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.

     

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).

     

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.

     

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.

     

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.

     

Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction

For each Gorenstein cover of degree we define a scheme and a generically finite map of degree called the discriminant of . Using this construction we deal with smooth degree covers with . Moreover we also generalize the trigonal construction of S. Recillas.

     

Lower bounds for dimensions of representation varieties

The set of -dimensional complex representations of a finitely generated group form a complex affine variety . Suppose that is such a representation and consider the associated representation on complex matrices obtained by following with conjugation of matrices. Then it is shown that the dimension of at is at least the difference of the complex dimensions of and . It is further shown that in the latter cohomology may be replaced by various proalgebraic groups associated to and .

     

A weak-type inequality for differentially subordinate harmonic functions

Assuming an extra condition, we decrease the constant in the sharp inequality of Burkholder for two harmonic functions and . That is, we prove the sharp weak-type inequality under the assumptions that , and the extra assumption that . Here is the harmonic measure with respect to and the constant is the one found by Davis to be the best constant in Kolmogorov's weak-type inequality for conjugate functions.

     

Singular limit of solutions of the porous medium equation with absorption

We prove that as the solutions of , , , , , , , converges in to the solution of the ODE , , where , , satisfies in for some function , , satisfying whenever for a.e. , for and for , where is a constant and is any measurable subset of .

     

The irrationality of

We shall show that the numbers and
are linearly independent over for any natural number . The key is to construct explicit Padé-type approximations using Legendre-type polynomials.

     

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 .

     

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.

     

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.

     

The trace of jet space

The classical Whitney extension theorem describes the trace of the space of -jets generated by functions from to an arbitrary closed subset . It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space of functions whose higher derivatives satisfy the Zygmund condition with majorant . The main result states that the vector function belongs to the corresponding trace space if the trace to every subset of cardinality , where , can be extended to a function and . The number generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset . The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.

     

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 .

     

Minimizing the Laplacian of a function squared with prescribed values on interior boundaries- Theory of polysplines

In this paper we consider the minimization of the integral of the Laplacian of a real-valued function squared (and more general functionals) with prescribed values on some interior boundaries , with the integral taken over the domain D. We prove that the solution is a biharmonic function in except on the interior boundaries , and satisfies some matching conditions on . There is a close analogy with the one-dimensional cubic splines, which is the reason for calling the solution a polyspline of order 2, or biharmonic polyspline. Similarly, when the quadratic functional is the integral of a positive integer, then the solution is a polyharmonic function of order for , satisfying matching conditions on , and is called a polyspline of order . Uniqueness and existence for polysplines of order , provided that the interior boundaries are sufficiently smooth surfaces and , is proved. Three examples of data sets possessing symmetry are considered, in which the computation of polysplines is reduced to computation of one-dimensional splines.

     

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 .

     

Geometry of families of nodal curves on the blown-up projective plane

Let be the projective plane blown up at generic points. Denote by the strict transform of a generic straight line on and the exceptional divisors of the blown-up points on respectively. We consider the variety of all irreducible curves in with nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For we give the complete answer concerning the existence of nodal curves in .