Suppose μ and ν are Borel measures on locally compact spaces X and Y, respectively. A product measure λ can be defined on the Borel sets of X x Y by the formula λ(M) = ∫ν(Mx) dμ, provided that vertical cross section measure ν(Mx) is a measurable function in x. Conditions are summarized for ν(Mx) to be measurable as a function in x, and examples are given in which the function ν(Mx) is not measurable. It is shown that a dense, countably compact set fails to be a Borel set if it contains no nonempty zero set.

We consider the following problem arising in agricultural statistics. Suppose that a large number of plants are set out on a regular grid, which may be triangular, square or hexagonal, and that among these plants, half are to be given one and half the other of two possible treatments. For the sake of statistical balance, we require also that, if one plant in every k plants has i of its immediate neighbours receiving the same treatment as itself, then k is constant over all possible values of i. For square and triangular grids, there exist balanced arrays of finite period in each direction, but for the hexagonal grid, we show that no such balanced array can exist. Several related questions are discussed.

Let k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v)–X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C2, PSL(2, 2n) or pΣL(2, 52a+1), where n≥2 and a ≥ 1.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20

Generalizations are proved for theorems of Carathéodory (1907), Kirchberger (1903) and Watson (1973), the theme of these results being how thickly the convex hull of a family of points is covered by simplexes whose vertices are chosen from the points of the family.

Subject classification (Amer. Math. Soc. (MOS) 1970): 52A35.

A recent result of Rogozin on the relative stability of a distribution function is extended, by giving equivalences for relative stability in terms of truncated moments of the distribution and in terms of the real and imaginary parts of the characteristic function. As an application, the known results on centering distributions in the domain of attraction of a stable law are extended to the case of stochastically compact distributions.

We show that the complete symmetric digraph DKn, n≧5, can be decomposed into each of the four oriented pentagons if and only if n ≡ 0 or 1 (mod 5).

For a class of primes p ≡ 1 (mod 20) for which 2 is a quintic nonresidue, unambiguous evaluations of parameters of bidecic Jacobi and Jacobsthal sums (mod p) are presented, in terms of the partition p = a2+5b2+5c2+5d2, ab = d2–c2–cd. Similar results for sums of others orders have been obtained by E. Lehmer and by K. S. Williams.

Subject classification (Amer. Math. Soc. (MOS) 1970): 10 G 05.

This paper extends results on product varieties of groups to inverse semigroups. We show that if is a variety of groups and any inverse semigroup variety, then ∘ is a variety. We give a characterization of the identities of∘in terms of the identities of and ofWe show that if does not contain the variety of all groups then it has uncountably many supervarieties. Finally we show that ifis another variety of groups then

Subject classifiaction (Amer. Math. Soc. (MOS) 1970): 20 M 05.

Let E, F be sequence spaces and A an infinite matrix that maps E to F. Sufficient conditions are given so that the transposed matrix maps Fβ to Eβ.

In this note we determine which of the trees homeomorphic to a star have a spectrum consisting entirely of integers. We also specify the integral double stars, and we consider the problem of trees with more complicated structure.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 05.

The extended diagram of a two-bridged knot is introduced, and it is shown how the coefficients of the Alexander polynomial of the knot may be read straight from this diagram. Using this result, it is shown by diagram manipulation that a conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is true at least for two-bridged knots (which are all alternating).

1980 Mathematics subject classification (Amer. Math. Soc.): 57 M 25.

I am grateful to W. Taylor for pointing out that Lemma 2.3 of Neumann (1978), which claimed that a variety and its finitary reduct have the same finite algebras, is incorrect. This weakens several results of the paper: the lattice homomorphism of Lemma 3.1 is not complete, Theorem 4.1 is false, and the Mal'cev conditions claimed to be strong in 6.1, 6.2 and 7.1 are not strong. Taylor (1980) shows one cannot improve on this.

A permutation π of the set {1, 2, …, n} is four-discordant if π(i) ≢ i, i+ 1, i + 2, i +3 (mod n) for 1 ≦i ≦ n. Generating functions for rook polynomials associated with four-discordant permutations are derived. Hit polynomials associated with four-discordant permutations are studied. Finally, it is shown that the leading coefficients of these rook polynomials form a “tribonacci” sequence which is a generalized Fibonacci sequence.

A class of cyclically presented groups with n generators and n Fibonacci type relations is discussed. Conditions are given for these groups to be finite and metacyclic. With these conditions the presentations are reduced to the standard form for metacyclic groups with trivial Schur multiplicator. This enables certain ismorphisms between the groups to be found.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 F 05.

It is proved that a finite metabelian group G is determined by its group ring KG where the ring K satisfies the following conditions.