It is wellknown that a compact embedded hypersurface of the Euclidean space withoutboundary is a round sphere if one of mean curvature functions is constant. Inthis note, we show that a compact embedded hypersurface of the Euclidean space(and other constant curvature spaces) without boundary is a round sphere if theratio of some two mean curvature functions isconstant.

1991 Mathematics Subject Classification 53C40, 53C20.

The univalent functions in the diagonalBesov space A_{p}, where 1<p<\infty ,are characterized in terms of the distance from the boundary of a point in theimage domain. Here A_{2} is the Dirichlet space. A consequenceis that there exist functions in A_{p},\ p>2, for which thearea of the complement of the image of the unit disc iszero.

1991 Mathematics Subject Classification 30C99, 46E35.

We extendsplitting theorems due to Zaicev and Duan proving the following result. LetG be a locally soluble FC-hypercentral group and letA be a periodic artinian ℤG-module. IfA has no finite ℤG-submodules then anyextension E of A by Gsplits conjugately over A.

1991 Mathematics Subject Classification 20F19.

Let L^2_a (D, d\sigmad\theta /2\pi ) be a complete weighted Bergman space on the open unitdisc D, where d\sigma is a positive finiteBorel measure on [0, 1). We show the following : when \phi isa continuous function on the closed unit disc \bar {D}, T_\phi is compact if and only if \phi = 0 on\partial D.

1991 Mathematics Subject Classification 47B35, 47B07.

In thispaper we study the problem of the existence on non-inner automorphisms for theclass of torsion-free supersolvable groups, answering a question raised byRobinson.

1991 Mathematics Subject Classification 20F16, 20F28.

We define the notion of lightcone Gauss maps,lightcone pedal curves and lightcone developables of spacelike curves inMinkowski 3-space and establish the relationships between singularities of theseobjects and geometric invariants of curves under the action of the Lorentzgroup.

Let X be an infinite,locally finite, almost transitive graph with polynomial growth. We show that sucha graph X is the inverse limit of an infinite sequence offinite graphs satisfying growth conditions which are closely related to growthproperties of the infinite graphX.

1991 Mathematics Subject Classification. Primary 05C25, Secondary 20F8.

In this paper, weshow new results on slant submanifolds of an almost contact metric manifold. Westudy and characterize slant submanifolds of K-contact andSasakian manifolds. We also study the special class of three-dimensional slantsubmanifolds. We give several examples of slantsubmanifolds.

1991 Mathematics Subject Classification 53C15, 53C40.

In this paper we deduce the existence of analyticstructure in a neighbourhood of a maximal ideal M in thespectrum of a commutative Banach algebra, A, from homologicalassumptions. We assume properties of certain of the cohomology groupsH^n(A,A/M), rather than the stronger conditions on thehomological dimension of the maximal ideal the first author has considered inprevious papers. The conclusion is correspondingly weaker: in the previous workone deduces the existence of a Gel'fand neighbourhood with analytic structure,here we deduce only the existence of a metric neighbourhood with analyticstructure. The main method is to consider products of certain co-cycles to deducefacts about the symmetric second cohomology, which is known to be related to thedeformation theory of algebras.

1991 Mathematics Subject Classification. 46J20, 46M20.

Inthis paper, groups are investigated in which all subgroups, all normal subgroups,or all characteristic subgroups have a proper supplement. This supplement can beeither an arbitrary subgroup, a normal or a characteristic subgroup, resulting innine classes of groups. Properties of these classes are studied such ascontainment and closure properties, and characterizations for several of theseclasses are given.

1991 Mathematics Subject Classification Primary 20E34, Secondary 20E15.

For eachm≥1, u_{m}(G) is defined to be theintersection of the normalizers of all the subnormal subgroups of defect at mostm in G. An ascending chain of subgroupsu_{m,i}(G) is defined by settingu_{m,i}(G)/u_{m,i−1}(G)=u_{m}(G/u_{m,i−1}(G)). Ifu_{m,n}(G)=G, for some integer n, them-Wielandt length of G is theminimal of such n.

In [3], Bryce examined thestructure of a finite soluble group with given m-Wielandtlength, in terms of its polynilpotent structure. In this paper we extend hisresults to infinite soluble groups.

1991 Mathematics SubjectClassification. 20E15, 20F22.

We obtain two refinements of the socalled local duality of ultrapowers, that is, the ultrapower version of thewell-known principle of local reflexivity.

1991 Mathematics Subject Classification. 46B07, 46B08.

Weprove that exponentially harmonic morphisms are precisely the Riemanniansubmersions with minimal fibres.

1991 Mathematics Subject Classification 58E20.

Given a full subcategory[Fscr] of a category [Ascr], the existenceof left [Fscr]-approximations (or[Fscr]-preenvelopes) completing diagrams in a unique way isequivalent to the fact that [Fscr] is reflective in[Ascr], in the classical terminology of categorytheory.

