Wild Recurrent Critical Points

It is conjectured that a rational map whose coefficients arealgebraic over Q_p has no wandering components of the Fatou set.Benedetto has shown that any counterexample to this conjecturemust have a wild recurrent critical point. We provide the firstexamples of rational maps whose coefficients are algebraic over Q_p and that have a (wild) recurrent critical point. In fact,it is shown that there is such a rational map in every one-parameterfamily of rational maps that is defined over a finite extensionof Q_p and that has a Misiurewicz bifurcation.     

Error Control with Polynomial Approximations

A detailed analysis is given of the accumulation of errors whichmay occur in evaluating a polynomial approximation to a givenfunction. Both backward recursion using untransformed Chebyshevexpansions and the much faster nested multiplication using thetransformed simple polynomial form are treated. Two types ofarithmetic are dealt with covering most current machines. Forthe case of polynomials with coefficients of the same sign orstrictly alternating signs, a situation which is of considerablepractical importance in polynomial approximation of mathematicalfunctions, we show that controlling relative error requiresthe ratios |y|_max/|y|_min and |y'|_max/|y|_min to be kept small.Experimental verification of these effects is given based onexpansions available in the literature or produced by the authorsfor the Bessel function I₀(x).     

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.     

Cerf Theory for Graphs

We develop a deformation theory for k-parameter families ofpointed marked graphs with fixed fundamental group F_n. Applicationsinclude a simple geometric proof of stability of the rationalhomology of Aut(F_n), computations of the rational homology insmall dimensions, proofs that various natural complexes of freefactorizations of F_n are highly connected, and an improvementon the stability range for the integral homology of Aut(F_n).     

A fully discrete stabilized finite-element method for the time-dependent Navier-Stokes problem

A fully discrete stabilized finite-element method is presentedfor the two-dimensional time-dependent Navier–Stokes problem.The spatial discretization is based on a finite-element spacepair (X_h, M_h) for the approximation of the velocity and thepressure, constructed by using the Q₁ – P₀ quadrilateralelement or the P₁ – P₀ triangular element; the time discretizationis based on the Euler semi-implicit scheme. It is shown thatthe proposed fully discrete stabilized finite-element methodresults in the optimal order error bounds for the velocity andthe pressure.     

A pointwise estimate on the 1-order approximation of G{varepsilon}x0

On the basis of Avellaneda & Hua-Lin (1991, Commun. PureAppl. Math., 44 897–910), a pointwise error estimate onthe 1-order approximation of Green function G_x0 defined in R²is shown at first. Then based on this estimate and using asymptoticexpansion method, an improved approximation of G_x0 and its pointwiseerror estimate are obtained.     

The Generation of Chisholm Rational Polynomial Approximants to Power Series in Two Variables

The generation of successive Chisholm rational polynomial approximantsf_m/_m of f(x, y), a power series in two variables, is discussed. A necessary and sufficient condition for the non-degeneracyof f_m/_m is given. It is shown that the non-degeneracy of thediagonal Pad? approximants of order m in each variable separatelyis a necessary condition for the non-degeneracy of f_m/_m. In the case of a symmetric function, it is proved that the Chisholmapproximant f_m/_m is symmetric and non-degenerate if and onlyif all the diagonal Pad? approximants of order up to m in onevariable are non-degenerate. The generation of successive Chisholmapproximants to symmetric functions is also considered. The computational scheme, called the prong method, extends tocover the computation of Chisholm approximants in N-variables(Chisholm & McEwan, 1974).     

