Let x = (x1, x2,…, xn) be a normal random vector with zero expectation vector and with a variance-covariance matrix which has 1 for its diagonal elements and ρ for its off-diagonal elements. Consider the quantity where.

If f(z) is a power series convergent for |z| < ρ, where ρ > 0, then f(z) is said to have a fixpoint of multiplir a1 at z = 0. In the (local) iteration of f(z) one studies the sequence {fn(z)}, n = 0, 1, 2, … in a neighbourhood of z = 0, fn(z) benig defined by For many values of the multiplier a1, including 0 < |a1| < 1 and |a1| > 1, the local iteration of f(z) is completely mastered by the introduction of Schröder's functional equation.

Let f1(x), f2(x), … be a sequence of functions belonging to the real or complex Banach space L, (see S. Banach: [1] (The results can be generalised to functions on any space that is the union of countably many sets of finite measure). We are concerned with various properties that such a sequence may have, and in particular with the more important kinds of convergence (strong, weak and pointwise). This article shows what relations connect the various properties considered; for instance that for strong convergence (i.e. ║fn — f║ → 0) it is necessary and sufficient firstly that the sequence should converge weakly (i.e. if g is bounded and measurable then f(fn(x) — f(x))g(x)dx → 0) and secondly that any sub-sequence should contain a sub-sub-sequence converging p.p. to f(x).

The fractional iteration of ex and solutions of the functional equation have frequently been discussed in literature. G. H. Hardy has shown (in [3], and in greater detail in [4]) that the asymptotic behaviour of the solutions of (1) cannot be expressed in terms of the logarithmico-exponential scale, although they are comparable with each member of the scale.1 Hence solutions of (1) provide a remarkably simple instance of functions whose manner of growth does not fit into the scale of L-functions but requires non-elementary orders of infinity for an accurate representation. This raises quite naturally the question whether there exists a most regularly growing solution of equation (1) which might serve as a prototype for this kind of growth. In a slightly more general context we may ask whether there exists a ‘best’ family of fractional iterates fσ(x), satisfying.

In the preceding paper 1) one of us has proposed a definition for the “best” or most regularly growing fractional iterates of logarithmico-exponential type functions. The definition was essentially based on two observations. First, that the functional equation has exactly one solution with the property that is totally monotonic for every x > 0. Secondly, that if f(x) is a logarithmicoexponential function such that then the Abel equation has (apart from the arbitrariness of an additive constant) exactly one solution with the property that exists) and the same is true for any reasonably well-behaved function with property (3) whose manner of growth does not transcend Hardy's scale of L-functions. Thus every such function has a uniquely determined family of fractional iterates given by and these fσ(x) may be regarded as the most regularly growing iterates of f(x).

In this paper the Clifford groups PCT(pm), p > 2, PCG(pm) and CS'(pm), and the factor groups ½CS' (pm), which were defined in Paper I of this series (Bolt, Room and Wall [1]), are considered as transformations of projective [pm–1] over the complex field, C. We note that the geometrical results are the same if any of the corresponding groups CT, CG or GG and CS, respectively, are considered instead.

As an illustration of the use of his identity [10], Spitzer [11] obtained the Pollaczek-Khintchine formula for the waiting time distribution of the queue M/G/1. The present paper develops this approach, using a generalised form of Spitzer's identity applied to a three-demensional random walk. This yields a number of results for the general queue GI/G/1, including Smith' solution for the stationary waiting time, which is established under less restrictive conditions that hitherto (§ 5). A soultion is obtained for the busy period distribution in GI/G/1 (§ 7) which can be evaluated when either of the distributions concerned has a rational characteristic function. This solution contains some recent results of Conolly on the quene GI/En/1, as well as well-known results for M/G/1.

Contrary to the general impression that variation principles are of purely theoretical interest, we show by means of examples that they can lead to considerable practical advantages in the solution of non-linear vibration problems. In this paper, we develop a variation principle for the period of a free oscillation, as a function of the average value of the Lagrangian over one period. Even extremely simple-minded approximations to the true motion lead to excellent values for the period. The stability of such free oscillations against small disturbances of the initial conditions is treated in a previous paper.

The spinor analysis of Infeld and van der Waerden [1] is particularly well suited to the transcription of given flat space wave equations into forms which constitute possible generalizations appropriate to Riemann spaces [2]. The basic elements of this calculus are (i) the skew-symmetric spinor Γμν, (ii) the hermitian tensor-spinor σκ(img)ν(generalized Pauli matrices), and (iii) the curvature spinor Ρμνκι. When one deals with wave equations in Riemann spaces V4 one is apt to be confronted with expressions of somewhat bewildering appearance in so far as they may involve products of a large number of σ-symbols many of the indices of which may be paired in all sorts of ways either with each other or with the indices of the components of the curvature spinors. Such expressions are generally capable of great simplification, but how the latter may be achieved is often far from obvious. It is the purpose of this paper to present a number of useful relations between basic tensors and spinors, commonly known relations being taken for granted [3], [4], [5]. That some of these new relations appear as more or less trivial consequences of elementary identities is largely the result of a diligent search for their simplest derivation, once they had been obtained in more roundabout ways.