Flow of a nematic liquid crystal near the leading edge of an infinite prism

Flow of a nematic liquid crystal in an infinite wedge boundedby sidewalls = ± (with no-slip condition) is considered.The fluid is contained in the region 0 r < , – and – < z < (0 ). The near-tip velocity fieldis assumed to have the form v_i(r, ) = rF_i()(i = r, , z) as rtends to zero. We investigate the dependence of eigenvalues and functions F_i() on the tilt angle, G(), between the directorfield and the plane z = c (c ) and on the included angle 2 of the wedge shaped prism. Two kinds of nematicliquid crystal are considered as examples: MBBA and PAA near25 °C and 125 °C, respectively. In general, when 0 <G() < /2 the liquid crystalline material is curvilinear anisotropicand no symmetry properties are found. Here all velocity fieldcomponents are coupled. This coupling reduces the magnitudeof the leading-order eigenvalue and the one with smallest realpart is purely real for any wedge included angle. However, complexeigenvalues can occur for the next eigenvalues ordered in termsof the magnitude of the real part. Thus, if we impose the appropriatebehaviour on the far velocity field so that it is orthogonalto the eigenvectors associated with the first real eigenvalues,the remaining flow fields may display eddies.     

Asymptotic results for bifurcations in pure bending of rubber blocks

The bifurcation of an incompressible neo-Hookean thick blockwith a ratio of thickness to length , subject to pure bending,is considered. The two incremental equilibrium equations correspondingto a nonlinear pre-buckling state of strain are reduced to afourth-order linear eigenproblem that displays a multiple turningpoint. It is found that for 0 < < , the block experiencesan Euler-type buckling instability which in the limit degeneratesinto a surface instability. Singular perturbation methods enableus to capture this transition, while direct numerical simulationscorroborate the analytical results.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

The effects of a complexation reaction on travelling wave-fronts in a quadratic autocatalytic system

A model which describes the effect that a complexation reactioncan have on the propagation of reaction fronts in a quadraticautocatalytic system is considered. An initial-value problemis set up, which involves the (dimensionless) parameters K,the equilibrium constant for the complexation reaction, and, the initial concentration of the complexing agent. This initial-valueproblem is analysed, with global existence and uniqueness beingestablished. Numerical integrations indicate the formation ofpermanent-form travelling waves at large times. The equationsthat govern the travelling waves in the model are treated indetail. It is determined that there is a minimum propagationspeed v_min lying in the range v₀ 2/(1 + ) v_min 2, with thevalue v₀ corresponding to the minimum speed derived from thelinearization of the travelling wave equations. The existenceof a curve is established, which divides the (K, ) parameter plane into two regions, one where v_min =v₀ and one where v_min > v₀ with waves propagating fasterthan their linearized speed in this region. The curve is determined numerically together with the dependenceof v_min on K and .     

Singular behaviour in a generalized boundary eigenvalue problem for annular plates in tension

A novel application of boundary-layer asymptotic techniquesto a generalized linear eigenvalue problem is presented. Ourinvestigation is concerned with a bifurcation equation thatgoverns the formation of wrinkles in thin annular plates subjectedto in-plane tensile loading on the inner boundary. If denotesthe ratio of the inner and outer radii of the annulus, thenthe critical wrinkling load satisfies = _C(), where the function_C is available only numerically. It is known that there is acritical value such that as but, until now, little has been understood about this singular behaviour. Asymptoticmethods enable us to capture accurately and describe the natureof this blow-up phenomenon which we show is sensitive to theforms of the boundary conditions imposed at the edges of theannular plate. Our analytical findings are complemented by aseries of comparisons with direct numerical simulations thatshed further light on the singular behaviour.     

Corrigendum: Elliptic Curves and Quadratic Recurrence Sequences

The statement of the numerical values and z₀ on page 167 of[1, Section 3] contained an error. The values that were actuallyused were (to nine decimal places): thesebeing shifted, by the periods 2₁ and 2₃ respectively, comparedwith the values given in [1] (with ₁ = 1.496729323 and ₃ = 1.225694691i).With ₀ = ₁ and (z) denoting the sigma function (z; g2, g3) withinvariants g2 = 4, g3 = –1 associated with the ellipticcurve given by equation (3.2), these values of and z₀ yield and the latter three values all agreewith those stated in the paper (apart from rounding down thelast digit in the imaginary part of A). 2000 Mathematics SubjectClassification 11B37 (primary), 33E05, 37J35 (secondary).     

