It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

A simple explanation of Pauli's principle was first given with the wave mechanics. Its interpretation in the new theory was that the wave functions of Schrödinger were antisymmetrical in all the electrons concerned. Thus when the interactions of the electrons may be neglected, the wave function (for a system of n electrons) can never be of the form

in nature, but only of the form

Hence σ, τ,…ω must be all distinct.

An interesting extension of the idea of convexity has been introduced by Valentine (2). He considered a plane set X such that for any three points a, b, c in X at least one of the segments [a, b], [b, c], [c, a] is contained in X, and showed that such a set can be regarded as the union of at most three convex sets. See also (3). The five pointed star (Fig. 1) is an example that shows that X may not be the union of fewer than three convex sets.

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

The notion of the order of convergence of iterative processes may well have been known to Gauss or even Newton, but to the author's knowledge it was first considered in detail in 1870 by Schröder(5). The subject was later investigated independently by Hartree(4) and Bodewig(1). Hartree alone mentions, briefly, the extension from one to several variables, but gives no detailed analysis. Domb and Fisher(2) consider a particular type of iterative process in several variables, which they call ‘degenerate’, in which all the variables tend to the same value. We shall follow essentially Hartree's treatment, in considering the solution of r simultaneous equations in r variables and applications to a generalization of the Newton-Raphson process.

We extend Thurston’s topological characterisation theorem for postcritically finite rational maps to a class of rational maps which have a fixed bounded type Siegel disk. This makes a small step towards generalizing Thurston’s theorem to geometrically infinite rational maps.

1. Introduction. In a principal ideal domain R, any two-sided ideal is of the form Rc = cR, i.e. it has an invariant element as generator, and the customary development of ideal theory in a principal ideal domain (cf. e.g. (10), ch. III) takes on a more transparent form when expressed in terms of invariant elements. Likewise, the one-sided bounded ideals may be studied in terms of their generators.

The iterative solution of Hallen's integral equation for the current distribution on a lossless dipole transmitting antenna is studied from first principles, and a general solution in terms of an expansion function is derived. In this new solution even the zero-order approximation gives a reasonable representation of both the quadrature and the in-phase current distributions. The choice of an expansion function is discussed. At each stage the approximations which must be made in order to obtain earlier iterative solutions are clearly described. Three low order solutions are derived in detail: the zero-order solution using both a constant expansion parameter and a simple expansion function, and the rst-order solution using a constant expansion parameter. The results are compared and special attention is paid to the very short antenna, although the results are good for lengths up to and including the half-wave dipole. The results are presented in such a way that the first-order admittance of any given antenna may be readily calculated.

In the following pages I have developed the theory of matrices by resolving them into parallel components arranged diagonally, rather than into the usual rows and columns. This treatment is natural in view of the fundamental fact that the resolution is undestroyed when matrices are formed into products (Theorem 2). It is closely related to the theory of continuants and of continued fractions. Certain features stand out in such a presentation—the distinction between the length and range of a diagonal (§ 4), that between regular and irregular diagonals (§ 6), and the use of equable partition (§ 7). The exact conditions for the existence of an rth root of a given singular matrix are examined in § 9 and summarized under the title, the condition of equability.

1. The object of the present paper is to obtain some finite integrals involving the Appell's function F4 and the H-function of Fox by utilizing the results recently given by Saxena, Sharma and Tranter respectively. The first three results proved in this paper are the extension of the results recently established by Saxena and Sharma in these proceedings. MacRobert's result follows as a very special case of one of our results. A few very interesting special cases of the main results have also been given.

The general solution of Chaplygin's equation for the motion of an inviscid compressible fluid is obtained for ‘simple wedge flows’. Some particular examples are given and a discrepancy in earlier Russian papers explained. After a discussion of the properties of the velocity potential and the expression for the physical coordinates in terms of the hodograph variables, a general theorem on sonic jets is proved, namely, that the physical changes due to the presence of solid boundaries in the flow are completed within a finite distance in those directions in which the flow becomes finally a sonic jet.

In this note we prove the following result due to Milnor ((5), Theorem 4·1).

Let p(X1, …, Xn) be a polynomial in the commuting indeterminates X1, …, Xn.

Define

The quadratic functional equation f(f(x)) *–Tf(x) + Dx = 0 is equivalent to the requirement that the graph be invariant under a certain linear map The induced projective map is used to show that the equation admits a rich supply of continuous solutions only when L is hyperbolic (T2 > 4D), and then only when T and D satisfy certain further conditions. The general continuous solution of the equation is given explicitly in terms of either (a) an expression involving an arbitrary periodic function, function additions, inverses and composites, or(b) suitable limits of such solutions.

A recently developed theory of the electromagnetic field in nearly periodic structures is used to calculate the field in an Alvarez type linear accelerator. The problem is reduced to the solution of the transmission line equations for the ‘voltage’ and ‘current’ of the dominant mode. Formulae are given for the coefficients in these equations, and also for the ratio of mode current to accelerating field, as functions of the cavity dimensions of the accelerator; numerical values are given for the Berkeley accelerator.

Suppose that Ω is a compact Hausdorff space with a preorder [les ] whose graph is closed, and let Ω∘ be an open subset of Ω. This paper provides conditions sufficient to allow every increasing bounded real continuous function on Ω∘ to be extended to an increasing real continuous function on Ω. These conditions are: (i) that Ω is a Stonian space, and (ii) that the set C↑(Ω, [les ]) of increasing real continuous functions on Ω is a regular Dedekind complete sublattice of C(Ω). Under these conditions it is also shown that C↑(Ω, [les ]) is generated by idempotents, and an extension theorem for idempotents is proved.

A basic problem of the theory of Markov processes is to find a semigroup S of operators on the bounded Borel measurable functions on ℝd, such that S is given by a transition function, and

for all f belonging to the domain of a given second-order elliptic differential operator D. A process is constructed from the transition function associated with S, and then one hopes to realize the process as one with continuous paths [2].

In [3, 4] we showed how the use of a random-walk analogue can be made to yield non-trivial information about the behaviour of certain trigonometric sums in one variable. Our aim here is to show how our method can be adapted to yield similar results for a broad class of trigonometric sums in several variables. Let

be a polynomial in v independent variables with integral coefficients. We choose integers n ≥ 0, d ≥ 1 and p ≥ 2 with p prime, and assume that f(x) has total degree ≤ d + 1. We shall consider the problem of obtaining non-trivial upper bounds for the absolute value of sums of the type

where P = {1, 2, …, p} and f is non-constant.