Let <G,+> be a group with identity 0 and let S be a semigroup of endomorphisms of G. The set Ms(G)={f:G→G; f(0)=0, fσ=σf, for all σ∈S} with the operations of unction addition and composition is a zero-symmetric near-ring with identity called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings have been studied for many classes of semigroups of endomorphisms. (See [8] and the references given there.) In this paper we continue these investigations into the structure of centralizer near-rings via our study of the relationship between distributive elements in Ms(G) and endomorphisms in Ms(G). More specifically, let N = Ms(G) and let Nd={f∈N; f(g1+g2)=fg1+fg2}, the set of distributive elements in N. Under the operation of function composition, Nd is a semigroup containing the identity map, id. Moreover, Nd contains as a submonoid = {α ∈ End G; ασ=σα for all σ∈S}. Here we determine for certain semigroups S, whether or not = Nd.

If u and v are vertices of the (finite, connected) graph Γ, let d(u, v) denote the length of the shortest path joining u to v in Γ. The graph Γ is said to be distance-transitive if whenever d(u, v) = d(u′, v′), there exists an automorphism g of Γ such that ug = u′ and if vg = v′. Distance-transitive graphs of valency 3 and 4 were originally classified [2, 11, 12, 13] by using a computer to generate all “feasible intersection arrays” (cf. [1, Chapter 20]). In both cases a classification has since been given by hand [4, 5]. Wecontinue this latter tradition and prove the following theorem—which was recently proved independently by Ivanov et al. using a computer [10].

The purpose of this paper is to discuss non-linear boundary value problems for elliptic systems of the type

where Ak is a second order uniformly elliptic operator and is such that the problem

has a one-dimensional space of solutions that is generated by a non-negative function. The boundary ∂G is supposed to be smooth and the functions gk, 1≦k≦m are defined on Ḡ×Rm and are continuously differentiate (usually, Bk represents Dirichlet or Neumann conditions and is the first eigenvalue associated with Ak and such boundary conditions).

A congruence relation θ on an algebra L is principal if there exist a, b)∈L such that θ is the smallest congruence relation for which (a, b)∈θ. The property that, for every algebra in a variety, the intersection of two principal congruences is again a principal congruence is one that is known to be shared by many varieties (see, for example, K. A. Baker [1]). One such example is the variety of Boolean algebras. De Morgan algebras are a generalization of Boolean algebras and it is the intersection of principal congruences in the variety of de Morgan algebras that is to be considered in this note.

Our paper [6] studied in some depth certain locally nilpotent skew linear groups, but our conclusions there left some obvious gaps. By means of a trick, which now seems obvious, but then did not, we are able to tidy up the situation very satisfactorily. This present paper should be viewed as a follow up to [6]. In particular we do not repeat the motivation, basic definitions and references to related work given here.

The following was conjectured in [6], where substantial steps were taken towards its solution.

In 1948 P. Lévy formulated the following theorem: If U is an open subset of the complex plane and f:U → ℂ is a nonconstant analytic function, then f maps a 2-dimensional Brownian motion Bt (up to the exit time from U) into a time changed 2-dimensional Brownian motion. A rigorous proof of this result first appeared in McKean [22]. This theorem has been used by many authors to solve problems about analytic functions by reducing them to problems about Brownian motion where the arguments are often more transparent. The survey paper [8] is a good reference for some of these applications. Lévy's theorem has been generalized, first by Bernard, Campbell, and Davie [5], and subsequently by Csink and Øksendal [7]. In Section 1 of this note we use these generalizations of Lévy's theorem to extend some results about BMO functions in the unit disc to harmonic morphisms in ℝn to holomorphic functions in ℂn and to analytic functions on Riemann surfaces. In Section 2, we characterize the domains in ℝn which have the property that the expected exit time of elliptic diffusions is uniformly bounded as a function of the starting point. This extends a result of Hayman and Pommerenke [15], and Stegenga [24] about BMO domains in the complex plane.

This paper concerns an application of an algorithm for the second derived factor group as described by Howse and Johnson in [3]. This algorithm has as its basis theFox derivative (see [1]), a mapping from the free group F to the group-ring ℤF, definedas follows: let X be a set of generators of a group G, and let w = y1…yk with each yi∈X±1.

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the form

where Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

Given any group G, its tensor square G ⊗ G is defined by the followingpresentation(see [[3]):

where g,g′,h,h′ range independently over G, and gh=ghg−1. In what follows, gg′ ⊗ gh is often written in the abbreviated form g(g′ ⊗ h).

