Let be a positive definite binary quadratic form with real coefficients and discriminant b2 − 4ac = −1.

Among such forms, let . The Epstein zeta function of f is denned to be

Rankin [7], Cassels [1], Ennola [5], and Diananda [4] between them proved that for every real s > 0,

We prove a corresponding result for theta functions. For real α > 0, let

This function satisfies the functional equation

(This may be proved by using the formula (4) below, and then twice applying the identity (8).)

The Dedekind eta-function is defined for any τ in the upper half-plane by

where x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a function

where N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ by

when a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier series

then we define a Hecke operator Tp by

where

and

In this paper we consider mappings induced by matrix multiplication which are defined on lattices of matrices whose coordinates come from a fixed orthomodular lattice L (i.e. a lattice with an orthocomplementation denoted by ′ in which a ≦ b ⇒ a ∨ (a′ ∧ b) = b). will denote the set of all m × n matrices over L with partial order and lattice operations defined coordinatewise. For conformal matrices A and B the (i,j)th coordinate of the matrix product AB is defined to be (AB)ij = Vk(Aik ∧ BkJ). We assume familiarity with the notation and results of [1]. is an orthomodular lattice and the (lattice) centre of is defined as , where we say that A commutes with B and write . In § 1 it is shown that mappings from into characterized by right multiplication X → XP (P ∈ ) are residuated if and only if p ∈ ℘ (). (Similarly for left multiplication.) This result is used to show the existence of residuated pairs. Hence, in § 2 we are able to extend a result of Blyth [3] which relates invertible and cancellable matrices (see Theorem 3 and its corollaries). Finally, for right (left) multiplication mappings, characterizations are given in § 3 for closure operators, quantifiers, range closed mappings, and Sasaki projections.

[7] Sands raised the following questions:

(1) Must a hereditary radical which is right strong be left strong?

(2) Must a right hereditary radical be left hereditary?

(3) (Example 6) Does there exist a right strong radical containing the prime radical β which is not left strong or hereditary?

Negative answers to questions (1) and (2) were given by Beidar [1].

In this paper we present different examples to answer (1) and (2), and we answer (3). We prove that the strongly prime radical defined in [4, 5] is right but not left strong. In the proof we use an example given by Parmenter, Passman and Stewart [6]. The same example and the strongly prime radical are used to answer (2) and (3).

All groups considered in the sequel are finite. Let (ℭ and denote the formations of groups which consist of collections of groups that respectively either split over each normal subgroup (nC-groups) or for which the groups do not possess nontrivial Frattini chief factors [8]. The purpose of this article is to develop and expand a concept that arises naturally with the residuals for these formations, namely each G-chief factor is non-complemented (Frattini). With respect to a solid set X of maximal subgroups, these properties are generalized respectively to so-called X-parafrattini (X-profrattini) normal subgroups for which each type is closed relative to products. The relationships among the unique maximal normal subgroups that result from these products, the solid set of maximal subgroups X, X-prefrattini subgroups, and the residuals of formations are explored. This leads to a well-defined collected of formations, the partially nonsaturated formations, with properties analogous to those which are totally non-saturated. In the development, attention is given to a set of maximal subgroups which is the image of a solid function defined on all groups, a weaker condition than that of a solid set. A result of particular interest answers affirmatively the long-standing conjecture that a non-trivial nC-group G is solvable if and only if each G-chief factor is complemented by a maximal subgroup. This will force a critical re-examination of the classification problem for nC-groups. Since the article continues the investigations on finite groups initiated in [2], a familiarity with that article is assumed. All other notation and terminology is from [6]. If M is a maximal subgroup of a group G and G/C or e G(M) is a monolithic primitive group, i.e. a group with a unique minimal normal subgroup, then M is called a monolithic maximal subgroupof G.

In a recent paper M. Cho [5] asked whether Taylor's joint spectrum σ(a1, …, an; X) of a commuting n-tuple (a1,…, an) of continuous linear operators in a Banach space X is contained in the closure V(a1, …, an; X)- of the joint spatial numerical range of (a1, …, an). Among other things we prove that even the convex hull of the classical joint spectrum Sp(a1, …, an; 〈a1, …, an〉), considered in the Banach algebra 〈a1, …, an〉, generated by a1, …, an, is contained in V(a1, …, an; X)-.

There are some simple facts which distinguish Lie-algebras over fields of prime characteristic from Lie-algebras over fields of characteristic zero. These are

(1) The degrees of the absolutely irreducible representations of a Lie-algebra of prime characteristic are bounded whereas, according to a theorem of H. Weyl, the degrees of the absolutely irreducible representations of a semi-simple Lie-algebra over a field of characteristic zero can be arbitrarily high.

(2) For each Lie-algebra of prime characteristic there are indecomposable representations which are not irreducible, whereas every indecomposable representation of a semi-simple Liealgebra over a field of characteristic zero is irreducible (cf. [4]).

(3) The quotient ring of the embedding algebra of a Lie-algebra over a field of primecharacteristic is a division algebra of finite dimension over its center, whereas this is not the case for characteristic zero. (cf. [4]).

(4) There are faithful fully reducible representations of every Lie-algebra of primecharacteristic, whereas for characteristic zero only ring sums of semi-simple Lie-algebras and abelian Lie-algebras admit faithful fully reducible representations (cf. [6], [2], [4]).

