The trivial observation that every automorphism of a group is determined by its restriction to a set of generators suggests the converse question: if X is a subset of a group G such that each automorphism of G is determined (or “almost” determined) by its restriction to X, to what extent is the structure of G governed by that of the subgroup which X generates? Is this subgroup in some sense necessarily “large” in G? If the index of the subgroup is used as a measure of largeness, then in the absence of additional hypotheses, the answer to the second question is generally “no”, the additive group of rationals with X = {1} being an obvious counterexample. (More confounding is the existence of uncountable torsion-free abelian groups for which inversion is the only non-trivial automorphism. See, for example, [2], [3], and [4].) However, under certain finiteness assumptions, it seems that some positive conclusions are obtainable. One such example will be considered here.

We consider in this paper the problem (1) where Ω is the unit ball in ℝN centred at the origin, 0 ≤ α < pN,β > 0, N ≥ 3. Suppose qϵ → q as ϵ → 0+ and qϵ, q satisfy, respectively, we investigate the asymptotic estimates of the ground-state solutions (uϵ, vϵ) of (1) as β → + ∞ with p, qϵ fixed. We also show the symmetry-breaking phenomenon with α, β fixed and qϵ → q as ϵ → 0+. In addition, the ground-state solution is non-radial provided that ϵ > 0 is small or β is large enough.

The Bass–Serre theorem supplies generators and relations for a group of automorphisms of a tree. Recently K. S. Brown [2] has extended the result to produce a presentation for a group of automorphisms of a simply connected complex, the extra ingredient being relations which come from the 2-cells of the complex. Suppose G is the group, K the complex and L the 1-skeleton of K. Then an extension of π1(L) by G acts on the universal covering space of L (which is of course a tree) and Brown's technique is to apply the work of Bass and Serre to this action. Our aim is to give a direct elementary proof of Brown's theorem which makes no use of covering spaces, deals with the Bass–Serre theorem as a special case and clarifies the roles played by the various generators and relations.

In Section 33 of [2], Bonsall and Duncan define an element t of a Banach algebra to act compactly on if the map a → tat is a compact operator on . In this paper, the arguments and technique of [1] are used to study this question for C*-algebras (see also [10]). We determine the elements b of a C*-algebra for which the maps a → ba, a → ab, a → ab + ba, a → bab are compact (respectively weakly compact), determine the C*-algebras which are compact in the sense of Definition 9, of [2, p. 177] and give a characterization of the C*-automorphisms of which are weakly compact perturbations of the identity.

A ring R is called right PCI if every proper cyclic right R-module is injective, i.e. if C is a cyclic right R-module then CR ≅ RR or CR is injective. By [2] and [3], if R is a non-artinian right PCI ring then R is a right hereditary right noetherian simple domain. Such a domain is called a right PCI domain. The existence of right PCI domains is guaranteed by an example given in [2]. As generalizations of right PCI rings, several classes of rings have been introduced and investigated, for example right CDPI rings, right CPOI rings (see [8], [6]). In Section 2 we define right PCS, right CPOS and right CPS rings and study the relationship between all these rings.

This paper gives examples to show that a polar operator is not necessarily AC and an AC operator is not necessarily polar.

In this note we consider a finite graph without loops and multiple edges. The colouring of a graph G in λ colours is the colouring of its vertices in such a way that no two of adjacent vertices have the same colours and the number of used colours does not exceed λ [1, 4]. Two colourings of graph G are called different if there exists at least one vertex which changes colour when passing from one colouring to another.

Let ω be a primitive cube root of unity. We define the cubic residue symbol (Legendre symbol) on ℤ[ω] as follows. Let πεℤ[ω] be a prime, (3, π)=1. For α ε ℤ[ω] such that (α, π)=1 we let be that third root of unity so that

1. Every family of subnormal operators in a Hilbert space fulfils the Halmos-Bram condition on a suitable dense subset of its domain [2], [3]. In [2] and [4] it is shown that the generalized commutation relation implies the Halmos-Bram condition for one operator. In this paper it is proved that the generalized commutation relation implies the Halmos-Bram condition for infinite families of operators (in a special case Jorgensen proved it in a different way for finite families of operators, see [2]) and as an example of the application of this property it is shown that every family of generalized creation operators in the Bargmann space of an infinite order, indexed by mutually orthogonal vectors from I2 is subnormal. See [1] for the definitions.

A description of regular group rings is well known (see [12]). Various authors have considered regular semigroup rings (see [17], [8], [10], [11], [4]). These rings have been characterized for many important classes of semigroups, although the general problem turns out to be rather difficult and still has not got a complete solution. It seems natural to describe the regular radical in semigroup rings for semigroups of the classes mentioned. In [10], the regular semigroup rings of commutative semigroups were described. The aim of the present paper is to characterize the regular radical ρ(R[S]) for each associative ring R and commutative semigroup S.

The definition of a power-free group will be found in [1]. It is a partial algebraic system which, roughly speaking, may be thought of as a group in which (with the exception of the identity) squares and higher powers of an element are not defined.

