Farrell sets for harmonic functions

Let denote a relatively closed subset of the unit ball of . The purpose of this paper is to characterize those sets which have the following property: any harmonic function on which satisfies on (where 0$">) can be locally uniformly approximated on by a sequence of harmonic polynomials which satisfy the same inequality on . This answers a question posed by Stray, who had earlier solved the corresponding problem for holomorphic functions on the unit disc.

     

A characterization of compactly generated metric groups

Recall that a topological group is: (a) -compact if where each is compact, and (b) compactly generated if is algebraically generated by some compact subset of . Compactly generated groups are -compact, but the converse is not true: every countable non-finitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. We prove that a metric group is compactly generated if and only if is -compact and for every open subgroup of there exists a finite set such that algebraically generates . As a corollary, we obtain that a -compact metric group is compactly generated provided that one of the following conditions holds: (i) has no proper open subgroups, (ii) is dense in some connected group (in particular, if is connected itself), (iii) is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element (eventually constant sequences are not excluded here). Therefore, a countable metric group can be generated by a (possibly eventually constant) sequence converging to its identity element in each of the cases (i), (ii) and (iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.

     

On polynomial products in nilpotent and solvable Lie groups

We are dealing with Lie groups which are diffeomorphic to , for some . After identifying with , the multiplication on can be seen as a map . We show that if is a polynomial map in one of the two (sets of) variables or , then is solvable. Moreover, if one knows that is polynomial in one of the variables, the group is nilpotent if and only if is polynomial in both its variables.

     

Perfect cliques and colorings of Polish spaces

A coloring of a set is any subset of , where 1$"> is a natural number. We give some sufficient conditions for the existence of a perfect -homogeneous set, in the case where is and is a Polish space. In particular, we show that it is sufficient that there exist -homogeneous sets of arbitrarily large countable Cantor-Bendixson rank. We apply our methods to show that an analytic subset of the plane contains a perfect -clique if it contains any uncountable -clique, where is a natural number or (a set is a -clique in if the convex hull of any of its -element subsets is not contained in ).

     

Une propriété de continuité du temps local

Let denote the local time (at 0) associated with a martingale . The aim of this note is to prove that the mapping is continuous from into weak-.

     

The Whitehead products and powers in -topology

We initiate a study of -topology. This is a modification of ordinary topology in which morphisms exist after smashing with a fixed space . Several classical topics are considered in this setting, inter alia Whitehead products, Hopf invariants and the -- sequence. The main emphasis is on detection of non-trivial elements in the -homotopy groups of spheres.

     

Approximation of measurable mappings by sequences of continuous functions

Let be a completely regular Hausdorff space, a positive, finite Baire measure on , and a separable metrizable locally convex space. Suppose is a measurable mapping. Then there exists a sequence of functions in which converges to a.e. . If the function is assumed to be weakly continuous and the measure is assumed to be -smooth, then a separability condition is not needed.

     

Imbeddings of free actions on handlebodies

Fix a free, orientation-preserving action of a finite group on a -dimensional handlebody . Whenever acts freely preserving orientation on a connected -manifold , there is a -equivariant imbedding of into . There are choices of closed and Seifert-fibered for which the image of is a handlebody of a Heegaard splitting of . Provided that the genus of is at least , there are similar choices with closed and hyperbolic.

     

-inner automorphisms of finite groups

We refer to an automorphism of a group as -inner if given any subset of with cardinality less than , there exists an inner automorphism of agreeing with on . Hence is 2-inner if it sends every element of to a conjugate. New examples are given of outer -inner automorphisms of finite groups for all natural numbers .

     

Discrete spectra of -algebras and complemented submodules in Hilbert -modules

Let be a -algebra and let be a full (right) Hilbert -module. It is shown that if the spectrum of is discrete, then every closed --submodule of is complemented in , and conversely that if is a -space and if every closed --submodule of is complemented in , then is discrete.

     

Bernstein-Walsh inequalities and the exponential curve in

It is shown that for the pluripolar set in there is a global Bernstein-Walsh inequality: If is a polynomial of degree on and on , this inequality gives an upper bound for which grows like . The result is used to obtain sharp estimates for .

     

On the Bieberbach conjecture and holomorphic dynamics

In this note we prove that when is a polynomial of degree with connected Julia set and when belongs to the filled-in Julia set , then . We also show that equality is achieved if and only if is a segment of which one extremity is . In that case, is conjugate to a Tchebycheff polynomial or its opposite. The main tool in our proof is the Bieberbach conjecture proved by de Branges in 1984.

     

Catégorie de Lusternik-Schnirelmann et genre des -espaces

Soit un espace ayant le type d'homotopie rationnelle d'un produit de sphères impaires. Si, pour tout nombre premier , la LS-catégorie de tous les -localisés de est majorée par , nous montrons que la LS-catégorie de est majorée par . Si est un élément dans le genre de Mislin de , nous en déduisons: . Dans le cas d'un -espace de rang 2, nous avons exactement , pour tout espace dans le genre de .

     

Deformations of minimal Lagrangian submanifolds with boundary

Let be a special Lagrangian submanifold of a compact Calabi-Yau manifold with boundary lying on the symplectic, codimension 2 submanifold . It is shown how deformations of which keep the boundary of confined to can be described by an elliptic boundary value problem, and two results about minimal Lagrangian submanifolds with boundary are derived using this fact. The first is that the space of minimal Lagrangian submanifolds near with boundary on is found to be finite dimensional and is parametrized over the space of harmonic 1-forms of satisfying Neumann boundary conditions. The second is that if is a symplectic, codimension 2 submanifold sufficiently near , then, under suitable conditions, there exists a minimal Lagrangian submanifold near with boundary on .

     

Modular group algebras of -separable -groups

Under the assumptions of MA and CH, it is proved that if is a field of prime characteristic and is an -separable abelian -group of cardinality , then an isomorphism of the group algebras and implies an isomorphism of and .

     

On stable equivalences of Morita type for finite dimensional algebras

In this paper, we assume that algebras are finite dimensional algebras with 1 over a fixed field and modules over an algebra are finitely generated left unitary modules. Let and be two algebras (where is a splitting field for and ) with no semisimple summands. If two bimodules and induce a stable equivalence of Morita type between and , and if maps any simple -module to a simple -module, then is a Morita equivalence. This conclusion generalizes Linckelmann's result for selfinjective algebras. Our proof here is based on the construction of almost split sequences.

     

Characterizations of paracompactness and Lindelöfness by the separation property

The separation property in our title is that, for two spaces and , any two disjoint closed copies of in are separated by open sets in . It is proved that a Tychonoff space is paracompact if and only if this separation property holds for the space and every Tychonoff space which is a perfect image of (where denotes the Stone-Cech compactification of ). Moreover, we give a characterization of Lindelöfness in a similar way under the assumption of paracompactness.

     

What is the Rees algebra of a module?

In this paper we show that the Rees algebra can be made into a functor on modules over a ring in a way that extends its classical definition for ideals. The Rees algebra of a module may be computed in terms of a ``maximal' map from to a free module as the image of the map induced by on symmetric algebras. We show that the analytic spread and reductions of can be determined from any embedding of into a free module, and in characteristic 0--but not in positive characteristic!--the Rees algebra itself can be computed from any such embedding.

     

Number of Sylow subgroups in -solvable groups

If is a finite group and is a prime number, let be the number of Sylow -subgroups of . If is a subgroup of a -solvable group , we prove that divides .