Let Nn be the number of real roots of a random algebraic equation The coefficients ξν are independent random variables identically distributed with expectation zero; the variance and third absolute moment are finite and non-zero. It is proved that

where εν tends to zero, but εν log n tends to infinity. The measure of the exceptional set tends to zero as n tends to infinity.

In a recent paper, (2), Baumslag showed that a wreath product of A by B is nilpotent if and only if A and B are nilpotent p-groups for the same prime p, with A of finite exponent and B finite. We shall calculate the (nilpotency) class of such groups when A and B are Abelian. This provides a lower bound for the class in the general case. We give a simple construction for a set of non-nilpotent metabelian groups which satisfy a finite Engel condition. With Engel class defined by Definition 6·1, we show that there are nilpotent groups of arbitrarily large nilpotency class for which the nilpotency class is equal to the Engel class.

The study of the divisibility of polynomials in a single variable over a field is essentially based on the division algorithm, which may be stated as follows: (A) Given polynomials a, b such that there exist polynomials q and a1, such that† .

Singular points on irreducible primals were investigated briefly by C. Segre(8), where the author classified multiple points by the nature of the nodal tangent cone. For surfaces the problem of classification was investigated by, amongst others, Du Val(1) and a complete classification of isolated double points of surfaces lying on non-singular threefolds was given by Kirby(5). In (3) we classified certain types of double points on algebraic primals in Sn. An isolated double point which after a finite number of resolutions gave rise to at most a finite number of isolated double points was called a double point of rank zero. We found that the only isolated double points of rank zero are those which are analogous to the binodes, unodes and exceptional unodes (2) of surfaces.

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

It is known that, for n ≤3, ‘ought to have’ can truthfully be replaced by ‘has’ (see (4), and (5), Cor. 6·6). In this paper, this conjecture will be proved for n ≥ 5. The only unsolved case then will be in dimension four.

The main aim of the present note is to evaluate certain definite integrals involving Meijer's G-function. The results proved are of general character and include as particular cases some of the results given earlier by MacRobert ((3), p. 81). For the definitions and properties of the G-functions see (1), §§5·3 and 5·3·1.

Consider the differential systems

where A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.

Here we shall mention only the results referring to stability. The definitions of the various quantities Tn, Sn, SNn, and the basic hypotheses made concerning their structure will be found in §§ 2·1, 3·1 or 4·1. For convenience we shall introduce some further terminology in this section. The single-server queues {SNn, Tn} arising in connexion with queues in series will be called the component queues, and the queue {Sn, sTn} implicit in the discussion of many-server queues will be called the consolidated queue. We have already in § 2.33 called the single-server queue {Sn, Tn} critical if E(S0-T0) = 0. We shall now call it subcritical if E(S0 − To) > 0 and supercritical if E(S0 − T0) < 0. A system of queues in series is subcritical if each component queue is subcritical, critical if (at least) one component queue is critical and the rest are subcritical, and supercritical if (at least) one component queue is supercritical. A many-server queue will be described in these terms according to the character of its consolidated queue. Finally, a single-server queue {Sn, Tn} will be said to be of type M if it has the property considered in Corollary 1 to Theorem 5: the sequences {Sn} and {Tn} are independent of each other, and one is composed of mutually independent non-constant random variables.

Single-server queues:

(i) Subcritical: stable (Theorem 3).

(ii) Supercritical: unstable (Theorem 2).

(iii) Critical: stable, properly substable, or unstable (examples in §2·33, including one due to Lindley); unstable if type M (Theorem 5, Corollary 1).

Queues in series:

(i) Subcritical: stable (Theorem 7).

(ii) Supercritical: unstable (Theorem 7).

(iii) Critical: stable, properly substable, or unstable, if the component queues are substable (examples in § 3·2); unstable if any component queue is unstable (Theorem 7), and in particular if any critical component queue is of type M (Theorem 7, Corollary).

Many-server queues:

(i) Subcritical: stable or properly substable (Theorem 8, and example in § 4·3).

(ii) Supercritical: unstable (Theorem 8).

(iii) Critical: stable, properly substable, or unstable, if consolidated queue is substable (examples in § 4·3); unstable if consolidated queue unstable (Theorem 8), and in particular if this is of type M (Theorem 8, Corollary).

From Lemma 1 it follows that none of these queues can be properly substable if all the servers are initially unoccupied.

The general relativistic field equations for regions of space containing electromagnetic fields but no matter are

,

where

,

and κ = 8ΠG/c4 is the gravitational constant. Here giκ is the metric tensor and Fiκ is the electromagnetic field tensor which satisfies Maxwell's equations if

and

,

where Ai is the electromagnetic four-potential.

The propagation of plane waves in a thermo-elastic body placed in a magnetic field has been investigated. It is shown that the problem can be reduced to one of pure thermo-elasticity, provided that the longitudinal elastic wave velocity is suitably increased in the case of infinite electrical conductivity. For finite electrical conductivity and low frequency, however, the thermo-elastic coupling factor is to be increased.

We consider the propagation of thermo-elastic Rayleigh waves in a half space, which has been subjected to large uniform extensions, at constant temperature, in three mutually perpendicular directions, one being perpendicular to the surface, and the extension ratios in the other two directions being equal. Rayleigh waves, assuming (a) isentropic conditions and (b) isothermal conditions, are studied, and the relation of the approximation (a) to the exact thermo-elastic theory is considered.

In this note we consider the axially symmetric indentation of a transversely isotropic layer resting on a rigid foundation. The corresponding half-space problems both for isotropic and transversely isotropic materials have been solved previously (see (l), (2), (3), p. 455) and, more recently, the present problem has been solved for isotropic materials in (4) and (5) using methods different from that given in this paper. In § 2 the problem is reduced to the solution of a Fredholm integral equation of the second kind. In § 3 and subsequently the case of incomplete penetration by the punch, when the radius of the circle of contact is less than that of the end face of the punch, is considered. Finally, in §4 the results are illustrated by considering a hemispherically ended punch.

Series which behave rather like Taylor's series at more than, one point are described and their convergence is discussed briefly.

Let

where z, zs are complex numbers and zs are the points, (the base points)about which a function is to be expanded. They need not be distinct. p{z) is a polynomial of degree n. in z. Assume first that z, zs are real and the zs are distinct. Then polynomials pn-1,t(z) may be chosen so that the series

has contact of order N at the points zs with any specified real function which has N derivatives at these points. This is proved merely by rearranging the simplest polynomial in z which has contact of order N with/(z) at z = zs. The term given by t = 0 in (2) allows the series to have the values f(zs) at z = zs; adding the term for t = 1 does not alter the values of the series at z = zs; it allows the series to have a derivative with values f′(zs) at z = z; similarly adding the term for t = 2 does not alter the value of the series or its first derivative at z = zs, it allows the series to have a second derivative with values f″{zs) at z = zs; and so on. If m of the zs coincide then f(z) has mN + m− 1 derivatives there and contact is of order mN + m− 1. If z, zs are complex the result is seen to be valid subject to the obvious generalization of ‘ contact of the ith order’ to complex variables and functions.

Using Newtonian theory of gravitation and postulating the existence of informational waves of gravitation, a time-varying gravitational potential is obtained. When a point mass is submitted to such a potential a precession of its orbit, of the right order of magnitude, is obtained. The correct value of the angle of deflexion of light by massive bodies can also be calculated.