On the pointwise maximum of convex functions

We study the conjugate of the maximum, , of and when and are proper convex lower semicontinuous functions on a Banach space . We show that on the bidual, , of provided that and satisfy the Attouch-Brézis constraint qualification, and we also derive formulae for and for the ``preconjugate' of .

     

Pyramidal vectors and smooth functions on Banach spaces

We prove that if , are Banach spaces such that has nontrivial cotype and has trivial cotype, then smooth functions from into have a kind of ``harmonic" behaviour. More precisely, we show that if is a bounded open subset of and is -smooth with uniformly continuous Fréchet derivative, then is dense in . We also give a short proof of a recent result of P. Hájek.

     

Strongly asymptotically stable Frobenius-Perron operators

Let be a -finite measure space and let be a Frobenius-Perron operator.

In 1997 Bartoszek and Brown proved that if overlaps supports and if there exists , 0$"> on , such that , then is (strongly) asymptotically stable.

In the note we prove that instead of assuming that 0$"> on , it is enough to assume that and . More precisely, we prove that is asymptotically stable if and only if overlaps supports and there exists , , , such that .

     

A counterexample for approximable functions

Let be the unit disk. We show that for some relatively closed set there is a function that can be uniformly approximated on by functions of , but such that cannot be written as , with and uniformly continuous on . This answers a question of Stray.

     

Fixed point and selection theorems in hyperconvex spaces

It is shown that a set-valued mapping of a hyperconvex metric space which takes values in the space of nonempty externally hyperconvex subsets of always has a lipschitzian single valued selection which satisfies for all . (Here denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded -lipschitzian self-mappings of is itself hyperconvex. Several related results are also obtained.

     

A co-Frobenius Hopf algebra with a separable Galois extension is finite

If is a co-Frobenius Hopf algebra over a field, having a Galois -object which is separable over , its ring of coinvariants, then is finite dimensional.

     

The Lindelöf property and fragmentability

Let be a compact Hausdorff space and the space of continuous real functions on . In this paper we prove that any -Lindelöf subset of which is compact for the topology of pointwise convergence on a dense subset is norm fragmented; i.e., each non-empty subset of it contains a non-empty -relatively open subset of small supremum norm diameter. Several applications are given.     

Composition operators on Dirichlet-type spaces

The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.

     

An -dimensional space that admits a Poincaré inequality but has no manifold points

For each integer we construct a compact, geodesic metric space which has topological dimension , is Ahlfors -regular, satisfies the Poincaré inequality, possesses as a unique tangent cone at almost every point, but has no manifold points.

     

Integral representation of linear functionals on spaces of unbounded functions

Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.

     

Statistical extensions of some classical Tauberian theorems

Hardy's well-known Tauberian theorem for Cesàro means says that if the sequence satisfies and , then . In this paper it is shown that the hypothesis can be replaced by the weaker assumption of the statistical limit: st-lim , i.e., for every , . Similarly, the ``one-sided' Tauberian theorem of Landau and Schmidt's Tauberian theorem for the Abel method are extended by replacing and with st-lim and st-lim , respectively. The Hardy-Littlewood Tauberian theorem for Borel summability is also extended by replacing , where is a continuous parameter, with , and further replacing it by -st-lim , where is the Borel matrix method.

     

Incompressible surfaces in handlebodies and closed 3-manifolds of Heegaard genus 2

In this paper, we shall prove that for any integer 0$">, 1) a handlebody of genus 2 contains a separating incompressible surface of genus , 2) there exists a closed 3-manifold of Heegaard genus which contains a separating incompressible surface of genus .

     

Semigroups and weights for group representations

Let be a finite group. Consider a pair of linear characters of subgroups of with and agreeing on . Naturally associated with is a finite monoid . Semigroup representation theory then yields a representation of . If is irreducible, we say that is a weight for . When the underlying field is the field of complex numbers, we obtain a formula for the character of in terms of and . We go on to construct weights for some familiar group representations.

     

Convergence of paths for pseudo-contractive mappings in Banach spaces

Let be a real Banach space, let be a closed convex subset of , and let , from into , be a pseudo-contractive mapping (i.e. for all and 1)$">. Suppose the space has a uniformly Gâteaux differentiable norm, such that every closed bounded convex subset of enjoys the Fixed Point Property for nonexpansive self-mappings. Then the path , , defined by the equation is continuous and strongly converges to a fixed point of as , provided that satisfies the weakly inward condition.

     

Betti numbers of modules of essentially monomial type

Let be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of -modules of essentially monomial type. These modules are defined with respect to various -regular sequences. For example, finite length modules of monomial type over regular local rings of dimension are modules of essentially monomial type with respect to -regular sequences of length . If a module is of essentially monomial type with respect to an -regular sequence of length , then the rank of its -th syzygy is at least and its -th Betti number is at least .

     

A note on Pontrjagin forms

Let be a principal bundle over a manifold of dimension , and let be its -dimensional Pontrjagin class. In this paper, we aim at answering the following question: Which representatives of the class can be realised as the Pontrjagin form of some connection on ?

     

The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

Irreducible restriction and zeros of characters

Let be a finite group, let be normal in and suppose that is an irreducible complex character of . Then is not irreducible if and only if vanishes on some coset of in .

     

Four-genus and four-dimensional clasp number of a knot

For a knot in the -sphere, by using the linking form on the first homology group of the double branched cover of the -sphere, we investigate some numerical invariants, -genus , nonorientable -genus and -dimensional clasp number , defined from the four-dimensional viewpoint. T. Shibuya gave an inequality , and asked whether the equality holds or not. From our result in this paper, we find that the equality does not hold in general.