It has been shown [1, Theorem 3.3] that the usual cancellation laws need not hold in a power-free group. When these laws do hold, the power-free group is called cancellative. In this paper we shall be solely concerned with cancellative power-free groups and the term ‘power-free group’ is to be understood to mean ‘cancellative power-free group’.

It is well known that if α and β are commuting *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 then M can be decomposed into a direct sum of subalgebras Mp and M(l − p) by a central projection p in M such that α = β on Mp and α = β-1 on M(1 − p) (see, for instance, [6], [7], [2]). Originally this equation arose in the Tomita-Takesaki theory (see, for example, [11]) in the form of one-parameter modular automorphism groups and later on it has been studied for arbitrary automorphisms and one-parameter groups of automorphisms on von Neumann algebras [7], [8], [9]. In the case of automorphism groups satisfying the above equation, one has a similar decomposition but this time without assuming the commutativity condition (cf. [7], [8]). For another relevant work on one-parameter groups of automorphisms which is close to our papers [7] and [8], we refer to Ciorănescu and Zsidó [1]. Regarding applications, this equation has been used for arbitrary automorphisms in the geometric interpretation of the Tomita-Takesaki theory [2] and in the case of automorphism groups it has been a fundamental tool in the generalization of the Tomita-Takesaki theory to Jordan algebras [3]. We may remark that the decomposition in the commuting case [6], [7] is much simpler than in the case of automorphism groups in the non-commutative situation [8]. In some cases one can obtain the decomposition for an arbitrary pair of automorphisms without assuming their commutativity but the problem in the general case has been unresolved. Recently we have shown that if α and β are *-automorphisms of a von Neumann algebra M satisfying the equation α + α-1 = β + β-1 (without assuming the commutativity of α and β) then there exists a central projection p in M such that α2= β2 on Mp and α2 = β−2 on M(l − p) [10].

Let V denote the class of discrete groups G which satisfy the following conditions (a), (b) and (c):

(a) G = (A * B; K = φ(H)) is the free product of two groups A and B with the subgroup H amalgamated.

(b) H does not contain the verbal subgroup A(X2) of A and K does not contain the verbal subgroup B(X2)of B.

Let M be a compact connected boundaryless surface and f: M → ℝ3 a smooth immersion transverse to a straight line L. Thus there is an even number p of points xεM such that f(x)εL. Under further transversality assumptions on f (see §3) there is a finite number q of points x of M such that the plane containing f(x) and L touches f(M) at f(x). These assumptions are mild in the sense that they hold for any f in an open dense subset of the space of smooth immersions under consideration. Suppose that the Gaussian curvature of f(M) is positive at q+ of these points and negative at q−, with q = q++ q−. Then

where e(M) denotes the Euler number of M.

In [13], Poincaré asked the following question: Which abstract groups can appear as monodromy groups [14] of second order, linear, homogeneous differential equations with meromorphic coefficients (which might depend on one or more parameters) on ℂ? In the present paper, we initiate a classification of monodromy groups of differential equations on compact Riemann surfaces of genus 1. We proceed as follows: Let

be the general Euler equation [1] which has two regular singular points at 0 and λ in the extended complex plane C. Further, let yu(z)(v – 1,2) be an arbitrary but fixed pair of linearly independent solutions to (1) valid in a neighborhood of some ordinary point. Analytic continuation of each solution along a closed loop A in C–{0, λ}, starting and ending at some fixed base point, produces a new solution yv,A (v = 1, 2) which can be expressed as

where the constants avlx (v1 ν = l,2) in C depend on the homotopy class [A] of A. Clearly, y1,A(z)/y2,A(z) = To(y1(z)/y2(z)) where the Möbius transformation T:w→ (a11W + a12)/(a21W + a22) depends on [A]. The set of all Mobius transformations T belonging to every possible closed loop A in C–{0,λ} forms a group G, called the monodromy group of the Euler equation (1). G is generated by the Mobius transformation belonging to a simple, closed loop Ao encircling 0. Hence, G is cyclic.

In this note we show that any derivation of a simple C*-algebra, whose range is not dense, is closable. We also derive a necessary and sufficient condition for a *-derivation of a C*-algebra, which is defined on the domain of a closed *- derivation, to be closed.

Let K be a field of characteristic different from 2, and a, b quadratically independent elements of K. Put J= K(√a, √b). In [4], Jensen and Yui discuss the question of quaternionic (Q8) extensions of J, and give a survey of known results. In [8], Ware discusses (among other things) some general conditions for, and relations between, the existence of Q8 and D4 (dihedral) extensions of K. A general theorem of Witt [9] says that J will have a quaternionic extension J(√u) if and only if there exists a 3 × 3 matrix P over K such that PPt = diag(a, b, 1/ab), and an appropriate value for u is given in terms of the entries of P. The problem of actually finding P in a particular case is not trivial.

We prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

We study a class of rings which are closely related to principal ideal domains, and prove in particular that finitely-generated projective modules over such rings are free. Examples include the ring of Lipschitz quaternions; Z[a½] with d = —3 or d = —7; and any subring R of M2(Z) such that R ⊇ M2(pZ) for some prime number/? and R/M2(pZ) is a field with p2 elements.

There does not exist an algorithm that can determine whether or not a group presented by commutators is a right-angled Artin group.