In the first part of the paper we establish, for a rather general[Ascr], the relationship between reflectivity and covariantfiniteness of [Fscr] in [Ascr], andgeneralize Freyd's adjoint functor theorem (for inclusion functors) to notnecessarily complete categories. Also, we study the good behaviour of reflectionswith respect to direct limits. Most results in this part are dualizable, thusproviding corresponding versions for coreflective subcategories.

In thesecond half of the paper we give several examples of reflective subcategories ofabelian and module categories, mainly of subcategories of the form Copres(M) and Add (M). The second case covers thestudy of all covariantly finite, generalized Krull-Schmidt subcategories of{\rm Mod}_{R}, and has some connections with the“pure-semisimple conjecture”.

1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

Inthis paper, we prove that if M^2 is a complete maximalspacelike surface of an anti-de Sitter space {\bfH}^{4}_{2}(c) with constant scalar curvature, then S=0,S={-10c\over 11}, S={-4c\over 3} or S=-2c,where S is the squared norm of the second fundamental form ofM^{2}. Also

(1) S=0 if and only if M^2 is the totallygeodesic surface {\bf H}^2(c);

(2) S={-4c\over3} if and only if M^2 is the hyperbolic Veronesesurface;

(3) S=-2c if and only if M^2 is the hyperboliccylinder of the totally geodesic

surface {\bfH}^{3}_{1}(c) of {\bfH}^{4}_{2}(c).

1991 Mathematics Subject Classifaction 53C40, 53C42.

Let A be anoetherian local ring, x a non-unit element of A,B=A/(x). Let E be the Koszul complex associated to anarbitrary set of generators of the ideal (x) of A. Assume thatH1(E) is a free B-module. Then Ais Gorenstein if and only if B isalso.

1991 Mathematics Subject Classification 13H10, 13D03.

In multiple-roll coaters thin liquid films are transferred from roll to roll by means of liquid ‘beads’ which occupy the small gaps between adjacent rolls. Double-Film-Fed (DFF) beads are those which feature two ingoing films instead of the usual one, and arise in the intermediate stages of certain types of roll coater. One of the ingoing films, h1, is supplied from the previous inter-roll gap while the other, h2, ‘returns’ from the subsequent gap. Such a flow is investigated here under the conditions of low flow rate, small capillary number and negligible gravity and inertia, using lubrication theory and finite element analysis. The thickness of film h1 is fixed independently, while that of h2 is specified as a fraction, ζ, of the film output on the same roll. This simple approach allows a degree of feedback between the output and input of the bead, and enables one to simulate different conditions in the subsequent gap. Predictions of outgoing film thicknesses made using the two models agree extremely well and show that, for each value of ζ < 1, one outgoing film thickness decreases monotonically with speed ratio, S, while the other features a maximum. Good agreement is also seen in the pressure profiles, which are entirely sub-ambient in keeping with the small capillary number conditions. The finite element solutions reveal that in the ‘zero-flux’ case (when ζ = 1) the flow structures are very similar to those seen in an idealized cavity problem. In the more general (ζ < 1) situation, as in single-film-fed meniscus roll coating, several liquid transfer-jets occur by which liquid is conveyed through the bead from one roll to the other. The lubrication model is used to calculate several critical flow rates at which the flow is transformed, and it is shown that when the total dimensionless flow rate through the bead exceeds 1/3, the downstream flow structure is independent of the relative sizes of the ingoing films.

A variety of mass preserving moving boundary problems for the thin film equation, ut = −(unuxxx)x, are derived (by formal asymptotics) from a number of regularisations, the case in which the substrate is covered by a very thin pre-wetting film being discussed in most detail. Some of the properties of the solutions selected in this fashion are described, and the full range of possible mass preserving non-negative solutions is outlined.

Free surface flows generated by a moving distribution of pressure are considered. The fluid consists of two superposed layers in a two-dimensional channel. The upper layer is inviscid and the lower layer, which is introduced as a damping mechanism, is modelled by the mathematically convenient lubrication equations. Numerical and analytical solutions are presented. Special attention is given to solutions for which there is a train of waves on each side of the distribution of pressure. It is shown that, depending on the values of the parameters, the short waves can appear on either side of the distribution of pressure.

We study the effect of van der Waals forces on globally energy minimizing profiles for liquid droplets which lie on a solid substrate in a vapour atmosphere and which are assumed to have a uniform cross-section. We prove that for repulsive van der Waals forces as well as for certain short range repulsive-long range attractive forces, there exists a unique globally minimizing profile. Although this profile necessarily contains vertical bounding segments, the height A of the vertical bounding segments can often be demonstrated to be order of magnitude smaller that the overall height B of the droplet. This is the case, in particular, when the droplet is sufficiently large, the Hamaker constant is sufficiently small, and the attractive forces are sufficiently mild. In the presence of repulsive forces only, A is on the order of angströms when B is on the order of millimeters, for realistic parameter values. Moreover, conditions are prescribed under which Young's law is satisfied to leading order despite the appearance of the vertical segments, when the contact angle is measured via an inscribed circle construction at a distance ξ0 from the edge of the droplet, where A [Lt] ξ0 [Lt] B.