Love waves with a transverse structure

For an arbitrary layered isotropic structure, new exact solutions of the elastodynamic problem for the propagation of surface waves are presented. These solutions describe waves with rectilinear wave fronts propagating at the phase velocities of common SH-polarized Love waves. They linearly depend on a lateral transverse variable and, in addition to being standardly SH-polarized, have a longitudinally polarized anomalous component. The construction uses the assumption of the existence of standard Love waves. It is based on a potential representation of the wavefield and is quite elementary.     

Invariants of Finite Group Schemes

Let G be a finite group scheme operating on an algebraic varietyX, both defined over an algebraically closed field k. The paperfirst investigates the properties of the quotient morphism X- X/G over the open subset of X consisting of points whose stabilizershave maximal index in G. Given a G-linearized coherent sheafon X, it describes similarly an open subset of X over whichthe invariants in the sheaf behave nicely in some way. The pointsin X with linearly reductive stabilizers are characterized inrepresentation theoretic terms. It is shown that the set ofsuch points is nonempty if and only if the field of rationalfunctions k(X) is an injective G-module. Applications of theseresults to the invariants of a restricted Lie algebra g operatingon the function ring k[X] by derivations are considered in thefinal section. Furthermore, conditions are found ensuring thatthe ring k[X]^g is generated over the subring of pth powers ink[X], where p=char,k>0, by a given system of invariant functionsand is a locally complete intersection.     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

On the denominators of rational points on elliptic curves

Let x(P) = A_P/B²_P denote the x-coordinate of the rational pointP on an elliptic curve in Weierstrass form. We consider whenB_P can be a perfect power or a prime. Using Faltings' theorem,we show that for a fixed f > 1, there are only finitely manyrational points P with B_P equal to an fth power. Where descentvia an isogeny is possible, we show that there are only finitelymany rational points P with B_P equal to a prime, that thesepoints are bounded in number in an explicit fashion, and thatthey are effectively computable. Finally, we prove a strongerversion of this result for curves in homogeneous form.     

Properties of a Semi-discrete Approximation to the Beam Equation

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx = 0, 0 < x < L, t > 0, where it is assumed that w and p are positive on the interval[O, L], is approximated by using the method of straight lines.The resulting approximation is a linear system of differentialequations with coefficient matrix S. The matrix S is studiedunder very general boundary conditions which result in a conservativesystem. In all cases the matrix S is either an oscillation matrixor possesses nearly all the properties of an oscillation matrix.     

Conditions for Superlinear Convergence in l1 and l{infty} Solutions of Overdetermined Non-linear Equations

Let n be the number of unknowns in an overdetermined systemof non-linear equations. If fewer than n equations are satisfiedin an l₁ solution or if fewer than (n+1) maximum residuals occurin an lsolution, then iterative methods of calculation convergesuperlinearly only if some second derivative information isused. This paper establishes some conditions on second derivativeestimates that are necessary and sufficient for superlinearconvergence.     

Minimal Determinants and Lattice Inequalities

Some results of P. McMullen on determinants of sublattices ofZ^d induced by rational subspaces are generalized to arbitrarylattices. As an application, we obtain an equality for the minimaldeterminants introduced by J. M. Wills, namely D_t(L) = D_d(L)D_d–1((L*).Using an inequality of Lagarias, Lenstra and Schnorr, we generalizetwo isoperimetric inequalities withlattice constraints by Bokowski,Hadwiger and Wills, and Hadwiger, respectively, to arbitrarylattices.     

Predictor-corrector Methods with Improved Absolute Stability Regions

Generalized predictor-corrector methods with an extended regionof absolute stability are studied. Choosing an extrapolation-predictorand a backward-differentiation-corrector, methods of ordersup to 6 are constructed. The real stability boundaries of thesemethods are of magnitude m², where m is the number of right-handside evaluations per integration step. The coefficients of themethods can be generated during the computation for arbitraryvalues of m. The storage requirements are limited and are independentof m.     

