Tangentially positive isometric actions and conjugate points

Let be a complete Riemannian manifold with no conjugate points and a principal -bundle, where is a Lie group acting by isometries and the smooth quotient with the Riemannian submersion metric.

We obtain a characterization of conjugate point-free quotients in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle , from which we then derive necessary conditions, involving and , for the quotient metric to be conjugate point-free, particularly for a reducible Riemannian manifold.

Let , with the Lie Algebra of , be the moment map of the tangential -action on and let be the canonical pseudo-Riemannian metric on defined by the symplectic form and the map , . First we prove a theorem, stating that if is not positive definite on the action vector fields for the tangential action along then acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on in terms of the positivity of the -length of their tangential lifts along certain canonical subsets of . We use this to derive some necessary conditions, on and , for actions to be tangentially positive on relevant subsets of , which we then apply to isometric actions on complete conjugate point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.

     

Weak amenability of triangular Banach algebras

Let and be unital Banach algebras, and let be a Banach -module. Then becomes a triangular Banach algebra when equipped with the Banach space norm . A Banach algebra is said to be -weakly amenable if all derivations from into its dual space are inner. In this paper we investigate Arens regularity and -weak amenability of a triangular Banach algebra in relation to that of the algebras , and their action on the module .

     

Hyperbolic unit groups

In this paper we study the groups whose integral group rings have hyperbolic unit groups . We classify completely the torsion subgroups of and the polycyclic-by-finite subgroups of the group . Finally, we classify the groups for which the boundary of has dimension zero.

     

Constants of derivations in polynomial rings over unique factorization domains

A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any -derivation of , where is a commutative field of characteristic zero, is a polynomial ring in one variable over . In this paper we give an elementary proof of this theorem and show that it remains true if we replace by any unique factorization domain of characteristic zero.

     

On the use of the topological degree theory in broken orbits analysis

Dynamical systems in are studied. Let be a bounded open set. We will be interested in those periodic orbits such that at least one of its points lies inside and at least one of its points lies outside ; the orbits with this property are called -broken. Information about the structure of the set of -broken orbits is suggested; results are formulated in terms of topological degree theory.

     

Computing the multiplicative group of residue class rings

Let be a global field with maximal order and let be an ideal of . We present algorithms for the computation of the multiplicative group of the residue class ring and the discrete logarithm therein based on the explicit representation of the group of principal units. We show how these algorithms can be combined with other methods in order to obtain more efficient algorithms. They are applied to the computation of the ray class group modulo , where denotes a formal product of real infinite places, and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields.

     

Algebraic gamma monomials and double coverings of cyclotomic fields

We investigate the properties of algebraic gamma monomials--that is, algebraic numbers which are expressible as monomials in special values of the classical gamma function. Recently Anderson has constructed a double complex , to compute , where is the universal ordinary distribution. We use the double complex to deduce explicit formulae for algebraic gamma monomials. We provide simple proofs of some previously known results of Deligne on algebraic gamma monomials. Deligne used the theory of Hodge cycles for his results. By contrast, our proofs are constructive and relatively elementary. Given a Galois extension , we define a double covering of to be an extension of degree , such that is Galois. We demonstrate that each class gives rise to a double covering of , by . When lifts a canonical basis element indexed by two odd primes, we show that this double covering can be non-abelian. However, if represents any of the canonical basis classes indexed by an odd squarefree positive integer divisible by at least four primes, then the Galois group of is abelian and hence . The may very well be a new supply of abelian units. The relevance of these units to the unit index formula for cyclotomic fields calls for further investigations.

     

Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros

Bounds for the distance between adjacent zeros of cylinder functions are given; and are such that ; stands for the th positive zero of the cylinder (Bessel) function , , .

These bounds, together with the application of modified (global) Newton methods based on the monotonic functions and , give rise to forward ( ) and backward ( ) iterative relations between consecutive zeros of cylinder functions.

The problem of finding all the positive real zeros of Bessel functions for any real and inside an interval , 0$">, is solved in a simple way.

     

The backward shift on Dirichlet-type spaces

We study the backward shift operator on Hilbert spaces (for ) which are norm equivalent to the Dirichlet-type spaces . Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on in terms of a weighted superposition of backward shifts.

