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 .

     

Homogeneous polynomials on strictly convex domains

We consider a circular, bounded, strictly convex domain with boundary of class . For any compact subset of we construct a sequence of homogeneous polynomials on which are big at each point of . As an application for any circular subset of type we construct a holomorphic function which is square integrable on and such that where denotes unit disc in .

     

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.

     

The Noether map II

Let be a faithful representation of a finite group . In this paper we proceed with the study of the image of the associated Noether map

In our 2005 paper it has been shown that the Noether map is surjective if is a projective -module. This paper deals with the converse. The converse is in general not true: we illustrate this with an example. However, for -groups (where is the characteristic of the ground field ) as well as for permutation representations of any group the surjectivity of the Noether map implies the projectivity of .

     

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,

     

Artinianess of graded local cohomology modules

Let be a Noetherian homogeneous ring with local base ring and let be a finitely generated graded -module. Let be the largest integer such that is not Artinian. We will prove that are Artinian for all and there exists a polynomial of degree less than such that for all . Let be the first integer such that the local cohomology module is not cofinite. We will show that for all the graded module is Artinian.

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

Linear bijections preserving the Hölder seminorm

Let be a compact metric space and let be a real number with The aim of this paper is to solve a linear preserver problem on the Banach algebra of Hölder functions of order from into We show that each linear bijection having the property that for every where

is of the form for every where with is a surjective isometry and is a linear functional.

     

Approximation of holomorphic maps with a lower bound on the rank

Let be a closed polydisc or ball in , and let be a quasi-projective algebraic manifold which is Zariski locally equivalent to , or a complement of an algebraic subvariety of codimension in such a manifold. If is an integer satisfying , then every holomorphic map from a neighborhood of to with rank at every point of can be approximated uniformly on by entire maps with rank at every point of .

     

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 .

     

Counting cusps of subgroups of

Let be a number field with real places and complex places, and let be the ring of integers of . The quotient has cusps, where is the class number of . We show that under the assumption of the generalized Riemann hypothesis that if is not or an imaginary quadratic field and if , then has infinitely many maximal subgroups with cusps. A key element in the proof is a connection to Artin's Primitive Root Conjecture.

     

On the characteristic polynomial of the almost Mathieu operator

Let be the rotation C*-algebra for angle . For with and relatively prime, is the sub-C*-algebra of generated by a pair of unitaries and satisfying . Let

be the almost Mathieu operator. By proving an identity of rational functions we show that for even, the constant term in the characteristic polynomial of is .

     

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 .

     

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 .

     

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 .

     

A non-Hausdorff model for the complement of a complexified hyperplane arrangement

Given a hyperplane arrangement in a real vector space , we introduce a real algebraic prevariety , and exhibit the complement of in the complexification of as the total space of an affine bundle over with fibers modeled on the dual vector space .

     

Solutions to arithmetic convolution equations

In the -algebra of arithmetic functions , endowed with the usual pointwise linear operations and the Dirichlet convolution, let denote the convolution power with factors . We investigate the solvability of polynomial equations of the form

with fixed coefficients . In some cases the solutions have specific properties and can be determined explicitly. We show that the property of the coefficients to belong to convergent Dirichlet series transfers to those solutions , whose values are simple zeros of the polynomial . We extend this to systems of convolution equations, which need not be of polynomial-type.

     

Cross -sections of star bodies and dual mixed volumes

In this paper, we establish an extension of Funk's section theorem. Our result has the following corollary: If is a star body in whose central -slices have the same volume (with appropriate dimension) as the central -slices of a centered body , then the dual quermassintegrals satisfy , for any , with equality if and only if . The case that is a centered body implies Funk's section theorem.

     

On sums of Darboux and nowhere constant continuous functions

We show that the property

(P)

for every Darboux function there exists a continuous nowhere constant function such that is Darboux

follows from the following two propositions:

(A)

for every subset of of cardinality there exists a uniformly continuous function such that ,

(B)

for an arbitrary function whose image contains a non-trivial interval there exists an of cardinality such that the restriction of to is uniformly continuous,

which hold in the iterated perfect set model.

     

A ``deformation estimate" for the Toeplitz operators on harmonic Bergman spaces

Let denote the open unit ball in for and the Lebesgue volume measure on . For , the (weighted) harmonic Bergman space is the space of all harmonic functions which are in . For , the Toeplitz operator is defined on by , where is the orthogonal projection of onto . In this note, we prove that for radial, .