In a previous paper the author and Demay advanced a model to explain the melt fracture instability observed when molten linear polymer melts are extruded in a capillary rheometer operating under the controlled condition that the inlet flow rate was held constant. The model postulated that the melts were a slightly compressible viscous fluid and allowed for slipping of the melt at the wall. The novel feature of that model was the use of an empirical switch law which governed the amount of wall slip. The model successfully accounted for the oscillatory behavior of the exit flow rate, typically referred to as the melt fracture instability, but did not simultaneously yield the fine scale spatial oscillations in the melt typically referred to as shark skin. In this note, a new model is advanced which simultaneously explains the melt fracture instability and shark skin phenomena. The model postulates that the polymer is a slightly compressible linearly viscous fluid but assumes no-slip boundary conditions at the capillary wall. In simple shear the shear stress τ and strain rate d are assumed to be related by d = Fτ, where F ranges between F2 and F1 > F2. A strain-rate dependent yield function is introduced and this function governs whether F evolves towards F2 or F1. This model accounts for the empirical observation that at high shears polymers align and slide more easily than at low shears, and explains both the melt fracture and shark skin phenomena.

Fibre drawing is an important industrial process for synthetic polymers and optical communications. In the manufacture of optical fibres, precise diameter control is critical to waveguide performance, with tolerances in the submicron range that are met through feedback controls on processing conditions. Fluctuations arise from material non-uniformity plague synthetic polymers but not optical silicate fibres which are drawn from a pristine source. The steady drawing process for glass fibres is well-understood (e.g. [11, 12, 20]). The linearized stability of steady solutions, which characterize limits on draw speed versus processing and material properties, is well-understood (e.g. [9, 10, 11]). Feedback is inherently transient, whereby one adjusts processing conditions in real time based on observations of diameter variations. Our goal in this paper is to delineate the degree of sensitivity to transient fluctuations in processing boundary conditions, for thermal glass fibre steady states that are linearly stable. This is the relevant information for identifying potential sources of observed diameter fluctuation, and for designing the boundary controls necessary to alter existing diameter variations. To evaluate the time-dependent final diameter response to boundary fluctuations, we numerically solve the model nonlinear partial differential equations of thermal glass fibre processing. Our model simulations indicate a relative insensitivity to mechanical effects (such as take-up rates, feed-in rates), but strong sensitivity to thermal fluctuations, which typically form a basis for feedback control.

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

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.