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.

     

The quantum analog of a symmetric pair: a construction in type

Let be the ideal in the enveloping algebra of generated by the maximal compact subalgebra of . In this paper we construct an analog of in the quantized enveloping algebra corresponding to a type diagram at generic . We find generators for and explicit bases for .

     

A -deformation of a trivial symmetric group action

Let be the arrangement of hyperplanes consisting of the reflecting hyperplanes for the root system . Let be the Varchenko matrix for this arrangement with all hyperplane parameters equal to . We show that is the matrix with rows and columns indexed by permutations with entry equal to where is the number of inversions of . Equivalently is the matrix for left multiplication on by

Clearly commutes with the right-regular action of on . A general theorem of Varchenko applied in this special case shows that is singular exactly when is a root of for some between and . In this paper we prove two results which partially solve the problem (originally posed by Varchenko) of describing the -module structure of the nullspace of in the case that is singular. Our first result is that

in the case that where Lie denotes the multilinear part of the free Lie algebra with generators. Our second result gives an elegant formula for the determinant of restricted to the virtual -module with characteristic the power sum symmetric function .

     

The behavior of the heat operator on weighted Sobolev spaces

Denoting by the heat operator in , we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which is an injection, a surjection, or an isomorphism. We also describe the range and kernel of in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in .

     

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).

     

Strongly self-absorbing -algebras

Say that a separable, unital -algebra is strongly self-absorbing if there exists an isomorphism such that and are approximately unitarily equivalent -homomorphisms. We study this class of algebras, which includes the Cuntz algebras , , the UHF algebras of infinite type, the Jiang-Su algebra and tensor products of with UHF algebras of infinite type. Given a strongly self-absorbing -algebra we characterise when a separable -algebra absorbs tensorially (i.e., is -stable), and prove closure properties for the class of separable -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

     

On differential equations for Sobolev-type Laguerre polynomials

The Sobolev-type Laguerre polynomials are orthogonal with respect to the inner product

where , and . In 1990 the first and second author showed that in the case and the polynomials are eigenfunctions of a unique differential operator of the form

where are independent of . This differential operator is of order if is a nonnegative integer, and of infinite order otherwise. In this paper we construct all differential equations of the form

where the coefficients , and are independent of and the coefficients , and are independent of , satisfied by the Sobolev-type Laguerre polynomials . Further, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise. Finally, we show that in the case and the polynomials are eigenfunctions of a linear differential operator, which is of order if is a nonnegative integer and of infinite order otherwise.

     

Length, multiplicity, and multiplier ideals

Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

     

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.

     

Rumely's local global principle for algebraic PC fields over rings

Let be a finite set of rational primes. We denote the maximal Galois extension of in which all totally decompose by . We also denote the fixed field in of elements in the absolute Galois group of by . We denote the ring of integers of a given algebraic extension of by . We also denote the set of all valuations of (resp., which lie over ) by (resp., ). If , then denotes the ring of integers of a Henselization of with respect to . We prove that for almost all , the field satisfies the following local global principle: Let be an affine absolutely irreducible variety defined over . Suppose that for each and for each . Then . We also prove two approximation theorems for .

     

Boundary value maps, Szegö maps and intertwining operators

We consider one series of unitarizable representations, the cohomological induced modules with dominant regular infinitesimal character. The minimal -type of determines a homogeneous vector bundle . The derived functor modules can be realized on the solution space of a first order differential operator on . Barchini, Knapp and Zierau gave an explicit integral map from the derived functor module, realized in the Langlands classification, into the space of smooth sections of the vector bundle . In this paper we study the asymptotic behavior of elements in the image of . We obtain a factorization of the standard intertwining opeartors into the composition of the Szegö map and a passage to boundary values.

     

A growth dichotomy for o-minimal expansions of ordered groups

Let be an o-minimal expansion of a divisible ordered abelian group with a distinguished positive element . Then the following dichotomy holds: Either there is a -definable binary operation such that is an ordered real closed field; or, for every definable function there exists a -definable with . This has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) -definable groups with underlying set .

     

Tauberian theorems and stability of solutions of the Cauchy problem

Let be a bounded, strongly measurable function with values in a Banach space , and let be the singular set of the Laplace transform in . Suppose that is countable and uniformly for , as , for each in . It is shown that

as , for each in ; in particular, if is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on , and it implies several results concerning stability of solutions of Cauchy problems.

     

Group actions on arrangements of linear subspaces and applications to configuration spaces

For an arrangement of linear subspaces in that is invariant under a finite subgroup of the general linear group we develop a formula for the -module structure of the cohomology of the complement . Our formula specializes to the well known Goresky-MacPherson theorem in case , but for the formula shows that the -module structure of the complement is not a combinatorial invariant. As an application we are able to describe the free part of the cohomology of the quotient space . Our motivating examples are arrangements in that are invariant under the action of by permuting coordinates. A particular case is the ``-equal' arrangement, first studied by Björner, Lovász, and Yao motivated by questions in complexity theory. In these cases and are spaces of ordered and unordered point configurations in many of whose properties are reduced by our formulas to combinatorial questions in partition lattices. More generally, we treat point configurations in and provide explicit results for the ``-equal' and the ``-divisible' cases.

     

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.

     

Invariant ideals and polynomial forms

Let denote the group algebra of an infinite locally finite group . In recent years, the lattice of ideals of has been extensively studied under the assumption that is simple. From these many results, it appears that such group algebras tend to have very few ideals. While some work still remains to be done in the simple group case, we nevertheless move on to the next stage of this program by considering certain abelian-by-(quasi-simple) groups. Standard arguments reduce this problem to that of characterizing the ideals of an abelian group algebra stable under the action of an appropriate automorphism group of . Specifically, in this paper, we let be a quasi-simple group of Lie type defined over an infinite locally finite field , and we let be a finite-dimensional vector space over a field of the same characteristic . If acts nontrivially on by way of the homomorphism , and if has no proper -stable subgroups, then we show that the augmentation ideal is the unique proper -stable ideal of when . The proof of this result requires, among other things, that we study characteristic division rings , certain multiplicative subgroups of , and the action of on the group algebra , where is the additive group . In particular, properties of the quasi-simple group come into play only in the final section of this paper.

     

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.

     

On zeta functions and Iwasawa modules

We study the relation between zeta-functions and Iwasawa modules. We prove that the Iwasawa modules for almost all determine the zeta function when is a totally real field. Conversely, we prove that the -part of the Iwasawa module is determined by its zeta-function up to pseudo-isomorphism for any number field Moreover, we prove that arithmetically equivalent CM fields have also the same -invariant.

     

Small subalgebras of Steenrod and Morava stabilizer algebras

Let (resp. be the subalgebra of the Steenrod algebra (resp. th Morava stabilizer algebra) generated by reduced powers , (resp. , . In this paper we identify the dual of (resp. , for with some Frobenius kernel (resp. -points) of a unipotent subgroup of the general linear algebraic group . Using these facts, we get the additive structure of for odd primes.