A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

Smooth diameter and eigenvalue rigidity in positive Ricci curvature

A recent injectivity radius estimate and previous sphere theorems yield the following smooth diameter sphere theorem for manifolds of positive Ricci curvature: For any given and there exists a positive constant 0$">such that any -dimensional complete Riemannian manifold with Ricci curvature , sectional curvature and diameter is Lipschitz close and diffeomorphic to the standard unit -sphere. A similar statement holds when the diameter is replaced by the first eigenvalue of the Laplacian.

     

Equivariant cohomology with local coefficients

We show that for a discrete group , the equivariant cohomology of a -space with -local coefficients is isomorphic to the Bredon-Illman cohomology of with equivariant local coefficients .

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

The f-depth of an ideal on a module

Let be an ideal of a Noetherian local ring and a finitely generated -module. The f-depth of on is the least integer such that the local cohomology module is not Artinian. This paper presents some part of the theory of f-depth including characterizations of f-depth and a relation between f-depth and f-modules.

     

Exactness of one relator groups

A discrete group is -exact if the reduced crossed product with converts a short exact sequence of --algebras into a short exact sequence of -algebras. A one relator group is a discrete group admitting a presentation where is a countable set and is a single word over . In this short paper we prove that all one relator discrete groups are -exact. Using the Bass-Serre theory we also prove that a countable discrete group acting without inversion on a tree is -exact if the vertex stabilizers of the action are -exact.

     

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group with principal isotropy group and cohomogeneity such that . Moreover, we prove that the Euler characteristic of a compact Riemannian manifold or with positive sectional curvature is positive if admits an effective isometric action of a torus , i.e., if the symmetry rank of is .

     

Averaging distances in finite dimensional normed spaces and John's ellipsoid

A Banach space has the average distance property (ADP) if there exists a unique real number such that for each positive integer and all in the unit sphere of there is some in the unit sphere of such that

A theorem of Gross implies that every finite dimensional normed space has the average distance property. We show that, if has dimension , then . This is optimal and answers a question of Wolf (Arch. Math., 1994). The proof is based on properties of the John ellipsoid of maximal volume contained in the unit ball of .

     

Almost positive curvature on the Gromoll-Meyer 7-sphere

D. Gromoll and W. Meyer have represented a certain exotic 7-sphere as a biquotient of the compact Lie group . Thus any invariant normal homogeneous metric on induces a metric of nonnegative sectional curvature on . We show that the simplest such metrics (except the bi-invariant one) induce metrics which have in fact strictly positive curvature outside a subset of with measure zero.

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.

     

Index of B-Fredholm operators and generalization of a Weyl theorem

The aim of this paper is to show that if and are commuting B-Fredholm operators acting on a Banach space , then is a B-Fredholm operator and , where means the index. Moreover if is a B-Fredholm operator and is a finite rank operator, then is a B-Fredholm operator and We also show that if is isolated in the spectrum of , then is a B-Fredholm operator of index if and only if is Drazin invertible. In the case of a normal bounded linear operator acting on a Hilbert space , we obtain a generalization of a classical Weyl theorem.     

Proof of the prime power conjecture for projective planes of order

Let be an abelian collineation group of order of a projective plane of order . We show that must be a prime power, and that the -rank of is at least if for an odd prime .

     

Metacompact subspaces of products of ordinals

Let be a subspace of the product of finitely many ordinals. is countably metacompact, and is metacompact iff has no closed subset homeomorphic to a stationary subset of a regular uncountable cardinal. A theorem generalizing these two results is: is -metacompact iff has no closed subset homeomorphic to a -stationary set where .

     

Abelian point stabilizers in transitive permutation groups

In this note we prove that if the point stabilizer in a transitive permutation group of degree is abelian, then the exponent of is less than . This extends an earlier result of Andrea Lucchini, who proved this in the case where is cyclic.

     

On normal operator exponentials

Suppose that and are bounded normal operators on a complex Hilbert space and that . In this paper some conditions implying are given.

     

Invariant ideals of abelian group algebras under the multiplicative action of a field. I

Let be a division ring and let be a finite-dimensional -vector space, viewed multiplicatively. If is the multiplicative group of , then acts on and hence on any group algebra . Our goal is to completely describe the semiprime -stable ideals of . As it turns out, this result follows fairly easily from the corresponding results for the field of rational numbers (due to Brookes and Evans) and for infinite locally-finite fields. Part I of this work is concerned with the latter situation, while Part II deals with arbitrary division rings.

     

A uniqueness result for harmonic functions

Let , , and suppose is harmonic in and on the closure of . If the gradient of vanishes continuously on a subset of of positive -dimensional Lebesgue measure and satisfies certain regularity conditions, then must be identically constant.

     

Signed sums of polynomial values

We give a generalization of Bleicher's result on signed sums of th powers. Let be an integral-valued polynomial of degree satisfying the necessary condition that there exists no integer 1$"> dividing the values for all integers . Then, for every positive integer and every integer , there are infinitely many integers and choices of such that

     

The exponent three class group problem for some real cyclic cubic number fields

We determine all the simplest cubic fields whose ideal class groups have exponent dividing , thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number and the determination by D. Byeon of all all the simplest cubic fields with class number . We prove that there are simplest cubic fields with ideal class groups of exponent (and simplest cubic fields with ideal class groups of exponent , i.e. with class number one).

     

On the comparison of the spaces

The notion of -variation and the space arise in the study of regularity properties of solutions to perturbed conservation laws. In this article we show that this notion is equivalent to variation in the regular sense, and therefore the space is the same as the space in the sense of Cesari-Tonelli. We also point out some connection between the space and the Favard classes for translation semigroups.