Asymptotically harmonic spaces in dimension 3

Let be a Hadamard manifold of dimension whose sectional curvature satisfies and whose curvature tensor satisfies for suitable constants and . We show that is of constant sectional curvature provided is asymptotically harmonic. This was previously only known if admits a compact quotient.

     

Volume preserving embeddings of open subsets of into manifolds

We consider a connected smooth -dimensional manifold endowed with a volume form , and we show that an open subset of of Lebesgue measure embeds into by a smooth volume preserving embedding whenever the volume condition is met.

     

A concrete description of -spaces as -spaces and its applications

We prove that for a compact Hausdorff space without isolated points, and are isometrically Riesz isomorphic spaces under a certain topology on . Moreover, is a closed subspace of . This provides concrete examples of compact Hausdorff spaces such that the Dedekind completion of is (= the set of all bounded real-valued functions on ) since the Dedekind completion of is ( and spaces as Banach lattices).

     

Non-cupping and randomness

Let be Martin-Löf-random. Then there is a promptly simple set such that for each Martin-Löf-random set , . When , one obtains a c.e. non-computable set which is not weakly Martin-Löf cuppable. That is, for any Martin-Löf-random set , if , then .

     

The spectrum of Schrödinger operators with positive potentials in Riemannian manifolds

Let be a noncompact complete Riemannian manifold. We consider the Schrödinger operator acting on , where is a nonnegative, locally integrable function on . We obtain some simple conditions which imply that , the bottom of the spectrum of , is strictly positive. We also establish upper and lower bounds for the counting function .

     

Special values of elliptic functions at points of the divisors of Jacobi forms

The main result of the paper gives an explicit formula for the sum of the values of even order derivatives with respect to of the Weierstrass -function for the lattice (where is in the upper half-plane) extended over the points in the divisor of (where is a meromorphic Jacobi form) in terms of the coefficients of the Laurent expansion of around .

     

Degenerating families of rank two bundles

We construct families of rank two bundles on , in characteristic two, where for , is a sum of line bundles, and is non-split. We construct families of rank two bundles on , in characteristic , where for , is a sum of line bundles, and is non-split.

     

On a subspace perturbation problem

We discuss the problem of perturbation of spectral subspaces for linear self-adjoint operators on a separable Hilbert space. Let and be bounded self-adjoint operators. Assume that the spectrum of consists of two disjoint parts and such that 0$">. We show that the norm of the difference of the spectral projections

for and is less than one whenever either (i) or (ii) and certain assumptions on the mutual disposition of the sets and are satisfied.

     

On the algebra range of an operator on a Hilbert -module over compact operators

Let be a Hilbert -module over the -algebra of all compact operators on a complex Hilbert space . Given an orthogonal projection , we describe the set for an arbitrary adjointable operator . The relationship between the set and the matricial range of is established.

     

Small covers of the dodecahedron and the -cell

Let be the right-angled hyperbolic dodecahedron or -cell, and let be the group generated by reflections across codimension-one faces of . We prove that if is a torsion free subgroup of minimal index, then the corresponding hyperbolic manifold is determined up to homeomorphism by modulo symmetries of .

     

Une propriété de continuité du temps local

Let denote the local time (at 0) associated with a martingale . The aim of this note is to prove that the mapping is continuous from into weak-.

     

On linear transformations preserving at least one eigenvalue

Let be an algebraically closed field and be a linear transformation. In this paper we show that if preserves at least one eigenvalue of each matrix, then preserves all eigenvalues of each matrix. Moreover, for any infinite field (not necessarily algebraically closed) we prove that if is a linear transformation and for any with at least an eigenvalue in , and have at least one common eigenvalue in , then preserves the characteristic polynomial.

     

Multi-dimensional versions of a theorem of Fine and Wilf and a formula of Sylvester

Let be vectors in which generate . We show that a body with the vectors as edge vectors is an almost minimal set with the property that every function with periods is constant. For the result reduces to the theorem of Fine and Wilf, which is a refinement of the famous Periodicity Lemma.

Suppose is not a non-trivial linear combination of with non-negative coefficients. Then we describe the sector such that every interior integer point of the sector is a linear combination of over , but infinitely many points on each of its hyperfaces are not. For the result reduces to a formula of Sylvester corresponding to Frobenius' Coin-changing Problem in the case of coins of two denominations.

     

Transferred Chern classes in Morava -theory

Let be a complex -plane bundle over the total space of a cyclic covering of prime index . We show that for the -th Chern class of the transferred bundle differs from a certain transferred class of by a polynomial in the Chern classes of the transferred bundle. The polynomials are defined by the formal group law and certain equalities in .

     

-additive families and the invariance of Borel classes

We prove that any -additive family of sets in an absolutely Souslin metric space has a -discrete refinement provided every partial selector set for is -discrete. As a corollary we obtain that every mapping of a metric space onto an absolutely Souslin metric space, which maps -sets to -sets and has complete fibers, admits a section of the first class. The invariance of Borel and Souslin sets under mappings with complete fibers, which preserves -sets, is shown as an application of the previous result.

     

A note on the imbedding theorem of Browder and Ton

The imbedding theorem of Browder and Ton states that for any real separable Banach space there exist a real separable Hilbert space and a compact linear injection such that is dense in We shall give a short and elementary new proof to this result. We also briefly discuss the corresponding result without the completeness assumption.

     

Isomorphisms of subalgebras of nest algebras

Let be a subalgebra of a nest algebra . If contains all rank one operators in , then is said to be large; if the set of rank one operators in coincides with that in the Jacobson radical of , is said to be radical-type. In this paper, algebraic isomorphisms of large subalgebras and of radical-type subalgebras are characterized. Let be a nest of subspaces of a Hilbert space and be a subalgebra of the nest algebra associated to (). Let be an algebraic isomorphism from onto . It is proved that is spatial if one of the following occurs: (1) () is large and contains a masa; (2) () is large and closed; (3) () is a closed radical-type subalgebra and ( is quasi-continuous (i.e. the trivial elements of are limit points); (4) () is large and one of and is not quasi-continuous.

     

Spreading of quasimodes in the Bunimovich stadium

We consider Dirichlet eigenfunctions of the Bunimovich stadium , satisfying . Write where is the central rectangle and denotes the ``wings,' i.e., the two semicircular regions. It is a topic of current interest in quantum theory to know whether eigenfunctions can concentrate in as . We obtain a lower bound on the mass of in , assuming that itself is -normalized; in other words, the norm of is controlled by times the norm in . Moreover, if is an quasimode, the same result holds, while for an quasimode we prove that the norm of is controlled by times the norm in . We also show that the norm of may be controlled by the integral of along , where is a smooth factor on vanishing at . These results complement recent work of Burq-Zworski which shows that the norm of is controlled by the norm in any pair of strips contained in , but adjacent to .