Rayleigh waves having generalised lateral dependence

It is shown that the surface-guided elastic waves found by Kiselevfor isotropic materials and having displacements depending linearlyupon the Cartesian coordinate orthogonal to the sagittal planemay be generalised in many ways. For surface waves on any anisotropichalf-space, a simple procedure applied to the displacementswithin the standard surface wave having dependence eⁱ, where k · x – t and k is the (surface) wave vector,yields displacements depending linearly upon the surface cartesiancoordinate orthogonal to the group velocity vector. Moreover,repeated application of this (differentiation) procedure yieldsa hierarchy of waves having algebraic dependence of successivelyincreasing degree. For isotropic materials, substantial simplificationand generalization are possible. Solutions of all algebraicdegrees have identical depth dependence. This allows the solutionsto be constructed iteratively and motivates a search for generalsolutions having depth dependence of the normal displacementthe same as in the standard surface wave. The procedure givesa new derivation of the solutions found by Achenbach havingamplitude of the normal displacement of the surface given byany solution to the two-dimensional Helmholtz equation. Furthermore,exploiting the scale invariance (a property of surface waveson any homogeneous half-space) shows that in every surface-guideddisturbance of an elastic half-space, the elevation of the freesurface is a solution of the wave equation in two dimensions(the membrane equation). Using the paraxial approximation tothe membrane equation, high-frequency Rayleigh waves propagatingas narrow beams are described in terms of a scalar Gaussianbeam.     

Mixed finite element analysis of a non-linear three-fields Stokes model

In this article, a mixed finite element analysis of the non-linearStokes problem with monotone constitutive laws is considered.We construct a new three-field model for incompressible fluidswhere the velocity u, the non-linear stress tensor = (|u|)u and the pressure p are the most relevant unknowns. We giveexistence and unicity results for the continuous problem andits approximation. Stable and optimal error estimates underminimal regularity assumptions are derived and numerical resultsare presented. Received 29 April 1999. Accepted 30 November 1999.     

Dynamic crack-tip singular stress in a cylindrically orthotropic elastic solid

Anti-plane wave motion is induced in a cylindrically orthotropicelastic solid containing a semi-infinite stress-free crack,by a point impulsive body force. First, the static version ofthe problem is solved for the fracture stress _z. Here, a globalsolution is obtained and then examined at the crack tip in orderto determine the nature of the spatial singularity. Next, thedynamic problem is treated and it is found that the dominantspatial singularity for _z at the crack tip is the same as inthe static case. However, the dynamic part of the stress intensityfactor, T, may introduce a further singularity. Several equivalentexpressions are presented for T, one of which is examined insome detail.     

