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.

     

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.

     

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 .

     

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.

     

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space . The length of a smooth curve , , is measured by means of , where denotes the usual norm of operators. The initial value problem is solved: for any isometry and each tangent vector at (which is an operator of the form with ) with norm less than or equal to , there exist curves of the form , with initial velocity , which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given a symmetric operator

find all possible extending to all , with . We also consider the problem of finding metric geodesics joining two given isometries and . It is well known that if there exists a continuous path joining and , then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining and .

     

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.

     

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.

     

Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space for and connected, nilpotent CW complexes of finite type with finite. For the component corresponding to a general homotopy class , we give a formula directly computable from the Sullivan model for . For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of and . When both and are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of explicitly in terms of the homomorphism induced by on rational cohomology.

     

Noncoherence of some rings of functions

Let , denote the unit disc and unit circle, respectively, in , with center 0. If , then let denote the set of complex-valued functions defined on that are analytic in , and continuous and bounded on . Then is a ring with pointwise addition and multiplication. We prove that if the intersection of with the set of limit points of is not empty, then the ring is not coherent.

     

On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem

in , on , where is a bounded open set in with smooth boundary and , are continuous functions on such that , , and for all . The main result of this paper establishes that any sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.

     

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.

     

A strong comparison principle for the -Laplacian

We consider weak solutions of the differential inequality of p-Laplacian type

such that on a smooth bounded domain in and either or is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that on the boundary of the domain we prove that , and assuming that on the boundary of the domain we prove unless . The novelty is that the nonlinearity is allowed to change sign. In particular, the result holds for the model nonlinearity with .

     

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 .

     

Form estimates for the -Laplacean

We consider the problem of establishing conditions on that ensure that the form associated with the -Laplacean is positive bounded below. It was shown recently by Fan, Zhang and Zhao that - unlike the constant case - this is not possible if has a strict extrema in the domain. They also considered the closely related problem of eigenvalue existence and estimates. Our main tool is the adaptation of a technique, employed by Protter for involving arbitrary vector fields. We also examine related results obtained by a variant of Picone Identity arguments. We directly consider problems in with and while we focus on Dirichlet boundary conditions we also indicate how our approach can be used in cases of mixed boundary conditions, of unbounded domains and of discontinuous Our basic criteria involve restrictions on and its gradient.

     

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 .

     

Dimension des familles de courbes lisses sur une surface quartique normale de

Dans cette note, on montre que les courbes, lisses connexes, de degré et genre , tracées sur une surface quartique normale variable de , et n'y étant pas intersection complète, forment des familles de dimensions . Cette majoration est la meilleure possible. Comme application on prouve que le schéma de Hilbert des courbes lisses connexes de de degré et genre est irréductible.

     

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.

     

Planar finitely Suslinian compacta

We show that a planar unshielded compact set is finitely Suslinian if and only if there exists a closed set and a lamination of such that is homeomorphic to . If is a continuum, the analogous statement follows from Carathéodory theory and is widely used in polynomial dynamics.