Proposition 3.1 of the paper Annihilators and the CS-condition, Glasgow Math. J. 40 (1998), 213–222, is incorrect as stated, and consequently the note added in proof is incorrect. Hence the question of Faith and Menal whether every strongly right Johns ring is quasi- Frobenius remains open. The problem is that the assumption that the left socle S1 and the right socle Sr are equal is not established. All we know is that Si⊆Sr = r(J) = l(J) by [8, Lemma 2.2]. We can prove the following result.

It is implicit in a result of Kapp and Schneider [3] that, if Sisa completely simple semigroup, then the lattice Λ(S) of congruences on S can be embedded in the product of certain sublattices. In this paper we consider the problem of embedding Λ(S) in a product of sublattices, when S is an arbitrary band of groups. The principal tool is the θ-relation of Reilly and Scheiblich [7]. The class of θ-modular bands of groups is definedby means of a type of modularity condition on Λ(S). It is shown that the θ-modular bands of groups are precisely those for which a certain function is an embedding of Λ(S) into a product of sublattices. The problem of embedding the inverse semigroup congruences into a certain product lattice is also considered.

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results. Let Proj (G, α) denote the set of irreducible projective characters of a group G with cocyle α. In previous papers (for exampe [2], [4], and [6]) numerous authors have considered the situation when Proj(G, α) = 1 or 2; such groups are said to be of α-central type or of 2α-central type, respectively. In particular in [4, Theorem A] the author showed that if Proj(G, α) = {ξ1, ξ2}, then ξ1(1)=ξ2(1). This result has recently been independently confirmed in [8, Corollary C].

1. The problem of determining the state of stress in the vicinity of a penny-shaped crack which is opened by thermal means has been considered by Olesiak and Sneddon [1]. In that paper no simple closed expressions were given either for the stress-intensity factor at the tip of the crack or for the normal component of the surface displacement. The purpose of this note is to show how such expressions may be derived.

Let X be a compact Hausdorff space, let C(X) denote the algebra of all continuous functions on X, let B be a Banach algebra, and let θ: → C(X) → B be a (possibly discontinuous) homomorphism with dense range. A classical theorem by W. G. Bade and P. C. Curtis ([2, Theorem 4.3]) describes in great detail the structure of θ we shall refer to this result as the Bade–Curtis theorem. Before we give a brief sketch of this theorem, we fix some notation. For Y ⊂ X let I(Y) and J(Y) denote the ideals of all functions in C(X) that vanish on Y and on a neighborhood of Y respectively; if Y = {x} for some x ɛ X, we write mx and Jx for I(Y) and J(Y) respectively. According to the Bade–Curtis theorem there is a finite set {x1,…, xn) ⊂ X, the so-called singularity set of θ, such that θ | ({x1, …, xn}) is continuous. As a consequence, the restriction of θ to the dense subalgebra of C (X) consisting of all those functions which are constant near each Xj (j = 1,…, n) is continuous, and extends to a continuous homomorphism θcont: C(X)→ B. Let θsing: = θ – θcont. Then θsing | I({x1,…, xn}) is a homomorphism onto a dense subalgebra of rad (B). θcont, and θsing are called the continuous and the singular part of θ respectively. Moreover, there are linear maps : C(X)⊒ B such that

(i)

(ii) is a homomorphism, and

Condition (iii) forces the homomorphisms to map into rad(B); such homomorphisms are called radical homomorphisms.

Throughout, (X, ≤ ) denotes a partially ordered set (p. o. set), where X is assumed to be finite. A subset Y of X is called a k-union if Y contains no chain of length K + 1. In particular, therefore, a 1-union is just an antichain; and it is readily seen that Y is a k-union if and only if it is a union of K antichains. (Dually, a subset Z of X is a k-counion if Z contains no antichain of length k + 1.) We denote by dk (X) the maximum cardinality of a k-union in X, with a similar notation for other p. o. sets. Now let be any partition of X into chains, and write

.

Let G be a finite linear group of degree n; that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients). We shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n -dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n. See [5].

In the analysis of mixed boundary value problems by Hankel transforms, one often encounters dual integral equations of the form

where I1 = (0, 1), I2 = (1, ∞); w1(x), w2(x) are weight functions, ψ(x) is the unknown function, and f(y), g(y) are functions continuously differentiate on I1 and I2 respectively. Many successful attempts have been made to solve (1.1) and (1.2). These are all discussed in a recent book by Sneddon [7]. As pointed out in a recent paper by Erdogan and Bahar [4], in mixed boundary value problems of semi-infinite domains involving more than one unknown function such as those arising in elastostatics, viscoelasticity, and electrostatics, the formulation will lead to a system of simultaneous dual integral equations which is a generalization of (1.1) and (1.2). These equations may be expressed as follows:

with i = 1, 2, …, n, where we use the notation

We prove the compactness of certain Hankel operators on weighted Bergman spaces of harmonic functions on the unit ball in Rn.

Let K be a quadratic number field with 2-class group of type (2,2). Thus if Sk is the Sylow 2-subgroup of the ideal class group of K, then Sk = ℤ/2ℤ × ℤ/2ℤ Let

K ⊂ K1 ⊂ K2 ⊂ K3 ⊂…

the 2-class field tower of K. Thus K1 is the maximal abelian unramified extension of K of degree a power of 2; K2 is the maximal abelian unramified extension of K of degree a power of 2; etc. By class field theory the Galois group Ga1 (K1/K) ≅ Sk ≅ ℤ/2ℤ × ℤ/2ℤ, and in this case it is known that Ga(K2/Kl) is a cyclic group (cf. [3] and [10]). Then by class field theory the class number of K2 is odd, and hence K2 = K3 = K4 = …. We say that the 2-class field tower of K terminates at K1 if the class number of K1 is odd (and hence K1 = K2 = K3 = … ); otherwise we say that the 2-class field tower of K terminates at K2. Our goal in this paper is to determine how likely it is for the 2-class field tower of K to terminate at K1 and how likely it is for the 2-class field tower of K to terminate at K2. We shall consider separately the imaginary quadratic fields and the real quadratic fields.