Correction: Abelian Varieties defined over their Fields of Moduli, I

Bull London Math. Soc, 4 (1972), 370–372. The proof of the theorem contains an error. Before giving acorrect proof, we state two lemmas. LEMMA 1. Let K/k be a cyclic Galois extension of degree m, let generate Gal (K/k), and let (A, I, ) be defined over K. Supposethat there exists an isomorphism :(A,I,) (A, I, ) over K suchthat v^m–1 ... = 1, where v is the canonical isomorphism(A^m, I^m, ^m) (A, I, ). Then (A, I, ) has a model over k, whichbecomes isomorphic to (A, I, ) over K. Proof. This follows easily from [7], as is essentially explainedon p. 371. LEMMA 2. Let G be an abelian pro-finite group and let : G Q/Z be a continuous character of G whose image has order p.Then either: (a) there exist subgroups G' and H of G such that H is cyclicof order p^m for some m, (G') = 0, and G = G' x H, or (b) for any m > 0 there exists a continuous character m ofG such that p^m _m = . Proof. If (b) is false for a given m, then there exists an element G, of order p^r for some r m, such that () ¦ 0. (Considerthe sequence dual to 0 Ker (p^m) G ^pm G). There exists an opensubgroup G_o of G such that (G₀) = 0 and has order p^r in G/G₀.Choose H to be the subgroup of G generated by , and then aneasy application to G/G₀ of the theory of finite abelian groupsshows the existence of G' (note that () ¦ 0 implies that is not a p-th. power in G). We now prove the theorem. The proof is correct up to the statement(iv) (except that (i) should read: F' k₁ F'ab). To removea minor ambiguity in the proof of (iv), choose to be an elementof Gal (F'_ab/k₂) whose image $$\stackrel{\&macr;}{\sigma}$$ in Gal (k₁/k₂) generates this last group. The error occursin the statement that the canonical map v : A^P A acts on pointsby sending a^p a; it, of course, sends a a. The proof is correct, however, in the case that it is possibleto choose so that ^p = 1 (in Gal (F'/k₂)). By applying Lemma 2 to G = Gal (F'_ab/k₂) and the map G Gal(k₁/k₂) one sees that only the following two cases have to beconsidered. (a) It is possible to choose so that ^pm = 1, for some m, andG = G' x H where G' acts trivially on k₁ and H is generatedby . (b) For any m > 0 there exists a field K, F'ab K k₁ k₂is a cyclic Galois extension of degree p^m. In the first case, we let K F'_ab be the fixed field of G'.Then (A, I, ), regarded as being defined over K, has a modelover k₂. Indeed, if m = 1, then this was observed above, butwhen m > 1 the same argument applies. In the second case, let : (A, I, ) (A$$\stackrel{\&macr;}{\sigma}$$, I$$\stackrel{\&macr;}{\sigma }$$, $$\stackrel{\&macr;}{\sigma}$$) be an isomorphism defined over k₁ and let v ... ^p–1 = µ(R). If is replaced by for some Aut_k1((A, I, )) then is replacedby ^P. Thus, as µ(R) is finite, we may assume that ^pm–1= 1 for some m. Choose K, as in (b), to be of degree p^m overk₂. Let _m be a generator of Gal (K/k₂) whose restriction tok₁ is $$\stackrel{\&macr;}{\sigma }$$. Then : (A, I, ) (A^{$$\stackrel{\&macr;}{\sigma }$$}, I^{$$\stackrel{\&macr;}{\sigma}$$}, ^{$$\stackrel{\&macr;}{\sigma }$$} = (A^{$$\stackrel{\&macr;}{\sigma}$$m}, I^{$$\stackrel{\&macr;}{\sigma }$$m}, ^{$$\stackrel{\&macr;}{\sigma}$$m} is an isomorphism defined over K and v ^mpm–1, ... ^m =^pm–1 = 1, and so, by) Lemma 1, (A, I, ) has a model overk₂ which becomes isomorphic to (A, I, over K. The proof may now be completed as before. Addendum: Professor Shimura has pointed out to me that the claimon lines 25 and 26 of p. 371, viz that µ(R) is a puresubgroup of _R^*t, does not hold for all rings R. Thus this condition,which appears to be essential for the validity of the theorem,should be included in the hypotheses. It holds, for example,if µ(R) is a direct summand of µ(F).     

Skew Derivations with Central Invariants

If is an automorphism and is a -derivation of a ring R, thenthe subring of invariants is the set R⁽⁾={rR|(r)=0}. The mainresult of this paper is ‘let R be a semiprime ring withan algebraic -derivation such that R⁽⁾ is central; then R iscommutative’. This theorem generalizes results on theinvariants of automorphisms and derivations and is proved byreducing down to the special cases of automorphisms and derivations.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Are All Uniform Algebras Amnm?

A Banach algebra a is AMNM if whenever a linear functional on a and a positive number satisfy |(ab)–(a)(b)|||a||·||b||for all a, b a, there is a multiplicative linear functional on a such that ||–||=o(1) as 0. K. Jarosz [1] asked whetherevery Banach algebra, or every uniform algebra, is AMNM. B.E. Johnson [3] studied the AMNM property and constructed a commutativesemisimple Banach algebra that is not AMNM. In this note weconstruct uniform algebras that are not AMNM. 1991 MathematicsSubject Classification 46J10.     

