Rings and distributive lattices can both be considered as semirings with commutative regular addition. Within this framework we can consider subdirect products of rings and distributive lattices. We may also require that the semirings with these restrictions are regarded as algebras with two binary operations and the unary operation of additive inversion (within the additive subgroup of the semiring). We can also consider distributive lattices with the two binary operations and the identity mapping as the unary operation. This makes it possible to speak of the join of ring varieties and distributive lattices. We restrict the ring varieties in order that their join with distributive lattices consist only of subdirect products. In certain cases these subdirect products can be obtained via a general construction of semirings by means of rings and distributive lattices.

In Part I it has been shown that, given a contact transformation, two equations

can be derived which lead to the compatible differential equations

It will be shown in the present communication that the necessary and sufficient condition that (1.3), (1.4) should be compatible is that

regarded as an equation in the non-commutative variables q, p which themselves satisfy the condition

We shall call functions satisfying this condition conjugate functions. From this point of view the method employed by Professor Whittaker in his original paper, involving the use of a contact transformation,, was really a particular method of generating conjugate functions. This powerful method may be supplemented and extended by the other methods developed in the following pages.

Since an n-gon is determined by 2n − 3 conditions, and n − i conditions are involved in its being equilateral, there are still in the case of an equilateral n-gon n − 2 conditions to be determined. These n − 2 conditions cannot all be given in terms of the angles, since an infinite number of n-gons may always be described similar to, out not necessarily congruent with, any given equilateral n-gon. Hence only n − 3 of the angles of an equilateral n-gon may be assigned arbitrarily, and there must therefore be 3 independent relations connecting the angles of any equilateral n-gon. These three conditions may be obtained by projecting the perimeter of the n-gon on any three lines.

Under the assumption of the Selberg conjecture I establish by means of the Selberg trace formula the following:

Theorem. Let Γ denote Γ(q) or Γ0(q), q square-free. Let Δq denote the Laplace operator on L2(Γ\H), and let Σq denote its discrete spectrum. Then there exists an absolute positive constant A such that for q≧A

A Banach space X is called an HT space if the Hilbert transform is bounded from Lp(X) into Lp(X), where 1 < p < ∞. We introduce the notion of an ACF Banach space, that is, a Banach space X for which we have an abstract M. Riesz Theorem for conjugate functions in Lp(X), 1 < p < ∞. Berkson, Gillespie and Muhly [5] showed that X ∈ HT ⇒ X ∈ ACF. In this note, we will show that X ∈ ACF ⇒ X ∈ UMD, thus providing a new proof of Bourgain's result X ∈ HT ⇒ X ∈ UMD.

In his recent book A History of the Conceptions of Limits and Fluxions in Great Britain from Neuton to Woodhouse (Chicago and London: The Open Court Publishing Company, 1919), as well as in a series of articles in the American Mathematical Monthly for 1915 on The History of Zeno's Arguments on Motion, Mr Cajori discusses certain aspects of the conception of a limit, and treats in considerable detail the controversy between Jurin and Robins that arose out of the publication of Berkeley's Analyst. I gave an account of the controversy in a paper that appears in Volume XVII. of our Proceedings, and as Mr Cajori's estimate of the respective merits of the contributions by Jurin and Robins differs greatly from mine, and as the conception of a limit is fundamental in modern mathematics I venture to draw the attention of the Society to the matter.

The following work is a sequel to three previous communications, and more particularly to the first. The present object is to shew the effect of repeated operation with the matrix differential operator , when it acts upon a scalar matrix formed from an n rowed determinant |xij|, or sums of principal minors, the n2 elements xij being treated as independent variables. Thus when z is a scalar quantity ω z means the matrix [∂z/∂xij], whose ijth element is the derivative.

All our rings will be commutative with identity not equal to zero. Also R will always denote a ring. is a filter of ideals of R if is a nonempty set of ideals of R satisfying: I∈ and J is an ideal of R with I⊂J, then J∈, and if I, J∈ then I∩J∈. A Gabriel topology of R is a filter of ideals of R satisfying: if J∈ and I is an ideal of R with (I:x)∈ for all x∈J, then I∈. See the B. Stenström text [6]. We say that a ring R is an FGC ring if every finitely generated R-module is a direct sum of cyclic R-modules. Use mspec R for the set of all maximal ideals of R.

Every holomorphic modular form of weight k > 2 is a sum of Poincaré series; see, for example, Chapter 5 of (5). In particular, every cusp form of even weight k ≧ 4 for the full modular group Γ(1) is a linear combination over the complex field C of the Poincaré series

.

Here mis any positive integer, z ∈ H ={z ∈ C: Im z>0} and

The summation is over all matrices

with different second rows in the (homogeneous) modular group, i.e. in SL(2, Z).The factor ½ is introducted for convenience.

Denoting the sum of the products of the first n natural numbers taken r at a time by the symbol G(n, r), I have shown that

with the initial values and

In general it was shown that

Defining G (x, r) by the fundamental relation:

for all values of x, we get

§ 1. Consider the system of n first order linear differential equations:

together with the n boundary conditions

where aij, bij are constants and where we assume for simplicity of notation that gii = 0.