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 .

     

Sur une question de capitulation

Let and be prime numbers such that and . Let , , and let be the 2-Hilbert class field of , the 2-Hilbert class field of and the Galois group of . The 2-part of the class group of is of type , so contains three extensions . Our goal is to study the problem of capitulation of the 2-classes of in , and to determine the structure of .

RSESUM´E. Soient et deux nombres premiers tels que et , , , , le 2-corps de classes de Hilbert de , le 2-corps de classes de Hilbert de et le groupe de Galois de . La 2-partie du groupe de classes de est de type , par suite contient trois extensions . On s'intéresse au problème de capitulation des 2-classes de dans , et à déterminer la structure de .

     

Absolutely continuous Jacobi operators

We show (among other results) that a symmetric Jacobi matrix whose diagonal is the zero sequence and whose super-diagonal 0$">satisfies , and has purely absolutely continuous spectrum when considered as a self-adjoint operator on .

     

The hit problem for the Dickson algebra

Let the mod 2 Steenrod algebra, , and the general linear group, , act on with in the usual manner. We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra is -decomposable in for arbitrary 2$">. This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in are the elements of Hopf invariant one and those of Kervaire invariant one.

     

A Pieri-type formula for the -theory of a flag manifold

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form or the form , and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the -Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.

     

Computing the multiplicative group of residue class rings

Let be a global field with maximal order and let be an ideal of . We present algorithms for the computation of the multiplicative group of the residue class ring and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group modulo , where denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

     

Solution of the quadratically hyponormal completion problem

For , let be a collection of () positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on to guarantee the existence of a quadratically hyponormal unilateral weighted shift with as the initial segment of weights. We prove that admits a quadratically hyponormal completion if and only if the self-adjoint matrix

is positive and invertible, where , , , , , and, for notational convenience, . As a particular case, this result shows that a collection of four positive numbers always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.

     

A note on triangular derivations of

For a field of characteristic zero, and for each integer , we construct a triangular derivation of whose ring of constants, though finitely generated over , cannot be generated by fewer than elements.

     

On the computation of all extensions of a -adic field of a given degree

Let be a -adic field. It is well-known that has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions of a given degree and discriminant.

     

A sharp inequality for the logarithmic coefficients of univalent functions

We prove the sharp inequality

for the logarithmic coefficients of a normalized univalent function in the unit disk.

     

Real -flats tangent to quadrics in

Let and denote the dimension and the degree of the Grassmannian , respectively. For each there are (a priori complex) -planes in tangent to general quadratic hypersurfaces in . We show that this class of enumerative problems is fully real, i.e., for there exists a configuration of real quadrics in (affine) real space so that all the mutually tangent -flats are real.

     

Generalized stationary random fields with linear regressions - an operator approach

Existence, -stationarity and linearity of conditional expectations of square integrable random sequences satisfying

for a real sequence is examined. The analysis is reliant upon the use of Laurent and Toeplitz operator techniques.

     

Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants

In the Laurent expansion

of the Riemann-Hurwitz zeta function, the coefficients are known as Stieltjes, or generalized Euler, constants. [When , (the Riemann zeta function), and .] We present a new approach to high-precision approximation of . Plots of our results reveal much structure in the growth of the generalized Euler constants. Our results when for , and when for (for such as 53/100, 1/2, etc.) suggest that published bounds on the growth of the Stieltjes constants can be much improved, and lead to several conjectures. Defining , we conjecture that is attained: for any given , for some (and similarly that, given and , is within of for infinitely many ). In addition we conjecture that satisfies for 1$">. We also conjecture that , a special case of a more general conjecture relating the values of and for . Finally, it is known that for . Using this to define for all real 0$">, we conjecture that for nonintegral , is precisely times the -th (Weyl) fractional derivative at of the entire function . We also conjecture that , now defined for all real arguments 0$">, is smooth. Our numerical method uses Newton-Cotes integration formulae for very high-degree interpolating polynomials; it differs in implementation from, but compares in error bounding to, Euler-Maclaurin summation based methods.

     

On the irreducibility of Hecke polynomials

Let be the characteristic polynomial of the th Hecke operator acting on the space of cusp forms of weight for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials . Using this criterion with some machine computation, we show that if there exists such that is irreducible and has the full symmetric group as Galois group, then the same is true of for each prime .

     

The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends

Given an integer , let be the space of complete embedded singly periodic minimal surfaces in , which in the quotient have genus zero and Scherk-type ends. Surfaces in can be proven to be proper, a condition under which the asymptotic geometry of the surfaces is well known. It is also known that consists of the -parameter family of singly periodic Scherk minimal surfaces. We prove that for each , there exists a natural one-to-one correspondence between and the space of convex unitary nonspecial polygons through the map which assigns to each the polygon whose edges are the flux vectors at the ends of (a special polygon is a parallelogram with two sides of length and two sides of length ). As consequence, reduces to the saddle towers constructed by Karcher.

     

On the polynomial representation for the number of partitions with fixed length

In this paper, it is shown that the number of partitions of a nonnegative integer with parts can be described by a set of polynomials of degree in , where denotes the least common multiple of the integers and denotes the quotient of when divided by . In addition, the sets of the polynomials are obtained and shown explicitly for and .

     

On the stability index of hypersurfaces with constant mean curvature in spheres

Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere . In this paper we prove that the weak index of any other compact constant mean curvature hypersurface in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to , with equality if and only if is a constant mean curvature Clifford torus with radius .

     

On the characterization of the kernel of the geodesic X-ray transform

Let be a compact manifold with boundary. We consider covariant symmetric tensor fields of order two that belong to a Sobolev space . We prove, under the assumption that the metric is simple, that solenoidal tensor fields that belong to the kernel of the geodesic X-ray transform are smooth up to the boundary. As a corollary we obtain that they form a finite-dimensional set in .

     

Obstructions to deformations of d.g. modules

Let be a field and . There exist a differential graded -module and various approximations to a differential on one of which gives a non-trivial deformation, another is obstructed, and another is unobstructed at order . The analogous problem in the category of -algebras in characteristic zero remains a long-standing open question.