On images of Borel measures under Borel mappings

Let and be metric spaces. We show that the tight images of a (fixed) tight Borel probability measure on , under all Borel mappings , form a closed set in the space of tight Borel probability measures on with the weak-topology. In contrast, the set of images of under all continuous mappings from to may not be closed. We also characterize completely the set of tight images of under Borel mappings. For example, if is non-atomic, then all tight Borel probability measures on can be obtained as images of , and as a matter of fact, one can always choose the corresponding Borel mapping to be of Baire class 2.

     

Character degree sets that do not bound the class of a -group

Suppose that we are given a set of powers of a prime and that . A technique is presented that enables the construction of a -group of specified nilpotence class such that its set of irreducible character degrees is exactly . If , then this can be done for and if , then the only requirement is .

     

Fourier asymptotics of Cantor type measures at infinity

Let be an integer and let . In this note we prove that for all ; if is odd and if is even This improves a classical result of Wiener and Wintner. We also give a necessary and sufficient condition for the product to approach zero at infinity.

     

Convexity numbers of closed sets in

For 2$"> let be the -ideal in generated by all sets which do not contain equidistant points in the usual metric on . For each 2$"> a set is constructed in so that the -ideal which is generated by the convex subsets of restricted to the convexity radical is isomorphic to . Thus is equal to the least number of convex subsets required to cover -- the convexity number of .

For every non-increasing function \aleph_0\}$"> we construct a model of set theory in which for each . When is strictly decreasing up to , uncountable cardinals are simultaneously realized as convexity numbers of closed subsets of . It is conjectured that , but never more than , different uncountable cardinals can occur simultaneously as convexity numbers of closed subsets of . This conjecture is true for and .     

-coincidences for maps of homotopy spheres into CW-complexes

Let be a finite group acting freely in a CW-complex which is a homotopy -dimensional sphere and let be a map of to a finite -dimensional CW-complex . We show that if , then has an -coincidence for some nontrivial subgroup of .

     

Anti-Wick quantization with symbols in spaces

We give a classification of pseudo-differential operators with anti-Wick symbols belonging to spaces: if the corresponding operator belongs to trace classes; if we get Hilbert-Schmidt operators; finally, if , the operator is compact. This classification cannot be improved, as shown by some examples.

     

Isomorphism of commutative group algebras of closed -local algebraically compact groups

Let be an abelian group and let be a field of 0$">. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the -isomorphism for some group yields provided is a closed -group or a -local algebraically compact group. In particular, this is the case when is closed -primary of arbitrary power, or is -local algebraically compact with cardinality at most and is in cardinality not exceeding . The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).

     

Anti-symplectic involutions with lagrangian fixed loci and their quotients

We study the lagrangian embedding as a fixed point set of anti-symplectic involution on a symplectic 4-manifold . Suppose the fixed loci of are the disjoint union of smooth Riemann surfaces ; then each component becomes a lagrangian submanifold. Furthermore, if one of the components is a Riemann surface of genus , then its quotient has vanishing Seiberg-Witten invariants. We will discuss some examples which allow an anti-symplectic involution with lagrangian fixed loci.

     

Helgason-Marchaud inversion formulas for Radon transforms

Let be either the hyperbolic space or the unit sphere , and let be the set of all -dimensional totally geodesic submanifolds of . For and , the totally geodesic Radon transform is studied. By averaging over all at a distance from , and applying Riemann-Liouville fractional differentiation in , S. Helgason has recovered . We show that in the hyperbolic case this method blows up if does not decrease sufficiently fast. The situation can be saved if one employs Marchaud's fractional derivatives instead of the Riemann-Liouville ones. New inversion formulas for , are obtained.

     

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 .

     

A Lindelöf space with no Lindelöf subspace of size

A consistent example of an uncountable Lindelöf (and hence normal) space with no Lindelöf subspace of size is constructed. It remains unsolved whether extra set-theoretic assumptions are necessary for the existence of such a space. However, our space has size and is a -space, i.e., 's are open, and for such spaces extra set-theoretic assumptions turn out to be necessary.

     

A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function at such a point where is smooth. We also consider the extension of the Zygmund classes and to the two-dimensional torus .

     

On syzygies of Segre embeddings

We study the syzygies of the ideals of the Segre embeddings. Let , ; we prove that the line bundle on the ( copies) satisfies Property of Green-Lazarsfeld if and only if . Besides we prove that if we have a projective variety not satisfying Property for some , then the product of it with any other projective variety does not satisfy Property . From this we also deduce other corollaries about syzygies of Segre embeddings.

     

Applications of a theorem of H. Cramér to the Selberg class

We prove two results on the nature of the Dirichlet coefficients of the -functions in the extended Selberg class . The first result asserts that if for some entire function of order 1 and finite type, then is constant. The second result states, roughly, that if are still the coefficients of some -function from , then with and . The proofs are based on an old result by Cramér and on the characterization of the functions of degree 1 of .

     

First stability eigenvalue characterization of Clifford hypersurfaces

The stability operator of a compact oriented minimal hypersurface is given by , where is the norm of the second fundamental form. Let be the first eigenvalue of and define . In 1968 Simons proved that for any non-equatorial minimal hypersurface . In this paper we will show that only for Clifford hypersurfaces. For minimal surfaces in , let denote the area of and let denote the genus of . We will prove that . Moreover, if is embedded, then we will prove that . If in addition to the embeddeness condition we have that , then we will prove that .

     

Local dual spaces of Banach spaces of vector-valued functions

We show that is a local dual of , and is a local dual of , where is a Banach space. A local dual space of a Banach space is a subspace of so that we have a local representation of in satisfying the properties of the representation of in provided by the principle of local reflexivity.

     

Consecutive numbers with the same Legendre symbol

Let be an odd prime, and be a complete set of residues . The goal of the paper is to determine all the values of such that or , where is the Legendre symbol.

     

On a problem of J. P. Williams

Let be the algebra of all bounded operators on a Hilbert space . Let be a continuous function on the closed disk and let

for some 0,$"> for all and all with . Then is differentiable on . The paper shows that the function may have a discontinuous derivative.

     

Locally finite dimensional shift-invariant spaces in

We prove that a locally finite dimensional shift-invariant linear space of distributions must be a linear subspace of some shift-invariant space generated by finitely many compactly supported distributions. If the locally finite dimensional shift-invariant space is a subspace of the Hölder continuous space or the fractional Sobolev space , then the superspace can be chosen to be or , respectively.