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.

     

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 .

     

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.

     

Polynomial continuity on

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. A Banach space has property (RP) if given two bounded sequences , we have that for every polynomial on whenever for every polynomial on ; i.e., the restriction of every polynomial on to each bounded set is uniformly sequentially continuous for the weak polynomial topology. We show that property (RP) does not imply that every scalar valued polynomial on must be polynomially continuous.

     

A note on

There is only one pair of non-real zeros of , and of , in the left half-plane. The Riemann Hypothesis implies that and have no zeros in the strip .

     

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.

     

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.

     

Kaehler structures on

Let be a compact connected semi-simple Lie group, let , and let be an Iwasawa decomposition. To a given -invariant Kaehler structure on , there corresponds a pre-quantum line bundle on . Following a suggestion of A.S. Schwarz, in a joint paper with V. Guillemin, we studied its holomorphic sections as a -representation space. We defined a -invariant -structure on , and let denote the space of square-integrable holomorphic sections. Then is a unitary -representation space, but not all unitary irreducible -representations occur as subrepresentations of . This paper serves as a continuation of that work, by generalizing the space considered. Let be a Borel subgroup containing , with commutator subgroup . Instead of working with , we consider , for all parabolic subgroups containing . We carry out a similar construction, and recover in the unitary irreducible -representations previously missing. As a result, we use these holomorphic sections to construct a model for : a unitary -representation in which every irreducible -representation occurs with multiplicity one.

     

Zero divisors and

Let be a discrete group, the group ring of over and the Lebesgue space of with respect to Haar measure. It is known that if is torsion free elementary amenable, and , then . We will give a sufficient condition for this to be true when , and in the case we will give sufficient conditions for this to be false when .

     

Point-spectrum-preserving elementary operators on

Let be an infinite dimensional separable Hilbert space over the complex field. Structure characterizations are given for some elementary operators on which preserve point spectrum.

     

Norm attaining functionals on

It is shown that for any infinite compact Hausdorff space , the Bishop-Phelps set in is of the first Baire category when has the supremum norm.

     

Exotic cohomology for

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

     

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.

     

Weak compactness in

We characterize weak compactness and weak conditional compactness of subsets of in terms of regular methods of summability. We also study when these results still hold using only convergence in the sense of Cesàro.

     

On polarized surfaces

Let be a smooth projective surface over and an ample Cartier divisor on . If the Kodaira dimension or , the author proved , where . If , then the author studied with . In this paper, we study the polarized surface with , , and .

     

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.

     

Some characterizations of

We show that a function on the unit disk extends continuously to , the maximal ideal space of iff it is uniformly continuous (in the hyperbolic metric) and close to constant on the complementary components of some Carleson contour.