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.     

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.     

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.     

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).     

The large deformation of nonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((₀, ₀), 0, P₀) in the solution-parameter space of ((₀, ₀),0, P₀) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((₀, ₀), 0, P₀) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P₀ and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((₀, ₀), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((₀, ₀),0).     

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 .     

Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximationof the following system of nonlinear degenerate parabolic equations: (u)/(t) + . (u² [(v)]) - (1)/(3) .(u³ w)= 0, w = - c u - u^-+ a u^-3 , (v)/(t) + . (u v [(v)]) - v - .(u² v w) = 0. The above models a surfactant-driven thin-film flow in the presenceof both attractive, a>0, and repulsive, >0 with >3,van der Waals forces; where u is the height of the film, v isthe concentration of the insoluble surfactant monolayer and(v):=1-v is the typical surface tension. Here 0 and c>0 arethe inverses of the surface Peclet number and the modified capillarynumber. In addition to showing stability bounds for our approximation,we prove convergence, and hence existence of a solution to thisnonlinear degenerate parabolic system, (i) in one space dimensionwhen >0; and, moreover, (ii) in two space dimensions if inaddition 7. Furthermore, iterative schemes for solving the resultingnonlinear discrete system are discussed. Finally, some numericalexperiments are presented.     

Fitted and Unfitted Finite-Element Methods for Elliptic Equations with Smooth Interfaces

This paper considers the finite-element approximation of theelliptic interface problem: -?(u) + cu = f in Rⁿ (n = 2 or3), with u = 0 on , where is discontinuous across a smoothsurface in the interior of . First we show that, if the meshis isoparametrically fitted to using simplicial elements ofdegree k - 1, with k 2, then the standard Galerkin method achievesthe optimal rate of convergence in the H¹ and L² norms overthe approximations _l⁴ of _l where _{l 2}. Second, since itmay be computationally inconvenient to fit the mesh to , weanalyse a fully practical piecewise linear approximation ofa related penalized problem, as introduced by Babuska (1970),based on a mesh that is independent of . We show that, by choosingthe penalty parameter appropriately, this approximation convergesto u at the optimal rate in the H¹ norm over _l⁴ and in the L²norm over any interior domain _l^* satisfying _l^* _l^** _l⁴ for somedomain _l^**. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton BN1 9QH     

Generators and defining relations for ring of invariants of commuting locally nilpotent derivations or automorphisms

Let A be an algebra over a field K of characteristic zero andlet ₁, ..., _sDer _K(A) be commuting locally nilpotent K-derivationssuch that _i(x_j) equals _ij, the Kronecker delta, for some elementsx₁, ..., x_sA. A set of generators for the algebra is found explicitly and a set of defining relationsfor the algebra A is described. Similarly, let ₁, ..., _s Aut_K(A)be commuting K-automorphisms of the algebra A is given suchthat the maps _i – id_A are locally nilpotent and _i (x_j)= x_j + _ij, for some elements x₁, ..., x_s A. A set of generatorsfor the algebra A: = {a A | ₁(a) = ... = _s(a) = a} is foundexplicitly and a set of defining relations for the algebra Ais described. In general, even for a finitely generated non-commutativealgebra A the algebras of invariants A and A are not finitelygenerated, not (left or right) Noetherian and a minimal numberof defining relations is infinite. However, for a finitely generatedcommutative algebra A the opposite is always true. The derivations(or automorphisms) just described appear often in many differentsituations (possibly) after localization of the algebra A.     

A Variation on a Theorem of Galvin and Hajnal

Let be a singular cardinal of regular uncountable cofinality. Let {(): < } be a continuous increasing sequence withlimit , and let =₍₎₊₍₎, < be regular cardinals. Let I be a normal ideal on , and assume that the reduced product_</I admits a cofinal -scale of ordinal functions. Then ₊, where =||||_I is the I-norm of .     

Gyration Numbers, Fixed Points and K2

We give a formula for the Boyle-Krieger gyration numbers ofan involution in the symmetry group Aut (_A) of an even subshiftof finite type _A in terms of the number of fixed points of in the periodic orbits of _A. A relationship between gyrationnumbers, the algebraic K-theory group K₂, and the tame symbolis also discussed.     

