In this note we prove some cyclic inequalities which are generalisations of known results. We shall assume throughout that ai+n = ai ≧ 0 for all i, that no denominator in the statement of a result vanishes and finally that p, m and q are positive integers. We shall also use A(i, m) to denote with the convention that A(i, 0) = 0. The most interesting of our results is probably Theorem 2 since, in the special case p = 1, m = 2, r = 0, it gives a lower bound of ⅓n for the Shapiro sum . Although it is by no means best possible, see (2), our method implicitly gives a really simple way of obtaining this lower bound which, incidentally, is an improvement on Rankin's original result (5).

The theorem which I propose to establish first attracted my attention while I was turning over the pages of a volume of Cayley's Collected Mathematical Papers (Cayley, 1). The enunciation of the theorem (with no attempt towards a proof) had been published earlier by Kirkman (3) in a lengthy paper on combinatorial analysis (one of the three-score papers of which Kirkman was the author); among the topics discussed in this paper was the enumeration of the total number of different ways D(r, k) in which a (convex) polygon of r sides can be dissected into k+l parts by drawing k non-intersecting diagonals (i.e., no two diagonals may cross each other except at a vertex or outside the polygon).

When P is joined to four points A, B, C, D coplanar with P, a pencil of four lines is formed whose cross ratio is constant if ABCD are collinear. If A, B, C, D are not in a line the cross ratio P(ABCD) has a value which in general varies with the position of P, but which should be known when P is given in position and also A, B, C, D. A simple expression for the cross ratio is given and its utility in locus problems is illustrated by a variety of simple examples, which in several cases furnish methods for constructing a general cubic curve, with or without double point, a trinodal quartic, etc.

In reducing some experiments, I noticed that the logarithm of 237 is about 2.37 …. Hence it occurred to me to find in what cases the figures of a number and of its common logarithm are identical:—i.e., to solve the equation

where m is any positive integer.

The possibilities under rearrangement of terms in a complex series were discussed by Levy and by Steinitz. A reference the Steinitz paper as first disposing of the questions raised made by Bieberbach. It is the purpose of the present paper give an independent treatment by methods somewhat resembling those of the Levy paper.

The following method of establishing the existence and properties of the Focal Circles of a Circular Cubic is, as far as I know, new, and it has the advantage of dispensing, almost entirely, explicitly with analysis, while many of the properties can be proved without using the complicated method of generating the curve given in Salmon. The results which I have arrived at in connexion with the Nodal Cubic and Cuspidal Cubic are not given in Salmon.

This paper is an attempt (I) to deduce from first principles the number of conditions required to determine a plane polygon of n sides; (II) thence to deduce the numbers for special cases; and (III) to discuss the effects of a redundancy and a deficiency in the number of conditions. An investigation of this kind should form an important as well as interesting accompaniment to the ordinary study of elementary geometry.

This paper is concerned with d.g. near-rings and their relationship to faithful d.g. near-rings. For general definitions and results, we refer to Pilz [5]. We use left near-rings where he uses right near-rings, but otherwise there is little difference. This work follows earlier work [3], [4] and Mahmood [2]. Before outlining the contents of the paper we present a précis of the definitions.

The use of inverse operators is only justifiable when it is obvious what direct operation of the calculus they symbolise.

The purpose of this note is to point out how the usual method of obtaining the integral of this differential equation can be shown as the result of direct operations.

In this paper we examine the dependence of the solutions of an evolution inclusion on a parameter λ We prove two dependence theorems. In the first the parameter appears only in the orientor field and we show that the solution set depends continuously on it for both the Vietoris and Hausdorff topologies. In the second the parameter appears also in the monotone operator. Using the notion of G-convergence of operators we prove that the solution set is upper semicontinuous with respect to the parameter. Both results make use of a general existence theorem which we also prove in this paper. Finally, we present two examples. One from control theory and the other from partial differential inclusions.

§ 1. Introduction. Little is known concerning the theory of resultants of equations other than in the complex number system. The cyclic number systems provide a simple example which is not a division algebra. In such a system with n units er. any number y ≡ y0 + y1e1 + y2e2 + … + yn−1en−1 has coefficients yr drawn from a field, and the units satisfy the product law:

During an investigation into the existence of Gauss-type quadrature formulae for the numerical solution of Fredholm integral equations with weakly singular kernels an intermediate result was found which is of independent interest.

Let be a class of finite groups. Then a c-group shall be a topological group which has a fundamental system of open neighbourhoods of the identity consisting of normal subgroups with -factor groups and trivial intersection. In this note we study groups which are existentially closed (e.c.) with respect to the class Lc of all direct limits of c-groups (where satisfies certain closure properties). We show that the so-called locally closed normal subgroups of an e.c. Lc-group are totally ordered via inclusion. Moreover it turns out that every ∀2-sentence, which is true for countable e.c. L-groups, also holds for e.c. Lc-groups. This allows it to transfer many known properties from e.c. L-groups to e.c. Lc-groups.

Fundamental statements for (associative) rings are that (a) the endomorphisms of each commutative group (U, +) form a ring and (b) eachring may be embedded in such a ring of endomorphisms. In order to generalise these theorems to groups and rings whose addition may not be commutative, one has to deal with partial endomorphisms. But thesering-theoretical Theorems 4a and 4b turn out to be specialisations of similarones for semi-near-rings, near-rings and semirings, developed here inSection 2 after some preliminaries on semi-near-rings in Section 1. A glance at Definition 1 and the ring-theoretical theorems and remarks at the end of Section 2 may give more orientation.

In (5) the author showed how to construct all inverse semigroups from their trace and semilattice of idempotents: the construction is by means of a family of mappings between ℛ-classes of the semigroup which we refer to as the structure mappings of the semigroup. In (7) (see also (8) and (9)) K. S. S. Nambooripad has adopted a similar approach to the structure of regular semigroups: he shows how to construct regular semigroups from their trace and biordered set of idempotents by means of a family of mappings between ℛ-classes and between ℒ-classes of the semigroup which we again refer to as the structure mappings of the semigroup. In the present paper we aim to provide a simpler set of axioms characterising the structure mappings on a regular semigroup than the axioms (R1)-(R7) of Nambooripad (9). Two major differences occur between Nambooripad's approach (9) and the approach adopted here: first, we consider the set of idempotents of our semigroups to be equipped with a partial regular band structure (in the sense of Clifford (3)) rather than a biorder structure, and second, we shall enlarge the set of structure mappings used by Nambooripad.

§1. Introduction. Care is needed in dealing with determinants whose elements are subject to experimental error, particularly when a determinant itself is small compared with its first minors. For, as these examples show, a relatively tiny error in one element may be responsible for a large error in the determinant