Actions of pointed Hopf algebras with reduced pi invariants

Let be an -module algebra, where is a pointed Hopf algebra acting on finitely of dimension . Suppose that for every nonzero -stable left ideal of . It is proved that if satisfies a polynomial identity of degree , then satisfies a polynomial identity of degree provided at least one of the following additional conditions is fulfilled:

1. is semiprime and is almost central in ,
2. is reduced.

If we also assume that is central, then satisfies the standard polynomial identity of degree , where is the greatest integer in .

     

Cofinality changes required for a large set of unapproachable ordinals below

In , assume that is a strong limit cardinal and . Let be the set of approachable ordinals less than . An open question of M. Foreman is whether can be non-stationary in some and preserving extension of . It is shown here that if is such an outer model, then is infinite, for each positive integer .

     

The problem of minimizing locally a functional around non-critical points is well-posed

In this paper, we prove the following general result: Let be a real Hilbert space and a functional, with locally Lipschitzian derivative.

Then, for each with , there exists such that, for every , the restriction of to the sphere has a unique global minimum toward which every minimizing sequence strongly converges.

     

The geography of symplectic -manifolds with an arbitrary fundamental group

In this article, for each finitely presented group , we construct a family of minimal symplectic -manifolds with which cover most lattice points with large in the region . Furthermore, we show that all these -manifolds admit infinitely many distinct smooth structures.

     

Banach-Stone theorem for Banach lattice valued continuous functions

Let and be compact Hausdorff spaces, be a Banach lattice and be an AM space with unit. Let be a Riesz isomorphism such that if and only if for each . We prove that is homeomorphic to and is Riesz isomorphic to . This generalizes some known results.

     

A new estimate in optimal mass transport

Let be a bounded Lipschitz regular open subset of and let be two probablity measures on . It is well known that if is absolutely continuous, then there exists, for every , a unique transport map pushing forward on and which realizes the Monge-Kantorovich distance . In this paper, we establish an bound for the displacement map which depends only on , on the shape of and on the essential infimum of the density .

     

On quasifree profinite groups

Recently, it has been shown by Harbater and Stevenson that a profinite group is free profinite of infinite rank if and only if is projective and -quasifree. The latter condition requires the existence of distinct solutions to certain embedding problems for . In this paper we provide several new non-trivial examples of -quasifree groups, projective and non-projective. Our main result is that open subgroups of -quasifree groups are -quasifree.

     

Decomposing representations of finite groups on Riemann-Roch spaces

If is a finite subgroup of the automorphism group of a projective curve and is a divisor on stabilized by , then we compute a simplified formula for the trace of the natural representation of on the Riemann-Roch space , under the assumption that is ``rational', is nonspecial, and the characteristic is ``good'. We discuss the partial formulas that result if is not rational.

     

Semiprime smash products and -stable prime radicals for PI-algebras

Assume that is a finite-dimensional Hopf algebra over a field and that is an -module algebra satisfying a polynomial identity (PI). We prove that if is semisimple and is -semiprime, then is semiprime. If is cosemisimple, we show that the prime radical of is -stable.

     

Transversals for strongly almost disjoint families

For a family of sets , and a set , is said to be a transversal of if and for each . is said to be a Bernstein set for if for each . Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets is said to admit a -transversal if can be written as such that each admits a transversal. We study the question of when an almost disjoint family admits a -transversal and related questions.

     

On the extensions of homogeneous polynomials

We investigate the problem of the uniqueness of the extension of -homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space that admits contractive projection of finite rank of at least dimension 2, for every there exists an -homogeneous polynomial on that has infinitely many extensions to . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice , for instance when satisfies an upper -estimate with constant one for some , any -homogeneous polynomial on attaining its norm at with a finite rank band projection , has a unique extension to its bidual . We apply these results in a class of Orlicz sequence spaces.

     

A new proof and generalizations of Gearhart's theorem

Let be a Hilbert space, let be the space of almost periodic functions from to , and let be a closed densely defined linear operator on . For a closed subset , let be the subspace of consisting of functions with spectrum contained in . We prove that the following properties are equivalent: (i) for every function there exists a unique mild solution of equation ; (ii) and . The case yields a new proof of the well-known Gearhart's spectral mapping theorem.

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

Hereditary and maximal crossed product orders

We first deal with classical crossed products , where is a finite group acting on a Dedekind domain and (the -invariant elements in ) a DVR, admitting a separable residue fields extension. Here is a 2-cocycle. We prove that is hereditary if and only if is semi-simple. In particular, the heredity property may hold even when is not tamely ramified (contradicting standard textbook references). For an arbitrary Krull domain , we use the above to prove that under the same separability assumption, is a maximal order if and only if its height one prime ideals are extended from . We generalize these results by dropping the residual separability assumptions. An application to non-commutative unique factorization rings is also presented.

     

Uniqueness for boundary blow-up problems with continuous weights

In this paper, we prove that for the problem in a bounded domain of has a unique positive solution with on . The nonnegative weight is continuous in , but is only assumed to verify a ``bounded oscillations" condition of local nature near , in contrast with previous works, where a definite behavior of near was imposed.

     

A structure theorem for quasi-Hopf comodule algebras

If is a quasi-Hopf algebra and is a right -comodule algebra such that there exists a morphism of right -comodule algebras, we prove that there exists a left -module algebra such that . The main difference when comparing to the Hopf case is that, from the multiplication of , which is associative, we have to obtain the multiplication of , which in general is not; for this we use a canonical projection arising from the fact that becomes a quasi-Hopf -bimodule.

     

Solutions for a nonlocal conservation law with fading memory

Global entropy solutions in for a scalar nonlocal conservation law with fading memory are constructed as the limits of vanishing viscosity approximate solutions. The uniqueness and stability of entropy solutions in are established, which also yield the existence of entropy solutions in while the initial data is only in . Moreover, if the memory kernel depends on a relaxation parameter and tends to a delta measure weakly as measures when , then the global entropy solution sequence in converges to an admissible solution in for the corresponding local conservation law.

     

Stable constant mean curvature hypersurfaces

Let be a Riemannian manifold with sectional curvatures uniformly bounded from below. When we prove that there are no complete (strongly) stable -hypersurfaces, without boundary, provided is large enough. In particular, we prove that there are no complete strongly stable -hypersurfaces in without boundary,

     

Non-reflexivity of the derivation space from Banach algebras of analytic functions

Let be an open connected subset of the plane, and let be a Banach algebra of analytic functions on . We show that the space of bounded derivations from into is not reflexive. We also obtain similar results when for .

     

A reduction theorem for the topological degree for mappings of class

The reduction theorem for the Leray-Schauder degree provides an efficient tool to calculate the value of the degree in a suitable invariant subspace. We shall prove how the calculation of the value of the topological degree for a mapping of class from a real separable reflexive Banach space into the dual space can be reduced into the calculation of degree of mapping from a closed subspace into Since the Leray-Schauder mappings are acting from to and we are dealing with mappings from to the standard `invariant subspace' condition must be replaced by a less obvious one.