The formula to be proved is

where R(z)>0.

The purpose of this note is to present certain aspects of the theory of spectral operators in Grothendieck spaces with the Dunford-Pettis property, briefly, GDP-spaces, thereby elaborating on the recent note [10].

For example, the sum and product of commuting spectral operators in such spaces are again spectral operators (cf. Proposition 2.1) and a continuous linear operator is spectral if and only if it has finite spectrum (cf. Proposition 2.2). Accordingly, if a spectral operator is of finite type, then its spectrum consists entirely of eigenvalues. Furthermore, it turns out that there are no unbounded spectral operators in such spaces (cf. Proposition 2.4). As a simple application of these results we are able to determine which multiplication operators in certain function spaces are spectral operators.

In this note we consider the problem of determining the stress on the boundary y = 0 of the elastic half-plane y ≡ 0 when there are prescribed body forces acting in the interior and the boundary is free from applied stress. Expressions for the components of stress at a general point of the half-plane when the imposed body force is concentrated at a single point have been derived by Melan [1], Sneddon [2] and Green [3], each author making use of a different method.

On p. 244 of Glasgow Math. J.27 (1985) on the right hand side of one of the 6 equations characterizing the 4 fixed points of the involution v a sign error has occurred.The relevant equation should read

y0y3y5y6=–1,

or the points would not lie on the curve.

Correcting the error unfortunately invalidates the model of an elliptic curve given in §6, which therefore has to be re-evaluated. First we find, in the notation of the paper,

2 f (x) = ((r + 1)/(R + 2))2.

1. A number of inclusion theorems have been given in connection with methods of summation which include the Riesz method (R, λ, κ). Lorentz [4, Theorem 10] gives necessary and sufficient conditions for a sequence to sequence regular matrix A = (an, v) to be such that A ⊃ (R, λ, 1)†. He imposes restrictions on the sequence { λn}, so that A does not include all Riesz methods of order 1. In Theorem 1 below, we generalize the Lorentz theorem by giving a condition without restriction on λn, If the matrix A is a series to sequence or series to function regular matrix, there do not appear to be any results concerning the general inclusion

A ⊃ (R, λ, κ).

However, when A is the Riemann method (ℜ, λ, μ), Russell [7], generalizing earlier results, has given sufficient conditions for (ℜ, λ, μ) ⊃ (R, λ, κ). Our Theorem 2 gives necessary and sufficient conditions for A ⊃ (R, λ, 1), where A satisfies the condition an, v → 1 (n →co, ν fixed). Thus Theorem 2 applies to any series to sequence regular matrix A. In Theorem 3 we give a further representation for matrices A which include (R, λ, 1), and finally make some remarks on the problem of characterizing matrices which include Riesz methods of any positive order κ.

Let Г(1) denote the homogeneous modular group of 2 × 2 matrices with integral entries and determinant 1. Let (1) be the inhomogeneous modular group of 2 × 2 integral matrices of determinant 1 in which a matrix is identified with its negative. (N), the principal congruence subgroup of level N, is the subgroup of (1) consisting of all T ∈ (1) for which T ≡ ± I (mod N), where N is a positive integer and I is the identity matrix. A subgroup of (1) is said to be a congruence group of level N if contains (N) and N is the least such integer. Similarly, we denote by Г(N) the principal congruence subgroup of level N of Г(1), consisting of those T∈(1) for which T ≡ I (mod N), and we say that a sub group of Г(1) is a congruence group of level N if contains Г (N) and N is minimal with respect to this property. In a recent paper [9] Rankin considered lattice subgroups of a free congruence subgroup of rank n of (1). By a lattice subgroup of we mean a subgroup of which contains the commutator group . In particular, he showed that, if is a congruence group of level N and if is a lattice congruence subgroup of of level qr, where r is the largest divisor of qr prime to N, then N divides q and r divides 12. He then posed the problem of finding an upper bound for the factor q. It is the purpose of this paper to find such an upper bound for q. We also consider bounds for the factor r.

In a recent paper [1], the author divided the semi-special permutations on [n] that are not linear into two classes. The first class consists of the semi-special permutations which, for all possible values of s, have s as a principal number and which induce modulo s the identity permutation. The second class consists of all the semi-special permutations, with principal number s, which induce modulo s linear permutations other than the identity, where again s takes all its possible values.

In the course of a study of commutator subgroups I. D. Macdonald [1] presented the free nilpotent group G4 of class 2 on 4 generators as an example of a nilpotent group whose commutator subgroup has elements that are not commutators. To demonstrate this he proceeded as follows: let G4 = 〈a1, a2, a3, a4〉 and put cij = [ai, aj] for 1 ≦ i < j ≦ 4. Then the relations in G4 are [cij, ak] = 1 for 1 ≦ i < j ≦ 4 and 1 ≦ k ≦ 4, and their consequences. Macdonald observed that an arbitrary commutator may be written as

which simplifies to

where δij = αiβj - αjβi The indices δij satisfy the relation

It follows that the element c13c24 in G′4 (for which δ12 = δ14 = δ23 = δ34 = 0 and δ13 = δ24 = 1) is not a commutator.

