In this paper we introduce the notion of normally ordered block-group as a natural extension of the notion of normally ordered inverse semigroup considered previously by the author. We prove that the class NOS of all normally ordered block-groups forms a pseudovariety of semigroups and, by using the Munn representation of a block-group, we deduce the decompositions in Mal'cev products NOS = EIPOI and NOS ∩ A = NPOI, where A, EI and N denote the pseudovarieties of all aperiodic semigroups, all semigroups with just one idempotent and all nilpotent semigroups, respectively, and POI denotes the pseudovariety of semigroups generated by all semigroups of injective order-preserving partial transformations on a finite chain. These relations are obtained after showing the equalities BG = EIEcom = NEcom, where BG and Ecom denote the pseudovarieties of all block-groups and all semigroups with commuting idempotents, respectively.

1. All operators considered in this paper are bounded operators on a Hilbert space. In case A and B are self-adjoint, certain conditions on A, B and their difference

assuring the unitary equivalence of Aand B,

have recently been obtained by Rosenblum [6] and Kato [2]. The present paper will consider the problem of investigating consequences of an assumed relation of type (2) for some unitary U together with an additional hypothesis that the difference H of (1) be non-negative, so that

First, it is easy to see that if only (2) and (3) are assumed, thereby allowing H = 0, relation (2) can hold for A arbitrary with U = I (identity) and B = A. If H = 0 in (3) is not allowed, however (an impossible assumption in the finite dimensional case, incidentally, since then the trace of H is zero and hence H = 0), it will be shown, among other things, that any unitary operator U for which (2) and (3) hold must have a spectrum with a positive measure (as a consequence of (i) of Theorem 2 below). Moreover A (hence B) cannot differ from a completely continuous operator by a constant multiple of the identity (Theorem 1). In case 0 is not in the point spectrum of H, then U is even absolutely continuous (see (iv) of Theorem 2). In § 4, applications to semi-normal operators will be given.

The purpose of this note is to establish the following

Theorem. The centre of a (left) hereditary local ring is either afield or a one-dimensional regular local ring.

Before starting the proof, it is necessary to explain the terminology. A ring R with an identity element is called a left local ring if the elements of R which do not have left inverses form a left ideal I. In these circumstances (see [1, Proposition 2.1, p. 147]), I is necessarily a two-sided ideal and it consists precisely of all the elements of R which do not have right inverses. Furthermore, every element of R which is not in I possesses a two-sided inverse. Thus there is, in fact, no difference between a left local ring and a right local ring and therefore one speaks simply of a local ring. In addition, I contains every proper left ideal and every proper right ideal. We may therefore describe I simply as the maximal ideal of R.

The commutator [a, b] of two elements a and b in a group G satisfies the identity

ab = ba[a, b].

The subgroups we study are contained in the commutator subgroup G′, which is the subgroup generated by all the commutators.

The group G is covered by a well-known set of normal subgroups, namely the normal closures {g}G of the cyclic subgroups {g} in G. In a similar way one may associate a subgroup K(g) with each element g, by defining K(g) to be the subgroup generated by the commutators [g, x] as x takes all values in G. These subgroups generate G′ (but do not cover G′ in general), and are normal in G in consequence of the identical relation

(A) [g, x]Y = [g, y]−1[g, xy]

holding for all g, x and y in G. (By ab we mean b−1ab.) It is easy to see that

{g}G = {g, K(g)}.

The integral function

is known as Airy's Integral since, when z is real, it is equal to the integral

which first arose in Airy's researches on optics. It is readily seen that w= Ai(z) satisfies the differential equation d2w/dz2 = zw, an equation which also has solutions Ai(ωz), Ai(ω2z), where ω is the complex cube root of unity, exp 2/3πi. The three solutions are connected by the relation.

We show that every Hilbert C$^*$-module $E$ is a JB$^*$-triple in a canonical way, establish an explicit expression for the holomorphic automorphisms of the unit ball of $E$, discuss the existence of fixed points for these automorphisms and give sufficient conditions for $E$ to have the density property.

We give explicit examples of asymmetric Riemann surfaces (that is, Riemann surfaces with trivial conformal automorphism group) for all genera g ≥ 3. The technique uses Schreier coset diagrams to construct torsion-free subgroups in groups of signature (0; 2,3,r) for certain values of r.

It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k).