Let Tn denote the full transformation semigroup on the finite set = {1, 2, … n}, that is the set of all mappings from to , with function composition as the semigroup operation. In this paper algorithms are introduced to solve equations such as axmb = c and ax = xb (a, b, c ∊ Tn), which employ a representation of members of Tn as special directed graphs.

A series ∑an is said to be summable (C, — 1) to s if it converges to s and nan = o(1) [8]. It is well known that this definition is equivalent to tn→s (n→∞), where tn = sn + nan, sn = ao + … + an. The series is summable | C, — 1 | to s if the sequence t = {tn} is of bounded variation (t ∈ B.V.), i.e. ∑ |; ▲tn |; = ∑ | tn - tn-1 | < ∞, and ∑ ▲tn = lim tn = s. An equivalent condition is ∑ | an |; < ∞, ∑an = s and ∑ | ▲(nan) | < ∞. For, suppose that ∑an = s | C, - 1 |. Since {sn} is the sequence of (C, 1)-means of {tn} and since | C, 0 | ⊂ | C, 1 |, we have ∑ | an | < ∞ and ∑an = s whence ∑ | ▲(nan) | < ∞. Conversely, ∑ | an | < ∞, ∑an = s and ∑ | ▲(nan) | < ∞ imply t ∈ B.V. and ∑▲n = s + lim nan. But lim nan = 0, since ∑ | an | < ∞.

The problem of solving the equation of thermal conduction for cases in which heat is generated in the interior of the medium under consideration arises frequently in physics and engineering. It occurs, for instance, when we consider the diffusion of heat in a solid undergoing radioactive decay (1) or which is absorbing radiation (2). Complications of a similar nature arise when there is a generation or absorption of heat in the solid as a result of a chemical change-for example, the hydration of cement (3). The particular case in which the rate of generation of heat is independent of the temperature arises in the theory of the ripening of apples and has been discussed by Awberry (4).

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Set

σem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

Let S be an inverse semigroup with semilattice of idempotents E, and let ρ be a congruence on S. Then ρ is said to be idempotent-determined [2], or I.D. for short, if (a, b) ∈ р and a∈E imply that b ∈ E. If, further, ρ is a group congruence, then clearly ρ is the minimum group congruence on S, and in this case S is said to be proper [8]. Let T = S/ρ.

Free monoids play a central role in the theory of formal languages. Their endomorphisms appear naturally in the context of deterministic OL-schemes which trace their origin to biology. Closely related to such a scheme is a DOL-system which consists of a triple (X, φ, w) where X is a finite set, φ is an endomorphism of the free monoid X* and w ∈ X. The associated language is defined as the set {w, φw, φ2w,…} called a DOL-language. For a full discussion of this subject, we recommend the book [2] by Herman and Rozenberg.

Let A be an n × n complex matrix and c = (c1… cn) єℂn. Define the c-numerical range of A to be the set is an orthonormal set in , where * denotes the conjugate transpose. Westwick [8[ proved that if c … cn are collinear, then Wc(A) is convex. (Poon [6] gave another proof.) But in general for n ≧3, Wc(A) may fail to be convex even for normal A (for example, see Marcus [4] or Lemma 3 in this note) though it is star-shaped (Tsing [7]). In the following, we shall assume that A is normal. Let W(A) = {diag UAU*: U is unitary}. Horn [3] proved that if the eigenvalues of A are collinear, then W(A) is convex. Au-Yeung and Sing [2] showed that the converse is also true. Marcus [4] further conjectured (and proved for n = 3) that if Wc(A) is convex for all cєℂn then the eigenvalues of A are collinear. Let λ = (λ1, …, λn єℂn. We denote by the vector λ1, …, λn and by [λ] the diagonal matrix with λ1, …, λn lying on its diagonal. Since, for any unitary matrix U,. Wc(A) = Wc (UAU*), the Marcus conjecture reduces to: if Wc([λ]) is convex for all c єℂn then λ1, … λn are collinear. For the case n = 3, Au-Yeung and Poon [1] gave a complete characterization on the convexity of the set Wc([λ]) in terms of the relative position of the points , where σ є S3 the permutation group of order 3. As an example they showed that if λ1, λ2, λ3 are not collinear, then is not convex (Lemma 3 in this note gives another proof). We shall show that for the case n = 4, is not convex if λ1, λ2. λ3. λ4 are not collinear. Thus for n = 3, 4 the Marcus conjecture is answered and improved.

Let ℋ be a complex Hilbert space and B(ℋ) the algebra of all bounded linear operators on ℋ. Let ℋ(ℋ) be the algebra of all compact operators of B(ℋ). For an operator T ε B(ℋ), let σ(T), σp(T), σπ(T) and πoo(T) denote the spectrum, the point spectrum, the approximate point spectrum and the set of all isolated eigenvalues of finite multiplicity of T, respectively. We denote the kernel and the range of an operator T by ker(T) and R(T), respectively. For a subset of ℋ, the norm closure of is denoted by . The Weyl spectrum ω(T) of T ε B(ℋ) is defined as the set

Let Sbe a compact Riemann surface of genus g ≥ 2 and σ an automorphism (conformal self-homeomorphism) of S of order n. Let S* = S/ « σ« have genus g*. In [5], Schoeneberg gave a sufficient condition that a fixed point P ∈ S of σ should be a Weierstrass point of S, i.e., that Sshould support a function that has a pole of order less than or equal to g at P and is elsewhere regular.