In a recent paper [1], Jones extended the definition of the convolution of distributions so that further convolutions could be defined. The convolution w1*w2 of two distributions w1 and w2 was defined as the limit ofithe sequence {wln*w2n}, provided the limit w exists in the sense that

for all fine functions ф in the terminology of Jones [2], where

w1n(x) = wl(x)τ(x/n), W2n(x) = w2(x)τ(x/n)

and τ is an infinitely differentiable function satisfying the following conditions:

(i) τ(x) = τ(—x),

(ii)0 ≤ τ (x) ≤ l,

(iii)τ (x) = l for |x| ≤ ½,

(iv) τ (x) = 0 for |x| ≥ 1.

The definition of Ωf,a on p. 57 should read a

In 1856 Hermite showed how to determine by purely rational operations the number of zeros of a given polynomial lying in a specified half plane [1]: one inspects the signature of a certain Hermitian form. This type of result is still of interest for practical applications, and several authors have provided alternatives to Hermite's original, highly computational proof (for example [2, 3]). Recently V. Pták and the author gave a simple matrix-theoretic proof and generalization of a class of Hermite-type theorems [4]. This class included the Schur-Cohn test for zeros in a circle, but not, to our regret, the original theorem of Hermite. The purpose of this note is to show that a slight modification of our method does indeed provide a simple proof of Hermite's theorem.

In his recent study of free inverse semigroups, Munn [2] introduced and used extensively the concept of a word-tree. In this note the number of such trees is found.

In this paper we use the Bourbaki conventions for rings and modules: all rings are associative but not necessarily commutative and have an identity element; all modules are unital.

§ 1. When a numerical method of obtaining an approximate solution of a linear differential equation is employed, the process involves two distinct types of approximation. The region of integration having been covered with a regular net, the differential equation and the appropriate boundary conditions are replaced by finite difference equations which are linear equations in the values of the dependent variable at the nodes of the net.

In two papers [3] and [4], the author has extended the inequality of Schur (Theorem 319 of [2]) to cases involving kernels which satisfy identities of the form

The purpose of this paper is to prove a general inequality, which includes the above and also the inequality of Young (Theorem 281 of [2]) as special cases. We shall give the results a general setting by considering functions defined on abstract measure spaces. From this we shall deduce an extension to n dimensions of the results given in [3], which also generalises a similar extension of the Schur inequality given by Stein and Weiss. In fact some cases of the other results given in [5] will follow directly from our theorem.

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε ℋU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the ℋW(p + 1/2) operator  by A = |A|1/2U |A|l/2 Let  = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.

The translational hull Ω(S) of a semigroup S plays an important role in the theory of ideal extensions of semigroups. In fact, every ideal extension of S by a semigroup T with zero can be constructed using a certain partial homomorphism of T\0 into Ω(S); a particular case of interest is when S is weakly reductive (see §4.4 of [3], [2], [7]). A theorem of Gluskin [6, 1.7.1] states that if S is a weakly reductive semigroup and a densely embedded ideal of a semigroup Q, then Q and Ω(S) are isomorphic. A number of papers of Soviet mathematicians deal with the abstract characteristic (abstract semigroup, satisfying certain conditions, isomorphic to the given semigroup) of various classes of (partial) transformation semigroups in terms of densely embedded ideals (see, e.g., [4]). In many of the cases studied, the densely embedded ideal in question is a completely 0-simple semigroup, so that Gluskin's theorem mentioned above applies. This enhances the importance of the translational hull of a weakly reductive, and in particular of a completely 0-simple semigroup. Gluskin [5] applied the theory of densely embedded ideals (which are completely 0-simple semigroups) also to semigroups and rings of endomorphisms of a linear manifold and to certain classes of abstract rings.

A Banach space E is said to have Property (w) if every operator from E into E' is weakly compact. This property was introduced by E. and P. Saab in [9]. They observe that for Banach lattices, Property (w) is equivalent to Property (V*), which in turn is equivalent to the Banach lattice having a weakly sequentially complete dual. Thus the following question was raised in [9].

Does every Banach space with Property (w) have a weakly sequentially complete dual, or even Property (V*)?

In this paper, we give two examples, both of which answer the question in the negative. Both examples are James type spaces considered in [1]. They both possess properties stronger than Property (w). The first example has the property that every operator from the space into the dual is compact. In the second example, both the space and its dual have Property (w). In the last section we establish some partial results concerning the problem (also raised in [9]) of whether (w) passes from a Banach space E to C(K, E).

Brian Hartley asked me whether a free (nilpotent of class 2 and exponent p2)-group of countable rank has a faithful linear representation of finite degree, p here being a prime of course. The answer is yes. The point is that this then yields via work of F. Leinen and M. J. Tomkinson, see [3,3.6] an image of a linear p-group, which is not even finitary linear. The question of which relatively free groups have faithful linear representations dates back at least to work of W. Magnus in the 1930's, see [4, pp. 33, 34 and the final comment on p. 40] for a discussion of this. Our construction, which works more generally, is a further contribution. We write ℜc for the variety of nilpotent groups of class at most c and (ℭ9 for the variety of groups of exponent dividing q.

Let M be a smooth m-dimensional submanifold in (m + d)-dimensional Euclidean space ℝm+d For x ∊ M and a non-zero vector X in TXM, we define the (d + l)-dimensional affine subspace E(x, X) ofℝm+d by