Computing Mordell-Weil ranks of cyclic covers of elliptic surfaces

We give explicit formulas for computing the Mordell-Weil ranks of the elliptic surfaces subject to some restrictions on the surface .

     

The hypersurface over a field of arbitrary characteristic

We develop techniques for computing the AK invariant of domains with arbitrary characteristic. As an example, we show that for any field the ring is not isomorphic to a polynomial ring over .

     

Corrigenda to ``New primitive whose degree is a Mersenne exponent,' and some new primitive pentanomials

We report an error in our previous paper [#!K1!#], where we announced that we listed all the primitive trinomials over of degree 859433, but there is a bug in the sieve. We missed the primitive trinomial and its reciprocal, as pointed out by Richard Brent et al. We also report some new primitive pentanomials.

     

Groups G2(p) as quotients of (2,3,7;2p)

It is proved that for any prime the group is a quotient of      

Seiberg-Witten invariants, orbifolds, and circle actions

The main result of this paper is a formula for calculating the Seiberg-Witten invariants of 4-manifolds with fixed-point-free circle actions. This is done by showing under suitable conditions the existence of a diffeomorphism between the moduli space of the 4-manifold and the moduli space of the quotient 3-orbifold. Two corollaries include the fact that } 1$"> -manifolds with fixed-point-free circle actions are simple type and a new proof of the equality . An infinite number of -manifolds with whose Seiberg-Witten invariants are still diffeomorphism invariants is constructed and studied.

     

Integer points on the curve

We completely solve diophantine equations of the form where is a positive integer, using a reduction to some quartic elliptic equations, which can be solved with well known methods.

     

Self-commutator approximants

This paper deals with minimizing , where is fixed, self-adjoint and , and where varies such that and , . (Here, , , denotes the von Neumann-Schatten class and its norm.) The upshot of this paper is that , , is minimized if, and for only if, , and that the map , , has a critical point at if and only if (with related results for normal if or ).

     

Extreme points of unit balls in Lipschitz function spaces

We give a new characterization of the set of all extreme points of the unit ball in the Banach space of all Lipschitz functions on a metric space This result is applied to get a total variation characterization of in the particular case when is a convex subset of a Banach space.

     

Commensurators of parabolic subgroups of Coxeter groups

Let be a Coxeter system, and let be a subset of . The subgroup of generated by is denoted by and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of in is the subgroup of in such that has finite index in both and . The subgroup can be decomposed in the form where is finite and all the irreducible components of are infinite. Let be the set of in such that for all . We prove that the commensurator of is . In particular, the commensurator of a parabolic subgroup is a parabolic subgroup, and is its own commensurator if and only if .

     

The group of Weierstrass points of a plane quartic with at least eight hyperflexes

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed number of hyperflexes in the moduli space of curves of genus 3.

     

On the range of the sum of monotone operators in general Banach spaces

The purpose of this paper is to generalize the Brézis-Haraux theorem on the range of the sum of monotone operators from a Hilbert space to general Banach spaces. The result obtained provides that the range is topologically almost equal to the sum where is a compatible topology in as proposed by Gossez. To illustrate the main result we consider some basic properties of densely maximal monotone operators.

     

Hahn-Banach extension operators and spaces of operators

Let be Banach spaces and let be closed operator ideals. Let be a Banach space having the Radon-Nikodým property. The main results are as follows. If is a Hahn-Banach extension operator, then there exists a set of Hahn-Banach extension operators , , such that , where . If is an ideal in for all equivalently renormed versions of , then there exist Hahn-Banach extension operators and such that .

     

New primitive whose degree is a Mersenne exponent

All primitive trinomials over with degree 859433 (which is the 33rd Mersenne exponent) are presented. They are and its reciprocal. Also two examples of primitive pentanomials over with degree 86243 (which is the 28th Mersenne exponent) are presented. The sieve used is briefly described.

     

Polynomial Pell's equation

Consider the polynomial Pell's equation , where is a monic polynomial in and . Then for , , and , a necessary and sufficient condition for the polynomial Pell's equation to have a nontrivial solution in is obtained.

     

Ideals of the cohomology rings of Hilbert schemes and their applications

We study the ideals of the rational cohomology ring of the Hilbert scheme of points on a smooth projective surface . As an application, for a large class of smooth quasi-projective surfaces , we show that every cup product structure constant of is independent of ; moreover, we obtain two sets of ring generators for the cohomology ring .

Similar results are established for the Chen-Ruan orbifold cohomology ring of the symmetric product. In particular, we prove a ring isomorphism between and for a large class of smooth quasi-projective surfaces with numerically trivial canonical class.

     

A dual finite element complex on the barycentric refinement

Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial cochain complex. We construct a new complex of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial chain complex on the original (unrefined) mesh, such that the duality is non-degenerate on for each . In particular is a space of -conforming vector fields which is dual to Raviart-Thomas -conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

     

Rational nodal curves with no smooth Weierstrass points

Let denote the rational curve with nodes obtained from the Riemann sphere by identifying 0 with and with for , where is a primitive th root of unity. We show that if is even, then has no smooth Weierstrass points, while if is odd, then has smooth Weierstrass points.

     

On quadratic fields with large 3-rank

Davenport and Heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields, which has been used to tabulate the latter. We give a simpler proof of their theorem then analyze and improve the table-building algorithm. It computes the multiplicities of the general cubic discriminants (real or imaginary) up to in time and space , or more generally in time and space for a freely chosen positive . A variant computes the -ranks of all quadratic fields of discriminant up to with the same time complexity, but using only units of storage. As an application we obtain the first real quadratic fields with , and prove that is the smallest imaginary quadratic field with -rank equal to .

     

Twin solutions to singular boundary value problems

In this paper we establish the existence of two nonnegative solutions to singular and singular focal boundary value problems. Our nonlinearity may be singular at , and/or .

     

Sur le rang du

On the rank of the -class group of . Let be a square-free positive integer and be a prime such that . We set , where or . In this paper, we determine the rank of the -class group of .

RÉSUMÉ. Soit , un corps biquadratique où ou bien un premier et étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du -groupe de classes de .