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.

     

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 .

     

Extremal vectors and invariant subspaces

For a bounded linear operator on Hilbert space we define a sequence of so-called minimal vectors in connection with invariant subspaces and show that this presents a new approach to invariant subspaces. In particular, we show that for any compact operator some weak limit of the sequence of minimal vectors is noncyclic for all operators commuting with and that for any normal operator , the norm limit of the sequence of minimal vectors is noncyclic for all operators commuting with . Thus, we give a new and more constructive proof of existence of invariant subspaces. The sequence of minimal vectors does not seem to converge in norm for an arbitrary bounded linear operator. We will prove that if belongs to a certain class of operators, then the sequence of such vectors converges in norm, and that if belongs to a subclass of , then the norm limit is cyclic.

     

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.

     

The Classification of the Simple Modular Lie Algebras: VI. Solving the Final Case

We investigate the structure of simple Lie algebras over an algebraically closed field of characteristic . Let denote a torus in the -envelope of in of maximal dimension. We classify all for which every 1-section with respect to every such torus is solvable. This settles the remaining case of the classification of these algebras.

     

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.

     

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 ?

     

Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary

Let be a second order elliptic differential operator on a Riemannian manifold with no zero order terms. We say that a function is -harmonic if . Every positive -harmonic function has a unique representation

where is the Martin kernel, is the Martin boundary and is a finite measure on concentrated on the minimal part of . We call the trace of on . Our objective is to investigate positive solutions of a nonlinear equation

for [the restriction is imposed because our main tool is the -superdiffusion, which is not defined for ]. We associate with every solution of (*) a pair , where is a closed subset of and is a Radon measure on . We call the trace of on . is empty if and only if is dominated by an -harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. In an earlier paper, we investigated the case when is a second order elliptic differential operator in and is a bounded smooth domain in . We obtained necessary and sufficient conditions for a pair to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators with bounded coefficients in an arbitrary bounded domain of , assuming only that the Martin boundary and the geometric boundary coincide.

     

Baire and

Let be a locally compact Hausdorff space and let be the Banach space of all bounded complex Radon measures on . Let and be the -rings generated by the compact subsets and by the compact subsets of , respectively. The members of are called Baire sets of and those of are called -Borel sets of (since they are precisely the -bounded Borel sets of ). Identifying with the Banach space of all Borel regular complex measures on , in this note we characterize weakly compact subsets of in terms of the Baire and -Borel restrictions of the members of . These characterizations permit us to give a generalization of a theorem of Dieudonné which is stronger and more natural than that given by Grothendieck.

     

Test ideals in quotients of -finite regular local rings

Let be an -finite regular local ring and an ideal contained in . Let . Fedder proved that is -pure if and only if . We have noted a new proof for his criterion, along with showing that , where is the pullback of the test ideal for . Combining the the -purity criterion and the above result we see that if is -pure then is also -pure. In fact, we can form a filtration of , that stabilizes such that each is -pure and its test ideal is . To find examples of these filtrations we have made explicit calculations of test ideals in the following setting: Let , where is either a polynomial or a power series ring and is generated by monomials and the are regular. Set . Then .

     

Similarity to a contraction, for power-bounded operators with finite peripheral spectrum

Suppose is a power-bounded linear opertor on a Hilbert space with finite peripheral spectrum (spectrum on the unit circle). Several sufficient conditions are given for to be similar to a contraction. A natural growth condition on the resolvent in half-planes tangent to the unit circle at the peripheral spectrum is shown to be equivalent to having an functional calculus, for some open polygon contained in the unit disc, which, in turn, is equivalent to being similar to a contraction with numerical range contained in a closed polygon in the closed unit disc. Having certain orbits of be square summable also implies that is similar to a contraction.

     

Picard groups and infinite matrix rings

We describe a connection between the Picard group of a ring with local units and the Picard group of the unital overring . Using this connection, we show that the three groups , , and are isomorphic for any unital ring . Furthermore, each element of arises from an automorphism of , which yields an isomorphsm between and . As one application we extend a classical result of Rosenberg and Zelinsky by showing that the group is abelian for any commutative unital ring .

     

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 .

     

Local Boundary Regularity of the Szego Projection and Biholomorphic Mappings of Non-Pseudoconvex Domains

It is shown that the Szeg\H{o} projection of a smoothly bounded domain , not necessarily pseudoconvex, satisfies local regularity estimates at certain boundary points, provided that condition holds for . It is also shown that any biholomorphic mapping between smoothly bounded domains extends smoothly near such points, provided that a weak regularity assumption holds for .

     

On transversality with deficiency and a conjecture of Sard

Let be a map between manifolds and a manifold. In this paper, by using the Sard theorem, we study the topological properties of the space of maps which satisfy a certain transversality condition with respect to in a weak sense. As an application, by considering the case where is a point, we obtain some new results about the topological properties of , where is the set of points of where the rank of the differential of is less than or equal to . In particular, we show a result about the topological dimension of , which is closely related to a conjecture of Sard concerning the Hausdorff measure of .

     

Stability of multiple-pulse solutions

In this article, stability of multiple-pulse solutions in semilinear parabolic equations on the real line is studied. A system of equations is derived which determines stability of -pulses bifurcating from a stable primary pulse. The system depends only on the particular bifurcation leading to the existence of the -pulses.

As an example, existence and stability of multiple pulses are investigated if the primary pulse converges to a saddle-focus. It turns out that under suitable assumptions infinitely many -pulses bifurcate for any fixed . Among them are infinitely many stable ones. In fact, any number of eigenvalues between 0 and in the right half plane can be prescribed.