We denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions, and by Fn an n-dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian (unitary resp.) if A = A* (AA*= identity matrix resp.). An n ×x n hermitian matrix A is said to be definite (semidefinite resp.) if uAu*vAv* ≥ 0 (uAu*vAv* ≧ 0 resp.) for all nonzero u and v in Fn. If A and B are n × n hermitian matrices, then we say that A and B can be diagonalized simultaneously into blocks of size less than or equal to m (abbreviated to d. s. ≧ m) if there exists a nonsingular matrix U with elements in F such that UAU* = diag{A1,…, Ak} and UBU* = diag{B1…, Bk}, where, for each i = 1, …, k, Ai and Bk are of the same size and the size is ≧ m. In particular, if m = 1, then we say A and B can be diagonalized simultaneously (abbreviated to d. s.).

We recall (cf. [2] Definitions 3.1 and 3.2, p. 322) that a bounded linear operator T on a Banach space ℵ into itself is said to be asymptotically quasi-compact if K(Tn)⅟n → 0 as n → ∞. where K(U) = inf ∥U–C∥ for every bounded linear operator U on ℵ into itself, the infimum being taken over all compact linear operators C on ℵ into itself. For a complex Banach space, this is equivalent (cf. [2], pp. 319, 321 and 326) to T being a Riesz operator.

In [1], we showed how a collection of physical operations or experiments could be represented by a nonempty set of nonempty sets satisfying certain conditions (irredundancy and coherence) and we called such sets . We also introduced “complete stochastic models” for the empirical universe of discourse represented by such a manual , namely, the so-called weight functions for . These weight functions form a convex set the extreme points of which are called pure weights. We also showed that there is a so-called logic ∏() affiliated with a manual and that each weight function for induces a state on this logic.

We shall solve the equation

where 0 <a<b, and f(x) is a continuous function on the interval (a, b).

Let(V, ≧, ‖ · ‖) be a Banach lattice, and denote V\{0} by V'. For the definition of a Banach lattice and other undefined terms used below, see Vulikh [4]. Leader [3] shows that, if norm convergence is equivalent to order convergence for sequences in V, then the norm is equivalent to an M-norm. By assuming the equivalence for nets in V we can strengthen this result.

A block design is a finite set p of elements called points, where |p| = v, together with certain distinguished subsets of p called blocks, such that

(i) each block contains k points,

(ii) each point is contained in r blocks, and

(iii) two distinct points are contained in precisely λ blocks.

Let E be a finite set and let = (Al, …, A2) be a family of subsets of E. A subset T of E iscalled a transversal of if there exists a bijection Φ: T → {l, …, n} such that x ∈ Aψ(x) for all x ∈ T. If I ⊆ {1, …, n}, we shall, for brevity, write

(and similarly for families denoted by other letters). The cardinal of a set S will be denoted by |S|. If λ is a non-negative integer, we define λS as S or Ø according as λ > 0 or λ = 0.

Congruences on a semigroup S such that the corresponding factor semigroups are of a special type have been considered by several authors. Frequently it has been difficult to obtain worthwhile results unless restrictions have been imposed on the type of semigroup considered. For example, Munn [6] has studied minimum group congruences on an inverse semigroup, R. R. Stoll [9] has considered the maximal group homomorphic image of a Rees matrix semigroup which immediately determines the smallest group congruence on a Rees matrix semigroup. The smallest semilattice congruence on a general or commutative semigroup has been studied by Tamura and Kimura [10], Yamada [12] and Petrich [8]. In this paper we shall study congruences ρ on a completely regular semigroup S such that S/ρ is a semilattice of groups. We shall call such a congruence an SG-congruence.

For each characteristic p, let Fp be the prime field and let Ώp be a fixed universal field which is algebraically closed and of infinite transcendence degree over Fp. When p = 0 we take Ώp = ℂ. Let F be a subfield of Ώp and let R be an integral domain whose quotient field is F. We abbreviate SL(2, R), PGL(2, R), PSL(2, R) to SL(R), PGL(R), PSL(R) respectively, and we cohsider PSL(R) as a group of projective transformations of the projective line ℘(Ώp) and of the “subline” ℘(F) ⊂ ℘(ΏP). The elements of PSL(R) are classified by the number of fixed points they have on ℘(F). If x ∈ PSL(R) has one such fixed point P, then P is the unique fixed point of x on ℘(ΏP) and x is called parabolic. All other x (except the identity E) have two distinct fixed points on ℘(Ώp) and x is called hyperbolic if these are on ℘(F), and elliptic otherwise. We put symbols for operators on the right.

