Using a classical result of Nagata, Achilles, Huneke and Vogel gave a criterion for the Stückrad-Vogel multiplicity to take the value one. We use Huneke's extension of Nagata's theorem to give a necessary condition for the Stückrad-Vogel multiplicity to have an arbitrary preassigned bound, under certain conditions. A usable criterion of multiplicity n results (given mild hypotheses). We also revisit some basic results in the Stückrad-Vogel theory in the light of the behaviour of tensor products of affine primary rings, and also revisit some arguments of Achilles, Huneke and Vogel from the point of view of fibre rings.

Let E be a free product of a finite number of cyclic groups, and S a normal subgroup of E such that E/S ≅ G is finite. For a prime p, Ŝ = S / S′Sp may be regarded as an -module. Whenever E is a free group, Ŝ is called a relation module (modulo p); in general we call Ŝ a relative relation module (modulo p). Gaschütz, Gruenberg and others have studied relation modules; the aim of this paper is to study relative relation modules.

A semiband is defined as a semigroup generated by idempotents. It is known that every finite semigroup is embeddable in a finite semiband. For a class C of semigroups and an integer n≧2, the number σC (n) is defined as the smallest k with the property that every semigroup of order n in the class C is embeddable in a semiband of order not exceeding k. It is shown that for the class Gp of groups σGp(n) = nq(ρGp(n)), where

and

Estimates are known (and are quoted) for the function q. Estimates are considered for the function pC for various C

It is shown also that if C0S, CS denote respectively the classes of completely 0-simple and completely simple semigroups, then

We characterize regular bisimple rings in terms of some perspectivity conditions on their lattices of principal right ideals. We also show that, if S is the multiplicative subsemigroup generated by all the idempotents of a regular bisimple ring R, then

(i) if R does not have an identity, then S = R and has depth 2;

(ii) if R does have an identity but is not a division ring, then S = {a∈R:a is neither left nor right invertible} ∪ {1} and has depth 3.

Although it is well-known that tempered distributions on ℝn can be expanded into series of Herrnite functions, it does not seem to be known, however, that expansions of this type are accessible through the elementary concept of orthonorma! expansions in Hilbert space. This approach is developed here complementing previous work on a Hilbert space approach to distributions. The basis of the development is the observation that the Hermite functions are a complete orthogonal set in each space of a certain scale of Sobolev type Hilbert spaces associated with the family of differential operators defined by

Here Ф denotes a smooth function with compact support. The setting is first developed in the one-dimensional case. By use of the usual multi-index notation this can be extended to the higher-dimensional case. As applications various imbedding results are derived. The paper concludes with a characterization of tempered distributions by convergent Hermite expansions.

In this paper we present three results about Arens regular bilinear operators. These are: (a). Let X, Y be two Banach spaces, K a compact Hausdorff space, µ a Borel measure on K and m: X × Y →ℂ a bounded bilinear operator. Then the bilinear operator defined by is regular iff m is regular, (b) Let (Xα), (Xα),(Zα) be three families of Banach spaces and let mα:Xα ×Yα→Zα, be a family of bilinear operators with supα∥mα∥<∞. Then the bilinear operator defined by is regular iff each mα, is regular, (c) Let X, Y have the Dieudonné property and let m:X × Y→Z be a bounded bilinear operator with m(X×Y) separable and such that, for each z′ in ext Z′1, z′∘m is regular. Then m is regular. Several applications of these results are also given.

An essentially bounded function on the unit circle gives a continuous linear functional on the Hardy space H1. In this paper we study when there exists at least one function which attains its norm. We apply the results to an interpolation problem, Hankel operators and a characterization of exposed points of the closed unit ball of H1.

It is known that for certain rings R (for example R = ℤ, the ring of rational integers) the group GL2(R) contains subnormal subgroups which have free, non-abelian quotients. When such a subgroup has finite index it follows that every countable group is embeddable in a quotient of GL2(R). (In this case GL2(R) is said to be SQ-universal.) In this note we prove that the existence of subnormal subgroups of GL2(R) with this property is a phenomenon peculiar to “n = 2”.

For a large class of rings (which includes all commutative rings) it is shown that, for all , no subnormal subgroup of En(R) has a free, non-abelian quotient, when n≧3. (En(R) is the subgroup of GLn(R) generated by the elementary matrices.) In addition it is proved that, if is an SRt-ring, for some t ≧ 2, then no subnormal subgroup of GLn(R) has a free, non-abelian quotient, when n ≧ max (t, 3). From the above these results are best possible since ℤ is an SR3 ring.

A concordance classification of links of , p < 1, is given in terms of an algebraically defined group, Φ±, which is closely related to Levine's algebraic knot concordance group. For p=1,Φ_ captures certain obstructions to two component links in S3 being concordant to boundary links, the generalized Sato-Levine invariants defined by Cochran. As a result, purely algebraic proofs of properties of these invariants are derived.

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

An action of a group G on a tree, and an associated Lyndon length function l, give rise to a hyperbolic length function L and a normal subgroup K having bounded action. The Theorem in Section 1 shows that for two Lyndon length functions l, l′ to arise from the same action of G on some tree, L = L′ and K = K′. Moreover for L non-abelian L = L′ implies K = K′. That this is not so for abelian L is shown in Section 2 where two examples of Lyndon length functions l, l′ on an H.N.N. group are given, with their associated actions on trees, for which L = L′ is abelian but K≠K′.

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

Our main result is that a “restricted Poincaré duality” property with respect to finite dimensional coefficient modules over a field holds for a certain class of groups which includes all soluble groups of finite Hirsch length. This relies on a generalisation to the given class of a module construction by Stammbach; an extension of his result on homological dimension to these groups is given. We also generalise the well-known result that torsion-free soluble groups of finite rank are countable.

Nonstandard methods and a flat integral representation are used to give a simple and intuitive proof of the large deviation principle for a Gaussian measure on a separable Hilbert space.

Conditions are given on two maximal monotone (multivalued) operators A and B which ensure that A + B is also maximal. One condition used is that ∥Bx∥≦k(∥x∥)Ax| +d|(A + B)x| + c(∥x∥) for every x∈D(A)⊆D(B), where 0≦k(r)<1, and c(r)≧0 are nondecreasing functions, and 0≦d≦1 is a constant. Here, for a set C, |C| denotes inf{∥y∥:y∈C}. This extends the well known result which has d = 0 (and is used in the proof here). The second part of the paper uses similar hypotheses to give conditions under which the range of the sum, R(A + B), has the same interior and same closure as the sum of the ranges, R(A) + R(B).

Consider the free monoid on a non-empty set P, and let R be the quotient monoid determined by the relations:

Let R have its natural involution * in which each element of P is Hermitian. We show that the Banach *-algebra ℓ1(R) has a separating family of finite dimensional *-representations and consequently is *-semisimple. This generalizes a result of B. A. Barnes and J. Duncan (J. Funct. Anal.18 (1975), 96–113.) dealing with the case where P has two elements.

Let E be an injective module over the commutative Noetherian ring A, and let a be an ideal of A. The A-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈N is ultimately constant. This result is analogous to a theorem of M. Brodmann that, if M is a finitely generated A-module, then the sequence of sets (AssA(M/αnM))n∈N is ultimately constant.

We calculate K0 of the rational group algebra of a certain crystallographic group, showing that it contains an element of order 2. We show that this element is the Euler class, and use our calculation to produce a whole family of groups with Euler class of order 2.