Asymptotics for minimal discrete energy on the sphere

We investigate the energy of arrangements of points on the surface of the unit sphere in that interact through a power law potential where and is Euclidean distance. With denoting the minimal energy for such -point arrangements we obtain bounds (valid for all ) for in the cases when and . For , we determine the precise asymptotic behavior of as . As a corollary, lower bounds are given for the separation of any pair of points in an -point minimal energy configuration, when . For the unit sphere in , we present two conjectures concerning the asymptotic expansion of that relate to the zeta function for a hexagonal lattice in the plane. We prove an asymptotic upper bound that supports the first of these conjectures. Of related interest, we derive an asymptotic formula for the partial sums of when (the divergent case).

     

Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials

We prove that for every rational map on the Riemann sphere , if for every -critical point whose forward trajectory does not contain any other critical point, the growth of is at least of order for an appropriate constant as , then . Here is the so-called essential, dynamical or hyperbolic dimension, is Hausdorff dimension of and is the minimal exponent for conformal measures on . If it is assumed additionally that there are no periodic parabolic points then the Minkowski dimension (other names: box dimension, limit capacity) of also coincides with . We prove ergodicity of every -conformal measure on assuming has one critical point , no parabolic, and . Finally for every -conformal measure on (satisfying an additional assumption), assuming an exponential growth of , we prove the existence of a probability absolutely continuous with respect to , -invariant measure. In the Appendix we prove also for every non-renormalizable quadratic polynomial with not in the main cardioid in the Mandelbrot set.

     

Boundary slopes of punctured tori in 3-manifolds

Let be an irreducible 3-manifold with a torus boundary component , and suppose that are the boundary slopes on of essential punctured tori in , with their boundaries on . We show that the intersection number of and is at most . Moreover, apart from exactly four explicit manifolds , which contain pairs of essential punctured tori realizing and 6 respectively, we have . It follows immediately that if is atoroidal, while the manifolds obtained by - and -Dehn filling on are toroidal, then , and unless is one of the four examples mentioned above.

Let be the class of 3-manifolds such that is irreducible, atoroidal, and not a Seifert fibre space. By considering spheres, disks and annuli in addition to tori, we prove the following. Suppose that , where has a torus component , and . Let be slopes on such that . Then . The exterior of the Whitehead sister link shows that this bound is best possible.

     

Derivations, isomorphisms, and second cohomology of generalized Witt algebras

Generalized Witt algebras, over a field of characteristic , were defined by Kawamoto about 12 years ago. Using different notations from Kawamoto's, we give an essentially equivalent definition of generalized Witt algebras over , where the ingredients are an abelian group , a vector space over , and a map which is linear in the first variable and additive in the second one. In this paper, the derivations of any generalized Witt algebra
, with the right kernel of being , are explicitly described; the isomorphisms between any two simple generalized Witt algebras are completely determined; and the second cohomology group for any simple generalized Witt algebra is computed. The derivations, the automorphisms and the second cohomology groups of some special generalized Witt algebras have been studied by several other authors as indicated in the references.

     

Weak*-closedness of subspaces of Fourier-Stieltjes algebras and weak*-continuity of the restriction map

Let be a locally compact group and the Fourier-Stieltjes algebra of . We study the problem of how weak*-closedness of some translation invariant subspaces of is related to the structure of . Moreover, we prove that for a closed subgroup of , the restriction map from to is weak*-continuous only when is open in .

     

Equivalence of norms on operator space tensor products of -algebras

The Haagerup norm on the tensor product of two -algebras and is shown to be Banach space equivalent to either the Banach space projective norm or the operator space projective norm if and only if either or is finite dimensional or and are infinite dimensional and subhomogeneous. The Banach space projective norm and the operator space projective norm are equivalent on if and only if or is subhomogeneous.

     

Comultiplications on free groups and wedges of circles

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group . We consider individual comultiplications on and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of . For a comultiplication of we define a subset of quasi-diagonal elements which is basic to our investigation of associativity. The subset can be determined algorithmically and contains the set of diagonal elements . We show that is a basis for the largest subgroup of on which is associative and that is a free factor of . We also give necessary and sufficient conditions for a comultiplication on to be a coloop in terms of the Fox derivatives of with respect to a basis of . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group on the set of comultiplications of . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.

     

Factorisation in nest algebras. II

The main result of this paper is Theorem 5, which provides a necessary and sufficient condition on a positive operator for the existence of an operator in the nest algebra of a nest satisfying (resp. . In Section 3 we give a new proof of a result of Power concerning outer factorisation of operators. We also show that a positive operator has the property that there exists for every nest an operator in satisfying (resp. ) if and only if is a Fredholm operator. In Section 4 we show that for a given operator in there exists an operator in satisfying if and only if the range of is equal to the range of some operator in . We also determine the algebraic structure of the set of ranges of operators in . Let be the set of positive operators for which there exists an operator in satisfying . In Section 5 we obtain information about this set. In particular we discuss the following question: Assume and are positive operators such that and belongs to . Which further conditions permit us to conclude that belongs to ?

