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-.

     

Small covers of the dodecahedron and the -cell

Let be the right-angled hyperbolic dodecahedron or -cell, and let be the group generated by reflections across codimension-one faces of . We prove that if is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold is determined up to homeomorphism by modulo symmetries of .

     

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.

     

On quasi-affine transforms of Read's operator

We show that C. J. Read's example of an operator on which does not have any non-trivial invariant subspaces is not the adjoint of an operator on a predual of . Furthermore, we present a bounded diagonal operator such that even though is unbounded, the operator is a bounded operator on with invariant subspaces, and is adjoint to an operator on .

     

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.

     

The number of connected components in double Bruhat cells for nonsimply-laced groups

We compute the number of connected components in a generic real double Bruhat cell for series and and an exceptional group .

     

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.

     

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.

     

Segal-Bargmann transforms of one-mode interacting Fock spaces associated with Gaussian and Poisson measures

Let and denote the Gaussian and Poisson measures on , respectively. We show that there exists a unique measure on such that under the Segal-Bargmann transform the space is isomorphic to the space of analytic -functions on with respect to . We also introduce the Segal-Bargmann transform for the Poisson measure and prove the corresponding result. As a consequence, when and have the same variance, and are isomorphic to the same space under the - and -transforms, respectively. However, we show that the multiplication operators by on and on act quite differently on .

     

The critical point equation on a three-dimensional compact manifold

On a compact -dimensional manifold , a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation (CPE), given by . It has been conjectured that a solution of the CPE is Einstein. Restricting our considerations to and assuming that there exist at least two distinct solutions of the CPE throughout the paper, we first prove that, if the second homology of vanishes, then is diffeomorphic to (Theorem 2). Secondly, we prove that the same conclusion holds if we have a lower Ricci curvature bound or the connectedness of a certain surface of (Theorem 3). Finally, we also prove that, if two connected surfaces of are disjoint, is isometric to a standard -sphere (Theorem 4).

     

-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 .

     

Extremal properties of the derivatives of the Newman polynomials

Let be a set of distinct positive numbers. The span of

over will be denoted by

Our main result of this note is the following.

Theorem. Suppose . Let be a non-negative integer. Then there are constants 0$"> and 0$"> depending only on , , and such that

where the lower bound holds for all and for all , while the upper bound holds when and and when , , and .

     

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 .

     

Univalent mappings and invariant subspaces of the Bergman and Hardy spaces

In both the Bergman space and the Hardy space , the problem of determining which bounded univalent mappings of the unit disk have the wandering property is addressed. Generally, a function in has the wandering property in , where denotes either or , provided that every -invariant subspace of is generated by the orthocomplement of within . It is known that essentially every function which has the wandering property in either space is the composition of a univalent mapping with a classical inner function, and that the identity mapping has this property in both spaces. Consequently, weak-star generators of also have the wandering property in both settings. The present paper gives a partial converse to this, and shows that in both settings there is a large class of bounded univalent mappings which fail to have the wandering property.

     

Volume preserving embeddings of open subsets of into manifolds

We consider a connected smooth -dimensional manifold endowed with a volume form , and we show that an open subset of of Lebesgue measure embeds into by a smooth volume preserving embedding whenever the volume condition is met.

     

A weakly Stegall space that is not a Stegall space

A topological space is said to belong to the class of Stegall (weakly Stegall) spaces if for every Baire (complete metric) space and minimal usco , is single-valued at some point of . In this paper we show that under some additional set-theoretic assumptions that are equiconsistent with the existence of a measurable cardinal there is a Banach space whose dual, equipped with the weak topology, is in the class of weakly Stegall spaces but not in the class of Stegall spaces. This paper also contains an example of a compact space such that belongs to the class of weakly Stegall spaces but does not.

     

Isomorphism of Borel full groups

Suppose that and are Polish groups which act in a Borel fashion on Polish spaces and . Let and denote the corresponding orbit equivalence relations, and and the corresponding Borel full groups. Modulo the obvious counterexamples, we show that .

     

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.

     

On a theorem of R. Steinberg on rings of coinvariants

Let be a representation of a finite group over the field . Denote by the algebra of polynomial functions on the vector space . The group acts on and hence also on . The algebra of coinvariants is , where is the ideal generated by all the homogeneous -invariant forms of strictly positive degree. If the field has characteristic zero, then R. Steinberg has shown (this is the formulation of R. Kane) that is a Poincaré duality algebra if and only if is a pseudoreflection group. In this note we explore the situation for fields of nonzero characteristic. We prove an analogue of Steinberg's theorem for the case and give a counterexample in the modular case when .