Holomorphic sections of pre-quantum line bundles on

Let be the Iwasawa decomposition of a complex connected semi-simple Lie group . Let be a parabolic subgroup containing , and let be its commutator subgroup. In this paper, we characterize the -invariant Kähler structures on , and study the holomorphic sections of their corresponding pre-quantum line bundles.

     

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.

     

Existence and uniqueness theorems for singular anisotropic quasilinear elliptic boundary value problems

On bounded domains we consider the anisotropic problems in with 1$"> and on and in with and on . Moreover, we generalize these boundary value problems to space-dimensions 2$">. Under geometric conditions on and monotonicity assumption on we prove existence and uniqueness of positive solutions.     

On Burgess's theorem and related problems

Let be a graph. We determine all graphs which are -like. We also prove that if are graphs, then in order that each -like continuum be -indecomposable for some it is necessary and sufficient that if is a graph, then is not -like for some integer with . This generalizes a well known theorem of Burgess.

     

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.

     

Connected sums of manifolds which induce approximate fibrations

Codimension-2 fibrators are -manifolds which automatically induce approximate fibration, in the following sense: given any proper mapping from an -manifold onto a -manifold such that each point-preimage is a copy of the codimension-2 fibrator, is necessarily an approximate fibration. In this paper, we give some answers to the following question: given an -manifold which is a nontrivial connected sum, when is a codimension-2 fibrator?

     

Dendrites and light open mappings

It is shown that a metric continuum is a dendrite if and only if for every compact space and for every light open mapping such that there is a copy of in for which the restriction is a homeomorphism. Another characterization of dendrites in terms of continuous selections of multivalued functions is also obtained.

     

Asymptotic behaviour of Castelnuovo-Mumford regularity

Let be a polynomial ring over a field. For a graded -module generated in degree at most , the Castelnuovo-Mumford regularity of each of (i) its symmetric power, (ii) its torsion-free symmetric power and (iii) the integral closure of its torsion-free symmetric power is bounded above by a linear function in with leading coefficient at most . For a graded ideal of , the regularity of is given by a linear function of for all sufficiently large . The leading coefficient of this function is identified.

     

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.

     

Between the Lindelöf property and countable tightness

We consider a class of compact spaces for which the space of probability Radon measures on has countable tightness in the topology. We show that that class contains those compact zero-dimensional spaces for which is weakly Lindelöf, and, under MA + CH, all compact spaces with having property (C) of Corson.

     

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.

     

The Mardesic factorization theorem for extension theory and c-separation

We shall prove a type of Mardesic factorization theorem for extension theory over an arbitrary stratum of CW-complexes in the class of arbitrary compact Hausdorff spaces. Our result provides that the space through which the factorization occurs will have the same strong countability property (e.g., strong countable dimension) as the original one had. Taking into consideration the class of compact Hausdorff spaces, this result extends all previous ones of its type. Our factorization theorem will simultaneously include factorization for weak infinite-dimensionality and for Property C, that is, for C-spaces.

A corollary to our result will be that for any weight and any finitely homotopy dominated CW-complex , there exists a Hausdorff compactum with weight and which is universal for the property and weight . The condition means that for every closed subset of and every map , there exists a map which is an extension of . The universality means that for every compact Hausdorff space whose weight is and for which is true, there is an embedding of into .

We shall show, on the other hand, that there exists a CW-complex which is not finitely homotopy dominated but which has the property that for each weight , there exists a Hausdorff compactum which is universal for the property and weight .

     

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.     

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.

     

Openness of induced projections

For continua and it is shown that if the projection has its induced mapping open, then is -smooth. As a corollary, a characterization of dendrites in these terms is obtained.

     

Curvature restrictions on convex, timelike surfaces in Minkowski 3-space

Suppose that and are Minkowski Gauss curvature and Minkowski mean curvature respectively on a timelike surface that is immersed in Minkowski 3-space . Suppose also that and that is complete as a surface in the underlying Euclidean 3-space . It is shown that neither nor can be bounded away from zero on such a surface .

     

On weighted polynomial approximation with monotone weights

We construct an even weight monotone on the right half line such that the logarithmic integral of the largest -convex minorant of converges and the polynomials are dense in .     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .