On a conjecture about MRA Riesz wavelet bases

Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .

     

Entire pluricomplex Green functions and Lelong numbers of projective currents

Let be a positive closed current of bidimension (1,1) and unit mass on the complex projective space . We prove that the set of points where has Lelong number larger than is contained in a complex line if , and for some complex line if . We also prove that in dimension 2 and if , then for some conic .

     

The weak Dirichlet problem for Baire functions

Let be a simplex and a compact subset of the set of all extreme points of . We show that any bounded function of Baire class on can be extended to a function of affine class on . Moreover, can be chosen in such a way that .

     

Hyperelliptic curves over of every -rank without extra automorphisms

We prove that for any pair of integers such that or 0$">, there exists a (hyper)elliptic curve over of genus and -rank whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties over of dimension and -rank such that .

     

Self-commutator approximants

This paper deals with minimizing , where is fixed, self-adjoint and , and where varies such that and , . (Here, , , denotes the von Neumann-Schatten class and its norm.) The upshot of this paper is that , , is minimized if, and for only if, , and that the map , , has a critical point at if and only if (with related results for normal if or ).

     

Unimodular functions and interpolating Blaschke products

The result by Bourgain that every unimodular function on the unit circle can be factored as with and Blaschke products can be improved. We show that the same result holds with and interpolating Blaschke products. This will at the same time be a refinement of Jones's result that every unimodular function can be approximated in the -norm by a ratio of interpolating Blaschke products.

     

A simple proof for folds on both sides in complexes of graph homomorphisms

In this paper we study implications of folds in both parameters of Lovász' Hom complexes. There is an important connection between the topological properties of these complexes and lower bounds for chromatic numbers. We give a very short and conceptual proof of the fact that if is a fold of , then Hom collapses onto Hom, whereas Hom collapses onto Hom.

We also give an easy inductive proof of the only nonelementary fact which we use for our arguments: if is a closure operator on , then collapses onto .

     

The Berry-Esseen bound for character ratios

Let be a partition of chosen from the Plancherel measure of the symmetric group , let be the irreducible character of the symmetric group parameterized by evaluated on the transposition , and let be the dimension of the irreducible representation parameterized by . Fulman recently obtained the convergence rate of for any in the central limit theorem for character ratios by developing a connection between martingale and character ratios, and he conjectures that the correct speed is . In this paper we confirm the conjecture via a refinement of Stein's method for exchangeable pairs.

     

On differentiability properties of typical continuous functions and Haar null sets

Let () be the set of all continuous functions on which have a derivative ( , respectively) at least at one point . B. R. Hunt (1994) proved that is Haar null (in Christensen's sense) in .

In the present article it is proved that neither nor its complement is Haar null in . Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.

     

Fixed points of univalent functions II

For a closed nowhere dense subset of a bounded univalent holomorphic function on is found such that equals the cluster set of its fixed points.

     

A minimal pair of -degrees

We construct a minimal pair of -degrees. We do this by showing the existence of an unbounded nondecreasing function which forces -triviality in the sense that is -trivial if and only if for all , .

     

Eventual arm and leg widths in cocharacters of P. I. Algebras

Given a p.i. algebra , we study which partitions correspond to characters with non-zero multiplicities in the cocharacter sequence of . We define the , the eventual arm width to be the maximal so that such can have parts arbitrarily large, and to be the maximum so that the conjugate could have arbitrarily large parts. Our main result is that for any , .

     

A dual graph construction for higher-rank graphs, and -theory for finite 2-graphs

Given a -graph and an element of , we define the dual -graph, . We show that when is row-finite and has no sources, the -algebras and coincide. We use this isomorphism to apply Robertson and Steger's results to calculate the -theory of when is finite and strongly connected and satisfies the aperiodicity condition.

     

A Beurling-Carleson set which is a uniqueness set for a given weighted space of analytic functions

Let be a sequence of positive real numbers. We define as the space of functions which are analytic in the unit disc , continuous on and such that

where is the Fourier coefficient of the restriction of to the unit circle . Let be a closed subset of . We say that is a Beurling-Carleson set if

where denotes the distance between and . In 1980, A. Atzmon asked whether there exists a sequence of positive real numbers such that for all and that has the following property: for every Beurling-Carleson set , there exists a non-zero function in that vanishes on . In this note, we give a negative answer to this question.

     

Lifts of - and -morphisms to -morphisms

Let be the Hochschild complex of cochains on and let be the space of multivector fields on . In this paper we prove that given any -structure (i.e. Gerstenhaber algebra up to homotopy structure) on , and any -morphism (i.e. morphism of a commutative, associative algebra up to homotopy) between and , there exists a -morphism between and that restricts to . We also show that any -morphism (i.e. morphism of a Lie algebra up to homotopy), in particular the one constructed by Kontsevich, can be deformed into a -morphism, using Tamarkin's method for any -structure on . We also show that any two of such -morphisms are homotopic.

     

Linear series over real and -adic fields

We note that the degeneration arguments given by the author in 2003 to derive a formula for the number of maps from a general curve of genus to with prescribed ramification also yields weaker results when working over the real numbers or -adic fields. Specifically, let be such a field: we see that given , , , and satisfying , there exists smooth curves of genus together with points such that all maps from to can, up to automorphism of the image, be defined over . We also note that the analagous result will follow from maps to higher-dimensional projective spaces if it is proven in the case , , and that thanks to work of Sottile, unconditional results may be obtained for special ramification conditions.

     

Degree of a holomorphic map between unit balls from to

Let be a rational proper holomorphic map between the unit ball in and the unit ball in Write

where and are holomorphic polynomials, with Recall that the degree of is defined by

   deg

In this paper, we give a bound estimate for the degree of improving the bound given by Forstneric (1989).

     

Smooth actions of

Given a smooth action of on a -dimensional differentiable manifold , for each we associate with ``almost all" oriented orbits of dimension an element of .

     

On the number of zeros of certain rational harmonic functions

Extending a result of Khavinson and Swiatek (2003) we show that the rational harmonic function , where is a rational function of degree 1$">, has no more than complex zeros. Applications to gravitational lensing are discussed. In particular, this result settles a conjecture by Rhie concerning the maximum number of lensed images due to an -point gravitational lens.