Non-hyperbolic boundary equilibrium bifurcations in planar Filippov systems: a case study approach

Boundary equilibrium bifurcations in piecewise smooth discontinuous systems are characterized by the collision of an equilibrium point with the discontinuity surface. Generically, these bifurcations are of codimension one, but there are scenarios where the phenomenon can be of higher codimension. Here, the possible collision of a non-hyperbolic equilibrium with the boundary in a two-parameter framework and the nonlinear phenomena associated with such collision are considered. By dealing with planar discontinuous (Filippov) systems, some of such phenomena are pointed out through specific representative cases. A methodology for obtaining the corresponding bi-parametric bifurcation sets is developed.

Euclidean geometric invariant of framed knots in manifolds

We present an invariant of a three dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. An important feature of our work is that we are not using any nontrivial representation of the manifold fundamental group or knot group.

Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds

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Horospheres in hyperbolic geometry

In this paper we investigate the role of horospheres in Integral Geometry and Differential Geometry. In particular we study envelopes of families of horocycles by means of “support maps”. We define invariant “linear combinations” of support maps or curves. Finally we obtain Gauss-Bonnet type formulas and Chern-Lashof type inequalities.

The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center

We prove that if f is a partially hyperbolic diffeomorphism on the compact manifold M with one dimensional center bundle, then the logarithm of the spectral radius of the map induced by f on the real homology groups of M is smaller or equal to the topological entropy of f. This is a particular case of the Shub's entropy conjecture, which claims that the same conclusion should be true for any C1 map on any compact manifold.

An extension of the Marsden-Ratiu reduction for Poisson manifolds

We propose a generalization of the reduction of Poisson manifolds by distributions introduced by Marsden and Ratiu. Our proposal overcomes some of the restrictions of the original procedure, and makes the reduced Poisson structure effectively dependent on the distribution. Different applications are discussed, as well as the algebraic interpretation of the procedure and its formulation in terms of Dirac structures.

Differential subcellular localization of phosphorylated neurofilament and tau proteins in degenerating neurons of the human entorhinal cortex.

A panel of novel monoclonal antibodies was tested on the human entorhinal cortex for the recognition of age- and disease-related changes of neurofilament proteins (NF). Several antibodies identified phosphorylated NF-H subunit, which occurred preferentially in those aged between 60 and 80 years and were localized in degenerating neurons. Such neurons also contained neurofibrillary tangles, but neurofilament aggregates did not co-localize with tangles, nor did the quantity nor the number of NF-positive neurons correlate with the severity of Alzheimer's disease. This points to a susceptibility of NF in a subset of neurons for phosphorylation- and metabolically related morphological changes during neurodegeneration.

On hyperbolic once-punctured-torus bundles IV: automata for lightning curves

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On uniform conjugators in torsion-free hyperbolic groups

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Moduli of flat conformal structures of hyperbolic type

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The non-linear Dirichlet problem in Hadamard manifolds

We prove existence theorems for the Dirichlet problem for hypersurfaces of constant special Lagrangian curvature in Hadamard manifolds. The first results are obtained using the continuity method and approximation and then refined using two iterations of the Perron method. The a-priori estimates used in the continuity method are valid in any ambient manifold.

Parallel spinors on pseudo-riemannian spin(q) manifolds

We study simply-connected irreducible non-locally symmetric pseudo-Riemannian Spin(q) manifolds admitting parallel quaternionic spinors.

Complex contact manifolds and S1 actions

We prove rigidity and vanishing theorems for several holomorphic Euler characteristics on complex contact manifolds admitting holomorphic circle actions preserving the contact structure. Such vanishings are reminiscent of those of LeBrun and Salamon on Fano contact manifolds but under a symmetry assumption instead of a curvature condition.

Lentiviral Vector Tropism In Degenerating Retinas

Introduction: In normal mice, lentiviral vector (LV) shows a great efficiency to infect the RPE cells, but transduces retinal neurons more efficiently during development. Here, we investigated the tropism of LV in the degenerating retina of mice, knowing that the retina structure changes during degeneration. We postulated that the viral transduction would be increased by the alteration of the interphotoreceptor matrix (IPM). We tested two different LV-pseudotypes using the VSVG and the Mokola envelopes. Methods: Subretinal injections were performed in wild-type (C57/Bl6) and rhodopsin knockout (Rho-/-) mice. We injected LV-VSVG-EFS-GFPII into 3.3-4.9 month old mice and LV-VSVG-Rho-GFP into 1-1.4 month old mice to target the photoreceptors (PR). LV-MOK-CMV-GFP was injected into 2.4-3.3 months old mice. We sacrificed the animals one week post injection, used immunohistochemistry to identify the transduced cells, and investigated the OLM integrity. Results: Using LV-VSVG-EFS-GFPII into 3.3-4.9 months mice, we observed significant retinal and RPE transduction in Rho-/- mice. However, the retinas showed transduction mainly at the injection's site. We mostly observed GFP+ cells having a Müller cell morphology. Using LV-MOK-CMV-GFP into 2.4-3.3 months mice, we evidenced the same pattern of viral infection, but with more Müller cells targeted by the virus. Using LV-VSVG-Rho-GFP into 1-1.4 month old mice, we don't note any difference between Rho-/- and wild-type mice for transduced cells. The IPM stained with ZO1 appears irregular into the 4.9 months old Rho-/- mice; for the youngest mice (Rho-/- and C57/Bl6), there is no modification of the IPM. Conclusion: The degeneration improves retinal cells transduction due to the alteration of the IPM in old Rho-/- mice. Müller cells seem (by morphological evidences) to be the principal cells expressing the transgene. The LV with Mokola envelope can transduce Müller cells in a degenerating retina with an intact IPM. In 1 month old mice, the degeneration doesn't enhance the transduction in rod PR probably because the IPM is not yet altered. The possibility to target photoreceptors at a later stage of the degeneration is under investigation.