In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n-designs, where a spherical n-design is a set of m points on the unit sphere S 2 ⊂ R 3 that gives an equal weight cubature rule (or equal weight numerical integration rule) on S 2 which is exact for spherical polynomials of degree n. (A sequence of m-point spherical n-designs X on S 2 is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X ∈ the minimum spherical distance between points is bounded from below by λ √ m .) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n 2 ), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S 2 .Keywords acceleration of convergence · Coulomb energy · Coulomb potential · equal weight cubature · equal weight numerical integration · orthogonal polynomials · sphere · spherical designs · well separated point sets on sphere Mathematics Subject Classifications (2000)Primary: 31C20 · 42C10 · Secondary: 41A55 · 42C20 · 65B10 · 65D32 Communicated by: Tomas Sauer.Dedicated to Edward B. Saff on the occasion of his 60th birthday.

The algorithm devised by Feige and Schechtman for partitioning higher dimensional spheres into regions of equal measure and small diameter is combined with David and Christ's construction of dyadic cubes to yield a partition algorithm suitable to any connected Ahlfors regular metric measure space of finite measure.

This paper adapts the techniques of finite element exterior calculus to study and discretize the abstract Hodge-Dirac operator, which is a square root of the abstract Hodge-Laplace operator considered by Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281-354]. Dirac-type operators are central to the field of Clifford analysis, where recently there has been considerable interest in their discretization. We prove a priori stability and convergence estimates, and show that several of the results in finite element exterior calculus can be recovered as corollaries of these new estimates.

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This paper examines a pair of bent functions on Z 2m 2 and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical basis for the real representation of the Clifford algebra R m,m . Some other necessary conditions are also briefly examined. IntroductionA recent paper [11] constructs a sequence of edge-coloured graphs ∆ m (m 1) with two edge colours, and makes the conjecture that for m 1, there is an automorphism of ∆ m that swaps the two edge colours. This conjecture can be refined into the following question. Question 1.1. Consider the sequence of edge-coloured graphs ∆ m (m 1) as defined in [11], each with red subgraph ∆ m [−1], and blue subgraph ∆ m [1]. For which m 1 is there an automorphism of ∆ m that swaps the subgraphs ∆ m [−1] and ∆ m [1]?Note that the existence of such an automorphism automatically implies that the subgraphs ∆ m [−1] and ∆ m [1] are isomorphic.Considering that it is known that ∆ m [−1] is a strongly regular graph, a more general question can be asked concerning such graphs.