Linear independence of time-frequency translates

The refinement equation plays a key role in wavelet theory and in subdivision schemes in approximation theory. Viewed as an expression of linear dependence among the time-scale translates of , it is natural to ask if there exist similar dependencies among the time-frequency translates of . In other words, what is the effect of replacing the group representation of induced by the affine group with the corresponding representation induced by the Heisenberg group? This paper proves that there are no nonzero solutions to lattice-type generalizations of the refinement equation to the Heisenberg group. Moreover, it is proved that for each arbitrary finite collection , the set of all functions such that is independent is an open, dense subset of . It is conjectured that this set is all of .

     

The statistics of continued fractions for polynomials over a finite field

Given a finite field of order and polynomials of degrees respectively, there is the continued fraction representation . Let denote the number of such pairs for which and for . We give both an exact recurrence relation, and an asymptotic analysis, for . The polynomial associated with the recurrence relation turns out to be of P-V type. We also study the distribution of . Averaged over all and as above, this presents no difficulties. The average value of is , and there is full information about the distribution. When is fixed and only is allowed to vary, we show that this is still the average. Moreover, few pairs give a value of that differs from this average by more than

     

Some remarks on the variation of curve length and surface area

Consider the curve , where is absolutely continuous on . Then has finite length, and if is the -neighborhood of in the uniform norm, we compare the length of the shortest path in with the length of . Our main result establishes necessary and sufficient conditions on such that the difference of these quantities is of order as . We also include a result for surfaces.

     

The second iterate of a map with dense orbit

Suppose that is a Hausdorff topological space having no isolated points and that is continuous. We show that if the orbit of a point under is dense in while the orbit of under is not, then the space decomposes into three sets relative to which the dynamics of are easy to describe. This decomposition has the following consequence: suppose that has dense orbit under and that the closure of the set of points of having odd period under has nonempty interior; then has dense orbit under .

     

Derivations with Engel conditions on multilinear polynomials

Let be a prime algebra over a commutative ring with unity and let be a multilinear polynomial over . Suppose that is a nonzero derivation on such that for all in some nonzero ideal of , with fixed. Then is central--valued on except when char and satisfies the standard identity in 4 variables.

     

A Wolff-Denjoy theorem for infinitely connected Riemann surfaces

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let be a non-parabolic Riemann surface with Martin boundary . Suppose each Martin function , , extends continuously to and vanishes there. We show that if is an endomorphism of and the iterates of converge to the point at infinity, then the iterates converge locally uniformly to a point in . As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on but none on itself.

     

Class numbers and Iwasawa invariants of quadratic fields

Let and be quadratic fields with 2 (mod 3) a positive integer. Let be the respective Iwasawa -invariants of the cyclotomic -extension of these fields. We show that if , then 3 does not divide the class number of and .

     

Congruence lattices of algebras--- the signed labelling

For an arbitrary algebra a new labelling, called the signed labelling, of the Hasse diagram of is described. Under the signed labelling, each edge of the Hasse diagram of receives a label from the set . The signed labelling depends completely on a subset of the unary polynomials of and its inspiration comes from semigroup theory. For finite algebras, the signed labelling complements the labelled congruence lattices of tame congruence theory (TCT). It provides a different kind of information about those algebras than the TCT labelling particularly with regard to congruence semimodularity. The main result of this paper shows that the congruence lattice of any algebra admits a natural join congruence, denoted , such that satisfies the semimodular law. In an application of that result, it is shown that for a regular semigroup , for which in , is actually a lattice congruence, coincides with , and satisfies the semimodular law.

     

-regular maps on smooth manifolds

A continuous map is said to be -regular if whenever are distinct points of , then are linearly independent over . For smooth manifolds we obtain new lower bounds on the minimum for which a -regular map can exist in terms of the dual Stiefel-Whitney classes of .

     

Homotopy periodicity and coherence

If is a periodic homotopy equivalence () or a homotopy idempotent (), the question arises whether this periodicity property can be achieved by a homotopy ``compatible with' . These coherence questions are answered.