On the classes of spaces

The two better-known ways of understanding the notion of local unconditional structure allow us to define successive extensions of the well-known class of the spaces of Lindenstrauss and Pelczynski. This paper also studies stability properties of these classes under ultrapowers, biduals and complemented subspaces.

     

A splitting theorem for degrees

We prove that, for any , and with _{T}A\oplus U$"> and r.e., in , there are pairs and such that ; ; and, for any and from and any set , if and , then . We then deduce that for any degrees , , and such that and are recursive in , , and is into , can be split over avoiding . This shows that the Main Theorem of Cooper (Bull. Amer. Math. Soc. 23 (1990), 151-158) is false.

     

Character degrees and nilpotence class of finite groups

Let be a finite set of powers of containing 1. It is known that for some choices of , if is a finite -group whose set of character degrees is , then the nilpotence class of is bounded by some integer that depends on , while for some other choices of such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set is class bounding if and only if . In this article we provide a new approach to this problem. Our main result shows the relevance of certain -adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets such that .

     

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.

     

The -automorphism method and noninvariant classes of degrees

A set of nonnegative integers is computably enumerable (c.e.), also called recursively enumerable (r.e.), if there is a computable method to list its elements. Let denote the structure of the computably enumerable sets under inclusion, . Most previously known automorphisms of the structure of sets were effective (computable) in the sense that has an effective presentation. We introduce here a new method for generating noneffective automorphisms whose presentation is , and we apply the method to answer a number of long open questions about the orbits of c.e. sets under automorphisms of . For example, we show that the orbit of every noncomputable ( i.e., nonrecursive) c.e. set contains a set of high degree, and hence that for all the well-known degree classes (the low c.e. degrees) and (the complement of the high c.e. degrees) are noninvariant classes.

     

Odd perfect numbers have a prime factor exceeding

It is proved that every odd perfect number is divisible by a prime greater than .

     

Characteristic numbers and cobordism classes of fiberings with fiber

Some groups of -dimensional unoriented cobordism classes represented by the total space of a fibering with the real projective space as fiber are determined.

     

The root lattice and Ramanujan's circular summation of theta functions

We relate a formula of Ramanujan on the circular summation of the th power of theta functions, , to the theta series of the root lattice . We then use properties of the lattice to show that includes an modular form when is an odd perfect square as well as to derive a very simple expression for .

     

Automorphisms and cohomology of switching classes

A cohomological sufficient condition is found for a switching class of graphs to have a member admitting the full group of automorphisms of the corresponding two-graph.     

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 .     

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}">

     

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

     

Every odd perfect number has a prime factor which exceeds

It is proved here that every odd perfect number is divisible by a prime greater than .

     

Powers of

We prove that the Cech-Stone remainder of the integers, , maps onto its square if and only if there is a nontrivial map between two of its different powers, finite or infinite. We also prove that every compact space that maps onto its own square maps onto its own countable infinite product.

     

On and discriminants

We modify the -conjecture for number fields in order to make the support (like the height) well-behaved under field extensions. We show further that the exponent 1$"> of the absolute value of the discriminant cannot be replaced by , and even that an arbitrarily large power of must be present.

     

Mirror symmetry and

We show that counting functions of covers of are equal to sums of integrals associated to certain `Feynman' graphs. This is an analogue of the mirror symmetry for elliptic curves.

     

On the primality of

For each prime , let be the product of the primes less than or equal to . We have greatly extended the range for which the primality of and are known and have found two new primes of the first form ( ) and one of the second (). We supply heuristic estimates on the expected number of such primes and compare these estimates to the number actually found.

     

On the structure of

We show that for with its geometrically induced structure maps is not an Hopf algebroid because neither the augmentation nor the coproduct are multiplicative. As a consequence the algebra structure of is slightly different from what was supposed to be the case. We give formulas for and and show that the inversion of the formal group of is induced by an antimultiplicative involution . Some consequences for multiplicative and antimultiplicative automorphisms of for are also discussed.

     

On the stability of the

We prove the stability in of the projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the projection in holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.

     

Cotlar-Stein lemma and the theorem

In this note we give a generalization of the Cotlar-Stein lemma and using this lemma we give a new proof of a special case of the theorem which, in general, was proved by David, Journé and Semmes.