30 research outputs found
An Electrostatics Problem on the Sphere Arising from a Nearby Point Charge
For a positively charged insulated d-dimensional sphere we investigate how
the distribution of this charge is affected by proximity to a nearby positive
or negative point charge when the system is governed by a Riesz s-potential
1/r^s, s>0, where r denotes Euclidean distance between point charges. Of
particular interest are those distances from the point charge to the sphere for
which the equilibrium charge distribution is no longer supported on the whole
of the sphere (i.e. spherical caps of negative charge appear). Arising from
this problem attributed to A. A. Gonchar are sequences of polynomials of a
complex variable that have some fascinating properties regarding their zeros.Comment: 44 pages, 9 figure
Riesz external field problems on the hypersphere and optimal point separation
We consider the minimal energy problem on the unit sphere in
the Euclidean space in the presence of an external field
, where the energy arises from the Riesz potential (where is the
Euclidean distance and is the Riesz parameter) or the logarithmic potential
. Characterization theorems of Frostman-type for the associated
extremal measure, previously obtained by the last two authors, are extended to
the range The proof uses a maximum principle for measures
supported on . When is the Riesz -potential of a signed
measure and , our results lead to explicit point-separation
estimates for -Fekete points, which are -point configurations
minimizing the Riesz -energy on with external field . In
the hyper-singular case , the short-range pair-interaction enforces
well-separation even in the presence of more general external fields. As a
further application, we determine the extremal and signed equilibria when the
external field is due to a negative point charge outside a positively charged
isolated sphere. Moreover, we provide a rigorous analysis of the three point
external field problem and numerical results for the four point problem.Comment: 35 pages, 4 figure
Minimum Riesz energy problems for a condenser with "touching plates"
Minimum Riesz energy problems in the presence of an external field are
analyzed for a condenser with touching plates. We obtain sufficient and/or
necessary conditions for the solvability of these problems in both the
unconstrained and the constrained settings, investigate the properties of
minimizers, and prove their uniqueness. Furthermore, characterization theorems
in terms of variational inequalities for the weighted potentials are
established. The results obtained are illustrated by several examples.Comment: 32 pages, 1 figur
Log-optimal (d+2)-configurations in d-dimensions
We enumerate and classify all stationary logarithmic configurations of d+2
points on the unit (d-1)-sphere in d-dimensions. In particular, we show that
the logarithmic energy attains its relative minima at configurations that
consist of two orthogonal to each other regular simplexes of cardinality m and
n. The global minimum occurs when m=n if d is even and m=n+1 otherwise. This
characterizes a new class of configurations that minimize the logarithmic
energy on the (d-1)-sphere for all d. The other two classes known in the
literature, the regular simplex and the cross polytope, are both universally
optimal configurations.Comment: 17 page
Energy bounds for codes and designs in Hamming spaces
We obtain universal bounds on the energy of codes and for designs in Hamming
spaces. Our bounds hold for a large class of potential functions, allow unified
treatment, and can be viewed as a generalization of the Levenshtein bounds for
maximal codes.Comment: 25 page
Ping Pong Balayage and Convexity of Equilibrium Measures
In this presentation we prove that the equilibrium measure of a finite union of intervals on the real line or arcs on the unit circle has convex density. This is true for both, the classical logarithmic case, and the Riesz case. The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of many “electrons” will have convex density on every subinterval. Applications to external field problems and constrained energy problems are presented
Energy bounds for spherical codes, test functions and LP optimality
We derive universal lower bounds for the potential energy of spherical codes, that are optimal in the framework of Delsarte-Yudin linear programming method. Our bounds are universal in the sense of both Levenshtein and Cohn-Kumar; i.e., they are valid for any choice of dimension and code cardinality and they apply to any absolutely monotone potential. We further discuss a characterization on when the lower bounds are LP-optimal, that is they are the best possible in terms of the linear programming approach. Finally, we present the analogous results for codes in projective spaces
Universal lower bounds for potential energy of spherical codes
Based upon the works of Delsarte-Goethals-Seidel, Levenshtein, Yudin, and Cohn-Kumar we derive universal lower bounds for the potential energy of spherical codes, that are optimal (in the framework of the standard linear programming approach) over a certain class of polynomial potentials whose degrees are upper bounded via a familiar formula for spherical designs. We classify when improvements are possible employing polynomials of higher degree. Our bounds are universal in the sense of Cohn and Kumar; i.e., they apply whenever the potential is given by an absolutely monotone function of the inner product between pairs of points
Orthogonal Polynomials on the Real Line and External Field Problem
Studying the asymptotics of orthogonal polynomials on the real line leads to an external field problem. We will introduce the problem and present applications. Weighted versions of the capacity, transfinite diameter, and Chebyshev constant will be considered
On a Classical Theorem of Potential Theory in the Complex Plane
We shall provide the detailed proof of the fundamental theorem in classical potential theory in the plane, that the capacity of a compact set, its transfinite diameter, and the Chebyshev constant are all equal (the theorem was introduced in my colloquium talk last Thursday). Certain applications will be provided