The relationship between certain non-associative algebras and the deterministic theory of population genetics was first investigated by Etherington (3)-(8), who defined the concepts of baric, train and special train algebras. Gonshor (10) dealt with, among other topics, algebras corresponding to autopolyploidy, on the assumption that chromosome segregation operated. In this paper [ discuss algebras corresponding to more general systems of inheritance among polyploids, which have been discussed without using algebras by Haldane (11), Geiringer (9), Moran (13) and Seyffert (16). These algebras are special cases of what I have defined as segregation algebras, and mixtures of them. All the algebras corresponding to a fixed ploidy have a relationship which I have called special isotopy. An example shows that algebras arise in other genetic systems which are not isotopic to segregation algebras.

In a recent paper (13), we introduced the class of strongly E-reflexive inversesemigroups. This class was shown to coincide with the class of those inverse semigroups which are semilattices of E-unitary inverse semigroups. In particular, therefore, E-unitary inverse semigroups and semilattices of groups are strongly E-reflexive, and in fact so are subdirect products of these two types of semigroups.

The conception of the integral of one function with respect to another was introduced by Stieltjes in his classical memoir on continued fractions. He denned the integral as

and gave the formula for integration by parts.

We may develop the idea of principal lines at any point on a curve of (n−1)-triple curvature geometrically in the following way:

Two consecutive points on the curve determine the tangent, three consecutive points the osculating points, four consecutive points the osculating 3-space and so on, at any point on the curve. At the same point we have an (n−1)-space perpendicular to the tangent and we shall call this space the first normal space at the point; the intersection of the first normal space with the osculating plane is a line which we shall name as the first normal at the point. Similarly all lines perpendicular to the osculating plane determine an (n−2)-space, the second normal space at the point, and the inter-section of this space with the osculating 3-space is the second normal at the point. Proceeding thus we have lastly the (n−1)th normal which is perpendicular to the osculating (n−1)-space at the point. We thus see that the rth normal lies in the osculating (r+1)-space and is perpendicular to r consecutive tangents. These n−1 normals with the tangent constitute the n principal lines at the point which are mutually orthogonal.

In the second part of Professor Chrystal's Algebra (Exer. V., No. 4) the following exercise is given:-

If 2xyz-x2-y2-z2+1=0, and x,y,z are all real, then all, or none, of the quantities x,y,z lie between-1 and + 1.

The function

is, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

Adapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.

AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

The known methods of “summing” divergent series, e.g. the means of Cesàro, Riesz, Borel, Lindelöf, Mittag-Leffler are particular cases of the transformation of a sequence (formed from the partial sums) by a T-matrix. An equivalent method is that of the transformation of the series by a γ-matrix, the fundamental properties of which have been proved by Carmichael, Perron and Bosanquet.

Let X be a topological space, E a real or complex topological vector space, and C(X, E) the vector space of all bounded continuous E-valued functions on X; when E is the real or complex field this space will be denoted by C(X). The notion of the strict topology on C(X, E) was first introduced by Buck (1) in 1958 in the case of X locally compact and E a locally convex space. In recent years a large number of papers have appeared in the literature concerned with extending the results contained in Buck's paper. In particular, a number of these have considered the problem of characterising the strictly continuous linear functional on C(X, E); see, for example, (2), (3), (4) and (8). In this paper we suppose that X is a completely regular Hausdorff space and that E is a Hausdorff topological vector space with a non-trivial dual E′. The main result established is Theorem 3.2, where we prove a representation theorem for the strictly continuous linear functionals on the subspace Ctb(X, E) which consists of those functions f in C(X, E) such that f(X) is totally bounded.

If we put θ = π/2 in the well known equation

we get eiπ/2 = i; and raising both sides to the power i,

A formally self-adjoint differential operator L is said to be of limit circle type at infinity if its highest order coefficient is zero-free and all solutions x of L(x) = 0 are square-integrable on [a, ∞). (We will drop reference to “at infinity” in what follows.)

For the second-order case

Dunford and Schwartz (3) p. 1409 prove that given

then L is of limit circle type if and only if

If A and B be two fixed points on a great circular arc and P a variable point on the arc, there are two and only two possible positions of the point P corresponding to a given valae of the ratio sinAP/sinBP, provided arcs measured in one direction from A or B be considered positive, and in the opposite direction negative; and these two points are antipodal.

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

This paper studies questions connected with when the Rees algebra of an ideal or the formring of an ideal is Gorenstein. The main results are for ideals of small analytic deviation, and for m-primary ideals of a regular local ring (R, m). The general point proved is that the Gorenstein property forces (and is sometimes equivalent to) lowering the reduction number of the ideal by one from the value predicted if one only assumes the Rees algebra or formring is Cohen–Macaulay.

We show that given a knot in a homology sphere there is a sequence of invariants with the property that if the nth invariant does not vanish, then this implies the existence of a family of irreducible representations of the fundamental group of the complement of the knot into SU(n).