§ 1. When it is necessary to find the factors of such an expression as x2 – 7x – 120 the difficulty for beginners lies in finding the pair of factors of 120 which satisfy the middle term: in this case so that their difference is 7. They may write down the pairs in order (1 × 120; 2 × 60; etc.) and then choose the suitable pair, but this method is often long; or they may guess until they chance to find the pair required. This is often done in so haphazard a fashion that much time is wasted, especially if the given expression have no rational factors.

What sort of an equation is

Write

and start with the equations

This last gives

and the system thus is

viz., this is a system of ordinary differential equations between the five variables θ, r, X, Y, Z: the system can therefore be integrated with 4 arbitrary constants, and these may be so determined that for the value β of θ, X, Y, Z shall be each = 0; and r shall have the value r0.

Theorem I. Let nαp denote

where Then for any change in the independent variable x,

say z = f(x), the coefficient of in is nαp.

Theorem: If any conic be inscribed in a given triangle and a confocal to it pass through the circumcentre, then the circle through the intersection of these two confocals touches the nine-points circle of the triangle.

Ever since I first became acquainted with Wilkinson's method of establishing the existence of the nine-points circle of a triangle (see Mackay's “Euclid,” Appendix to Bk. IV., Prop. 2, the lettering of which I have followed in the first three sections of this paper), its simple and fundamental character has pleased me. I propose to point out first that this method yields probably the most elementary proof of the concurrence of the perpendiculars from the vertices, and then, after restating the investigation of the nine-points circle, to sketch some generalizations.

It is shown that a ring R is right noetherian if and only if every cyclic right R-module is injective or a direct sum of a projective module and a noetherian module.

Diophantus (Heath, 1885, page 122) is said to have the porism—“The diffeience of two cubes can be expressed as the sum of two cubes.”

For a pair of continuous linear operators T and S on complex Banach spaces X and Y, respectively, this paper studies the local spectral properties of the commutator C(S, T) given by C(S, T)(A): = SA−AT for all A∈L(X, Y). Under suitable conditions on T and S, the main results provide the single valued extension property, a description of the local spectrum, and a characterization of the spectral subspaces of C(S, T), which encompasses the closedness of these subspaces. The strongest results are obtained for quotients and restrictions of decomposable operators. The theory is based on the recent characterization of such operators by Albrecht and Eschmeier and extends the classical results for decomposable operators due to Colojoară, Foiaş, and Vasilescu to considerably larger classes of operators. Counterexamples from the theory of semishifts are included to illustrate that the assumptions are appropriate. Finally, it is shown that the commutator of two super-decomposable operators is decomposable.

The question having been proposed to me as a puzzle: To arrange eight men on a chess-board, so that no two of them shall be in the same line,—that is to say, that no two are to be in the same column, nor in the same row, nor in the same diagonal line,—I succeeded before very long in solving it by finding the annexed arrangement. (Fig. 45.)

Throughout this paper all near-rings will be left distributive. We shall denote the zero-symmetric part of a near-ring N by N0. The fact that the near-rings under consideration may not be zero-symmetric has important consequences for what follows, particularly the results of the last section.

These notes are intended to be read in connexion with Dr A. C. Aitken's paper, Proc. Edinburgh Math. Soc. (2) 1 (1929), 199-203. It is proposed to show (by a simple line of direct algebraic demonstration which is also applicable to the original formula) that Aitken's Theorem can be extended to the Everett types, i.e. the types which include two sets of terms—one set involving u (0) and the resultant of generalised operations on u (0), and the other set involving u (1) and the resultant of similar operations on u (1).

Euler's Theorem may be looked upon as the result of a certain operator acting on a special kind of function. This function may depend on any number of variables, but for convenience it is usual to consider three, viz., x, y, z. A function is homogeneous in x, y, z and of the nth degree if it can be put into the form

Let the equation of the four point system be

where h is a variable parameter and a, b, g, f and c are constants. The four fixed points are thus given by and . Since the quadrilateral is real g2>ac and f2>bc: it is convex if and have the same sign, i.e. if ab is positive, and concave if ab is negative.

The Cartan matrix C of a left artinian ring A, with indecomposable projectives P1,…,Pn and corresponding simples Si=Pi/JPi, is an n×n integral matrix with entries Cij, the number of copies of the simple sj which appear as composition factors of Pi. A relationship between the invertibility of this matrix (as an integral matrix) and the finiteness of the global dimension has long been known: gl dim A < ∞⇒det C = ± 1 (Eilenberg [3]). More recently Zacharia [9] has shown that gl dim A ≦ 2⇒det C = 1, and in fact no rings of finite global dimension are known with det C = −1. The converse, det C = l⇒gl dim A < ∞, is false, as easy examples show ([[1) or [3]). However if A is left serial, gl dim A < ∞iff det C = l [1]. If A = ⊕n ≧ 0 An is ℤ-graded and the radical J = ⊕n ≧ 0 An, Wilson [8] calls such rings positively graded. Here there is a graded Cartan matrix with entries from ℤ[X] and gl dim A < ∞⇒det = 1 and, hence, det C = l [8, Prop. 2.2].

Thus (1 + x)n − 1 is divisible by x.

Hence (1 + x)n = 1 + terms in x and its powers.