Piecewise Absolutely Continuous Cocycles Over Irrational Rotations

For an irrational rotation of the circle group T=R/Z and apiecewise absolutely continuous function f:TR, the unitary operatorVh(x)=e^2if(x)h(x+) on L²(T) is studied. It is shown that iff has a single discontinuity with non-integer jump then V is-weakly mixing for some with 0<||<1. In particular Vhas continuous singular spectrum. The property of -weak mixing(with possible change of the value of , 0<||<1) holdsfor all irrational rotations and, given , is stable under perturbationsof f by functions with sufficiently small O(1/n)-norm. On theother hand, there exists a piecewise linear function f withtwo non-integer jumps such that the spectrum of V is continuoussingular for one value of and Lebesgue for another.     

Calculs De Facteurs Epsilon De Paires Pour GLn Sur Un Corps Local, I

Soient F un corps commutatif localement compact non archimédienet un caractère additif non trivial de F. Soient unereprésentation du groupe de Weil–Deligne de F,et sa contragrédiente. Nous calculons le facteur (, , ). De manière analogue, nous calculons le facteur (x, , ) pour toute représentationadmissible irréductible de GL_n(F). En conséquence,si F est de caractéristique nulle et si et se correspondentpar la correspondance de Langlands construite par M. Harris,ou celle construite par les auteurs, alors les facteurs (, , s) et (x, , s) sont égaux pour tout nombre complexe s. Let F be a non-Archimedean local field and a non-trivial additivecharacter of F. Let be a representation of the Weil–Delignegroup of F and its contragredient representation. We compute (, , ). Analogously, we compute (x, , ) for all irreducible admissible representations of GL_n(F).Consequently, if F has characteristic zero, and , correspondvia the Langlands correspondence established by M. Harris orthe correspondence constructed by the authors, then we have(, , s) = (x, , s) for all sC. 1991 Mathematics Subject Classification22E50.     

Asymptotically Almost Periodic Solutions of Inhomogeneous Cauchy Problems on The Half-Line

Let u be a bounded, uniformly continuous, mild solution of aninhomogeneous Cauchy problem on R₊: u'(t)+Au(t)+(t)(t0). Supposethat u has uniformly convergent means, (A)iR is countable, and is asymptotically almost periodic. Then u is asymptoticallyalmost periodic. Related results have been obtained by Ruessand V, and by Basit, using different methods. A direct proofis given of a Tauberian theorem of Batty, van Neerven and Räbiger,and applications to Volterra equations are discussed. 1991 MathematicsSubject Classification 34C28, 44A10, 47D03.     

Metric Entropy of Convex Hulls in Hilbert Spaces

We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n^–. Then,for each n N, the convex hull of X can be covered by 2ⁿ ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(^–1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(^–2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46.     

A bifurcation problem governed by the boundary condition II

In this work we consider the problem u = a(x)u^p in on , where is a smooth bounded domain, isthe outward unit normal to , is regarded as a parameter and0 < p < 1. We consider both cases where a(x) > 0 in or a(x) is allowed to vanish in a whole subdomain ₀ of . Ourmain results include existence of non-negative non-trivial solutionsin the range 0 < < ₁, where ₁ is characterized by meansof an eigenvalue problem, uniqueness and bifurcation from infinityof such solutions for small , and the appearance of dead coresfor large enough .     

Finite Element Approximation of a Rigid Punch Indenting a Membrane

Optimal order H¹ and L error bounds are obtained for a continuouspiecewise linear finite element approximation of an obstacleproblem, where the obstacle's height as well as the contactzone, _c, are a priori unknown. The problem models the indentationof a membrane by a rigid punch. For R², given ,g R⁺ and an obstacle defined over E we consider the minimization of |v|²_1,+gµover (v, µ) H¹₀() x R subject to v+µ on E. In additionwe show under certain nondegeneracy conditions that dist (_c,^h_c)Ch ln 1/h, where ^h_c is the finite element approximation to_c. Finally we show that the resulting algebraic problem canbe solved using a projected SOR algorithm.