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.

     

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 local irreducibility of the spectrum

Let be the algebra of complex matrices, and for denote by and the spectrum and spectral radius of respectively. Let be a domain in containing 0, and let be a holomorphic map. We prove: (1) if for , then for ; (2) if for , then again for . Both results are special cases of theorems expressing the irreducibility of the spectrum near .

     

Perturbations and Weyl's theorem

A Banach space operator is completely hereditarily normaloid, , if either every part, and (also) for every invertible part , of is normaloid or if for every complex number every part of is normaloid. Sufficient conditions for the perturbation of by an algebraic operator to satisfy Weyl's theorem are proved. Our sufficient conditions lead us to the conclusion that the conjugate operator satisfies -Weyl's theorem.

     

Radical and cyclotomic extensions of the rational numbers

A radical extension of the rational numbers is a field generated by an element having a power in , and a cyclotomic extension is an extension generated by a root of unity. We show that a radical extension that is almost Galois over is almost cyclotomic. More precisely, we prove that if is radical with Galois closure , then contains a cyclotomic field such that the degree is bounded above by an almost linear function of . In particular, if is Galois, it contains a cyclotomic field such that .

     

Extensions of orthosymmetric lattice bimorphisms revisited

This note furnishes an example illustrating the following two facts. On the one hand, there exist Archimedean Riesz spaces and with Dedekind-complete and an orthosymmetric lattice bimorphism with lattice bimorphism extension which is not orthosymmetric, where denotes the Dedekind-completion of . On the other hand, there is an associative -multiplication in the same Archimedean Riesz space which extends to a -multiplication in which is not associative. The existence of such an example provides counterexamples to assertions in Toumi, 2005.

     

On quasi-isometric embeddings of Lamplighter groups

We denote by the Lamplighter group of a finite group . In this article, we show that if and are two finite groups with at least two elements, then there exists a quasi-isometric embedding from to . We also prove that the quasi-isometry group of contains all finite groups. We then show that the group of automorphisms of has infinite index in .

     

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.

     

Some Hopf Galois structures arising from elementary abelian -groups

Let be an odd prime, , the elementary abelian -group of rank , and let be the group of principal units of the ring . If is a Galois extension with Galois group , then we show that for , the number of Hopf Galois structures on afforded by -Hopf algebras with associated group is greater than , where .

     

Exponential growth of Lie algebras of finite global dimension

Let be a connected finite type graded Lie algebra. If dim and gldim , then log index . If, moreover, , then for some ,    dim where log index as

     

On the degree of Hilbert polynomials associated to the torsion functor

Let be a local, Noetherian ring and an ideal. A question of Kodiyalam asks whether for fixed , the polynomial giving the th Betti number of has degree equal to the analytic spread of minus one. Under mild conditions on , we show that the answer is positive in a number of cases, including when is divisible by or is an integrally closed -primary ideal.

     

On the topology of the boundary of a basin of attraction

Suppose is a continuous flow on a locally compact metrizable space and is an (asymptotically stable) attractor. Let be the boundary of the basin of attraction of . In the present paper it will be shown how the Conley index of plays an important role in determining the topological nature of and allows one to obtain information about the global dynamics of in .

     

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 .

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

Equilibrium points of logarithmic potentials on convex domains

Let be a convex domain in . Let be summable constants and let . If the converge sufficiently rapidly to from within an appropriate Stolz angle, then the function has infinitely many zeros in . An example shows that the hypotheses on the are not redundant and that two recently advanced conjectures are false.

     

-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Let or , where is the algebra of a bounded linear operator acting on the Hilbert space , and is the set of self-adjoint operators in . Denote the numerical range of by It is shown that a surjective map satisfies

if and only if there is a unitary operator such that has the form

where is the transpose of with respect to a fixed orthonormal basis. In other words, the map or is a -isomorphism on and a Jordan isomorphism on . Moreover, if has finite dimension, then the surjective assumption on can be removed.

     

Large cardinals with few measures

We show, assuming the consistency of one measurable cardinal, that it is consistent for there to be exactly many normal measures on the least measurable cardinal . This answers a question of Stewart Baldwin. The methods generalize to higher cardinals, showing that the number of strong compactness or supercompactness measures on can be exactly if is a regular cardinal. We conclude with a list of open questions. Our proofs use a critical observation due to James Cummings.

     

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.

     

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 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.