     

A new degree bound for vector invariants of symmetric groups

Let be a commutative ring, a finitely generated free -module and a finite group acting naturally on the graded symmetric algebra . Let denote the minimal number , such that the ring of invariants can be generated by finitely many elements of degree at most .

For and , the -fold direct sum of the natural permutation module, one knows that , provided that is invertible in . This was used by E. Noether to prove if .

In this paper we prove for arbitrary commutative rings and show equality for a prime power and or any ring with . Our results imply

for any ring with .

     

Morita equivalence for crossed products by Hilbert -bimodules

We introduce the notion of the crossed product of a -algebra by a Hilbert -bimodule . It is shown that given a -algebra which carries a semi-saturated action of the circle group (in the sense that is generated by the spectral subspaces and ), then is isomorphic to the crossed product . We then present our main result, in which we show that the crossed products and are strongly Morita equivalent to each other, provided that and are strongly Morita equivalent under an imprimitivity bimodule satisfying as Hilbert -bimodules. We also present a six-term exact sequence for -groups of crossed products by Hilbert -bimodules.

     

Generalized Hestenes' Lemma and extension of functions

Suppose we have an -jet field on which is a Whitney field on the nonsingular part of . We show that, under certain hypotheses about the relationship between geodesic and euclidean distance on , if the field is flat enough at the singular part , then it is a Whitney field on (the order of flatness required depends on the coefficients in the hypotheses). These hypotheses are satisfied when is subanalytic. In Section II, we show that a function on can be extended to one on if the differential goes to faster than the order of divergence of the principal curvatures of and if the first covariant derivative of is sufficiently flat. For the general case of functions with , we give a similar result for in Section III.

     

Widths of Subgroups

We say that the width of an infinite subgroup in is if there exists a collection of essentially distinct conjugates of such that the intersection of any two elements of the collection is infinite and is maximal possible. We define the width of a finite subgroup to be . We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic -manifolds satisfy the -plane property for some .

     

An intersection number for the punctual Hilbert scheme of a surface

We compute the intersection number between two cycles and of complementary dimensions in the Hilbert scheme parameterizing subschemes of given finite length of a smooth projective surface . The -cycle corresponds to the set of finite closed subschemes the support of which has cardinality 1. The -cycle consists of the closed subschemes the support of which is one given point of the surface. Since is contained in , indirect methods are needed. The intersection number is , answering a question by H. Nakajima.

     

Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Let denote the classical equilibrium distribution (of total charge ) on a convex or -smooth conductor in with nonempty interior. Also, let be any th order ``Fekete equilibrium distribution' on , defined by point charges at th order ``Fekete points'. (By definition such a distribution minimizes the energy for -tuples of point charges on .) We measure the approximation to by for by estimating the differences in potentials and fields,

both inside and outside the conductor . For dimension we obtain uniform estimates at distance from the outer boundary of . Observe that throughout the interior of (Faraday cage phenomenon of electrostatics), hence on the compact subsets of . For the exterior of the precise results are obtained by comparison of potentials and energies. Admissible sets have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions of equal point charges on . In that context there is an important open problem for the sphere which is discussed at the end of the paper.

     

Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups which can act properly discontinuously, isometrically, and cocompactly on hyperbolic -space in terms of the combinatorics of the action of on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the -sphere.

     

Approximation by polynomials with nonnegative coefficients and the spectral theory of positive operators

For a compact subset of symmetric with respect to conjugation and a continuous function, we obtain sharp conditions on and that insure that can be approximated uniformly on by polynomials with nonnegative coefficients. For a real Banach space, a closed but not necessarily normal cone with , and a bounded linear operator with , we use these approximation theorems to investigate when the spectral radius of belongs to its spectrum . A special case of our results is that if is a Hilbert space, is normal and the 1-dimensional Lebesgue measure of is zero, then . However, we also give an example of a normal operator (where is unitary and ) for which and .

     

Poisson transforms on vector bundles

Let be a connected real semisimple Lie group with finite center, and a maximal compact subgroup of . Let be an irreducible unitary representation of , and the associated vector bundle. In the algebra of invariant differential operators on the center of the universal enveloping algebra of induces a certain commutative subalgebra . We are able to determine the characters of . Given such a character we define a Poisson transform from certain principal series representations to the corresponding space of joint eigensections. We prove that for most of the characters this map is a bijection, generalizing a famous conjecture by Helgason which corresponds to the trivial representation.

     

On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with and nonbirational bicanonical map. Our main result is that, if is such a surface and if is minimal with no pencil of curves of genus , then is the symmetric product of a curve of genus , and therefore and . Furthermore we obtain some results towards the classification of minimal surfaces with . Such surfaces have , and we show that if and only if is the symmetric product of a curve of genus . We also classify the minimal surfaces with with a pencil of curves of genus , proving in particular that for those one has .