Oldroyd (1) considered non-Newtonian liquids for which the stress tensor sik and the rate of strain tensor eik=½(υk,i+υi,k)are related as follows

Here υi denotes the velocity vector, p an isotropic pressure, gik the metric tensor, and t time; η0 is a constant having the dimensions of viscosity and λ1, λ2, μ0, ν1, ν2, are constants having the dimension of time. Covariant suffixes are written below, contravariant above and the usual summation convention is employed. A suffix i following a comma indicates covariant differentiation with respect to the space variable xi.

Let γn denote the value of

where n is a definite integer; and let γ denote the limit of

when the integer k is indefinitely increased. It is known that the expansion of γn – γ in ascending powers of 1/n is

where B1, B3, B5… are the numbers of Bernoulli. The series (3) is, however, divergent, as B2r+1 not only increases indefinitely with r, but bears† an infinite ratio to B2r–1 in this case. It is proposed to find by elementary methods the expansion of γn – γ up to the term in nr and to estimate the error (of order l/nr+1) made in omitting further terms of series (3). I shall take the case of r = 9, but the process is quite general.

Let p be an odd prime and let GL(2, p) denote the general linear group of invertible 2x2 matrices with entries in the field of p elements. The groupPGL(2, p) is the factor of GL(2,p) by its centre and has derived group PSL(2,p) with derived factor C2, the cyclic group of order 2.

The paper studies dense Q-subalgebras of Banach and C*-algebras. It proves that the domain D(δ) of a closed unbounded derivation δ of a Banach unital algebra A automatically contains the identity and is a Q-subalgebra of A, so that SpA(x) = SpD(δ)(x) for all x∈D(δ). The paper shows that every finite-dimensional semisimple representation of a Q-subalgebra is continuous. It also shows that if π is an injective *-homomorphism of a dense locally normal Q*-subalgebra B of a C*-algebra, then ‖x‖≦‖π(x)‖ for all x∈B. The paper studies the link between closed ideals of a Banach algebra A and of its dense subalgebra B. In particular, if A is a C*-algebra and B is a locally normal *-subalgebra of A, then I→I∩B is a one-to-one mapping of the set of all closed two-sided ideals in A onto the set of all closed two-sided ideals in B and .

In (2), Holcombe investigated near-rings of zero-preserving mappings of a group Γ which commute with the elements of a semigroup S of endomorphisms of Γ and examined the question: under what conditions do near-rings of this type have near-rings of right quotients which are 2-primitive with minimum condition on right ideals? In the first part of this paper (§2) we investigate further properties of near-rings of this type. The second part of the paper (§3) deals with those near-rings which have semisimple near-rings of right quotients. Our results here are analogous to those of Goldie (1); in particular, with a suitable definition of finite rank we prove that a near-ring which has a semisimple near-ring of right quotients has finite rank

The communication on this subject, as originally made to the society, consisted of a series of theorems, giving (1) expressions for the radii of a great many sets of circles, (2) identities connecting several sets of these radii, and (3) miscellaneous identities closely related thereto. As, however, the paper culminated in a general theorem which may be looked upon as fundamental, and the proof of which makes evident the mode of arriving at the said expressions for radii, and as the relations connecting sets of radii are easily found when attention has been directed to their existence, I have thought it best to print little more than the fundamental theorem and a few auxiliary notes.

It is known that while the Shannon and the Rényi entropies are additive, the measure entropy of degree β proposed by Havrda and Charvat (7) is non-additive. Ever since Chaundy and McLeod (4) considered the following functional equation

which arose in statistical thermodynamics, (1.1) has been extensively studied (1, 5, 6, 8). From the algebraic properties of symmetry, expansibility and branching of the entropy (viz. Shannon entropy Hn, etc.) one obtains the sum representation

which with the property of additivity yields the functional equation (1.1), (9, 10).

In many problems of physics, even in widely different branches of the subject, the relation satisfied by the variables is expressible by means of a linear differential equation of the second order. In general, “initial” conditions have also to be satisfied. If the equation truly represents the physical conditions in, for example, some case of motion, and if no state of instability exists, the solution must be unique. But it is impossible in any case to say with absolute certainty that the representation is strict. The possible error depends on the error which may be made in observation or experiment, and on the number of independent observations or experiments the results of which have been used as the basis of the “law” expressed by the equation.

The classification of the normal subgroups of the infinite general linear group GL(Ω, R) has received much attention and has been studied in, for example, (6), (4) and (2). The main theorem of (6) gives a complete classification of the normal subgroups of GL(Ω, R) when R is a division ring, while the results of (2) require that R satisfies certain finiteness conditions. The object of this paper is to produce a classification, along the lines of that given by Wilson in (7) or by Bass in (3) in the finite dimensional case, that does not require any finiteness assumptions. However, when R is Noetherian, the classification given here reduces to that given in (2).

There are two planes, called “parabolic planes,” through eachline on the cubic surface cutting the surface in residual conics touching the line in question at two points called the “parabolic points” of that line.

The aim of this note is to present two observations about the classical Franklin system. First we show that the Franklin system, when considered in the space generated by special atoms (as defined and studied by Soares de Souza in [11] and ]12]) is an unconditional basis equivalent to the unit vector basis in l1. In our second result we give conceptually simpler proofs and some extensions of the results of Bočkariov's [1] about the conjugate Franklin system.

Strongly prime rings were introduced by Handelman and Lawrence [5] and in [2] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we extend the concept of strongly prime to near-rings. We show that the class M of distributively generated near-rings is a special class in the sense of Kaarli [6]. We also show that if N is any distributively generated near-ring, then UM(N), UM denotes the upper radical determined by the class M, coincides with the intersection of all the strongly prime ideals of N.

This paper concludes the investigation of axially symmetric stress distributions in elastic solids containing penny-shaped cracks, commenced in previous papers (1), (2), by considering the stress distribution in a circular beam containing a crack opened by internal pressure or by uniform tension. The method of analysis, developed in the previous papers, is to first seek a representation of the displacement at a point of the beam as a sum of two terms, one of which is a representation of the displacement due to the crack in an otherwise unbounded infinite solid whilst the second is a general representation of the displacement in an undamaged beam, and then to show that this representation satisfies the conditions on the crack and the curved surface of the beam provided an unknown function occurring in it is the solution of a certain Fredholm integral equation. This equation holds whatever the ratio of the radius of the crack to that of the beam, but is most readily solved by iteration when this ratio is small, this solution being a perturbation on that for a crack in an infinite solid.

The following theorems in ring theory are well-known:

1. Let R be a ring. If e is a unique left identity, then e is also a right identity.

2. If R is a ring with more than one element such that aR = R for every nonzero element a ε R, then R is a division ring.

3. A ring R with identity e ≠ 0 is a division ring if and only if it has no proper right ideals.

Let ∑an be a given infinite series and {λn} a non-negative, strictly increasing, monotonic sequence, tending to infinity with n. We write, for w > λ0,

and, for r>0, we write

is known as the Riesz sum of “ type ” λn and “ order ” r, and

is called the Riesz mean of type λn and order r.