Isolated invariant curves of a foliation

We bound the equisingularity type of the set of isolated separatrices of a holomorphic foliation of in terms of the Milnor number of . This result gives a bound for the degree of an algebraic invariant curve of a foliation of in terms of the degree of , provided that all the branches of are isolated separatrices.

     

The real rank zero property of crossed product

Let be a unital -algebra, and let be a -dynamical system with abelian and discrete. In this paper, we introduce the continuous affine map from the trace state space of the crossed product to the -invariant trace state space of . If is of real rank zero and is connected, we have proved that is homeomorphic. Conversely, if is homeomorphic, we also get some properties and real rank zero characterization of . In particular, in that case, is of real rank zero if and only if each unitary element in with the form can be approximated by the unitary elements in with finite spectrum, where , , and if moreover is a unital inductive limit of the direct sums of non-elementary simple -algebras of real rank zero, then the above can be cancelled.

     

The form sum and the Friedrichs extension of Schrödinger-type operators on Riemannian manifolds

We consider , where is a Riemannian manifold (not necessarily complete), and is the scalar Laplacian on . We assume that , where and ( is a constant) are real-valued, and is semibounded below on . Let be the Friedrichs extension of . We prove that the form sum coincides with the self-adjoint operator associated to the closure of the restriction to of the sum of two closed quadratic forms of and . This is an extension of a result of Cycon. The proof adopts the scheme of Cycon, but requires the use of a more general version of Kato's inequality for operators on Riemannian manifolds.

     

On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups and , and a -adapted ring (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings and is monomial, i.e., maps class sums in to class sums in up to multiplication with roots of unity. As a consequence, and have identical character tables if and only if the centers of their integral group rings and are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

     

Associated primes of local cohomology modules

Let be an ideal of a commutative Noetherian ring and a finitely generated -module. Let be a natural integer. It is shown that there is a finite subset of , such that is contained in union with the union of the sets , where and . As an immediate consequence, we deduce that the first non- -cofinite local cohomology module of with respect to has only finitely many associated prime ideals.

     

Minimal sufficiency of order statistics in convex models

Let be a convex and dominated statistical model on the measurable space , with minimal sufficient, and let . Then , the -algebra of all permutation invariant sets belonging to the -fold product -algebra , is shown to be minimal sufficient for the corresponding model for independent observations, .

The main technical tool provided and used is a functional analogue of a theorem of Grzegorek (1982) concerning generators of .

     

``Lebesgue measure' on , II

Let be the set of real numbers, and define . We construct a complete measure space where the -algebra contains the Borel subsets of , and is a translation-invariant measure such that for any measurable rectangle , if , then , where is Lebesgue measure on . The measure is not -finite. We prove three Fubini theorems, namely, the Fubini theorem, the mean Fubini-Jensen theorem, and the pointwise Fubini-Jensen theorem. Finally, as an application of the measure , we construct, via selfadjoint operators on , a ``Schrödinger model' of the canonical commutation relations: , , .

     

Poincaré duality algebras and rings of coinvariants

Let be a faithful representation of a finite group over the field . Via the group acts on and hence on the algebra of homogenous polynomial functions on the vector space . R. Kane (1994) formulated the following result based on the work of R. Steinberg (1964): If the field has characteristic 0, then is a Poincaré duality algebra if and only if is a pseudoreflection group. The purpose of this note is to extend this result to the case (i.e. the order of is relatively prime to the characteristic of ).

     

Estimates of Gromov's box distance

In 1999, M. Gromov introduced the box distance function on the space of all mm-spaces. In this paper, by using the method of T. H. Colding, we estimate and , where is the -dimensional unit sphere in and is the -dimensional complex projective space equipped with the Fubini-Study metric. In particular, we give the complete answer to an exercise of Gromov's green book. We also estimate from below, where is the special orthogonal group.

     

Cohomological dimension of certain algebraic varieties

Let be an ideal of a commutative Noetherian ring . For finitely generated -modules and with , it is shown that . Let be a finitely generated module over a local ring such that . Using the above result and the notion of connectedness dimension, it is proved that Here denotes the connectedness dimension of the topological space . Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.

     

A criterion for correct solvability of the Sturm-Liouville equation in the space

We consider an equation

where , and By a solution of equation (1), we mean any function such that and equality (1) holds almost everywhere on In this paper, we obtain a criterion for the correct solvability of (1) in ,

     

Nonabelian theta functions of positive genus

Let be a smooth projective irreducible curve over of genus and let be a set of distinct points on . We fix a nonnegative integer and denote by the moduli space of parabolic semistable vector bundles of rank on with trivial determinant and fixed parabolic structure of type at , where each weight is in . On there is a canonical line bundle , whose global sections are called generalized parabolic -theta functions of order . In this paper we prove the existence of such nonzero nonabelian theta functions, thus establishing a part of higher genus generalizations of the celebrated saturation conjectures.

     

Some results on the Hochschild cohomology of group algebras

In this paper we investigate structure of the second cohomology of a discrete group . First, for a -set we show that an isomorphism of vector spaces from onto exists, where is the set of orbits of . Next we define the notion of pseudoderivation and apply it for the calculation of .

     

Pointwise convergence of bounded cascade sequences

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function , a cascade sequence is constructed by the iteration where is defined by In this paper, under a condition that the sequence is bounded in , we prove that the following three statements are equivalent: (i) converges . (ii) For , there exist a positive constant and a constant such that (iii) For some converges in . An example is presented to illustrate our result.

     

On two problems of Erdos and Hechler: New methods in singular madness

For an infinite cardinal , denotes the set of all cardinalities of nontrivial maximal almost disjoint families over .

Erdos and Hechler proved in 1973 the consistency of for a singular cardinal and asked if it was ever possible for a singular that , and also whether for every singular cardinal .

We introduce a new method for controlling for a singular and, among other new results about the structure of for singular , settle both problems affirmatively.

     

A linear function associated to asymptotic prime divisors

Let be a Noetherian standard -graded ring and finitely generated, -graded -modules. Let be finitely many homogeneous ideals of . We show that there exist linear functions such that the associated primes over of and are stable whenever satisfies and , respectively.

     

Spaces of type and the Hankel convolution

In this paper we introduce new function spaces that are denoted by , -1/2$"> and and that are spaces of type where the Hankel convolution and the Hankel transformation are defined. The spaces will play the same role in the Hankel setting that the spaces play in the theory of Fourier transformation.     

Holomorphic perturbation of Fourier coefficients

Let be the unit circle, let be a Banach space continuously embedded in and suppose that is a Banach -module under convolution. We show that if and is holomorphic in a neighbourhood of with and then

     

Numerical peak points and numerical Šilov boundary for holomorphic functions

In this paper, we characterize the numerical and numerical strong-peak points for when is the complex space or . We also prove that for all is the numerical Šilov boundary for

     

Gradient ranges of bumps on the plane

For a -smooth bump function we show that the gradient range is the closure of its interior, provided that admits a modulus of continuity satisfying as . The result is a consequence of a more general result about gradient ranges of bump functions of the same degree of smoothness. For such bump functions we show that for open sets , either the intersection is empty or its topological dimension is at least two. The proof relies on a new Morse-Sard type result where the smoothness hypothesis is independent of the dimension of the space.