Linking, Legendrian Linking and Causality

The set N of all null geodesics of a globally hyperbolic (d+ 1)-dimensional spacetime (M, g) is naturally a smooth (2d– 1)-dimensional contact manifold. The sky of an eventx in M is the subset X of N consisting of all null geodesicsthrough x, and is an embedded Legendrian submanifold of N diffeomorphicto S^{(d – 1)}. It was conjectured by Low that for d = 2two events x and y are causally related if and only if X andY are linked (in an appropriate sense). We use the contact structureand knot polynomial calculations to prove this conjecture incertain particular cases, and suggest that for d = 3 smoothlinking should be replaced with Legendrian linking.     

Structural Projections on JBW*-Triples

A linear projection R on a Jordan*-triple A is said to be structuralprovided that, for all elements a, b and c in A, the equality{Rab Rc} = R{a Rbc} holds. A subtriple B of A is said to becomplemented if A = B + Ker(B), where Ker(B) = {aA: {B a B}= 0}. It is shown that a subtriple of a JBW*-triple is complementedif and only if it is the range of a structural projection. A weak* closed subspace B of the dual E* of a Banach space Eis said to be an N*-ideal if every weak* continuous linear functionalon B has a norm preserving extension to a weak* continuous linearfunctional on E* and the set of elements in E which attain theirnorm on the unit ball in B is a subspace of E. It is shown thata subtriple of a JBW*-triple A is complemented if and only ifit is an N*-ideal, from which it follows that complemented subtriplesof A are weak* closed, and structural projections on A are weak*continuous and norm non-increasing. It is also shown that everyN*-ideal in A possesses a triple product with respect to whichit is a JBW*-triple which is isomorphic to a complemented subtripleof A.     

A Class of Pencils of Matrices

A linear machine is one in which the time dependent input yis related to the output z by P(D). z = S(D). y where P andS are polynomials in D = d/dt with constant coefficients. Fornumerical computation it is necessary to replace this relationby a set of simultaneous first order differential equationsand this paper shows how to construct such equations by methodswhich extend the results of Gilder (1961). Attention is restrictedto those sets of equations that are of a special form (see (1))which is characterized by the matrix operating on the dependentvariables. This matrix forms a pencil, being linear in D, andthree theorems are given to show how such matrix pencils maybe constructed from the polynomials. The theorems also statethat any matrix pencil with the required properties can be transformedinto the canonical forms given in the theorems by pre- and post-multiplicationby suitable constant non-singular matrices. Thus the variablesof any set of equations having the required properties are linearcombinations of the variables of the equations given by thetheorems. In the paper it is assumed that the degree of P(D)is greater than that of S(D), as otherwise z would be replacedby z₁+Q(D) . y, where Q is the quotient of S(D)/P(D). Also,as the algebriac manipulations are independent of the natureof the polynomials, D is replaced by an indeterminate x andthe coefficients considered to be from an arbitrary field. Fortechnical reasons we rename y and z, y_o and y_n–_m respectively.     

Analytic Tridiagonal Reproducing Kernels

The paper characterizes the reproducing kernel Hilbert spaceswith orthonormal bases of the form {(a_n,0+a_n,1z+...+a_n,Jz^J)zⁿ,n 0}. The primary focus is on the tridiagonal case where J= 1, and on how it compares with the diagonal case where J =0. The question of when multiplication by z is a bounded operatoris investigated, and aspects of this operator are discussed.In the diagonal case, M_z is a weighted unilateral shift. Itis shown that in the tridiagonal case, this need not be so,and an example is given in which the commutant of M_z on a tridiagonalspace is strikingly different from that on any diagonal space.     

On the Structure of Minimal Left Ideals in the Largest Compactification of a Locally Compact Group

This paper is centred around a single question: can a minimalleft ideal L in G^LUC, the largest semi-group compactificationof a locally compact group G, be itself algebraically a group?Our answer is no (unless G is compact). In deriving this conclusion,we obtain for nearly all groups the stronger result that nomaximal subgroup in L can be closed. A feature of our work isthat completely different techniques are required for the connectedand totally disconnected cases. For the former, we can relyon the extensive structure theory of connected, non-compact,locally compact groups to derive the solution from the commutativecase, using some reduction lemmas. The latter directly involvestopological dynamics; we construct a compact space and an actionof G on it which has pathological properties. We obtain otherresults as tools towards our main goal or as consequences ofour methods. Thus we find an extension to earlier work on therelationship between minimal left ideals in G^LUC and H^LUC whenH is a closed subgroup of G with G/H compact. We show that thedistal compactification of G is finite if and only if the almostperiodic compactification of G is finite. Finally, we use ourmethods to show that there is no finite subset of G^LUC invariantunder the right action of G when G is an almost connected groupor an IN-group.     