In the usual definition of an inductive limit of locally convex spaces, one is given a linear space E, a family (Eα) of locally convex spaces and a set (iα) of linear maps from Eα into E. Garling in [2] studies an extension of this, looking at absolutely convex subsets Sα of Eα and restrictions jα of iα to such sets. If, in the definition of Garling [2, p. 3], each Sα is instead a balanced semiconvex set, then the finest linear (not necessarily locally convex) topology on E for which the maps ja are continuous, will be referred to as the generalized *-inductive limit topology of the semiconvex sets. This topology is our object of study in the present paper; we find applications in the closed graph theorem.

In a recent paper [6], this author has extended the method of the kernel function [1] to the boundary value problems of the generalized axially symmetric potentials

This method can also be applied to a more general class of singular differential equations, namely

or, equivalently,

We shall derive in the sequel explicit formulas for the Dirichlet problems of (1.1) in the first quadrant of the x-y plane in terms of sufficiently smooth boundary data, and obtain an error-bound for their approximate solutions. We shall also indicate how the Neumann problem can be solved.

We prove under quite general assumptions the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Neumann boundary conditions.

Denote by $(R,\cdot)$ the multiplicative semigroup of an associative algebra $R$ over an infinite field, and let $(R,\circ)$ represent $R$ when viewed as a semigroup via the circle operation $x\circy=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $R$. Namely, we prove that the following conditions on $R$ are equivalent: the semigroup $(R,\circ)$ satisfies an identity; the semigroup $(R,\cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $R$ satisfies the Engel condition. When $R$ is finitely generated these conditions are each equivalent to $R$ being upper Lie nilpotent.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25

A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

The Atiyah–Drinfeld–Hitchin–Manin–Nahm (ADHMN) construction of magnetic monopoles is given in terms of the (normalizable) solutions of an associated Weyl equation. We focus here on solving this equation directly by algebro-geometric means. The (adjoint) Weyl equation is solved using an ansatz of Nahm in terms of Baker–Akhiezer functions. The solution of Nahm's equation is not directly used in our development.

Following the notation in Faudree and Schelp [3], we write G → (F, H) to mean that every 2-coloring of E(G), the edge set of G, contains a green (the first color) F or a red (the second color) H. Then the Ramsey number r(F, H) of two graphs F and H with no isolated vertices has been defined as the minimum p such that Kp → (F, H).

Let A and B be semisimple Banach algebras, and let M1(A) (resp. M1(B)) be the algebra of left multipliers on A (resp. B). Suppose that A is an abstract Segal algebra in B. We find conditions on A and B which imply that M1(A) is topologically algebra isomorphic to M1(B). As a special case we obtain the result of [8] which states that if A is an A*-algebra that is a*-ideal in its B*-algebra completion B and A2 is dense in A then M1(A) is topologically algebra isomorphic to M1(B). We make an application of our main result to right complemented Banach algebras.

For a series Σan with partial sums An=a0 + a1 + … + an(n ≥ 0), supposed to be real in this note, we define, in a generally accepted notation ([2], pp. 7, 9, 94–98), the following transforms:

(C, α) sequence-transform,

(H, k)sequence-transform (k = 0, 1, 2, …):

(A: C, α)fnunction-transform,

We prove that there are only $O(H^{3+\epsilon})$ quartic integer polynomials with height at most $H$ and a Galois group which is a proper subgroup of $S_4$. This improves in the special case of degree four a bound by Gallagher that yielded $O(H^{7/2} \log H)$.

In a recent paper [5] Tits proves that a linear group over a field of characteristic zero is either solvable-by-finite or else contains a non-cyclic free subgroup. In this note we determine all the infinite irreducible solvable-by-finite subgroups of GL(2, F), where F is an algebraically closed field of characteristic zero. (Every reducible subgroup of GL(2, F) is metabelian.) In addition, we prove that an irreducible subgroup of GL(2, F) has an irreducible solvable-by-finite subgroup if and only if it contains an element of zero trace.

A sequence a1 < a2 < … of positive integers is said to be primitive if no element of the sequence divides any other. The study of primitive sequences arose naturally out of investigations into the subject of abundant numbers, where sequences each of whose elements is of the form , the pi being fixed primes, are of particular importance. Such a sequence is said to be built up from the primes p1…pr. Thus Dickson [1], in an early paper on abundant numbers, proved that a primitive sequence built up from a fixed set of primes is necessarily finite.