The

ROSSER and SCHOENFELD have used the fact that the first 3,500,000 zeros of the RIEMANN zeta function lie on the critical line to give estimates on and . With an improvement of the above result by BRENTet al., we are able to improve these estimates and to show that the prime is greater than for . We give further results without proof.

     

Support functionals and smooth points in abstract spaces

By presenting some properties of support functionals in abstract spaces, we get some sufficient and necessary conditions for smooth points in abstract (function) spaces. Moreover, the notion of the smallest support semi-norm is introduced and an explicit form for this functional in abstract function spaces is also given.

     

coefficients

Let be a system of real smooth vector fields, satisfying Hörmander's condition in some bounded domain (). We consider the differential operator

where the coefficients are real valued, bounded measurable functions, satisfying the uniform ellipticity condition:

for a.e. , every , some constant . Moreover, we assume that the coefficients belong to the space VMO (``Vanishing Mean Oscillation'), defined with respect to the subelliptic metric induced by the vector fields . We prove the following local -estimate:

for every , . We also prove the local Hölder continuity for solutions to for any with large enough. Finally, we prove -estimates for higher order derivatives of , whenever and the coefficients are more regular.

     

Existence and uniqueness for a degenerate parabolic equation with -data

In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in ,

where a is a Carathéodory function satisfying the classical Leray-Lions hypothesis, is the Neumann boundary operator associated to , the gradient of and is a maximal monotone graph in with .

     

Determining the small solutions to -unit equations

In this paper we generalize the method of Wildanger for finding small solutions to unit equations to the case of -unit equations. The method uses a minor generalization of the LLL based techniques used to reduce the bounds derived from transcendence theory, followed by an enumeration strategy based on the Fincke-Pohst algorithm. The method used reduces the computing time needed from MIPS years down to minutes.

     

Willmore Functionals

We prove some integral inequalities for immersed tori in the three sphere. The functionals considered are generalizations of the Willmore functional.

     

A wild minimal plane in

The main object of this article is to construct a complete minimal immersed plane in whose closure has nonempty interior but it is not dense in the whole space. Furthermore, its Gaussian curvature is bounded.

     

Baire and

Let be a locally compact Hausdorff space and let be the Banach space of all bounded complex Radon measures on . Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called Baire sets of and those of are called -Borel sets of (since they are precisely the -bounded Borel sets of ). Identifying with the Banach space of all Borel regular complex measures on , in this note we characterize weakly compact subsets of in terms of the Baire and -Borel restrictions of the members of . These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

     

An uncertainty inequality involving norms

We derive a sharp uncertainty inequality of the form

with . As a consequence of this inequality we derive an upper bound for the so-called Laue constant, that is, the infimum of the functional , taken over all with (). Precisely, we obtain that which improves a previous bound of T. Gneiting.

     

The Diophantine equation

If and are given positive integers with , then we show that the equation of the title possesses at most one solution in positive integers . Moreover, we give an explicit characterization of this solution, when it exists, in terms of fundamental units of associated quadratic fields. The proof utilizes estimates for linear forms in logarithms of algebraic numbers in conjunction with properties of Pellian equations and the Jacobi symbol and explicit determination of the integer points on certain elliptic curves.

     

Free

It has been conjectured that every free algebraic action of the additive group of complex numbers on complex affine three space is conjugate to a global translation. The main result lends support to this conjecture by showing that the morphism to the variety defined by the ring of invariants of the associated action on the coordinate ring is smooth. As a consequence, the graph morphism is an open immersion, and simple proofs of certain cases of the conjecture are obtained.

     

A global correspondence between CMC-surfaces in

We show there is a global correspondence between branched constant mean curvature (i.e. CMC-) immersions in and pairs of non-conformal harmonic maps into in the same associated family. Furthermore, we give two applications.

     

A characterization of

This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.

     

Operator ideal norms on

Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

     

Tautness and complete hyperbolicity of domains in

We prove that the existence of a local peak holomorphic function at each boundary point of an unbounded domain and at infinity implies the complete hyperbolicity of this domain, and we give a link between local tautness and global tautness of a domain. We end the note with some examples of taut and complete hyperbolic domains arising from the study of domains with noncompact automorphisms group.

     

A counterexample machine

We present a general method for constructing families of measure preserving transformations which are and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two transformations which are weakly isomorphic but not isomorphic, and a transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.

     

Non-integral toroidal surgery on hyperbolic knots in

It is shown that a hyperbolic knot in admits at most one non-integral Dehn surgery producing a toroidal manifold.

     

approximations of inertial manifolds via finite differences

We construct an inertial manifold for the evolution equation as a limit of the inertial manifolds for the difference approximations of the Trotter-Kato type and show that this limit is taken in a topology.

     

Hamiltonian stationary normal bundles of surfaces in

A surface in has Hamiltonian stationary normal bundle if and only if it is either minimal, a part of a round sphere, or a part of a cone with vertex angle .