Semilocal and related classes of group rings have been investigated by many authors (cf. [10]). In particular, the following results have been obtained.

Theorem A[4,10]. Let K be a field and G a group.

(i) If ch K = 0, then K[G] is semilocal if and only if G is finite.

(ii) If ch K = p>0 and G is locally finite, then K[G] is semilocal if and only if G contains a p-subgroup of finite index.

In the case of semigroup rings some stronger conditions have been studied. Munn examined the semisimple artinian situation [6]. Zelmanov showed that if K[G]is artinian then G must be finite [11].

S(X) is the semigroup of all continuous self maps of the topological space X and for any semigroup S, Cong(S) will denote the complete lattice of congruences on S. Cong(S) has a zero Z and a unit U. Specifically, Z = {(a, a):a ∈ S} and U = S × S. Evidently, Z and U are distinct if S has at least two elements. By a proper congruence on S we mean any congruence which differs from each of these. Since S(X) has more than one element when X is nondegenerate, we will assume without further mention that the spaces we discuss in this paper have more than one point. We observed in [4] that there are a number of topological spaces X such that S(X) has a largest proper congruence, that is, Cong(S(X)) has a unique dual atom which is greater than every other proper congruence on S(X). On the other hand, we also found out in [5] that it is also common for S(X) to fail to have a largest proper congruence. We will see that the situation is quite different at the other end of the spectrum in that it is rather rare for S(X) not to have a smallest proper congruence. In other words, for most spaces X, Cong(S(X)) has a unique atom which is smaller than every other proper congruence.

A group G is the product of its subgroups A and B if G equals the set AB = (ab | a ∊ A, b ∊ B). A subgroup H of G is called prefactorized if it is the product of a subgroup of A and a subgroup of B; thus H is prefactorized if and only if H = (H ∩ A)(H ∩ B). A prefactorized subgroup H of G is factorized if it contains A ∩ B. if H is any subgroup of G = AB, then the intersection X ofall factorized subgroups of G containing H is itself factorized; see for example [2, Lemma 1.1.2]. This subgroup, which is evidently the smallest factorized subgroup of G which contains H, is called the factorizer of H in G = AB.

The nonabelian tensor square G⊗G of a group G is generated by the symbols g⊗h, g, h ∈ G, subject to the relations

,

for all g, g′, h, h′ ∈ G, where The tensor square is a special case of the nonabelian tensor product which has its origins in homotopy theory. It was introduced by R. Brown and J. L. Loday in [4] and [5], extending ideas of Whitehead in [6].

Let f(x, y, z, t) satisfy the equation

For certain purposes, particularly in connection with the propagation of a boundary of fusion etc., it is of interest to discover solutions of (1) which permit the equation:

to be solved explicitly in the form:

This suggests the examination of solutions of the type

where

and f, ϕ, ψ are functions to be determined. To save repetition, Roman capitals denote arbitrary constants throughout.

In this paper we give new information about the conjugacy vector of the group , the Sylow p-subgroup of GL(n, q) consisting of the upper unitriangular matrices. The first two components of this vector are given in [4]. Here, we obtain the third component, that is, the number of conjugacy classes whose centralizer has qn+l elements. Besides, we give the whole set of numbers which compose this vector:

A ring R is called radical if it coincides with its Jacobson radical, which means that Rforms a group under the operation a ° b = a + b + ab for all a and b in R. This group is called the adjoint group R° of R. The relation between the adjoint group R° and the additive group R+ of a radical rin R is an interesting topic to study. It has been shown in [1] that the finiteness conditions “minimax”, “finite Prufer rank”, “finite abelian subgroup rank” and “finite torsionfree rank” carry over from the adjoint group to the additive group of a radical ring. The converse is true for the minimax condition, while it fails for all the other above finiteness conditions by an example due to Sysak [6] (see also [2, Theorem 6.1.2]). However, we will show that the converse holds if we restrict to the class of nil rings, i.e. the rings R such that for any a є R there exists an n = n(a) with an = 0.

In the equation

dis any positive integer which is not a perfect square. For convenience we shall consider only those solutions of (1) for which x and yare both positive. All the others can be obtained from these. In fact, it is well known that if (x0, y0) is the minimum positive integer solution of (1), then all integer solutions (x, y) are given by

and, in particular, all positive integer solutions are given by

The first formula to be proved is

where p ≧ q + 1, | amp z | < л, R(k±n + αr)>0, r = l, 2, …, p. For other values of p and q the result is valid if the integral is convergent. A second formula is given in § 3.

The following formulae are required in the proof:

where R(z);>0, (1);

where R(α)>0, | amp z | < л, (2);

where the contour starts from -∞ on the ξ-axis, passes round the origin in the positive direction, and ends at -∞ on the ξ-axis, the initial value of amp ζ being - л, (3).