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A trickle infection experiment was undertaken to study in detail the population dynamics of Oesophagostomum dentatum in pigs. Three groups of 32 pigs were inoculated via the feed twice weekly with 100 (Group A), 1000 (Group B) or 10000 (Group C) O. dentatum infective larvae (L3). Five pigs from each group were killed 2, 4, 8, 12, 16, and 20 weeks after the first inoculation (p.i.) to determine their worm burdens. Weekly faecal egg counts were determined. At slaughter, worms were counted, differentiated according to sex and developmental stage, and their length measured. Faecal egg counts ranked with dose rate until week 15, but later were more variable. The proportion of the total number of L3 administered which were recovered at slaughter inversely ranked with dose rate. In group C it decreased over time, whereas in groups A and B there was no consistent pattern. Worm fecundities (epg/female) in groups A and B were higher than in group C. The lengths of the female worms increased over time, whereas the lengths of the male worms remained approximately constant from week 8. The study suggests reduced establishment of incoming larvae and lower fecundity of the female worms at high dose levels.

The effect of two types of diet on populations of Ascaris suum and Oesophagostomum dentatum in experimentally infected pigs Parasitology (1995), 111, 395–402Part of the reference list that was omitted from p. 402 appears below.

Let Zn denote the integer lattice in Rn, let A be a non-singular n × n matrix and ʗ ∈ Rn. Then G = AZn + ʗ is called a grid (non-homogeneous lattice) and its determinant det G is defined to be |det A|.

A new arithmetic invariant E(f) is defined for integral binary quadratic forms f. It has the property that, denoting by fm the norm-form of a quadratic number field Q(m), E(fm) < 1 if and only if Q(m) has class number one.

A convex polytope D (α) was defined in Barnes (1978) as the set of Minkowski-reduced forms with prescribed diagonal coefficients α1, α2,…αn. A local minimum of the determinant D(f) over D(α) must occur at a vertex of D(α). Here a criterion is obtained for a given vertex to provide a local minimum, completely analogous to Voronoï's criterion for a perfect form to be extreme.

Minkowski established a lower bound for the determinant D of a Minkowski-reduced quadratic form in terms of the product of its diagonal coefficients aii (i = 1, …n). Oppenheim and Barnes found, for n = 3 and n = 4 respectively, the precise minimum of D in terms of the aii; in each case the minimum is a polynomial in the aii. Here it is shown that no such result exists when n = 5; however a polynomial in all,…, a55 is determined which gives the minimum of D when a55 is sufficiently large.

A fundamental region for the reduction of positive quaternary quadratic forms is exhibited. It is a convex polyhedral cone with twelve edges in the 10-dimensional space of quaternary quadratic forms.

Forms which are reduced in the sense of Minkowski satisfy the “fundamental inequality” a11a22 hellipann≤λnD; the best possible value of λn is known for n≤5. A more precise result for the minimum value of D in terms of the diagonal coefficients has been stated by Oppenheim for ternary forms. The corresponding precise result for quaternary forms is established here by considering a convex polytope D(α), defined as the intersection of the cone of reduced forms with the hyperplanes aii = αi (i = 1, hellip n).

Let f(x, y, z) be an indefinite ternary quadratic form of signature (2, 1) and determinant d ≠ 0. Davenport [3] has shown that there exist integral x, y, z with, the equality sign being necessary if and only if f is a positive multiple of f1(x, y, z) = x2 + yz.

Let be a positive definite quadratic form of determinant D, and let M be the minimum of f(x) for integral x ≠ 0. Then we set and the maximum being over all positive forms f in n variables. f is said to be extreme if y γn(f) is a local maximum for varying f, absolutely extreme if y γ(f) is an absolute maximum, i.e. if y γ(f) = γn.

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

All extreme lattice-coverings ofn–space by spheres are known forn≦ 4; see for example [3]. Only one class of extreme covering is known for largen, namely that corresponding to the quadratic formthis was first shown to be extreme for alln≧ 2 by Bleicher [2].

It is well known that the problem of determining the most economical covering ofn-dimensional Eucidean space, by equal spheres whose centres form a lattice, may be formulated in terms of positive definite quadratic forms, as follows: Letbe positive definite, andFor real α, set(the minimum being taken over integralx)

A positive quadratic form, of determinantand minimum M for integral, is said to be extreme if the ratiois a (local) maximum for small variations in the coefficients.Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.

Let θ be an algebraic number of degree n, with conjugates θ(1), …, θ(n), where θ(1), …, θ(r) are real and θ(r+j), θ(r+s+j) are complex conjugates for j = 1, …, s. [Here r ≥ 0, s ≥ 0, r + 2s = n.] Let ω1, …, ωn be a basis for the integers of k(θ), and set

If f(x) is a real indefinite quadratic form in n variables with determinant d ≠ 0, we set for any real α.