Conformal metrics and Ricci tensors in the pseudo-Euclidean space

We consider constant symmetric tensors on , , and we study the problem of finding metrics conformal to the pseudo-Euclidean metric such that . We show that such tensors are determined by the diagonal elements and we obtain explicitly the metrics . As a consequence of these results we get solutions globally defined on for the equation Moreover, we show that for certain unbounded functions defined on , there are metrics conformal to the pseudo-Euclidean metric with scalar curvature .

     

Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

Subsemivarieties of -algebras

A variety is a class of Banach algebras , for which there exists a family of laws such that is precisely the class of all Banach algebras which satisfies all of the laws (i.e. for all , . We say that is an -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras , for which there exists a family of homogeneous laws such that is precisely the class of all Banach algebras , for which there exists 0$"> such that for all homogeneous polynomials , , where . However, there is no variety between the variety of all -algebras and the variety of all -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.

     

On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

FS-property for -algebras

A -algebra is said to have the FS-property if the set of all self-adjoint elements in has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal -tensor products even in case of separable nuclear -algebras.

     

On classification of polarized varieties with non-integral nef values

Let be an -dimensional normal projective variety with only Gorenstein, terminal, -factorial singularities. Let be an ample line bundle on . Let denote the nef value of . The classification of via the nef value morphism is given for the situations when satisfies or .

     

A bifurcation result for harmonic maps from an annulus to with not symmetric boundary data

We consider the problem of minimizing the energy of the maps from the annulus to such that is equal to for , and to , for , where is a fixed angle.

We prove that the minimum is attained at a unique harmonic map which is a planar map if , while it is not planar in the case \pi^2$">.

Moreover, we show that tends to as , where minimizes the energy of the maps from to , with the boundary condition , .

     

The Dirichlet-Jordan test and multidimensional extensions

If is a foliation of an open set by smooth -dimensional surfaces, we define a class of functions , supported in , that are, roughly speaking, smooth along and of bounded variation transverse to . We investigate geometrical conditions on that imply results on pointwise Fourier inversion for these functions. We also note similar results for functions on spheres, on compact 2-dimensional manifolds, and on the 3-dimensional torus. These results are multidimensional analogues of the classical Dirichlet-Jordan test of pointwise convergence of Fourier series in one variable.

     

Stability of wavelet frames and Riesz bases, with respect to dilations and translations

We consider the perturbation problem of wavelet frame (Riesz basis) about dilation and translation parameters and . For wavelet functions whose Fourier transforms have small supports, we give a method to determine whether the perturbation system is a frame (Riesz basis) and prove the stability about dilation parameter on Paley-Wiener space. For a great deal of wavelet functions, we give a definite answer to the stability about translation . Moreover, if the Fourier transform has small support, we can estimate the frame bounds about the perturbation of translation parameter . Our methods can be used to handle nonhomogeneous frames (Riesz basis).

     

Une preuve courte du principe de Selberg pour un groupe -adique

In 1992, Blanc and Brylinski showed the following property for a -adic group , called the ``abstract Selberg principle': the orbital integrals on conjugacy classes of non-compact elements of the Hattori rank of a finitely generated projective smooth representation of vanish. The proof is by explicit computations of ``low' level ( and cyclic and Hochschild cohomologies. Here we intend to show that this property is actually a direct consequence of two facts: Clozel's integration formula (which leads us to assume the defining characteristic to be zero) and the triviality of the action of unramified characters on the of (which is also proven here, using a standard -theoretic argument due to Grothendieck).

     

Normality and exceptional values of derivatives

We show that a family of functions meromorphic in some domain is normal, if for all the derivative omits the value and if the values that can take at the zeros of satisfy certain restrictions. As an application we obtain a new proof of a theorem of Langley which classifies the functions meromorphic in the plane such that and have no zeros.

     

Elements in a commutative Banach algebra determining the norm topology

For an element of a commutative complex Banach algebra we investigate the following property: every complete norm on making the multiplication by from to itself continuous is equivalent to .

     

Single elements of finite CSL algebras

An element of an (abstract) algebra is a single element of if and imply that or . Let be a real or complex reflexive Banach space, and let be a finite atomic Boolean subspace lattice on , with the property that the vector sum is closed, for every . For any subspace lattice the single elements of Alg are characterised in terms of a coordinatisation of involving . (On separable complex Hilbert space the finite distributive subspace lattices which arise in this way are precisely those which are similar to finite commutative subspace lattices. Every distributive subspace lattice on complex, finite-dimensional Hilbert space is of this type.) The result uses a characterisation of the single elements of matrix incidence algebras, recently obtained by the authors.

     

The mixed Hodge structure on the fundamental group of a punctured Riemann surface

Given a compact Riemann surface of genus and distinct points and on , we consider the non-compact Riemann surface with basepoint . The extension of mixed Hodge structures associated to the first two steps of is studied. We show that it determines the element in , where represents the canonical divisor of as well as the corresponding extension associated to . Finally, we deduce a pointed Torelli theorem for punctured Riemann surfaces.

     

Lifting wreath product extensions

Let and be finite groups and let be a hilbertian field. We show that if has a generic extension over and satisfies the arithmetic lifting property over , then the wreath product of and also satisfies the arithmetic lifting property over . Moreover, if the orders of and are relatively prime and is abelian, then any extension of by (which is necessarily a semidirect product) has the arithmetic lifting property.

     

Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

An order characterization of commutativity for -algebras

In this paper, we investigate the problem of when a -algebra is commutative through operator-monotonic increasing functions. The principal result is that the function is operator-monotonic increasing on a -algebra if and only if is commutative. Therefore, -algebra is commutative if and only if in for all positive elements in .

     

Linear maps preserving ideals of -algebras

We show that unital self-adjoint linear bijections of matrix algebras, type factors and abelian -algebras preserving maximal left ideals are isomorphisms and we show that a unital continuous linear map of a -algebra that maps the minimal left ideal into itself is the identity map.     

A new construction of semi-free actions on Menger manifolds

A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .

     

Separated sequences in nonreflexive Banach spaces

We prove that there is 1$"> such that the unit ball of any nonreflexive Banach space contains a -separated sequence. The supremum of these constants is estimated from below by and from above approximately by . Given any 1$">, we also construct a nonreflexive space so that if the convex hull of a sequence is sufficiently close to the unit sphere, then its separation constant does not exceed .