-adic formal series and primitive polynomials over finite fields

In this paper, we investigate the Hansen-Mullen conjecture with the help of some formal series similar to the Artin-Hasse exponential series over -adic number fields and the estimates of character sums over Galois rings. Given we prove, for large enough , the Hansen-Mullen conjecture that there exists a primitive polynomial over of degree with the -th ( coefficient fixed in advance except when if is odd and when if is even.

     

Automatic continuity of -derivations on -algebras

Let be a -algebra acting on a Hilbert space , let be a linear mapping and let be a -derivation. Generalizing the celebrated theorem of Sakai, we prove that if is a continuous -mapping, then is automatically continuous. In addition, we show the converse is true in the sense that if is a continuous --derivation, then there exists a continuous linear mapping such that is a --derivation. The continuity of the so-called - -derivations is also discussed.

     

On maximal operators on -spheres in

A. Magyar's result on -bounds for a family of operators on -spheres () in is improved to match the corresponding theorem for -spheres.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Packing spheres and fractal Strichartz estimates in for

We prove an estimate for the spherical average operator in if . This leads to a lower bound for the Hausdorff dimension of unions of certain collections of spheres and to a Strichartz-type estimate for solutions of the wave equation.

     

Lehmer -tuples

Generalizing a classical problem of Lehmer, in this paper we provide an asymptotic result for the number of Lehmer -tuples.

     

Whittaker models for , a -adic field

Let be a non-archimedean local field with odd residual characteristic. Let be the group We construct explicit Whittaker models for any supercuspidal representation of with positive level.

     

A van der Corput lemma for the -adic numbers

We prove a version of van der Corput's lemma for polynomials over the -adic numbers.

     

Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

Spectral pictures of and

The spectral pictures of products and of Banach space operators are compared; in particular when one of them is `of index zero'.

     

A short proof of the Zeilberger-Bressoud -Dyson theorem

We give a formal Laurent series proof of Andrews's -Dyson Conjecture, first proved by Zeilberger and Bressoud.

     

Symmetric Bush-type Hadamard matrices of order exist for all odd

Using reversible Hadamard difference sets, we construct symmetric Bush-type Hadamard matrices of order for all odd integers .

     

Sharply -transitive groups with point stabilizer of exponent or

Using the fact that all groups of exponent are nilpotent, we show that every sharply -transitive permutation group whose point stabilizer has exponent or is finite.

     

Pseudocompact spaces and -spaces

Let be a completely regular Hausdorff space, and let be the space of continuous real-valued functions on endowed with the compact-open topology. We find various equivalent conditions for to be a -space, resolving an old question of Jarchow and consolidating work by Jarchow, Mazon, McCoy and Todd. Included are analytic characterizations of pseudocompactness and an example that shows that, for , Grothendieck's -spaces do not coincide with Jarchow's -spaces. Any such example necessarily answers a thirty-year-old question on weak barrelledness properties for , our original motivation.

     

Borel cardinalities below

The Borel cardinality of the quotient of the power set of the natural numbers by the ideal of asymptotically zero-density sets is shown to be the same as that of the equivalence relation induced by the classical Banach space . We also show that a large collection of ideals introduced by Louveau and Velickovic, with pairwise incomparable Borel cardinality, are all Borel reducible to . This refutes a conjecture of Hjorth and has facilitated further work by Farah.

     

Extending into isometries of

In this paper we generalize a result of Hopenwasser and Plastiras (1997) that gives a geometric condition under which into isometries from to have a unique extension to an isometry in . We show that when and are separable reflexive Banach spaces having the metric approximation property with strictly convex and smooth and such that is a Hahn-Banach smooth subspace of , any nice into isometry has a unique extension to an isometry in .

     

An ultrafilter with property

Let be a -supercompact cardinal. We show that carries a normal ultrafilter with a property introduced by Menas. With it we give a transparent proof of Kamo's theorem that carries a normal ultrafilter with the partition property.

     

-theory of -pseudo-differential algebras

We are concerned with the so-called -pseudo-differential calculus. We describe the spectrum of the unital and commutative -algebra given by the norm closure of the space of -order pseudo-differential operators modulo compact operators; other related algebras are also considered. Finally, their -theory is computed.

     

Codes over and and Hermitian lattices over imaginary quadratic fields

We introduce a family of bi-dimensional theta functions which give uniformly explicit formulae for the theta series of hermitian lattices over imaginary quadratic fields constructed from codes over and , and give an interesting geometric characterization of the theta series that arise in terms of the basic strongly modular lattice . We identify some of the hermitian lattices constructed and observe an interesting pair of nonisomorphic 3/2 dimensional codes over that give rise to isomorphic hermitian lattices when constructed at the lowest level 7 but nonisomorphic lattices at higher levels. The results show that the two alphabets and are complementary and raise the natural question as to whether there are other such complementary alphabets for codes.

     

Functions -orthogonal with respect to their own zeros

In 1939, G. H. Hardy proved that, under certain conditions, the only functions satisfying

where the are the zeros of , are the Bessel functions. We replace the above integral by the Jackson -integral and give the -analogue of Hardy's result.