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It is shown that (1) every infinite-dimensional Banach space admits aC 1 Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces.

We consider a self-adjoint operator defined by a bidimensional linear system. We extend the Ishii-Pastur-Kotani theory that allows us to identify the absolutely continuous spectrum. From here we deduce that for almost everyE with null Lyapunov exponent the real projective flow admits absolutely continuous invariant measures with square integrable density function.

Classically a colored tree is associated to any p-adic groups of rank one. For some of these, subgroups acting simply transitively on vertices of given color are constructed. In fewer case, the same can be done for edges.

Suppose we are given a computably enumerable object. We are interested in the strength of oracles that can compute an object that approximates this c.e. object. It turns out that in many cases arising from algorithmic randomness or computable analysis, the resulting highness property is either close to, or equivalent to being PA complete. We examine, for example, majorizing a c.e. martingale by an oracle-computable martingale, computing lower bounds for two variants of Kolmogorov complexity, and computing a subtree of positive measure with no dead-ends of a given $$\prod _1^0$$ tree of positive measure. We separate PA completeness from the latter property, called the continuous covering property. We also separate the corresponding principles in reverse mathematics.

We study the tropical Dolbeault cohomology of non-archimedean curves as defined by Chambert-Loir and Ducros. We give a precise condition for when this cohomology satisfies Poincaré duality. The condition is always satisfied when the residue field of the non-archimedean base field is the algebraic closure of a finite field. We also show that for curves over fields with residue field ℂ, the tropical (1, 1)-Dolbeault cohomology can be infinite dimensional. Our main new ingredient is an exponential type sequence that relates tropical Dolbeault cohomology to the cohomology of the sheaf of harmonic functions. As an application of our Poincaré duality result, we calculate the dimensions of the tropical Dolbeault cohomology, called tropical Hodge numbers, for (open subsets of) curves.

In this paper we study integral extensions of noncommutative rings. To begin, we prove that finite subnormalizing extensions are integral. This is done by proving a generalization of the Paré-Schelter result that a matrix ring is integral over the coefficient ring. Our methods are similar to those of Lorenz and Passman, who showed that finite normalizing extensions are integral. As corollaries we note that the (twisted) smash product over the restricted enveloping algebra of a finite dimensional restricted Lie algebra is integral over the coefficient ring and then prove a Going Up theorem for prime ideals in these ring extensions. Next we study automorphisms of rings. In particular, we prove an integrality theorem for algebraic automorphisms. Combining group gradings and actions, we show that if a ringR is graded by a finite groupG, andH is a finite group of automorphisms ofR that permute the homogeneous components, with the order ofH invertible inR, thenR is integral overR 1 , the fixed ring of the identity component. This, in turn, is used to prove our final result: Suppose that ifH is a finite dimensional semisimple cocommutative Hopf algebra over an algebraically closed field of positive characteristic. IfR is anH-module algebra, thenR is integral overR H , its subring of invariants.

In this work, we study the stability of Hopf vector fields on Lorentzian Berger spheres as critical points of the energy, the volume and the generalized energy. In order to do so, we construct a family of vector fields using the simultaneous eigenfunctions of the Laplacian and of the vertical Laplacian of the sphere. The Hessians of the functionals are negative when they act on these particular vector fields and then Hopf vector fields are unstable. Moreover, we use this technique to study some of the open problems in the Riemannian case.