Nests with the partial factorization property

It is proved that a nest on a separable complex Hilbert space has the left (resp. right) partial factorization property, which means that for every invertible operator from onto a Hilbert space there exists an isometry (resp. a coisometry) from into such that both and are in the associated nest algebra if and only if it is atomic (resp. countable).

     

Instability of statistical factor analysis

Factor analysis, a popular method for interpreting multivariate data, models the covariance among variables as being due to a small number (, ) of hidden variables. A factor analysis of can be thought of as an ordered or unordered collection, , of linearly independent lines in . Let be the collection of data sets for which is defined. The ``singularities' of are those data sets, , in the closure, , at which the limit, , does not exist. is unstable near its singularities.

Let be the direct sum of the lines in . determines a -plane bundle, , over a subset, , of . If 1$"> and is rich enough, ordered or, at least if or 3, unordered, must have a singularity at some data set in . The proofs are applications of algebraic topology. Examples are provided.

     

Conformal metrics and Ricci tensors on the sphere

We consider tensors on the unit sphere , where , is the standard metric and is a differentiable function on . For such tensors, we consider the problems of existence of a Riemannian metric , conformal to , such that , and the existence of such a metric that satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability, and we construct the solutions when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor , 0 $">, has no solution on if and only metrics homothetic to admit as a Ricci tensor. We also show that if , then equation has no solution , conformal to on , and only metrics homothetic to are solutions to this equation when . Infinitely many solutions, globally defined on , are obtained for the equation

where . The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

     

Devaney's chaos implies existence of -scrambled sets

Let be a complete metric space without isolated points, and let be a continuous map. In this paper we prove that if is transitive and has a periodic point of period , then has a scrambled set consisting of transitive points such that each is a synchronously proximal Cantor set, and is dense in . Furthermore, if is sensitive (for example, if is chaotic in the sense of Devaney), with being a sensitivity constant, then this is an -scrambled set.

     

Hecke algebras for the basic characters of the unitriangular group

Let denote the unitriangular group of degree over the finite field with elements. In a previous paper we obtained a decomposition of the regular character of as an orthogonal sum of basic characters. In this paper, we study the irreducible constituents of an arbitrary basic character of . We prove that is induced from a linear character of an algebra subgroup of , and we use the Hecke algebra associated with this linear character to describe the irreducible constituents of as characters induced from an algebra subgroup of . Finally, we identify a special irreducible constituent of , which is also induced from a linear character of an algebra subgroup. In particular, we extend a previous result (proved under the assumption where is the characteristic of the field) that gives a necessary and sufficient condition for to have a unique irreducible constituent.

     

Vector measure Banach spaces containing a complemented copy of

Let a Banach space and a -algebra of subsets of a set . We say that a vector measure Banach space has the bounded Vitaly-Hahn-Sacks Property if it satisfies the following condition: Every vector measure , for which there exists a bounded sequence in verifying for all , must belong to . Among other results, we prove that, if is a vector measure Banach space with the bounded V-H-S Property and containing a complemented copy of , then contains a copy of .

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

Dimension distortion of hyperbolically convex maps

In this note, we provide an answer to a question of D. Mejia and Chr. Pommerenke, by constructing a hyperbolically convex subdomain of the unit disc so that the conformal map from to maps a set of dimension 0 on to a set of dimension

     

Hochschild cohomology of Frobenius algebras

Let be a field, a finite-dimensional Frobenius -algebra and , the Nakayama automorphism of with respect to a Frobenius homomorphism . Assume that has finite order and that has a primitive -th root of unity . Consider the decomposition of , obtained by defining , and the decomposition of the Hochschild cohomology of , obtained from the decomposition of . In this paper we prove that and that if the decomposition of is strongly -graded, then acts on and .

     

Strong comparison principle for solutions of quasilinear equations

Let , , be a bounded smooth connected open set and be a map satisfying the hypotheses (H1)-(H4) below. Let with , in and with be two weak solutions of

Suppose that in . Then we show that u_1$"> in under the following assumptions: either u_1$"> on , or on and in . We also show a measure-theoretic version of the Strong Comparison Principle.

     

A characterization of Möbius transformations

Let be an integer and let be a domain of . Let be an injective mapping which takes hyperspheres whose interior is contained in to hyperspheres in . Then is the restriction of a Möbius transformation.

     

Total curvatures of a closed curve in Euclidean -space

A classical result by J. W. Milnor states that the total curvature of a closed curve in the Euclidean -space is the limit of the total curvatures of polygons inscribed in . In the present paper a similar geometric interpretation is given for all total curvatures , .

     

Hilbert-Samuel coefficients and postulation numbers of graded components of certain local cohomology modules

Let be a Noetherian homogeneous ring with one-dimensional local base ring . Let be an -primary ideal, let be a finitely generated graded -module and let . Let denote the -th local cohomology module of with respect to the irrelevant ideal 0} R_n$"> of . We show that the first Hilbert-Samuel coefficient of the -th graded component of with respect to is antipolynomial of degree in . In addition, we prove that the postulation numbers of the components with respect to have a common upper bound.

     

Hilbert coefficients and the associated graded rings

Let be a -dimensional Cohen-Macaulay local ring with infinite residue field. Let be an -primary ideal of . In this paper, we prove that if for some minimal reduction of , then depth .

     

Control of radii of convergence and extension of subanalytic functions

Let : denote a real analytic function on an open subset of , and let denote the points where does not admit a local analytic extension. We show that if is semialgebraic (respectively, globally subanalytic), then is semialgebraic (respectively, subanalytic) and extends to a semialgebraic (respectively, subanalytic) neighbourhood of . (In the general subanalytic case, is not necessarily subanalytic.) Our proof depends on controlling the radii of convergence of power series centred at points in the image of an analytic mapping , in terms of the radii of convergence of at points , where denotes the Taylor expansion of at .

     

Hardy's inequality on exterior domains

Let be a smooth exterior domain in and . We prove that when , Hardy's inequality is valid on .

     

Grekos' S function has a linear growth

An exact additive asymptotic basis is a set of nonnegative integers such that there exists an integer with the property that any sufficiently large integer can be written as a sum of exactly elements of . The minimal such is the exact order of (denoted by ). Given any exact additive asymptotic basis , we define to be the subset of composed with the elements such that is still an exact additive asymptotic basis. It is known that is finite.

In this framework, a central quantity introduced by Grekos is the function defined as the following maximum (taken over all bases of exact order ):

In this paper, we introduce a new and simple method for the study of this function. We obtain a new estimate from above for which improves drastically and in any case on all previously known estimates. Our estimate, namely , cannot be too far from the truth since verifies . However, it is certainly not always optimal since . Our last result shows that is in fact a strictly increasing sequence.

     

On the algebra of functions -extendable for each finite

For each positive integer we construct a -function of one real variable, the graph of which has the following property: there exists a real function on which is -extendable to , for each finite, but it is not -extendable.

     

The Russo-Dye Theorem in a weakly closed -module

Suppose that is admissible. It is shown that the convex hull of unitary elements of a weakly closed -module contains the whole unit ball of if and only if and for any 0$">, 0$">.

     

Cohomology of symplectic reductions of generic coadjoint orbits

Let be a generic coadjoint orbit of a compact semi-simple Lie group . Weight varieties are the symplectic reductions of by the maximal torus in . We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of , and it makes explicit the dependence on and a parameter in .