A generalization of Carleman's uniqueness theorem and a discrete Phragmén-Lindelöf theorem

Let be a Borel measure on and be its moments. T. Carleman found sharp conditions on the magnitude of for to be uniquely determined by its moments. We show that the same conditions ensure a stronger property: if are the moments of another measure, with then the measure is supported on the interval This result generalizes both the Carleman theorem and a theorem of J. Mikusi\'{n}ski. We also present an application of this result by establishing a discrete version of a Phragmén-Lindelöf theorem.

     

Bounded sequence-to-function generalized Hausdorff transformations

Georgakis (1988) obtained the norm of the transformation

considered as an operator from the sequence space , with weights
to , with weight . As corollaries he obtained inequality statements for Borel and generalized Abel transformations. He also obtained the best constants possible for several weighted norm inequalities of Hardy and Littlewood. In this paper Georgakis' results are extended to the Endl generalized Hausdorff matrices.

     

Discriminants of convex curves are homeomorphic

For a given real generic curve let denote the ruled hypersurface in consisting of all osculating subspaces to of codimension 2. In this note we show that for any two convex real projective curves and the pairs and are homeomorphic.

     

On scrambled sets and a theorem of Kuratowski on independent sets

The measure of scrambled sets of interval self-maps was studied by many authors, including Smítal, Misiurewicz, Bruckner and Hu, and Xiong and Yang. In this note, first we introduce the notion of ``-chaos" which is related to chaos in the sense of Li-Yorke, and we prove a general theorem which is an improvement of a theorem of Kuratowski on independent sets. Second, we apply the result to scrambled sets of higher dimensional cases. In particular, we show that if a map of the unit -cube is -chaotic on , then for any there is a map such that and are topologically conjugate, and has a scrambled set which has Lebesgue measure 1, and hence if , then there is a homeomorphism with a scrambled set satisfying that is an -set in and has Lebesgue measure 1.

     

Complemented isometric copies of in dual Banach spaces

Let be a real or complex Banach space and . Then contains a -complemented, isometric copy of if and only if contains a -complemented, isometric copy of if and only if contains a subspace -asymptotic to .

     

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex in . It is proved that such an expansion is uniformly summable on the simplex for any continuous function if and only if . Moreover, it is shown that means define a positive linear polynomial identity, and the index is sharp in the sense that means are not positive for .

     

Centroaffine surfaces in -geodesics

For (positive) definite surfaces in there is a canonical choice of a centroaffine normal plane bundle, which induces a centroaffine invariant Ricci-symmetric connection . We classify all surfaces in with planar -geodesics. It turns out that the resulting class of surfaces is umbilical with projectively flat induced connection and flat normal plane bundle.

     

Bordism of two commuting involutions

In this paper we obtain conditions for a Whitney sum of three vector bundles over a closed manifold, , to be the fixed data of a -action; these conditions yield the fact that if is the fixed data of a -action, where is the trivial one dimensional bundle, then the same is true for . The results obtained, together with techniques previously developed, are used to obtain, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.

     

Metrizability of sequential topological groups with point-countable -networks

We prove that a Hausdorff sequential topological group with a point-countable -network is metrizable iff its sequential order is less than . In the non Hausdorff case metrizability may be replaced by -locally finite base.

     

On the cohomology of regular differential forms and dualizing sheaves

If is a local Dedekind scheme and is a projective Cohen-Macaulay variety of relative dimension , then is torsionfree if and only if is arithmetically Cohen-Macaulay for a suitable embedding in . If is regular then is torsionfree whenever the multiplicity of the special fibre is not a multiple of the characteristic of the residue class field.

     

Mean exit time from convex hypersurfaces

L. Karp and M. Pinsky proved that, for small radius , the mean exit time function of an extrinsic -ball in a hypersurface is bounded from below by the corresponding function defined on an extrinsic -ball in . A counterexample given by C. Mueller proves that this inequality doesn't holds in the large. In this paper we show that, if is convex, then the inequality holds for all radii. Moreover, we characterize the equality and show that analogous results are true in the sphere.

     

Extended Cesàro operators on mixed norm spaces

We define an extended Cesàro operator with holomorphic symbol in the unit ball of as

where is the radial derivative of . In this paper we characterize those for which is bounded (or compact) on the mixed norm space .

     

Critical points of real entire functions and a conjecture of Pólya

Let be a nonconstant real entire function of genus and assume that all the zeros of are distributed in some infinite strip , . It is shown that (1) if has nonreal zeros in the region , and has nonreal zeros in the same region, and if the points and are located outside the Jensen disks of , then has exactly critical zeros in the closed interval , (2) if is at most of order , , and minimal type, then for each positive constant there is a positive integer such that for all has only real zeros in the region , and (3) if is of order less than , then has just as many critical points as couples of nonreal zeros.

     

Derived lengths and character degrees

Let be a finite solvable group. Assume that the degree graph of has exactly two connected components that do not contain . Suppose that one of these connected components contains the subset , where and are coprime when . Then the derived length of is less than or equal to .

     

Some results on finite Drinfeld modules

Let be a global function field, a degree one prime divisor of and let be the Dedekind domain of functions in regular outside . Let be the Hilbert class field of , the integral closure of in . Let be a rank one normalized Drinfeld -module and let be a prime ideal in . We explicitly determine the finite -module structure of . In particular, if , is an odd prime number and is the Carlitz -module, then the finite -module is always cyclic.

     

Embeddings of -dimensional separable metric spaces into the product of Sierpinski curves

We give a short proof of the following fact: the set of embeddings of any -dimensional separable metric space into a certain -dimensional subset of the -product of Sierpinski curves is residual in .

     

Interpolation spaces between the Lipschitz class and the space of continuous functions

It is shown that the complex interpolation spaces and do not coincide with or and also that the couple is not a Calderón couple. Similar results are also obtained for the couples and when .

     

Chains of strongly non-reflexive dual groups of integer-valued continuous functions

Answering a question of Eklof-Mekler (Almost free modules, set-theoretic methods, North-Holland, Amsterdam, 1990), we prove: (1) If there exists a non-reflecting stationary set of consisting of ordinals of cofinality for each , then there exist abelian groups such that and for each . (2) There exist abelian groups such that for each and for each . The groups are the groups of -valued continuous functions on a topological space and their dual groups.

     

Dunford-Pettis composition operators on in several variables

A bounded composition operator on , where is the unit ball in , is Dunford-Pettis if and only if the radial limit function of takes values on the unit sphere only on a set of surface measure zero. A similar theorem holds on bounded strongly pseudoconvex domains with smooth boundary.

     

On the multiplicities of the zeros of Laguerre-Pólya functions

We show that all the zeros of the Fourier transforms of the functions , , are real and simple. Then, using this result, we show that there are infinitely many polynomials such that for each the translates of the function

generate . Finally, we discuss the problem of finding the minimum number of monomials , , which have the property that the translates of the functions , , generate , for a given .