry and anomaly cancelation. It turns out that the most interesting coset is $SU(3)/U(1)^2$. This space supports a large number of vector bundles which lead to consistent heterotic vacua, some of them with three chiral families.Comment: 32 pages, reference adde

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is ``rotating''--i.e., is such that the stationary Killing field is not everywhere normal to the horizon--must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, $P$. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.Comment: 24 pages, no figures, v2: footnotes and references added, v3: numerous minor revision

Conformally parametrized surfaces associated with CP^(N-1) sigma models

Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in R^(N^2-1). The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the CP^2 model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras.Comment: 32 page

Euclidean versus hyperbolic congestion in idealized versus experimental networks

This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum length routing. More specifically, it is shown that, in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is independent of the radius R whereas, in a Euclidean ball, the same proportion scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as the square of the volume, whereas the same traffic load scales as the volume to the power (n+1)/n in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel data base and synthetic Euclidean lattice networks. It is further conjectured that for networks that do not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of maximum traffic is near the center of mass of the network.Comment: 23 pages, 4 figure

Topological and geometrical restrictions, free-boundary problems and self-gravitating fluids

Let (P1) be certain elliptic free-boundary problem on a Riemannian manifold (M,g). In this paper we study the restrictions on the topology and geometry of the fibres (the level sets) of the solutions f to (P1). We give a technique based on certain remarkable property of the fibres (the analytic representation property) for going from the initial PDE to a global analytical characterization of the fibres (the equilibrium partition condition). We study this analytical characterization and obtain several topological and geometrical properties that the fibres of the solutions must possess, depending on the topology of M and the metric tensor g. We apply these results to the classical problem in physics of classifying the equilibrium shapes of both Newtonian and relativistic static self-gravitating fluids. We also suggest a relationship with the isometries of a Riemannian manifold.Comment: 36 pages. In this new version the analytic representation hypothesis is proved. Please address all correspondence to D. Peralta-Sala