1. This note takes its origin in a remark by Brauer (1) and Perfect (5): Let A be a square complex matrix of order n whose characteristic roots are α1,…, αn. If X1 is a characteristic column vector with associated root α and k is any row vector, then the characteristic roots of A + X1 k are α1 + KX1, α2, …, αn. Recently, Goddard (2) extended this result as follows: If x1; …, xr are linearly independent characteristic column vectors associated with the characteristic roots α1, …, αr of the matrix A, whose elements lie in any algebraically closed field, then any characteristic root of Λ + KX is also a characteristic root of A + XK, where K is an arbitrary r × n matrix, X = (x1, …, xr) and Λ = diag (α1, …, αr). We shall prove some theorems of which these and other well-known results are special cases.
The fact that the prime ideal associated with a given irreducible algebraic variety has a finite basis is a pure existence theorem. Only in a few isolated particular cases has the base for the ideal been found, and there appears to be no general method for determining the base which can be carried out in practice. Hilbert, who initiated the theory, proved that the prime ideal defining the ordinary twisted cubic curve has a base consisting of three quadrics, and contributions to the ideal theory of algebraic varieties have been made by König, Lasker, Macaulay and, more recently, by Zariski. A good summary, from the viewpoint of a geometer, is given by Bertini [(1), Chapter XII]. However, the tendency has been towards the development of the pure theory. In the following paper we actually find the bases for the prime ideals associated with certain classes of algebraic varieties. The paper falls into two parts. In Part I there is proved a theorem (the Principal Theorem) of wide generality, and then examples are given of some classes of varieties satisfying the conditions of the theorem. In Part II we find the base for the prime ideals associated with Veronesean varieties and varieties of Segre. The latter are particularly interesting since they represent (1, 1), without exception, the points of a multiply-projective space.
In this paper formulae are developed for the first and second focal lengths, and the positions of the first and second principal planes of a type of electrostatic lens which has been the subject of study (mostly experimental) in several previous papers. The lens, which is commonly used in electron optical devices, lends itself to a theoretical study, although this does not appear to have been attempted before. It consists of two equal semi-infinite cylinders placed end to end so that their axes coincide and the ends are separated by a small gap. If the cylinders are at potentials V1 and V2 and we write σ = V2/V1, the system behaves as an electron lens when σ > 0 and as an electron mirror when σ < 0. In the latter case some experimental results have been given by Nicoll(1) who also studied the focusing action in the case σ > 0 and, in particular, the formation of intermediate images when σ ≪ 1 and when σ ≫ 1. But for the precise formulation of the relationship between σ and the number of cross-overs a theoretical study, based on the paraxial equation, would be necessary. The problem will be indicated below. An experimental determination of the lens characteristics for values of σ from about 2 to 15 and for several gap widths has been made by Spangenberg(2), whose results will be compared with those obtained in the present paper. The two-cylinder lens has also been studied by Klemperer and Wright(3) using an experimental and a numerical (trigonometrical) method, and some crude analytical results have been given by Gray(4).
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