A fixed-point theorem for usco maps

The familiar fixed-point theorem of Kakutani is strengthened by weakening the hypotheses on the set-valued mapping. Applications are made for and decompositions of compact metric spaces.

     

A uniform

A uniform estimate of Bessel functions is obtained, which is used to get a characterization of the measures on the unit sphere of in terms of the mixed norm of the Fourier transform of the measures.

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

A rigidity theorem for the Clifford tori in

Let be the unit hypersphere in the 4-dimensional Euclidean space defined by . For each with , we denote by the Clifford torus in given by the equations and . The Clifford torus is a flat Riemannian manifold equipped with the metric induced by the inclusion map . In this note we prove the following rigidity theorem: If is an isometric embedding, then there exists an isometry of such that . We also show no flat torus with the intrinsic diameter is embeddable in except for a Clifford torus.

     

Noncommutative spaces

Let be a von Neumann algebra with a faithful, finite, normal tracial state , and let be a finite, maximal subdiagonal algebra of . Let be the closure of in the noncommutative Lebesgue space . Then possesses several of the properties of the classical Hardy space on the circle, including a commutant lifting theorem, some results on Toeplitz operators, an factorization theorem, Nehari's Theorem, and harmonic conjugates which are bounded.

     

Subaveraging estimate for

Let be a smooth real hypersurface of and a compact submanifold of . We generalize a result of A. Boggess and R. Dwilewicz giving, under some geometric conditions on and , an estimate of the submeanvalue on of any function on a neighbourhood of , by the norm of on a neighbourhood of in .

     

A critical metric for the

The -norm of the curvature tensor

defines a Riemannian functional on the space of metrics. This work exhibits a metric on which is of Berger type but not of constant ricci curvature, and yet is critical for .

     

Grothendieck's theorem on non-abelian and local-global principles

A theorem of Grothendieck asserts that over a perfect field of cohomological dimension one, all non-abelian -cohomology sets of algebraic groups are trivial. The purpose of this paper is to establish a formally real generalization of this theorem. The generalization - to the context of perfect fields of virtual cohomological dimension one - takes the form of a local-global principle for the -sets with respect to the orderings of the field. This principle asserts in particular that an element in is neutral precisely when it is neutral in the real closure with respect to every ordering in a dense subset of the real spectrum of . Our techniques provide a new proof of Grothendieck's original theorem. An application to homogeneous spaces over is also given.

     

On proofs of the general density theorem

We show that if is a compact manifold, then there is a residual subset of the set of homeomorphisms on with the property that if , then the periodic points of are dense in its chain recurrent set. This result was first announced in [4], but a flaw in that argument was noted in [1], where a different proof was given. It was recently noted in [5] that this new argument only serves to show that the density of periodic points in the chain recurrent set is generic in the closure of the set of diffeomorphisms. We show how to patch the original argument from [4] to prove the result.

     

A Groenewold-Van Hove Theorem for

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold which is irreducible on the su(2) subalgebra generated by the components of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra consisting of polynomials in . Furthermore, we show that the maximal Poisson subalgebra of containing that can be so quantized is just that generated by .

     

-inverse limit stability theorem

We prove that if an endomorphism satisfies weak Axiom A and the no-cycles condition then is -inverse limit stable. This result is a generalization of Smale's -stability theorem from diffeomorphisms to endomorphisms.

     

Products of images

Let be the \u{C}ech-Stone remainder . We show that there exists a large class of images of such that whenever is a subset of of cardinality at most the continuum, then is again an image of . The class contains all separable compact spaces, all compact spaces of weight at most and all perfectly normal compact spaces.

     

A group of paths in

We define a group structure on the set of compact ``minimal' paths in . We classify all finitely generated subgroups of this group : they are free products of free abelian groups and surface groups. Moreover, each such group occurs in . The subgroups of isomorphic to surface groups arise from certain topological -forms on the corresponding surfaces. We construct examples of such -forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using we construct a non-polygonal tiling problem in , that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group has applications to combinatorial tiling problems of the type: given a set of tiles and a region , can be tiled by translated copies of tiles in ?

     

The boundedness of Riesz

Let be a finite nonzero Borel measure in satisfying for all and and some . If the Riesz -transform

is essentially bounded, then is an integer. We also give a related result on the -boundedness.

     

A new bound for the smallest

Let denote the number of primes and let denote the usual integral logarithm of . We prove that there are at least integer values of in the vicinity of with . This improves earlier bounds of Skewes, Lehman, and te Riele. We also plot more than 10000 values of in four different regions, including the regions discovered by Lehman, te Riele, and the authors of this paper, and a more distant region in the vicinity of , where appears to exceed by more than . The plots strongly suggest, although upper bounds derived to date for are not sufficient for a proof, that exceeds for at least integers in the vicinity of . If it is possible to improve our bound for by finding a sign change before , our first plot clearly delineates the potential candidates. Finally, we compute the logarithmic density of and find that as departs from the region in the vicinity of , the density is , and that it varies from this by no more than over the next integers. This should be compared to Rubinstein and Sarnak.

     

Prescribing Gaussian curvature on

We derive a sufficient condition for a radially symmetric function which is positive somewhere to be a conformal curvature on . In particular, we show that every nonnegative radially symmetric continuous function on is a conformal curvature.

     

Kernel of locally nilpotent

In this paper we study the kernel of a non-zero locally nilpotent -derivation of the polynomial ring over a noetherian integral domain containing a field of characteristic zero. We show that if is normal then the kernel has a graded -algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in , and, conversely, the symbolic Rees algebra of any unmixed height one ideal in can be embedded in as the kernel of a locally nilpotent -derivation of . We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general.

     

A model for invertible composition operators on

A model is obtained for invertible hyperbolic and parabolic composition operators on . This model shows that the adjoints of these composition operators are similar to block Toeplitz matrices constructed with weighted bilateral shifts and rank one operators.