A counterexample machine

We present a general method for constructing families of measure preserving transformations which are and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two transformations which are weakly isomorphic but not isomorphic, and a transformation with no roots. Ornstein's isomorphism theorem says families of Bernoulli shifts cannot have these properties. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations.

     

Checking the odd Goldbach conjecture up to

Vinogradov's theorem states that any sufficiently large odd integer is the sum of three prime numbers. This theorem allows us to suppose the conjecture that this is true for all odd integers. In this paper, we describe the implementation of an algorithm which allowed us to check this conjecture up to .

     

-analyticity

Let be a finite-dimensional commutative algebra over and let , and be the ring of -differentiable functions of class , the ring of real analytic mappings with values in and the ring of -analytic functions, respectively, defined on an open subset of . We prove two basic results concerning -differentiability and -analyticity: ) , ) if and only if is defined over .

     

A uniform

A uniform estimate of Bessel functions is obtained, which is used to get a characterization of the measures on the unit sphere of in terms of the mixed norm of the Fourier transform of the measures.

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

Enriched -Partitions

An (ordinary) -partition is an order-preserving map from a partially ordered set to a chain, with special rules specifying where equal values may occur. Examples include number-theoretic partitions (ordered and unordered, strict or unrestricted), plane partitions, and the semistandard
tableaux associated with Schur's -functions. In this paper, we introduce and develop a theory of enriched -partitions; like ordinary -partitions, these are order-preserving maps from posets to chains, but with different rules governing the occurrence of equal values. The principal examples of enriched -partitions given here are the tableaux associated with Schur's -functions. In a sequel to this paper, further applications related to commutation monoids and reduced words in Coxeter groups will be presented.

     

A counterexample concerning the relation between decoupling constants and -constants

For Banach spaces and and a bounded linear operator
we let such that

for all finitely supported and all , where is the sequence of Haar functions. We construct an operator , where is superreflexive and of type 2, with such that there is no constant with

In particular it turns out that the decoupling constants , where is the identity of a Banach space , fail to be equivalent up to absolute multiplicative constants to the usual -constants. As a by-product we extend the characterization of the non-superreflexive Banach spaces by the finite tree property using lower 2-estimates of sums of martingale differences.

     

-analogue triangular numbers and distance geometry

The so-called ``-identities' play a major role in classical combinatorics. Most of them can be viewed as arising somehow in the context of hypergeometric series. Here we present a ``sum of squares' identity involving -analogues of the triangular numbers that, by contrast, arises in the context of distance geometry.

     

-regular Witt rings

We improve Kula's bounds on the size of possible -regular Witt rings.

     

Criteria for -continuity

Bernoullicity is the strongest mixing property that a measure-theoretic dynamical system can have. This is known to be intimately connected to the so-called metric on processes, introduced by Ornstein. In this paper, we consider families of measures arising in a number of contexts and give conditions under which the measures depend -continuously on the parameters. At points where there is -continuity, it is often straightforward to establish that the measures have the Bernoulli property.

     

Noncommutative spaces

Let be a von Neumann algebra with a faithful, finite, normal tracial state , and let be a finite, maximal subdiagonal algebra of . Let be the closure of in the noncommutative Lebesgue space . Then possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an factorization theorem, Nehari's Theorem, and harmonic conjugates which are bounded.

     

A critical metric for the

The -norm of the curvature tensor

defines a Riemannian functional on the space of metrics. This work exhibits a metric on which is of Berger type but not of constant ricci curvature, and yet is critical for .

     

On nonlinear -widths

For characterization of best nonlinear approximation, DeVore,
Howard, and Micchelli have recently suggested the nonlinear -width of a subset in a normed linear space . We proved by a topological method that for and the well-known Aleksandrov -width in a Banach space the following inequalities hold: . Let be the unit ball of Besov space , of multivariate periodic functions. Then for approximation in , with some restriction on and , we established the asymptotic degree of these -widths: .

     

Finite and -resolvability

A topological space is -resolvable if has disjoint dense subsets. In this paper, we prove that if is -resolvable for each positive integer , then is -resolvable.

     

A criterion for splitting -algebras in tensor products

The goal of the paper is to prove the following theorem: if , are unital -algebras, simple and nuclear, then any -subalgebra of the -tensor product of and , which contains the tensor product of with the scalar multiples of the unit of , splits in the -tensor product of with some -subalgebra of .

     

A group of paths in

We define a group structure on the set of compact ``minimal' paths in . We classify all finitely generated subgroups of this group : they are free products of free abelian groups and surface groups. Moreover, each such group occurs in . The subgroups of isomorphic to surface groups arise from certain topological -forms on the corresponding surfaces. We construct examples of such -forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using we construct a non-polygonal tiling problem in , that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group has applications to combinatorial tiling problems of the type: given a set of tiles and a region , can be tiled by translated copies of tiles in ?

     

A non-homogeneous zero-dimensional is a group

We provide an example of a zero-dimensional (separable metric) absolute Borel set which is not homogeneous, but whose square admits the structure of a topological group. We also construct a zero-dimensional absolute Borel set such that is a homogeneous non-group but is a group. This answers questions of Arhangel'skii and Zhou.

     

-complete sequences of integers

An infinite sequence is -complete if every sufficiently large integer is the sum of such that no one divides the other. We investigate -completeness of sets of the form and with nonnegative.