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.

     

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 .

     

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.

     

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.

     

Block bases of the Haar system as complemented subspaces of

It is shown that the span of , where is the Haar system in and the canonical basis of , is well isomorphic to a well complemented subspace of . As a consequence we get that there is a rearrangement of the (initial segments of the) Haar system in , any block basis of which is well isomorphic to a well complemented subspace of .

     

Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

Let generate a tight affine frame with dilation factor , where , and sampling constant (for the zeroth scale level). Then for , oversampling (or oversampling by ) means replacing the sampling constant by . The Second Oversampling Theorem asserts that oversampling of the given tight affine frame generated by preserves a tight affine frame, provided that is relatively prime to (i.e., ). In this paper, we discuss the preservation of tightness in oversampling, where (i.e., and ). We also show that tight affine frame preservation in oversampling is equivalent to the property of shift-invariance with respect to of the affine frame operator defined on the zeroth scale level.

     

Directional convexity of level lines for functions convex in a given direction

Let be the class of functions which are holomorphic and convex in direction in the unit disk , i.e. the domain is such that the intersection of and any straight line is a connected or empty set. In this note we determine the radius of the biggest disk with the property that each function maps this disk onto the convex domain in the direction .

     

On a question of B. H. Neumann

The automorphism group of a free group acts on the set of generating -tuples of a group . Higman showed that when , the union of conjugacy classes of the commutators and is an orbit invariant. We give a negative answer to a question of B.H. Neumann, as to whether there is a generalization of Higman's result for .

     

The Nevanlinna counting functions for Rudin's orthogonal functions

and denote the Hardy spaces on the open unit disc . Let be a function in and . If is an inner function and , then is orthogonal in . W.Rudin asked if the converse is true and C. Sundberg and C. Bishop showed that the converse is not true. Therefore there exists a function such that is not an inner function and is orthogonal in . In this paper, the following is shown: is orthogonal in if and only if there exists a unique probability measure on [0,1] with supp such that for nearly all in where is the Nevanlinna counting function of . If is an inner function, then is a Dirac measure at .

     

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 ).

     

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.

     

A non-standard proof of the Briançon-Skoda theorem

Using a tight closure argument in characteristic and then lifting the argument to characteristic zero with the aid of ultraproducts, I present an elementary proof of the Briançon-Skoda Theorem: for an -generated ideal of , the -th power of its integral closure is contained in . It is well-known that as a corollary, one gets a solution to the following classical problem. Let be a convergent power series in variables over which vanishes at the origin. Then lies in the ideal generated by the partial derivatives of .

     

Geometric properties coded in the long-time asymptotics for the heat equation on

This paper investigates connections between the long-time asymptotics of heat distribution on a body in , and various geometric properties of , starting from an initially constant heat distribution supported on . We use combinatorial and differential geometric methods. We begin the paper with a result in .

     

Mbekhta's subspaces and a spectral theory of compact operators

Let be an operator on an infinite-dimensional complex Banach space. By means of Mbekhta's subspaces and , we give a spectral theory of compact operators. The main results are: Let be compact. . The following assertions are all equivalent: (1) 0 is an isolated point in the spectrum of (2) is closed; (3) is of finite dimension; (4) is closed; (5) is of finite dimension; . sufficient conditions for to be an isolated point in ; . sufficient and necessary conditions for to be a pole of the resolvent of .

     

An inverse problem for an inhomogeneous conformal Killing field equation

Let be a Riemannian metric defined on a bounded domain with boundary and let be a vector field on satisfying . We show that if is a gradient field of a solution to the equation on , then both inner products and are uniquely determined by the restriction of the tensor to the gradient field , where is the Lie derivative of the metric tensor under the vector field and . This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.

     

On the correspondence of representations between and division algebras

For a division algebra over a -adic field we prove that depth is preserved under the correspondence of discrete series representations of and irreducible representations of by proving that an explicit relation holds between depth and conductor for all such representations. We also show that this relation holds for many (possibly all) discrete series representations of

     

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 .

     

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if , are finite modules over a Gorenstein local ring of codimension at most , then the vanishing of for is equivalent to the vanishing of for . Furthermore, if has no embedded deformation, then such vanishing occurs if and only if or has finite projective dimension.

     

-hyponormality is not translation-invariant

In this note we provide an example of a semi-hyponormal Hilbert space operator for which is not -hyponormal for some and all .

     

Characterizing nearly simple chain domains

G. Puninski, using model theoretical methods, showed that if a chain domain is nearly simple, then for any nonzero elements in , the Jacobson radical of . Here, an algebraic proof is given for this result, exceptional chain domains are characterized, and it is shown that , the lattice generated by all proper nonzero left and right ideals, is a direct product of two linearly ordered sets if is nearly simple. In a certain sense this property characterizes nearly simple chain domains among all integral domains.