The tensor product A ⊗ B of the distributive lattices A and B was first investigated by Fraser in [4] and [5]. In this paper, we present some results relevant to the structure and construction of this tensor product. In particular, we establish a sufficient condition for join-irreducibility in the tensor product and show that this condition characterizes join-irreducibility in the case that A and B satisfy the descending chain condition. We also show that if A and B satisfy the descending chain condition then so does A ⊗ B; this insures the compact generation of A ⊗ B by its join-irreducibles. We conclude with some examples and applications of our results to the tensor product of finite distributive lattices.

We show that the zeta function of a regular graph admits a representation as a quotient of a determinant over a L2-determinant of the combinatorial Laplacian.

[Of the following properties, some must have been long known, but I do not remember any early statement of them. The 1st, 10th, 15th, 16th, 19th, 20th, and 22nd are given, mostly without proof, in an article by Vecten in Gergonne's Annales de Matliématiques, vol. VII., p. 321 (1817); the first parts of the 3rd and 4th are proposed for proof by William Godward in The Gentleman's Diary for 1837, p. 48, and proved by him and others in the Diary for 1838, pp. 40-41; the first parts of the 5th, 6th, 12th, and 14th are given in M'Dowell's Exercises on Euclid and in Modern Geometry, §§ 27, 28, 29 (1863); the 9th in Milne's Weekly Problem Papers, p. 135 (1885); the first part of the 8th in Vuibert's Journal de Mathématiquts Elémentaires, 12° Année, p. 18 (1887); the 39th in the Journal de Mathématiques Elémentaires, edited by De Longchamps, 3° série,tome I.,p. 234 (1887). The others are believed to be new.

The roots of the equation in λ

were named by Sylvester the latent roots of the determinant | a11a22 … ann |. So early as 1852, Sylvester showed that if any determinant D is given, we can at once write down a determinant whose latent roots are the squares of the latent roots of D: this determinant is in fact the square of D, the process of squaring being performed by multiplying rows into columns : so that, e.g. if the latent roots of are λ1 and λ2, then the roots of are and . Spottiswoode had also shown in 1851 that the latent roots of the reciprocal of a determinant are the reciprocals of the latent roots of the determinant itself. Both these theorems were soon found to be particular cases of a general theorem which was enunciated by Sylvester thus: The latent roots of any function of a matrix are respectively the same functions of the latent roots of the matrix itself.

If, in any triangle ABC, the angle A > the angle B, and lines AD and BE be drawn so that the angles BAD and DAC are, respectively, greater than the angles ABE and EBC, then is BE > AD. (Fig. 6.)

Proof: Draw AD′ and AD″ making angles BAD′ and D″AC, respectively, equal to the angles ABE and EBC.

Let l be an odd prime and let A be a commutative ring containing 1/l. Let K*(A;Z/lv) denote the mod lv algebraic K-theory of A [3]. As explained in [4] there exists a “Bott element” βv∈K21v–1(l–1)(Z[1/l];Z/lv) and, using the K-theory product we may, following [16, Part IV], form

which is defined as the direct limit of iterated multiplication by βv. There is a canonical localisation map

The Klein–Gordon equation

Ω being a constant of dimensions [time]-1 and c being a constant velocity, appears in nuclear physics (1) and, when the Laplacian operator is twodimensional, in the theory of long gravity waves on a rotating earth (2). If Ω is zero it reduces to the wave equation

In the following paper I propose to give a short account of Dr Ernst Kötter's purely geometrical theory of the algebraic plane curves. This theory is developed in a treatise which, in 1886, gained the prize of the Berlin Royal Academy; but the contents of my paper are also partly drawn from a course of lectures delivered by Dr Kötter in the University of Berlin, W.S. 1887–88.

I send a note on the following problem, a solution of which was requested of me by one of the tutors at King's College, Cambridge.

We are given a quadrilateral of four jointed bars ABCD (fig. 83). The bar CD being held fast, find the tangent to the locus of P, the intersection of DA, CB in any position; and verify the following construction for the radius of curvature of the path of P:—

Let PQ be the third diagonal, draw through P a perpendicular to PQ meeting BA, CD in L and L′; through L and L′ draw parallels to PQ meeting AD in M and M′; through M and M′ draw perpendiculars to AD meeting the normal at P in O and O′; then will

Mr Crawford's note on this subject, read at a recent meeting, reminds me of a method I gave to a class four or five years ago.

The continued fractions treated in this paper are of the general form

where s0, s1, s2, … are real integers (positive, negative, or zero). An arbitrary real number can, of course, be developed in such a fraction in an infinite variety of ways. The continued fractions discussed here have a number of striking properties and present numerous contrasts with the ordinary continued fraction usually employed.

§ 1. Introduction. The algebra of quantum mechanics is characterized by the fact that the variables p, q obey all the laws of ordinary algebra except that multiplication is non-commutative and instead there exists a relation of the form

where c is a real or complex scalar constant and is thus commutative with both p and q.

We prove that in a residuated regular semigroup the elements of the form and are idempotents, and derive some consequences of this fact. In particular, we show how the maximality of such idempotents is related to the semigroup being naturally ordered, and obtain from this a characterisation of the boot-lace semigroup of [2].

What is here printed contains merely a list of the memoirs and treatises that may perhaps be found useful for one who wishes to trace the progress of the mathematical theory of heat beyond the stage at which Fourier left it. As discussions of the Fourier series and integrals occur in almost every treatise on the Integral Calculus, I have omitted reference to these. Similar considerations have led me to omit references to the discussion of differential equations, except where these specially dealt with the problem of the conduction of heat.

In this article we consider sums S(t) = Σn ψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.

In a previous paper, entitled the “Vibrations of a Particle about a Position of Equilibrium,” by the author in collaboration with Professor E. B. Ross (Proc. Edin. Math. Soc., XXXIX, 1921, pp. 34–57), a particular dynamical system having two degrees of freedom was chosen and solutions of the corresponding differential equations were obtained in terms of periodic series and also in terms of elliptic functions. It was shown that for certain values of the frequencies of the principal vibrations, the periodic series become divergent, whereas the elliptic function solution continues to give finite results.