The Emden-Fowler equation x″(t) + a(t)|x|γ sgn x = 0, t ≧ 0, is said to be in the sublinear or superlinear case according to whether γ < 1 or γ > 1. Conditions on a(t) are given to ensure local uniqueness of solutions in the sublinear case and continuability of solutions in the superlinear case. Boundedness of solutions is also studied.

A notion of a normal autometrized algebra is introduced which generalises the concepts of Boolean geometry. Brouwerian geometry, autometrized lattice ordered groups, semi-Brouwerian geometry, etc. The notions of ideals and congruence relations are introduced in normal autometrized algebras and a one to one correspondence between ideals and congruence relations is established. Some other common properties of the above geometries are also obtained for normal autometrized algebras.

In this paper the following result is proved. Suppose there exists a C-matrix of order n + 1. Then if n≡1 (mod 4) there exists a Hadamard matrix of order 2nr(n + 1), while if n≡3 (mod 4) there exists a Hadamard matrix of order nr(n + 1) for all r ≧0. If n≡1 (mod 4) is a prime power, the method is adapted to prove the existence of a Hadamard matrix of the Williamson type, of order 2nr(n + 1), for all r ≧0.

The powerful starter-adder theorems for constructing Howell Designs are improved and consequently many types of Howell Designs that previously could only be constructed by multiplicative techniques are shown amenable to a modified starter-adder method. The existence question for Howell Designs of many new types H(s, 2n) is settled affirmatively. For prime powers pn, p ≧ 7, we reduce the entire existence question for designs of type H*(pn, 2r), pn + 1 ≦ 2r ≦ 2pn to the corresponding question for designs of type H*(p, 2m), p + 1 ≦ 2m≦2p. If these designs exist, s has no prime divisors ≤ 7 and t odd is “close” to 1, a design H * (s, s + t) is shown to exist.