Boundary spectra of uniform Frostman Blaschke products

A closed set in the unit circle is the boundary spectrum of a uniform Frostman Blaschke product if and only if is nowhere dense in .

     

Isomorphism of Borel full groups

Suppose that and are Polish groups which act in a Borel fashion on Polish spaces and . Let and denote the corresponding orbit equivalence relations, and and the corresponding Borel full groups. Modulo the obvious counterexamples, we show that .

     

Sharp Marchaud and converse inequalities in Orlicz spaces

For spaces on , and , sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on , and ) for which is convex for some , , where is the Orlicz function. Sharp converse inequalities for such spaces are deduced.

     

Helly-type theorems for hollow axis-aligned boxes

A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

     

Algebraic reflexivity of linear transformations

Let be the set of all linear transformations from to , where and are vector spaces over a field . We show that every -dimensional subspace of is algebraically -reflexive, where denotes the largest integer not exceeding , provided is less than the cardinality of .

     

The Postnikov Tower and the Steenrod problem

The Steenrod problem asks: given a -module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a -module is -realizable if and only if it is -realizable for all -Sylow subgroups , for all primes .

     

Varieties with a reducible hyperplane section whose two components are hypersurfaces

We classify smooth complex projective varieties of dimension admitting a divisor of the form among their hyperplane sections, both and of codimension in their respective linear spans. In this setting, one of the following holds: 1) is either the Veronese surface in or its general projection to , 2) and is contained in a quadric cone of rank or , 3) and .

     

Minimal rank and reflexivity of operator spaces

Let be an -dimensional space of linear operators between the linear spaces and over an algebraically closed field . Improving results of Larson, Ding, and Li and Pan we show the following.

Theorem. Let be a basis of . Assume that every nonzero operator in has rank larger than . Then a linear operator belongs to if and only if for every , is a linear combination of .

     

Irreducible characters which are zero on only one conjugacy class

Suppose that is a finite solvable group which has an irreducible character which vanishes on exactly one conjugacy class. Then we show that has a homomorphic image which is a nontrivial -transitive permutation group. The latter groups have been classified by Huppert. We can also say more about the structure of depending on whether is primitive or not.

     

Dimension distortion of hyperbolically convex maps

In this note, we provide an answer to a question of D. Mejia and Chr. Pommerenke, by constructing a hyperbolically convex subdomain of the unit disc so that the conformal map from to maps a set of dimension 0 on to a set of dimension

     

Transversals for strongly almost disjoint families

For a family of sets , and a set , is said to be a transversal of if and for each . is said to be a Bernstein set for if for each . Erdos and Hajnal first studied when an almost disjoint family admits a set such as a transversal or Bernstein set. In this note we introduce the following notion: a family of sets is said to admit a -transversal if can be written as such that each admits a transversal. We study the question of when an almost disjoint family admits a -transversal and related questions.

     

Pointwise convergence of bounded cascade sequences

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function , a cascade sequence is constructed by the iteration where is defined by In this paper, under a condition that the sequence is bounded in , we prove that the following three statements are equivalent: (i) converges . (ii) For , there exist a positive constant and a constant such that (iii) For some converges in . An example is presented to illustrate our result.

     

Hypercyclicity in omega

A sequence of operators is said to be hypercyclic if there exists a vector , called hypercyclic for , such that is dense. A hypercyclic subspace for is a closed infinite-dimensional subspace of, except for zero, hypercyclic vectors. We prove that if is a sequence of operators on that has a hypercyclic subspace, then there exist (i) a sequence of one variable polynomials such that is hypercyclic for every fixed and (ii) an operator that maps nonzero vectors onto hypercyclic vectors for .

We complement earlier work of several authors.

     

Non-zero boundaries of Leibniz half-spaces

It is proved that for any , there exists a norm and two points , in such that the boundary of the Leibniz half-space has non-zero Lebesgue measure. When , it is known that the boundary must have zero Lebesgue measure.

     

Perturbing a product of stable flows

Suppose that and are axiom A flows with attractors and . Then the attractor for the product flow on the product manifold is no longer hyperbolic (although there is a hyperbolic action of ).

It is easy to see that the attractor cannot explode but we show here that it cannot implode: for any flow sufficiently close to any attractor whose basin is not too thin is -dense in .

     

Transference for amenable actions

Suppose acts amenably on a measure space with quasi-invariant -finite measure . Let be an isometric representation of on and a finite Radon measure on . We show that the operator has -operator norm not exceeding the -operator norm of the convolution operator defined by . We shall also prove an analogous result for the maximal function associated to a countable family of Radon measures .

     

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions of the Bunimovich stadium , satisfying . Write where is the central rectangle and denotes the ``wings,' i.e., the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in as . We obtain a lower bound on the mass of in , assuming that itself is -normalized; in other words, the norm of is controlled by times the norm in . Moreover, if is an quasimode, the same result holds, while for an quasimode we prove that the norm of is controlled by times the norm in . We also show that the norm of may be controlled by the integral of along , where is a smooth factor on vanishing at . These results complement recent work of Burq-Zworski which shows that the norm of is controlled by the norm in any pair of strips contained in , but adjacent to .

     

On the degree of Hilbert polynomials associated to the torsion functor

Let be a local, Noetherian ring and an ideal. A question of Kodiyalam asks whether for fixed , the polynomial giving the th Betti number of has degree equal to the analytic spread of minus one. Under mild conditions on , we show that the answer is positive in a number of cases, including when is divisible by or is an integrally closed -primary ideal.

     

A new estimate in optimal mass transport

Let be a bounded Lipschitz regular open subset of and let be two probablity measures on . It is well known that if is absolutely continuous, then there exists, for every , a unique transport map pushing forward on and which realizes the Monge-Kantorovich distance . In this paper, we establish an bound for the displacement map which depends only on , on the shape of and on the essential infimum of the density .

     

Steinhaus tiling problem and integral quadratic forms

A lattice in is said to be equivalent to an integral lattice if there exists a real number such that the dot product of any pair of vectors in is an integer. We show that if and is equivalent to an integral lattice, then there is no measurable Steinhaus set for , a set which no matter how translated and rotated contains exactly one vector in .