The modular group algebra problem for metacyclic -groups

It is shown that the isomorphism type of a metacyclic -group is determined by its group algebra over the field of elements. This completes work of Baginski. It is also shown that, if a -group has a cyclic commutator subgroup , then the order of the largest cyclic subgroup containing is determined by .

     

Multiplicative -quotients

Let be the Dedekind -function. In this work we exhibit all modular forms of integral weight , for positive integers and and arbitrary integers , such that both and its image under the Fricke involution are eigenforms of all Hecke operators. We also relate most of these modular forms with the Conway group via a generalized McKay-Thompson series.

     

The

In this paper we analyze the localization of , the fiber of the double suspension map , with respect to . If four cells at the bottom of , the th extended power spectrum of the Moore spectrum, are collapsed to a point, then one obtains a spectrum . Let be the James-Hopf map followed by the collapse map. Then we show that the secondary suspension map has a lifting to the fiber of and this lifting is shown to be a -periodic equivalence, hence an -equivalence.

     

On -amicable pairs

Let denote Euler's totient function, i.e., the number of positive integers and prime to . We study pairs of positive integers with such that for some integer . We call these numbers -amicable pairs with multiplier , analogously to Carmichael's multiply amicable pairs for the -function (which sums all the divisors of ).

We have computed all the -amicable pairs with larger member and found pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other -amicable pairs can be associated. Among these pairs there are so-called primitive -amicable pairs. We present a table of the primitive -amicable pairs for which the larger member does not exceed . Next, -amicable pairs with a given prime structure are studied. It is proved that a relatively prime -amicable pair has at least twelve distinct prime factors and that, with the exception of the pair , if one member of a -amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Finally, analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive -amicable pairs with larger member , the largest pair consisting of two 46-digit numbers.

     

Calibrated thin

Let be a compact metric space, and let be a calibrated thin -ideal. Then is . This solves an open problem, which was posed by Kechris, Louveau and Woodin. Using our result we obtain a new proof of Kaufman's theorem concerning -sets and -sets.

     

Gaps in

For a partial order , let denote the statement that for every -increasing -sequence there is a -decreasing -sequence on top of such that is an -gap in . The main result of this paper is that . It is also shown, as a corollary, that but .

     

On pasting -weights

Theorem 3 gives a condition when two -weights can be ``pasted' together to yield another -weight. It is subsequently used in Example 6 to give an example that shows that a necessary condition by Gohberg, Krupnik, and Spitkovsky is not sufficient.

     

without sharps

We show that the supremum of the lengths of prewellorderings of the reals can be , with inaccessible to reals, assuming only the consistency of an inaccessible.

     

-norm uniform distribution

In this paper, the -norm uniform distribution, which is a generalization of the uniform distribution studied by Cambanis, Huang, and Simon (1981), is defined for any . Then its marginal distributions and order statistics are studied.

     

Steiner systems

It is proved that there are precisely 4204 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

It is further proved that there are precisely 38717 pairwise non-isomorphic Steiner systems invariant under the group and which can be constructed using only short orbits.

     

Irreducible representations of the Cuntz algebra

In this paper, we establish formulas for the configuration of a special class of irreducible representations of the Cuntz algebra , . These irreducible representations arise as subrepresentations of naturally occurring representations of acting in and arise from consideration of multiresolution wavelet filters.

     

The Diophantine equation

An effective method is derived for solving the equation of the title in positive integers and for given completely, and is carried out for all . If is of the form , then there is the solution , ; in the above range, except for with solution , , there are no other solutions.

     

Exotic cohomology for

We show that for the mod group cohomology of is not detected on diagonal matrices.

     

Finding finite -sequences faster

A -sequence is a sequence of positive integers such that the sums , , are different. When is a power of a prime and is a primitive element in then there are -sequences of size with , which were discovered by R. C. Bose and S. Chowla.

In Theorem 2.1 I will give a faster alternative to the definition. In Theorem 2.2 I will prove that multiplying a sequence by integers relatively prime to the modulus is equivalent to varying . Theorem 3.1 is my main result. It contains a fast method to find primitive quadratic polynomials over when is an odd prime. For fields of characteristic 2 there is a similar, but different, criterion, which I will consider in ``Primitive quadratics reflected in -sequences', to appear in Portugaliae Mathematica (1999).

     

The irrationality of

We shall show that the numbers and
are linearly independent over for any natural number . The key is to construct explicit Padé-type approximations using Legendre-type polynomials.