Invariant Curves by Vector Fields on Algebraic Varieties

If C is a reduced curve which is invariant by a one-dimensionalfoliation F of degree d_F on the projective space then it isshown that d_F–1+a is a bound for the quotient of the twocoefficients of the Hilbert–Samuel polynomial for C, wherea is an integer obtained from a concrete problem of imposingsingularities to projective hypersurfaces, and so a bound isobtained for the degree of C when it is a complete intersection.Concrete values of a can be derived for several interestingapplications. The results are presented in the form of intersection-theoreticalinequalities for one-dimensional foliations on arbitrary smoothalgebraic varieties.     

Spectral Synthesis and Operator Synthesis for Compact Groups

Let G be a compact group and C(G) be the C^*-algebra of continuouscomplex-valued functions on G. The paper constructs an imbeddingof the Fourier algebra A(G) of G into the algebra V(G) = C(G)^hC(G)(Haagerup tensor product) and deduces results about parallelspectral synthesis, generalizing a result of Varopoulos. Itthen characterizes which diagonal sets in G x G are sets ofoperator synthesis with respect to the Haar measure, using thedefinition of operator synthesis due to Arveson. This resultis applied to obtain an analogue of a result of Froelich: atensor formula for the algebras associated with the pre-ordersdefined by closed unital subsemigroups of G.     

Chebyshev Approximation by A + B* log (1 + CX). II

The set of all first degree polynomials must be added to theset of approximations of the form a + b log (1 + cx) in orderthat a best Chebyshev approximation exist to all continuousfunctions on [0, ]. Best approximations in this augmented familyof approximations are characterized by alternation of theirerror curve and are unique. The Chebyshev operator is continuousat f if f is an approximant or f has a non-constant best approximation.     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

Morita Equivalence of Operator Algebras and Their C*-Envelopes

If two operator algebras A and B are strongly Morita equivalent(in the sense of [5]), then their C*-envelopes C*(A) and C*(B)are strongly Morita equivalent (in the usual C*-algebraic sensedue to Rieffel). Moreover, if Y is an equivalence bimodule fora (strong) Morita equivalence of A and B, then the operation,Y_hA–, of tensoring with Y, gives a bijection between theboundary representations of C*(A) for A and the boundary representationsof C*(B) for B. Thus the ‘noncommutative Choquet boundaries’of Morita equivalent A and B are the same. Other important objectsassociated with an operator algebra are also shown to be preservedby Morita equivalence, such as boundary ideals, the Shilov boundaryideal, Arveson's property of admissability, and the latticeof C*-algebras generated by an operator algebra. 1991 MathematicsSubject Classification 47D25, 46L05, 46M99, 16D90.     

A new transform method for evolution partial differential equations

We introduce a new transform method for solving initial-boundary-valueproblems for linear evolution partial differential equationswith spatial derivatives of arbitrary order. This method isillustrated by solving several such problems on the half-line{t > 0, 0 < x < }, and on the quarter-plane {t >0, 0 < x_j < , j = 1, 2}. For equations in one space dimensionthis method constructs q(x, t) as an integral in the complexk-plane involving an x-transform of the initial condition anda t-transform of the boundary conditions. For equations in twospace dimensions it constructs q(x₁, x₂, t) as an integral inthe complex (k₁, k₂)-planes involving an (x₁, x₂)-transformof the initial condition, an (x₂, t)-transform of the boundaryconditions at x₁ = 0, and an (x₁, t)-transform of the boundaryconditions at x₂ = 0. This method is simple to implement andyet it yields integral representations which are particularlyconvenient for computing the long time asymptotics of the solution.     

The A-Complexity of a Space

Suppose that A is a pointed CW-complex. The paper looks at howdifficult it is to construct an A-cellular space B from copiesof A by repeatedly taking homotopy colimits; this is determinedby an ordinal number called the complexity of B. Studying thecomplexity leads to an iterative technique, based on resolutions,for constructing the A-cellular approximation CW_A(X) of an arbitraryspace X.