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the type

where Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

For each finite group G, let G denote the set of all normal subgroups of the modular group Γ = PSL2(ℤ) with quotient group isomorphic to G; since Γ is finitely generated, the number NG = |G| of such subgroups is finite. We shall be mainly concerned with the case where G is the linear fractional group PSL2(q) over the Galois field GF(q), in which case we shall write (q) and N(q) for G and NG; for q>3, PSL2(q) is simple, so the elements of (q) will be maximal normal subgroups of Γ.

Tate cohomology of finite groups [5] is very good at emphasising periodic cohomological behaviour and hence at the study of free actions on spheres [8]. Tate cohomology of spaces was introduced by Swan [10] for finite dimensional spaces to systematically ignore free actions, and hence to simplify various arguments in fixed point theory.

The infinite continued fraction

in which

is periodic with period l and is equal to a quadratic surd if and only if the partial quotients, ak, are integers or rational numbers [1]. We shall also assume that they are positive. The transformation discussed below applies only to pure periodic fractions where n is zero.

Throughout this paper H will denote a complex separable Hilbert space and L(H) denotes the algebra of all bounded linear operators on H. If T lies in L(H), its spectrum σ(T) is the set of all complex numbers z such zI–T is not invertible in L(H) and its compression spectrum σcomp(T) is the set of all complex numbers z such that the range (zI-T)(H) is not dense in H ([3, p. 240]). This paper is concerned with the Sturm–Liouville operator problem

where λ is a complex parameter and X(t), Q, Ei, Fi for i = l,2, and t∈[0,a], are bounded operators in L(H). For the scalar case, the classical Sturm-Liouville theory yields a complete solution of the problem, see [4], and [7]. For the finite-dimensional case, second order operator differential equations are important in the theory of damped oscillatory systems and vibrational systems ([2, 6]). Infinite-dimensional differential equations occur frequently in the theory of stochastic processes, the degradation of polymers, infinite ladder network theory in engineering [1, 17], denumerable Markov chains, and moment problems [10, 20]. Sturm-Liouville operator problems have been studied by several authors and with several techniques ([12, 13, 14, 15, 16]).

In his thesis, A. A. Hussein Omar, motivated by the study of possible shapes of generic Dirichlet regions for a surface group, made a detailed study for g = 2,3 of the groups generated by pairs (μ, τ) of regular (i.e. fixed-point-free) permutations of order 2,3 respectively and of degree n = 6(2g − 1), such that μ ْ τ is an n-cycle. He observed that, for g = 2,3, precisely one pair generates what he calls a superimprimitive group, and raised the question whether such pairs exist for all g, and, if so, whether they areunique. Our main result is that they do always exist, but that, for large values of g, theyare far from unique. (For details and some motivation for the notation, see [4, 5].)

It is of some interest to the theory of locally convex *-algebras to know under what conditions such an algebra A is a pre-C*-algebra (the topology of A can be described by a submultiplicative norm such that ‖x*x‖ = ‖x‖2, ∀x∈A). We recall that a locally convex *-algebra is a complex *-algebra A with the structure of a Hausdorff locally convex topological vector space such that the multiplication is separately continuous, and the involution is continuous.

The aim in this paper is to indicate one way of interpreting covariants of a binary quintic F geometrically, interpretations having been found recently [7] for its quadratic covariant 2C2, called Γ in [7], and its invariant I4. The symbol dCn will denote a covariant of order n in the binary variables x, y and degree d in the coefficients of F, Id being used in preference to dI0 for invariants. The sum d + n is 4 for both 2C2 and I4, and no other covariant affords as small a sum; so it is natural to have begun by interpreting these two and to use them as auxiliaries in interpreting others.

Let G denote a relatively free group of a finite or countably infinite rank with a fixed set of free generators x1,x2,…,G′ the commutator subgroup, and V a verbal subgroup belonging to G′. Following H. Neumann [6] we shall use the vector representation for endomorphisms of G. Vector v = (ν1, ν2,…) represents an endomorphism v such that xiv = νi for all i. The identity map is represented by l=(x1,x2…). We need also thetrivial endomorphism 0 = (e, e,…). The length of vectors is equal to the rank of G. We shall consider the near-ring of vectors, with addition and multiplication given below u + v=(ulν1, u2ν2,…) where uiνi; is a product in G, and uv = (u1v, u2v,…) where uiv isthe image of ui, under the endomorphism v. There is only one distributivity law (u + v)w =uw + vw.

Our starting point is the differential equation

where A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],

for all , b∈C\{0}, at least outside a set of r of finite measure.

In the paper referred to above (see [1]), the following corrections should be made.

(1) Hypothesis 2.2 should read:

(2) The set E⊂ℝn should be defined as

(3) F0(υ)=υ+x should read F(υ)=υ–x.

(4) The unique solution of F0(υ)=0 is υ=x (not υ= –x as printed).

(5) Page 345, line 14: