On the number of generators of the torsion module of differentials

In this paper we study the (minimum) global number of generators of the torsion module of differentials of affine hypersurfaces with only isolated singularities. We show that for reduced plane curves the torsion module of differentials can be generated by at most two elements, whereas for higher codimensions there is no universal upper bound. We then proceed to give explicit examples. In particular (when ) , we give examples of a reduced hypersurface with a single isolated singularity at the origin in that require

generators for the torsion module, Torsion .

     

Discrete logarithms in free groups

For the free group on generators we prove that the discrete logarithm is distributed according to the standard Gaussian when the logarithm is renormalized appropriately.

     

A simple proof of some generalized principal ideal theorems

Using symmetric algebras we simplify and slightly strengthen the Bruns-Eisenbud-Evans ``generalized principal ideal theorem' on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwiecinski, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

     

On the connectivity of the Julia set of a finitely generated rational semigroup

In this paper we show that the Julia set of a finitely generated rational semigroup is connected if the union of the Julia sets of generators is contained in a subcontinuum of . Under a nonseparating condition, we prove that the Julia set of a finitely generated polynomial semigroup is connected if its postcritical set is bounded.

     

Uniformly perfect sets, rational semigroups, Kleinian groups and IFS's

We show that the Julia set of a non-elementary rational semigroup is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of . This also proves that the limit set of a non-elementary Möbius group is uniformly perfect when there is a uniform bound on the Lipschitz constants of the generators of the group and this implies that the limit set of a finitely generated non-elementary Kleinian group is uniformly perfect.

     

Birational automorphisms of quartic Hessian surfaces

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice of rank 26 and signature . The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondo. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.

     

Isomorphism problems and groups of automorphisms for generalized Weyl algebras

We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of by finding sets of generators for the group of automorphisms.

     

Simple complete Boolean algebras

For every regular cardinal there exists a simple complete Boolean algebra with generators.

     

Frames, modular functions for shift-invariant subspaces and FMRA wavelet frames

We introduce the concept of the modular function for a shift-invariant subspace that can be represented by normalized tight frame generators for the shift-invariant subspace and prove that it is independent of the selections of the frame generators for the subspace. We shall apply it to study the connections between the dimension functions of wavelet frames for any expansive integer matrix and the multiplicity functions for general multiresolution analysis (GMRA). Given a frame mutiresolution analysis (FMRA), we show that the standard construction formula for orthonormal multiresolution analysis wavelets does not yield wavelet frames unless the underlying FMRA is an MRA. A modified explicit construction formula for FMRA wavelet frames is given in terms of the frame scaling functions and the low-pass filters.

     

Some Ideals with Large Projective Dimension

For an ideal in a polynomial ring over a field, a monomial support of is the set of monomials that appear as terms in a set of minimal generators of . Craig Huneke asked whether the size of a monomial support is a bound for the projective dimension of the ideal. We construct an example to show that, if the number of variables and the degrees of the generators are unspecified, the projective dimension of grows at least exponentially with the size of a monomial support. The ideal we construct is generated by monomials and binomials.

     

Slopes of vector bundles on projective curves and applications to tight closure problems

We study different notions of slope of a vector bundle over a smooth projective curve with respect to ampleness and affineness in order to apply this to tight closure problems. This method gives new degree estimates from above and from below for the tight closure of a homogeneous -primary ideal in a two-dimensional normal standard-graded algebra in terms of the minimal and the maximal slope of the sheaf of relations for some ideal generators. If moreover this sheaf of relations is semistable, then both degree estimates coincide and we get a vanishing type theorem.

     

Determining the -Sylow subgroup of an elliptic curve over a finite field

In this paper we describe an algorithm that outputs the order and the structure, including generators, of the -Sylow subgroup of an elliptic curve over a finite field. To do this, we do not assume any knowledge of the group order. The results that lead to the design of this algorithm are of inductive type. Then a right choice of points allows us to reach the end within a linear number of successive halvings. The algorithm works with abscissas, so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field. Efficient methods for this computation determine the efficiency of our algorithm.

     

Characteristic classes and transfer relations in cobordism

Decompositions of products of the Ray elements by free generators of small dimensions in the symplectic cobordism ring are obtained. In particular it is stated that most of the -dimensional generators, for small, after multiplication by the Ray elements , , land in the ideal generated by Ray elements of low dimension.

     

Growth of fundamental groups and isoembolic volume and diameter

Some properties of fundamental groups of Riemannian manifolds will be studied without a lower bound assumption on Ricci curvature. The main method is to relate the local packing to global packing instead of using the Bishop-Gromov relative volume comparison. This method allows us to control the volume growth of the universal cover and yields bounds on the number of generators of in terms of some isoembolic geometric invariants of .

     

The universal central extension of the three-point loop algebra

We consider the three-point loop algebra,

where denotes a field of characteristic 0 and is an indeterminate. The universal central extension of was determined by Bremner. In this note, we give a presentation for via generators and relations, which highlights a certain symmetry over the alternating group . To obtain our presentation of , we use the realization of as the tetrahedron Lie algebra.

     

Symmetric numerical semigroups with arbitrary multiplicity and embedding dimension

We construct symmetric numerical semigroups for every minimal number of generators and multiplicity , . Furthermore we show that the set of their defining congruence is minimally generated by elements.

     

A group theoretic criterion for property FA

We give group-theoretic conditions on a set of generators of a group which imply that admits no non-trivial action on a tree. The criterion applies to several interesting classes of groups, including automorphism groups of most free groups and mapping class groups of most surfaces.

     

Poly-log diameter bounds for some families of finite groups

Fix a prime and an integer with . Define the family of finite groups

for . We will prove that there exist two positive constants and such that for any and any generating set ,

when is the diameter of the finite group with respect to the set of generators . It is defined as the maximum over of the length of the shortest word in representing .

This result shows that these families of finite groups have a poly-logarithmic bound on the diameter with respect to any set of generators. The proof of this result also provides an efficient algorithm for finding such a poly-logarithmic representation of any element.

     

Weakly abelian lattice-ordered groups

Every nilpotent lattice-ordered group is weakly Abelian; i.e., satisfies the identity . In 1984, V. M. Kopytov asked if every weakly Abelian lattice-ordered group belongs to the variety generated by all nilpotent lattice-ordered groups [The Black Swamp Problem Book, Question 40]. In the past 15 years, all attempts have centred on finding counterexamples. We show that two constructions of weakly Abelian lattice-ordered groups fail to be counterexamples. They include all preiously considered potential counterexamples and also many weakly Abelian ordered free groups on finitely many generators. If every weakly Abelian ordered free group on finitely many generators belongs to the variety generated by all nilpotent lattice-ordered groups, then every weakly Abelian lattice-ordered group belongs to this variety. This paper therefore redresses the balance and suggests that Kopytov's problem is even more intriguing.

     

Mordell's exponential sum estimate revisited

The aim of this paper is to extend recent work of S. Konyagin and the author on Gauss sum estimates for large degree to the case of `sparse' polynomials. In this context we do obtain a nearly optimal result, improving on the works of Mordell and of Cochrane and Pinner. The result is optimal in terms of providing some power gain under conditions on the exponents in the polynomial that are best possible if we allow arbitrary coefficients. As in earlier work referred to above, our main combinatorial tool is a sum-product theorem. Here we need a version for product spaces for which the formulation is obviously not as simple as in the -case.

Again, the method applies more generally to provide nontrivial bounds on (possibly incomplete) exponential sums involving exponential functions. At the end of the paper, some applications of these are given to issues of uniform distribution for power generators in cryptography.