Equations which may be regarded as extensions of the dual series equations discussed by Noble (1) and the present author (2) are the triple series equations of the first kind

and the triple series equations of the second kind

where f, f1, g, g1h and h1 are all known functions,

is the Jacobi polynomial (3).

Let S be a finite semigroup. Consider the set p(S) of all elements of S which can be represented as a product of all the elements of S in some order. It is shown that p(S) is contained in the minimal ideal M of S and intersects each maximal subgroup H of M in essentially the same way. The main result shows that p(S) intersects H in a union of cosets of H′.

The numerical solution of Integral equations with variable upper limits has been investigated by Professor Whittaker. In this investigation the nucleus, supposed to be given numerically by a table of single entry, is replaced by an approximate expression consisting of a finite number of terms, each term involving an exponential or simply a power of the variable, and then the solution is found as an analytical expression from which its numerical values may be computed. The numerical solution of integral equations with fixed limits has been discussed by H. Bateman.

It is evident in the first place as is pointed out by Steiner that the conic will be a parabola if, and cannot be a parabola unless the point at infinity on one range correspond to the point at infinity on the other, that is, the two ranges must be similar. This is the converse of the well-known proposition that a movable tangent to a parabola divides two fixed tangents similarly.

In (2) Bruhat has developed a theory of differentiable functions and distributions on a locally compact group in order to apply it to the study of the irreducible representations of the p-adic groups. Later, Whyburn (8) defined differentiable forms on a locally compact group and proved an analog of the de Rham theorem concerningthe relationship between the Čech cohomology and the De Rham cohomology. In (4) Ihave introduced the notions of “generalised manifold” (roughly speaking a projective limit of smooth manifolds) and of “differentiable forms” on it, extending some of the results due to Bruhat and Whyburn.

The general heat equation is defined as

where v is a fixed positive number and α is a fixed number. If v = α = 0, then (1.1) reduces to the ordinary heat equation

where u(x,t) is regarded as the temperature at a point x at time t, in an infinite insulated rod extended along the x-axis in the xt-plane. If we set , then (1.1) becomes

Let $H$ be a full Hilbert bimodule over a $C^*$-algebra $A$. We show that the Cuntz–Pimsner algebra associated to $H$ is exact if and only if $A$ is exact. Using this result, we give alternative proofs for exactness of reduced amalgamated free products of exact $C^*$-algebras. In the case in which $A$ is a finite-dimensional $C^*$-algebra, we also show that the Brown–Voiculescu topological entropy of Bogljubov automorphisms of the Cuntz–Pimsner algebra associated to an $A,A$ Hilbert bimodule is zero.

AMS 2000 Mathematics subject classification: Primary 46L08. Secondary 46L09; 46L54

The relationship between the prime ideals and the primal ideals of a Banach algebra is investigated. It is shown that the closure of the prime radical of a Banach algebra may be properly contained in the intersection of the closed primal ideals of the algebra.

Let G be a locally compact connected topological group. Let Aut0G be the identity component of the group of all bi-continuous automorphisms of G topologized by Birkhoff topology. We give a necessary and sufficient condition for Aut0G to be locally compact.

The problem of deciding which graded polynomial algebras over the field of p elements can occur as the -cohomology of a space has played a central rôle in the development of algebraic topology beginning as early as 1950. In the case where the polynomial generators do not occur in dimensions divisible by p, Adams and Wilkerson [1] have given a complete solution by showing that the spaces constructed by Clark and Ewing [3] suffice to realize all such algebras as -cohomology rings. The main result of Adams and Wilkerson for odd primes can be stated as follows.

Whereas pseudovarieties of commutative semigroups are known to be finitely based, the globals of monoidal pseudovarieties of commutative semigroups are shown to be finitely based (or of finite vertex rank) if and only if the index is 0, 1 or $\omega$. Nevertheless, on these pseudovarieties, the operation of taking the global preserves decidability. Furthermore, the gaps between many of these globals are shown to be big in the sense that they contain chains which are order isomorphic to the reals.

AMS 2000 Mathematics subject classification: Primary 20M07; 20M05

The inequality

which is valid for positive, non-null f, g in the spaces Lp(0, ∞), Lq(0, ∞), where p > 1, (1/p)+(1/q) = 1, is a well-known generalisation of the classical inequality of Hilbert (see for instance Chapter 9 of Hardy, Littlewood, and Polya (1)).

The present account is an application of the principles of combinantal forms and Schur function analysis given in a previous paper (A), the references therein being henceforward denoted by A1 to A9, and the complete irreducible system of invariants of three quaternary quadrics will now be obtained from the complete system (not necessarily itself irreducible) derived by Turnbull (A5, p. 483). This latter system comprises 47 invariants, viz. 15, 1, 6, 6, 1, 15 and 3 members of total degrees 4, 6, 8, 10, 12, 14 and 18 respectively in the coefficients of the quadrics. It will be proved that all of these are irreducible except for the one of degree 12 and the three of degree 18, the former being of especial interest as it is a real combinant and moreover, involves unusual features in the proof of its reduction and also in the derivation of the form expressing it in terms of irreducible invariants.

In this note we are concerned with the permutability of congruence relations on semilattices and lattices with pseudocomplementation. There are some results in the literature along these lines. For example, in (8) H. P. Sankappanavar characterises those pseudocomplemented semilattices whose congruence lattice is modular and employs the result in conjunction with the well-known fact that algebras with permuting congruences are congruence-modular to characterise those pseudocomplemented semilattices with permuting congruences. Our first result is a direct, short proof of his result. In (2), J. Berman shows that for all congruences on a distributive lattice L with pseudocomplementation to permute it is necessary and sufficient that D(L), the dense filter of L, be relatively complemented. Our second result is a generalisation of that result to an important equational class of lattices with pseudocomplementation which properly contains the modular lattices with pseudocomplementation.

The object of this note is to deduce from first principles the probability differential of Pearson's x2, and to account for “degrees of freedom.”

It is proposed here to consider the sequence un determined by the relation

where, in particular,

and initially u1 = θ1. The following is the main result to be proved.