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.

     

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 .

     

Positive definite spherical functions on Olshanskii domains

Let be a simply connected complex Lie group with Lie algebra , a real form of , and the analytic subgroup of corresponding to . The symmetric space together with a -invariant partial order is referred to as an Olshanskii space. In a previous paper we constructed a family of integral spherical functions on the positive domain of . In this paper we determine all of those spherical functions on which are positive definite in a certain sense.

     

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.

     

Spherical functions and conformal densities on spherically symmetric -spaces

Let be a -space which is spherically symmetric around some point and whose boundary has finite positive dimensional Hausdorff measure. Let be a conformal density of dimension on . We prove that is a weak limit of measures supported on spheres centered at . These measures are expressed in terms of the total mass function of and of the dimensional spherical function on . In particular, this result proves that is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.

     

On locally linearly dependent operators and derivations

The first section of the paper deals with linear operators , , where and are vector spaces over an infinite field, such that for every , the vectors are linearly dependent modulo a fixed finite dimensional subspace of . In the second section, outer derivations of dense algebras of linear operators are discussed. The results of the first two sections of the paper are applied in the last section, where commuting pairs of continuous derivations of a Banach algebra such that is quasi-nilpotent for every are characterized.

     

Building blocks for Quadratic Julia sets

We obtain results on the structure of the Julia set of a quadratic polynomial with an irrationally indifferent fixed point in the iterative dynamics of . In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set : there exists a nowhere dense subcontinuum such that , is the union of the impressions of a minimally invariant Cantor set of external rays, contains the critical point, and contains both the Cremer point and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and contains no periodic points. In both cases, the Julia set is the closure of a skeleton which is the increasing union of countably many copies of the building block joined along preimages of copies of a critical continuum containing the critical point. In addition, we prove that if is any polynomial of degree with a Siegel disk which contains no critical point on its boundary, then the Julia set is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

     

Compatible complex structures on almost quaternionic manifolds

On an almost quaternionic manifold we study the integrability of almost complex structures which are compatible with the almost quaternionic structure . If , we prove that the existence of two compatible complex structures forces to be quaternionic. If , that is is an oriented conformal 4-manifold, we prove a maximum principle for the angle function of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure on the twistor space of an almost quaternionic manifold and show that is a complex structure if and only if is quaternionic. This is a natural generalization of the Penrose twistor constructions.

     

Characterizations of spectra with -injective cohomology which satisfy the Brown-Gitler property

We work in the stable homotopy category of -complete connective spectra having mod homology of finite type. means cohomology with coefficients, and is a left module over the Steenrod algebra .

A spectrum is called spacelike if it is a wedge summand of a suspension spectrum, and a spectrum satisfies the Brown-Gitler property if the natural map is onto, for all spacelike .

It is known that there exist spectra satisfying the Brown-Gitler property, and with isomorphic to the injective envelope of in the category of unstable -modules.

Call a spectrum standard if it is a wedge of spectra of the form , where is a stable wedge summand of the classifying space of some elementary abelian -group. Such spectra have -injective cohomology, and all -injectives appear in this way.

Working directly with the two properties of stated above, we clarify and extend earlier work by many people on Brown-Gitler spectra. Our main theorem is that, if is a spectrum with -injective cohomology, the following conditions are equivalent:

(A) there exist a spectrum whose cohomology is a reduced -injective and a map that is epic in cohomology, (B) there exist a spacelike spectrum and a map that is epic in cohomology, (C) is monic in cohomology, (D) satisfies the Brown-Gitler property, (E) is spacelike, (F) is standard. ( is reduced if it has no nontrivial submodule which is a suspension.)

As an application, we prove that the Snaith summands of are Brown-Gitler spectra-a new result for the most interesting summands at odd primes. Another application combines the theorem with the second author's work on the Whitehead conjecture.

Of independent interest, enroute to proving that (B) implies (C), we prove that the homology suspension has the following property: if an -connected space admits a map to an -fold suspension that is monic in mod homology, then is onto in mod homology.

     

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?

     

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 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 .

     

Low-dimensional linear representations of

We classify all complex representations of the automorphism group of the free group of dimension Among those representations is a new representation of dimension which does not vanish on the group of inner automorphisms.

     

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.

     

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 .

     

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 .

     

The nilpotence height of for odd primes

K. G. Monks has recently shown that the element has nilpotence height in the mod Steenrod algebra. Here the method and result are generalized to show that for an odd prime the element has nilpotence height in the mod Steenrod algebra.

     

Vertex operators for twisted quantum affine algebras

We construct explicitly the -vertex operators (intertwining operators) for the level one modules of the classical quantum affine algebras of twisted types using interacting bosons, where for (), for , for (), and for (). A perfect crystal graph for is constructed as a by-product.