Iterating the Cesàro operators

The discrete Cesàro operator associates to a given complex sequence the sequence , where . When is a convergent sequence we show that converges under the sup-norm if, and only if, . For its adjoint operator , we establish that converges for any .

The continuous Cesàro operator, , has two versions: the finite range case is defined for and the infinite range case for . In the first situation, when is continuous we prove that converges under the sup-norm to the constant function . In the second situation, when is a continuous function having a limit at infinity, we prove that converges under the sup-norm if, and only if, .

     

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 .

     

Asymptotic cones and Assouad-Nagata dimension

We prove that the dimension of any asymptotic cone over a metric space does not exceed the asymptotic Assouad-Nagata dimension of . This improves a result of Dranishnikov and Smith (2007), who showed for all separable subsets of special asymptotic cones , where is an exponential ultrafilter on natural numbers.

We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.

     

On the triple jump of the set of atoms of a Boolean algebra

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let be a computable Boolean algebra with infinitely many atoms and be the Turing degree of the atom relation of . If is a c.e. degree such that , then there is a computable copy of where the atom relation has degree . In particular, for every c.e. degree , any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree .

     

On the - boundedness of operators

We prove that if is in , is a Banach space, and is a linear operator defined on the space of finite linear combinations of -atoms in with the property that

then admits a (unique) continuous extension to a bounded linear operator from to . We show that the same is true if we replace -atoms by continuous -atoms. This is known to be false for -atoms.

     

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 .

     

Global co-stationarity of the ground model from a new countable length sequence

Suppose are models of ZFC with the same ordinals, and that for all regular cardinals in , satisfies . If contains a sequence for some ordinal , then for all cardinals in with regular in and , is stationary in . That is, a new -sequence achieves global co-stationarity of the ground model.

     

A sheaf of Hochschild complexes on quasi-compact opens

For a scheme , we construct a sheaf of complexes on such that for every quasi-compact open , is quasi-isomorphic to the Hochschild complex of (Lowen and Van den Bergh, 2005). Since is moreover acyclic for taking sections on quasi-compact opens, we obtain a local to global spectral sequence for Hochschild cohomology if is quasi-compact.

     

Representation of measures with polynomial denseness in , , and its application to determinate moment problems

It has been proved that algebraic polynomials are dense in the space , , iff the measure is representable as with a finite non-negative Borel measure and an upper semi-continuous function such that is a dense subset of the space    as equipped with the seminorm . The similar representation ( ) with the same and ( , and is also a dense

subset of ) corresponds to all those measures (supported by ) that are uniquely determined by their moments on ( ). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach has also been examined.

     

Enriched Reedy categories

We define the notion of an enriched Reedy category and show that if is a -Reedy category for some symmetric monoidal model category and is a -model category, the category of -functors and -natural transformations from to is again a model category.

     

Self-commutators of automorphic composition operators on the Dirichlet space

Let be a conformal automorphism on the unit disk and be the composition operator on the Dirichlet space induced by . In this article we completely determine the point spectrum, spectrum, essential spectrum and essential norm of the operators and self-commutators of , which expose that the spectrum and point spectrum coincide. We also find the eigenfunctions of the operators.

     

On the best Hölder exponent for two dimensional elliptic equations in divergence form

We obtain an estimate for the Hölder continuity exponent for weak solutions to the following elliptic equation in divergence form:

where is a bounded open subset of and, for every , is a symmetric matrix with bounded measurable coefficients. Such an estimate ``interpolates' between the well-known estimate of Piccinini and Spagnolo in the isotropic case , where is a bounded measurable function, and our previous result in the unit determinant case . Furthermore, we show that our estimate is sharp. Indeed, for every we construct coefficient matrices such that is isotropic and has unit determinant, and such that our estimate for reduces to an equality, for every .

     

On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups and , and a -adapted ring (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings and is monomial, i.e., maps class sums in to class sums in up to multiplication with roots of unity. As a consequence, and have identical character tables if and only if the centers of their integral group rings and are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

     

Dieudonné rings associated with

We use Dieudonné theory for periodically graded Hopf rings to determine the Dieudonné ring structure of the -graded Morava -theory , with an odd prime, when applied to the -spectrum (and to ). We also expand these results in order to accomodate the case of the full Morava -theory .

     

A Hodge decomposition interpretation for the coefficients of the chromatic polynomial

Let be a simple graph with nodes. The coloring complex of , as defined by Steingrimsson, has -faces consisting of all ordered set partitions, in which at least one contains an edge of . Jonsson proved that the homology of the coloring complex is concentrated in the top degree. In addition, Jonsson showed that the dimension of the top homology is one less than the number of acyclic orientations of .

In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of into components . We go on to prove that the dimensions of the Hodge pieces of the homology are equal to the absolute values of the coefficients of the chromatic polynomial of . Specifically, if we write , then .

     

Free modules of relative invariants and some rings of invariants that are Cohen-Macaulay

Let be a faithful representation of a finite group and a linear character. We study the module of -relative invariants. We prove a modular analogue of result of R. P. Stanley and V. Reiner in the case of nonmodular reflection groups to the effect that these modules are free on a single generator over the ring of invariants . This result is then applied to show that the ring of invariants for is Cohen-Macaulay. Since the Cohen-Macaulay property is not an issue in the nonmodular case (it is a consequence of a theorem of Eagon and Hochster), this would seem to be a new way to verify the Cohen-Macaulay property for modular rings of invariants. It is known that the Cohen-Macaulay property is inherited when passing from the ring of invariants of to that of a pointwise stabilizer of a subspace . In a similar vein, we introduce for a subspace the subgroup of elements of having as an eigenspace, and prove that Cohen-Macaulay implies is also.

     

Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let and be complete local noetherian rings with maximal ideals and such that is isomorphic to for every . Does it follow that and are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

     

On Lelong-Bremermann Lemma

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma:

Let be a continuous plurisubharmonic function on a Stein manifold of dimension Then there exists an integer , natural numbers , and analytic mappings such that the sequence of functions

converges to uniformly on each compact subset of .

In the case when is a domain in the complex plane, it is shown that one can take in the theorem above (Section 3); on the other hand, for -circular plurisubharmonic functions in the statement of this theorem is true with (Section 4). The last section contains some remarks and open questions.

     

Polynomials with roots in for all

Let be a monic polynomial in with no rational roots but with roots in for all , or equivalently, with roots mod for all . It is known that cannot be irreducible but can be a product of two or more irreducible polynomials, and that if is a product of irreducible polynomials, then its Galois group must be a union of conjugates of proper subgroups. We prove that for any , every finite solvable group that is a union of conjugates of proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with ) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of .