The purpose of this paper is to expand upon the results obtained in [4]. We consider the set H of differential equations

where p and r are continuous real-valued functions of period ω (ω being fixed throughout). The equation (1.1) is denoted by P or (p,r), and we regard H as the set of pairs of continuous functions of period ω.On H we define a norm:

In an earlier paper (5) a description was given in set-theoretic terms of the semigroup generated by the idempotents of a full transformation semigroup , one of the results being that if X is finite then every element of that is not bijective is expressible as a product of idempotents. In view of this it was natural to ask whether by analogy every singular square matrix is expressibleas a product of idempotent matrices. This is indeed the case, as was shown by J. A. Erdos (2). Magill (6) has considered products of idempotents in thesemigroup of all continuous self-maps of a topological space X, but a comparable characterization of products of idempotents in this case appears to be extremelydifficult, and no solution is available yet.

Any property of a curve which depends solely upon its form may be styled intrinsic. Thus the circle of curvature at a point, the relation between s and ψ, the envelope of the normals, the envelope of straight lines making a constant angle with the curve, etc., are all intrinsic properties. It is proposed to investigate a few of these properties in this paper.

This paper was based mainly on the results of an investigation which will appear in full in the Transactions of the Royal Society of Edinburgh. Incidentally, however, it led to a discussion of the question:—Find the law of density of a planet's atmosphere, supposing Boyle's law to be true for all pressures, and the temperature to be uniform throughout.

I.—Take the well-known identity

Now if we can transform 4wz into a square we shall have two square numbers whose sum is a square. This will be effected by taking w = p2, z = q2, for then 4wz = 4p2q2 = (2pq)2 and we have

See Mathematical Magazine, Vol. II., No. 5, p.

Theorem. There exist non-Abelian finitely presented lattice-ordered groups which are totally ordered. This disproves a previous conjecture of the author [5]

We describe a class of soluble groups with a finite complete rewriting system which includes all the soluble groups known to have such a system. It is an open question, related to deep questions in the theory of groups, whether it includes all soluble groups with such a system.

In this paper we shall discuss the positive roots of the equation

where Iq is the modified Bessel function of the first kind. By means of a recurrence relation for Iq(r) [2, (5.7.9)], equation (1.1a) can also be written in the form

This paper deals with certain expansions of analytic functions in series of polynomials. Explicit forms for the polynomials are given in terms of the coefficients of the Taylor's expansion and of prescribed positive constants hk. Under suitable conditions, to be presently discussed, the series converge to the function in the Borel region.

The tensor product of semilattices has been studied in [2], [3] and [5]. A survey of this work is given in [4]. Although a number of problems were settled completely in these papers, the question of the associativity of the tensor product was only partially answered. In the present paper we give a complete solution to this problem.

For a fixed integer q≧2, every positive integer where each ar(q, k) ∈ {0, 1, 2, …, q–1}. The sum of digits function α(q, k) = behaves rather erratically but on averaging has a uniform behaviour. In particular if A(q, n) = , where n > 1, then it is well known that A(q, n)∼½ ((q – 1)/log q) n log n as n→∞. For even values of q, a lower bound is now given for the difference ½S(q, n) = A(q, n)–½(q–1)[logn/logq] n, where [log n/log q] denotes the greatest integer ≦ log n/log q, complementing an earlier result for odd values of q.

The excentric angle notation in the ellipse is extremely useful, and in part we can replace it by the hyperbolic sine and cosine in connection with the hyperbola.

Take the hyperbola x2/a2/-y2/b2/= 1, then the coordinates of any point on it may be written acoshφ, bsinhφ, for cosh2φ–sinh2φ=1. The objection to its use in all cases is that the hyperbolic cosine of an angle is always positive, so that (acoshφ, bsinhφ) can only represent any point on the branch on the positive side of the axis of y, for any point on the other branch we must take its coordinates as (–acoshφ, bsinhφ).

Let θ be an irrational number in (0,1) and f a real-valued continuous function on the 1-torus T. Letφθ, f be a Furstenberg transformation on the 2-torus T2 defined by for any (t, s) ∈ T2, where ρ is a non-zero integer, and we identify a function on T or T2 with a function on R or R2 with period 1, respectively. Let Aθ, f be the crossed product associated with φθ, f. In this paper we wiil compute the positive cone of the K0-group of Aθ, f.

Stewart, a Scotch geometrician, gave in 1763 the demonstration of the following theorem: “If we divide the base of a triangle into two segments by a straight line going through the vertex, the sum of the squares of the two sides, multiplied each by the non-adjacent segment, is equal to the product of the base multiplied by the square of the straight line plus the rectangle contained by the two segments.”

The paper was mainly an account of the abacus, as used in China and Japan. The instrument was shown, and the various operations of addition, subtraction, multiplication, division, and extraction of square and cube roots, were illustrated. The multiplication and division tables were fully described, the latter being especially interesting. The historic development of the abacus in the East was also touched upon. A full account of the Chinese and Japanese abacus will be found in a paper by the author, entitled, “The Abacus, in its Historic and Scientific Aspects,” published in the Transactions of the Asiatic Society of Japan (vol. XIV., 188G). A copy of this paper is in the library of the Edinburgh Mathematical Society.

Let umn, vmn be functions of m and n, vmn being real and positive for all positive values of m and n. Suppose that either vmn increases steadily to infinity with n, or that umn both tend to zero (the latter steadily) as n → ∞, for any fixed value of m. Denote by wmn, and assume that wmn exists for every value of m, being denoted by lm. Then from Stolz' extension of a result proved by Cauchy, and an allied theorem, we have , for all values of m. It follows from Pringsheim's Theorem that if the double limit of exists, being l, then lm → l as m → ∞.

The purpose of this work is to establish a priori $C^{2,\alpha}$ estimates for mesh function solutions of nonlinear difference equations of positive type in fully nonlinear form on a uniform mesh, where the fully nonlinear finite difference operator $\F$ is concave in the second-order variables. The estimate is an analogue of the corresponding estimate for solutions of concave fully nonlinear elliptic partial differential equations. We use the results for the special case that the operator does not depend explicitly upon the independent variables (the so-called frozen case) established in part I to approach the general case of explicit dependence upon the independent variables. We make our approach for the diagonal case via a discretization of the approach of Safonov for fully nonlinear elliptic partial differential equations using the discrete linear theory of Kuo and Trudinger and an especially agreeable mesh function interpolant provided by Kunkle. We generalize to non-diagonal operators using an idea which, to the author’s knowledge, is novel. In this paper we establish the desired Hölder estimate in the large, that is, on the entire mesh $n$-plane. In a subsequent paper a truly interior estimate will be established in a mesh $n$-box.

AMS 2000 Mathematics subject classification: Primary 35J60; 35J15; 39A12. Secondary 39A70; 39A10; 65N06; 65N22; 65N12