Ultrafilters on -their ideals and their cardinal characteristics

For a free ultrafilter on we study several cardinal characteristics which describe part of the combinatorial structure of . We provide various consistency results; e.g. we show how to force simultaneously many characters and many -characters. We also investigate two ideals on the Baire space naturally related to and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.

     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

Products on -modules

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of -modules such as , and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over that are concentrated in degrees divisible by ; this guarantees that various obstruction groups are trivial. We extend these results to the cases where or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising -local -modules as -modules.

     

On the diophantine equation

In this paper we prove that the equation , , , , , has only the solutions and with is a prime power. The proof depends on some new results concerning the upper bounds for the number of solutions of the generalized Ramanujan-Nagell equations.

     

Decomposition of

For any locally compact group , let and be the Fourier and the Fourier-Stieltjes algebras of , respectively. is decomposed as a direct sum of and , where is a subspace of consisting of all elements that satisfy the property: for any and any compact subset , there is an with and such that is characterized by the following: an element is in if and only if, for any there is a compact subset such that for all with and . Note that we do not assume the amenability of . Consequently, we have for all if is noncompact. We will apply this characterization of to investigate the general properties of and we will see that is not a subalgebra of even for abelian locally compact groups. If is an amenable locally compact group, then is the subspace of consisting of all elements with the property that for any compact subset , .

     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

``Best possible' upper and lower bounds for the zeros of the Bessel function

Let denote the -th positive zero of the Bessel function . In this paper, we prove that for and , 2, 3, ,

These bounds coincide with the first few terms of the well-known asymptotic expansion

as , being fixed, where is the -th negative zero of the Airy function , and so are ``best possible'.

     

Conjugacy Classes of

Let be a quadratic extension of a global field , of characteristic not two, and the integral closure in of a Dedekind ring of -integers in . Then is isomorphic to the spinorial kernel for an indefinite quadratic -lattice of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups of primitive odd binary hermitian matrices under the action of .

     

The problem on domains with piecewise smooth boundaries with applications

Let be a bounded domain in such that has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

where is a smooth -closed form with coefficients up to the bundary of , and . In particular, Equation (0.1) is solvable with smooth up to the boundary (for appropriate degree if satisfies one of the following conditions:

i)

is the transversal intersection of bounded smooth pseudoconvex domains.

ii)

where is the union of bounded smooth pseudoconvex domains and is a pseudoconvex convex domain with a piecewise smooth boundary.

iii)

where is the intersection of bounded smooth pseudoconvex domains and is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for on domains with piecewise smooth boundaries in a pseudoconvex manifold.

     

On the non-vanishing of cubic twists of automorphic -series

Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.

     

Left-symmetric algebras for

We study the classification problem for left-symmetric algebras with commutation Lie algebra in characteristic . The problem is equivalent to the classification of étale affine representations of . Algebraic invariant theory is used to characterize those modules for the algebraic group which belong to affine étale representations of . From the classification of these modules we obtain the solution of the classification problem for . As another application of our approach, we exhibit left-symmetric algebra structures on certain reductive Lie algebras with a one-dimensional center and a non-simple semisimple ideal.

     

Quasitriangular + small compact = strongly irreducible

Let be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let be a positive number. In this article, we prove that the perturbation of by a compact operator with can be strongly irreducible if is a quasitriangular operator with the spectrum connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero.

Suppose that is a bounded linear operator acting on a separable infinite dimensional Hilbert space with connected. Let be given. Is there a compact operator with such that is strongly irreducible?

     

Deformations of dihedral -group extensions of fields

Given a -Galois extension of number fields we ask whether it is a specialization of a regular -Galois cover of . This is the ``inverse" of the usual use of the Hilbert Irreducibility Theorem in the Inverse Galois problem. We show that for many groups such arithmetic liftings exist by observing that the existence of generic extensions implies the arithmetic lifting property. We explicitly construct generic extensions for dihedral -groups under certain assumptions on the base field . We also show that dihedral groups of order and have generic extensions over any base field with characteristic different from .

     

On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order

This paper is concerned with the multiplicity of radially symmetric solutions to the Dirichlet problem

on the unit ball with boundary condition on . Here is a positive function and is a function that is superlinear (but of subcritical growth) for large positive , while for large negative we have that , where is the smallest positive eigenvalue for in with on . It is shown that, given any integer , the value may be chosen so large that there are solutions with or less interior nodes. Existence of positive solutions is excluded for large enough values of .

     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

On minimal parabolic functions and time-homogeneous parabolic -transforms

Does a minimal harmonic function remain minimal when it is viewed as a parabolic function? The question is answered for a class of long thin semi-infinite tubes of variable width and minimal harmonic functions corresponding to the boundary point of ``at infinity.' Suppose is the width of the tube units away from its endpoint and is a Lipschitz function. The answer to the question is affirmative if and only if . If the test fails, there exist parabolic -transforms of space-time Brownian motion in with infinite lifetime which are not time-homogenous.

     

Combinatorial families that are exponentially far from being listable in Gray code sequence

Let be a collection of subsets of . In this paper we study numerical obstructions to the existence of orderings of for which the cardinalities of successive subsets satisfy congruence conditions. Gray code orders provide an example of such orderings. We say that an ordering of is a Gray code order if successive subsets differ by the adjunction or deletion of a single element of . The cardinalities of successive subsets in a Gray code order must alternate in parity. It follows that if is the difference between the number of elements of having even (resp. odd) cardinality, then is a lower bound for the cardinality of the complement of any subset of which can be listed in Gray code order. For , the collection of -blockfree subsets of is defined to be the set of all subsets of such that if and . We will construct a Gray code order for . In contrast, for we find the precise (positive) exponential growth rate of with as . This implies is far from being listable in Gray code order if is large. Analogous results for other kinds of orderings of subsets of are proved using generalizations of . However, we will show that for all , one can order so that successive elements differ by the adjunction and/or deletion of an integer from . We show that, over an -letter alphabet, the words of length which contain no block of consecutive letters cannot, in general, be listed so that successive words differ by a single letter. However, if and or if and , such a listing is always possible.

     

Capacity convergence results and applications to a Berstein-Markov inequality

Given a sequence of Borel subsets of a given non-pluripolar Borel set in the unit ball in with , we show that the relative capacities converge to if and only if the relative (global) extremal functions () converge pointwise to (). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support in guaranteeing that the pair satisfy a Bernstein-Markov inequality. This implies that the orthonormal polynomials associated to may be used to recover the global extremal function .

     

Classification of one -type representations

Suppose is a simple reductive -adic group with Weyl group . We give a classification of the irreducible representations of which can be extended to real hermitian representations of the associated graded Hecke algebra . Such representations correspond to unitary representations of which have a small spectrum when restricted to an Iwahori subgroup.

     

Galois coverings of selfinjective algebras by repetitive algebras

In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form where is the repetitive algebra of an artin algebra and is an admissible group of automorphisms of . If is of finite global dimension, then the stable module category of finitely generated -modules is equivalent to the derived category of bounded complexes of finitely generated -modules. For a selfinjective artin algebra , an ideal and , we establish a criterion for to admit a Galois covering with an infinite cyclic Galois group . As an application we prove that all selfinjective artin algebras whose Auslander-Reiten quiver has a non-periodic generalized standard translation subquiver closed under successors in are socle equivalent to the algebras , where is a representation-infinite tilted algebra and is an infinite cyclic group of automorphisms of .