The algebra

We study a class of endomomorphism algebras of certain -permutation modules over the Hecke algebra of type , whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed.

     

The Diophantine equation

Cao has recently proved that, subject to a certain condition on the odd prime , the equation has no solutions in positive integers and , provided also that . It is the object of this note to remove this restriction, and to provide a simple self-contained proof.

     

The action of

We prove that the orbit equivalence relation of the canonical action of on the subsets of is a universal countable Borel equivalence relation.     

Coloring

If and , then define the graph to be the graph whose vertex set is with two vertices being adjacent iff there are distinct such that . For various and and various , typically or , the graph can be properly colored with colors. It is shown that in some cases such a coloring can also have the additional property that if is an isometric embedding, then the restriction of to is a bijection onto .

     

From

Let be a space of finite type. Set as usual, and define the mod support of by for 0.$"> Call sparse if there is no with

Then we show the relation for any finite type space with being sparse.

As a special case, we have and the main theorem of Ravenel, Wilson and Yagita is also generalized in terms of the mod support.

     

again

It is shown that if, for an entire function,

where , then

In the proof, the zeros of the function are redistributed to minimize the large values of .

     

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.

     

Murley Groups and the Torsion-Freeness of Ext

This article has been retracted.     

The diffeomorphism type of certain

In this note we show that the unit tangent bundle of is diffeomorphic to the total space of a certain principal -bundle over , solving a problem of James and Whitehead.

     

Estimates of

We extend a result of Ramaré and Rumely, 1996, about the Chebyshev function in arithmetic progressions. We find a map such that and 0)}$">, whereas is a constant. Now we are able to show that, for ,

and, for ,

\frac{x}{2\ln x}.\end{displaymath}">

     

On contractible

We present a very short proof of a well-known result, that for each there exists a contractible -dimensional compactum, non-embeddable into .

     

Embeddings of

We show that there is only one embedding of in at the prime , up to self-maps of . We also describe the effect of the group of self-equivalences of at the prime on this embedding and then show that the Friedlander exceptional isogeny composed with a suitable Adams map is an involution of whose homotopy fixed point set coincide with

     

The finite vertex-primitive and vertex-biprimitive

A complete classification is given for finite vertex-primitive and vertex-biprimitive -transitive graphs for . The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group , and an infinite family of graphs of valency 5 admitting projective symplectic groups with prime and (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.

     

and applications

We study an analogue of the classical theory of weights in without assuming that the underlying measure is doubling. Then, we obtain weighted norm inequalities for the (centered) Hardy-Littlewood maximal function and corresponding weighted estimates for nonclassical Calderón-Zygmund operators. We also consider commutators of those Calderón- Zygmund operators with bounded mean oscillation functions (), extending the main result from R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. Finally, we study self-improving properties of Poincaré-B.M.O. type inequalities within this context; more precisely, we show that if is a locally integrable function satisfying for all cubes , then it is possible to deduce a higher integrability result for , assuming a certain simple geometric condition on the functional .     

Connecting Lemmas

Like the closing lemma, the connecting lemma is of fundamental importance in dynamical systems. Hayashi recently proved the connecting lemma for stable and unstable manifolds of a hyperbolic invariant set. In this paper, we prove several very general connecting lemmas. We simplify Hayashi's proof and extend the results to more general cases.

     

Lipscomb's space

It is known that Lipscomb's space can be imbedded in Hilbert's space . Let be the imbedded version of endowed with the -induced topology. We show how to construct as the attractor of an iterated function system containing an infinite number of affine transformations of . In this way we answer an open question of J.C. Perry.