     

Uniform Sobolev inequalities and absolute continuity of periodic operators

We establish certain uniform inequalities for a family of second order elliptic operators of the form on the -torus, where and is a symmetric, positive definite matrix with real constant entries. Using these Sobolev type inequalities, we obtain the absolute continuity of the spectrum of the periodic Dirac operator on with singular potential. The absolute continuity of the elliptic operator div on with a positive periodic scalar function is also studied.

     

Condensations of projective sets onto compacta

For a coanalytic-complete or -complete subspace of a Polish space we prove that there exists a continuous bijection of onto the Hilbert cube . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

     

Branch structure of -holomorphic curves near periodic orbits of a contact manifold

Let be a three-dimensional contact manifold, and a finite-energy pseudoholomorphic map from the punctured disc in that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into , where denotes a rational ellipsoid (contact structure induced by the standard complex structure on ), as well as contact structures arising from non-standard circle-fibrations of the three-sphere.

     

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.     

Zero product preserving maps of operator-valued functions

Let be locally compact Hausdorff spaces and , be Banach algebras. Let be a zero product preserving bounded linear map with dense range. We show that is given by a continuous field of algebra homomorphisms from into if is irreducible. As corollaries, such a surjective arises from an algebra homomorphism, provided that is a -algebra and is a semi-simple Banach algebra, or both and are -algebras.

     

The autohomeomorphism group of the Cech-Stone compactification of the integers

It is shown to be consistent that there is a nontrivial autohomeomorphism of , yet all such autohomeomorphisms are trivial on a dense -ideal. Furthermore, the cardinality of the autohomeomorphism group of can be any regular cardinal between and . The model used is one due to Velickovic in which, coincidentally, Martin's Axiom also holds.

     

Dynamical systems disjoint from any minimal system

Furstenberg showed that if two topological systems and are disjoint, then one of them, say , is minimal. When is nontrivial, we prove that must have dense recurrent points, and there are countably many maximal transitive subsystems of such that their union is dense and each of them is disjoint from . Showing that a weakly mixing system with dense periodic points is in , the collection of all systems disjoint from any minimal system, Furstenberg asked the question to characterize the systems in . We show that a weakly mixing system with dense regular minimal points is in , and each system in has dense minimal points and it is weakly mixing if it is transitive. Transitive systems in and having no periodic points are constructed. Moreover, we show that there is a distal system in .

Recently, Weiss showed that a system is weakly disjoint from all weakly mixing systems iff it is topologically ergodic. We construct an example which is weakly disjoint from all topologically ergodic systems and is not weakly mixing.

     

Borel complexity of the space of probability measures

Using a technique developed by Louveau and Saint Raymond, we find the complexity of the space of probability measures in the Borel hierarchy: if is any non-Polish Borel subspace of a Polish space, then , the space of probability Borel measures on with the weak topology, is always true , where is the least ordinal such that is .

     

Compact nilmanifolds with nilpotent complex structures: Dolbeault cohomology

We consider a special class of compact complex nilmanifolds, which we call compact nilmanifolds with nilpotent complex structure. It is shown that if is a compact nilmanifold with nilpotent complex structure, then the Dolbeault cohomology is canonically isomorphic to the -cohomology of the bigraded complex of complex valued left invariant differential forms on the nilpotent Lie group .

     

Countable dense homogeneity of definable spaces

We investigate which definable separable metric spaces are countable dense homogeneous (CDH). We prove that a Borel CDH space is completely metrizable and give a complete list of zero-dimensional Borel CDH spaces. We also show that for a Borel the following are equivalent: (1) is in , (2) is CDH and (3) is homeomorphic to or to . Assuming the Axiom of Projective Determinacy the results extend to all projective sets and under the Axiom of Determinacy to all separable metric spaces. In particular, modulo a large cardinal assumption it is relatively consistent with ZF that all CDH separable metric spaces are completely metrizable. We also answer a question of Stepr ns and Zhou, by showing that is not CDH.

     

Semiregular automorphisms of cubic vertex transitive graphs

It is shown that for a connected cubic graph , a vertex transitive group contains a large semiregular subgroup. This confirms a conjecture of Cameron and Sheehan (2001).