Leopold Flatto

A limit theorem for random coverings of a circle

LetNα, m equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that limα→0P(Nα,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutels resultE(Nα,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Eulers constant.

The converse of Gauss's theorem for harmonic functions

where da(v) denotes the element of area on the unit sphere and w, denotes the total area of the unit sphere; I is any positive number such that the sphere of radius t and center x lies inside R. (For a proof, see [6], p. 223). Conversely if f(x) is continuous and (1.1) holds, then f(x) is harmonic ([6], 224). In this paper we concern ourselves with the following question. To what extent can Condition 1.1) be relaxed without invalidating the conclusion that f(x) be harmonic ? We assume that R is the whole space En as the results are somewhat simpler to state for this case. We study the following two problems.

A quadrature formula of degree three

Abstract Let R be a region in n -space and Q a linear quadrature formula for R of the form (f)= ∑ r=1 k r f(x r ) . It is known that if Q(ƒ) = ∝ R ƒ whenever ƒ is a polynomial of degree 3 or lower, then k ⩾ n + 1. It is known that the minimum possible value of k depends on the region R , being 2 n for the n -cube and n + 2 for the n -simplex ( n > 1). In 1956 Hammer and Stroud conjectured that k ⩾ n + 2 for every R , when n > 1. In this paper we construct an R , and a Q with the required property, with k = n + 1.