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.     

Thermodynamic Inequalities in Continuum Mechanics

The constitutive relations for the transport of heat, stress,electric charge, etc., in a continuum must be chosen so thatthe second law of thermodynamics is not violated; the constraintstake the form of inequalities, typically requiring the entropygenerated within a material element to be non-negative. Thepaper is concerned with this concept—its history, thephysical principles on which it depends, how to apply it whensecond-order or non-linear effects are important and how itis widely misused in modern continuum mechanics. The history is reduced to the contributions of five leadingthermodynamicists—Clausius, Maxwell, Gibbs, Boltzmannand Duhem. The object here was to try to discover which formof the inequality one should regard as being fundamental. Oneimportant conclusion is that entropy S must be defined simultaneouslywith the identification of the inequality, and that in generalthis cannot be done until the constitutive equations are known.The empirical element enters with the notion of irreversibility,which is given a precise meaning with the aid of the motionreversed parity (x), a variable x having = +1 or = –1if, when time and motions are reversed, x x or x –x.The macroscopic parity of x, _*(x), is obtained by first replacingx by the constitutive equation for x. The entropy production rate has both irreversible (f) and reversible(r) parts. It is shown that the reciprocal relations followfrom the requirement that the macroscopic parity of (i) mustbe +1. Continuum thermodynamics is based on various principles extractedfrom theory developed for uniform systems, the example chosento illustrate the ideas being the simple monatomic gas. Second-orderconstitutive relations are introduced, and the expressions forentropy and its production rate per unit volume, , obtained.It is shown that the stability condition 0 cannot, in general,be satisfied merely by imposing constraints on the constitutiverelations. To second-order = ₁ + ₂, where ₁ is the usual bilinearform, and the terms in ₂ have an additional derivative. Thesecond-order term ₂ can have both signs, and is not dissipative.The relation between this fact and the frame-dependence of constitutiverelations is explained. The final section illustrates the errors frequently found inthe thermodynamic arguments appearing in books and papers onrational continuum mechanics. The principle of these is that 0 is interpreted as being a constraint on the constitutiverelations alone. Another is the idea that the balance equationscan be set aside as constraints by regarding them as mere definitionsof a heat source and a body force, an error based partly onthe misconception that constitutive relations should be frame-indifferent.Finally, an inequality due to Glansdorff & Prigogine isexamined and found to be in error.     

On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel (t)= [sum ]_j=1 exp(-t_j) for small positive t, where {_j} _j=1 arethe eigenvalues of the negative Laplacian -_n = -[sum ]ⁿ_k=1 (/x^k)²in Rⁿ (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains _i with smooth boundaries _i (i = 1,...,m), where smoothfunctions Y_i (i = 1,...,m) are assuming the Robin boundary conditions(n_i + Y_i) = 0 on _i. Here /n_i denote differentiations along theinward-pointing normals to _i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.     

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).     

On a theorem of K. Schmidt

Let Y be a locally compact group, Aut(Y) be the group of topologicalautomorphisms of Y and (Y) be the set of continuous positivedefinite functions on Y which have unit value at the identity.A function (Y²) is said to be of product type if there aresuch functions _j (Y) that (u, v) = ₁(u)₂(v). Define the mappingT: Y² Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), whereA_j Aut(Y), and assume that T is a one-to-one transform. K.Schmidt proved: (i) if both (u, v) and (T(u, v)) are of producttype, then the functions _j are infinitely divisible; (ii) ifY is Abelian, both (u, v) and (T(u, v)) are of product type,and (u, v) 0, then the functions _j are Gaussian. We show thatstatement (i), generally, is not valid, but K. Schmidt's proofholds true if (u, v) 0. We also give another proof of statement(ii). Our proof uses neither the Levy–Khinchin formulafor a continuous infinitely divisible positive definite functionnor (i) on which K. Schmidt's proof is based.     

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.     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.