We exhibit a large class K* of real 2 × 2 matrices of determinant ±1 such that, for nearly all A and B in K*, the group generated by A and B1 (the transpose of B) is the free product of the cyclic groups It is shown that K* contains all matrices of determinant ±1 with integer entries satisfying |b| > |a|, |c|, |d|. This gives a generalization of a theorem of Goldberg and Newman [2]. We also prove related results concerning the dominance of b and the discreteness of the free products .

Let f be an analytic function on a connected open set Ώ in the complex plane. Then, for given a, b ∈ Ώ, the equation

need not have a solution z ∈ Ώ. As a matter of fact, this would happen with each locally one-to-one analytic function which is not one-to-one on Ώ. But if we fix a a ∈ Ώ, then, for all b sufficiently close to a, (1) is solvable for z. This is an easy consequence of the Open Mapping Theorem applied to f'. For, assuming that f' is non-constant (otherwise, (1) holds for all a, b, z ∈Ώ), the Open Mapping Theorem tells us that f'(Ώ), the image under f' of Ώ, is an open neighbourhood of f'(a); so it is a direct consequence of the definition of f'(a) that there exists δ > 0 such that 0 < |b – a| < δ implies (f(b) – f(a))/(b – a) ∈f'(Ώ). A stronger statement has been obtained by J. M. Robertson [1, p. 329], who has shown that

and, if f''(a) ≠ 0, then

.

In [1], we introduced the notion of multiplicative forms on associative algebras of finite rank over integral domains D, and obtained a complete classification when D ⊆, the complex field. We propose here to remove the hypothesis of associativity, using a refinement of the technique of Schafer [2]. In [l], it was noted that multiplicative forms extend uniquely under the adjunction of an identity when is associative but not unitary; this appears difficult to verify in the general case, so that some mild restriction on is required. We shall assume that is biregular, that is that contains elements eL, eR such that the linear maps x eL x and x xeR, are bijective on We can then (§1) reduce the biregular case to the unitary case, which is handled in §2.

In order to study an arbitrary sequence of modules and homomorphisms, we propose a definition of “homology” modules, or what we call quasi-homology modules, for such a sequence. Then we seek partial analogues of the universal coefficient theorems to make some propaganda for the notion.

In their paper “ The enumeration of tree-like polyhexes”, Harary and Read [6] consider structures obtained by assembling hexagons subject to certain restrictions. Their problem is introduced as a simplified hexagonal cell-growth problem.

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 [1], Pall proved an interesting result on a certain class of 2 × 2 integral matrices. He showed that the semigroup of 2 × 2 matrices of determinant 1 and non-negative entries contains exactly 2 primes , and every other non-unit is expressible uniquely as products of these primes. Before formally stating this result, we need some notation. Let Gn denote the semigroup of n × n matrices with determinant 1 and nonnegative integral entries, In the n × n identity matrix, the n × n matrix with a 1 as its (i, j) element and zeros elsewhere, and let . When the dimension is clear, we shall drop the superscripts.

In 1960 Ericksen [1] introduced a simple theory of anisotropic fluids. This theory differs from the classical theory of fluids in that the deformation of the material is no longer solely described by the usual vector displacement field but requires in addition the specification of a further vector field di, termed the director. Moreover, corresponding to this increased kinematic flexibility new types of stress, body force and inertia are introduced. Leslie [2], adopting the conservation laws of [1], formulated constitutive equations similar to those considered by Ericksen and discussed the thermodynamical restrictions imposed by the Clausius–Duhem inequality. Here we shall consider the case in which at each point the director is constrained to remain a unit vector. Then the usual interpretation is to regard di as indicating a single preferred direction in the material (see for example [3]). It is thought that the physical applications of this theory are likely to lie in such areas as polymeric fluids and suspensions.

A permutation group is quasiregular if it acts regularly on each of its orbits (i.e. the stabiliser of an element fixes every other element in its orbit). So, in particular, any permutation representation of an abelian or hamiltonian group must be quasiregular.

In a recent paper, Boudjelkha and Diaz [1] have considered the solutions of the Dirichlet problems for Laplace's equation

and for Helmholtz's equation

in¼–ℝn. The purpose of this brief note is to show that their formulae may be derived easily by the use of the theory of multiple Fourier transforms.

A Banach space X is said to have property (PROXBID) if the canonical image of X in its bidual X** is proximal. In other words, if J: X → X** is the canonical embedding, then it is required that every element of X** have at least one best approximation (i.e., nearest point) from the closed subspace J(X). We show below that, if X is the space of (real or complex) continuous functions on a compact set, or the space of (real or complex) continuous functions that vanish at infinity on a locally compact set, then X has property (PROXBID). At this point we should mention the existence of a variety of examples [2, 8] of Banach spaces which lack property (PROXBID).