On the index and spectrum of differential operators on

If is an system of differential operators on having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that is Fredholm from to for all or no value We prove that both the index (when defined) and the spectrum of are independent of

     

Universal spaces for almost -dimensionality

We find universal functions for the class of lower semi-continuous (LSC) functions with at most -dimensional domain. In an earlier paper we proved that a space is almost -dimensional if and only if it is homeomorphic to the graph of an LSC function with an at most -dimensional domain. We conclude that the class of almost -dimensional spaces contains universal elements (that are topologically complete). These universal spaces can be thought of as higher-dimensional analogues of complete Erdos space.

     

A remark on irregularity of the -Neumann problem on non-smooth domains

It is an observation due to J. J. Kohn that for a smooth bounded pseudoconvex domain in there exists such that the -Neumann operator on maps (the space of -forms with coefficient functions in -Sobolev space of order ) into itself continuously. We show that this conclusion does not hold without the smoothness assumption by constructing a bounded pseudoconvex domain in , smooth except at one point, whose -Neumann operator is not bounded on for any .

     

On frames for countably generated Hilbert -modules

Let be a countably generated Hilbert -module over a -algebra We prove that a sequence is a standard frame for if and only if the sum converges in norm for every and if there are constants such that for every We also prove that surjective adjointable operators preserve standard frames. A class of frames for countably generated Hilbert -modules over the -algebra of all compact operators on some Hilbert space is discussed.

     

Embeddings of -dimensional separable metric spaces into the product of Sierpinski curves

We give a short proof of the following fact: the set of embeddings of any -dimensional separable metric space into a certain -dimensional subset of the -product of Sierpinski curves is residual in .

     

The -local -homology of

In this paper, we introduce a Hopf algebra, developed by the author and André Henriques, which is usable in the computation of the -homology of a space. As an application, we compute the -homology of in a manner analogous to Mahowald and Milgram's computation of the -homology .

     

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 .

     

On -supplemented maximal and minimal subgroups of Sylow subgroups of finite groups

This paper proves: Let be a saturated formation containing . Suppose that is a group with a normal subgroup such that .

(1) If all maximal subgroups of any Sylow subgroup of are -supple- mented in , then ;

(2) If all minimal subgroups and all cyclic subgroups with order 4 of are -supplemented in , then .

     

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 .

     

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 .

     

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.

     

On the local Hölder continuity of the inverse of the -Laplace operator

We prove an interpolation type inequality between , and spaces and use it to establish the local Hölder continuity of the inverse of the -Laplace operator: , for any and in a bounded set in .

     

spaces associated with the sections

spaces associated with the sections are introduced. It is shown that some properties which hold for the classical space related to the balls (or cubes) remain valid for the space related to the sections.

     

A note on -intersection bodies

The concept of an intersection body is central for the dual Brunn-Minkowski theory and has also played an important role in the solution of the Busemann-Petty problem. A more general concept of -intersection bodies is related to the generalization of the Busemann-Petty problem. In this note, we compare classes of -intersection bodies for different and examine the conjecture that these classes increase with . In particular, we construct a -intersection body that is not a -intersection body.

     

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 exceptional eigenvalues of the Laplacian for

An explicit Dirichlet series is obtained, which represents an analytic function of in the half-plane except for having simple poles at points that correspond to exceptional eigenvalues of the non-Euclidean Laplacian for Hecke congruence subgroups by the relation for . Coefficients of the Dirichlet series involve all class numbers of real quadratic number fields. But, only the terms with for sufficiently large discriminants contribute to the residues of the Dirichlet series at the poles , where is the multiplicity of the eigenvalue for . This may indicate (I'm not able to prove yet) that the multiplicity of exceptional eigenvalues can be arbitrarily large. On the other hand, by density theorem the multiplicity of exceptional eigenvalues is bounded above by a constant depending only on .

     

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.

     

A sharp result on -covers

Let be a finite system of residue classes which forms an -cover of (i.e., every integer belongs to at least members of ). In this paper we show the following sharp result: For any positive integers and , if there is such that the fractional part of is , then there are at least such subsets of . This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to -covers of the integral ring of any algebraic number field with a power integral basis.

     

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.

     

Local existence of -sets, projective tensor products, and Arens regularity for

Theorem. If are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets such that (i) is either a Kronecker set or (ii) for some , is a translate of a -set all of whose elements have order , and (iii) is isomorphic to the projective tensor product .

This extends what was previously known for groups such as or for the case to the general locally compact abelian group. Old results concerning the local existence of Kronecker and -sets are improved.