A note on least-squares mixed finite elements in relation to standard and mixed finite elements

^** Email: brandts{at}science.uva.nl The least-squares mixed finite-element method for second-orderelliptic problems yields an approximation u_h V_h H₀¹() of thepotential u together with an approximation p_{h h} H(div ; )of the vector field p = – Au. Comparing u_h with the standardfinite-element approximation of u in V_h, and p_h with the mixedfinite-element approximation of p, it turns out that they arehigher-order perturbations of each other. In other words, theyare ‘superclose’. Refined a priori bounds and superconvergenceresults can now be proved. Also, the local mass conservationerror is of higher order than could be concluded from the standarda priori analysis.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

Generalized Alternating Polynomials, some Properties and Numerical Applications

In the present paper we introduce the generalized alternatingpolynomials , the coefficients of which are defined together with the parameter W_n by the linear system where T_n(x) = cos (n arc cos x), is a set of n+2 distinct points in the interval [–1, 1], and fis a continuous function on [–1, 1], For the set of nodesx_k = cos [k/(n+1)] the w_n-polynomials coincide with the polynomialsof equiamplitude alternation introduced by de La Vall?e-Poussinand discussed in the literature earlier (Eterman, Malozemov,Meinardus, Cheney & Rivlin, Phillips & Taylor, Brutman,and others). It is shown that the generalized alternating polynomials arerelated to the polynomials of interpolation through the Lanczoseconomization process. Some approximation properties of w_n -polynomialsand W_n -parameters are studied. The application of w_n -polynomialsto function approximation and to the estimate of remainder termsfor quadrature formulas is discussed.     

A Conjecture on the Hall Topology for the Free Group

The Hall topology for the free group is the coarsest topologysuch that every group morphism from the free group onto a finitediscrete group is continuous. It was shoen by M.Hall Jr thatevery finitely generated subgroup of the free group is closedfor this topology. We conjecture that if H₁, H₂,...,H_n are finitelygenerated subgroups of the free group, then the product H₁ H₂...H_n is closed. We discuss some consequences of this conjecture.First, it would give a nice and simple algorithm to computethe closure of a given rational subset of the free group. Next,it implies a similar conjecture for the free monoid, which inturn is equivalent to a deep conjecture on finite semigroupsfor the solution of which J. Rhodes has offered $100. We hopethat our new conjecture will shed some light on Rhodes' conjecture.     

Oscillation Properties of a Semi-discrete Approximation to the Beam Equation with a Second Order Term

The solution of the equation w(x)u_tt+[p(x)u_xx]_xx–[p(x)u_x]_x=0, 0< x < L, t > 0, where it is assumed that w, p,and q are positive on the interval [0, L], is approximated bythe method of straight lines. The resulting approximation isa linear system of differential equations with coefficient matrixS. The matrix S is studied under a variety of boundary conditionswhich result in a conservative system. In all cases the matrixS is shown to be similar to an oscillation matrix.     

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.     

Prime Divisors of the Number of Rational Points on Elliptic Curves with Complex Multiplication

Let E/Q be an elliptic curve. For a prime p of good reduction,let E(F_p) be the set of rational points defined over the finitefield F_p. Denote by (#E(F_p)) the number of distinct prime divisorsof #E(F_p). For an elliptic curve with complex multiplication,the normal order of (#E(F_p)) is shown to be log log p. The normalorder of the number of distinct prime factors of the exponentof E(F_p) is also studied. 2000 Mathematics Subject Classification11N37, 11G20.     

Numerical Methods for y' =f(x, y) via Rational Approximations for the Cosine

To investigate stability and phase lag, a numerical method isapplied to the test equation y' = –²y. Frequently, thecharacteristic equation of the resulting recurrence relationhas the form ²– 2R_nm(v²) + 1 = 0, where v = h, with hthe steplength, and R_nm(v²) is a rational approximation forcos v. In this paper, properties of such approximations areused to provide a general framework for the study of stabilityintervals and orders of dispersion of a variety of one- andtwo-step methods. Upper bounds on the intervals of periodicityof explicit methods with maximum order of dispersion are established.It is shown that the order of dispersion of a P-stable method,for given n and m, cannot exceed 2m; a consequence is that,of the Pad? approximants for cos v, only the [0/2m] approximantshave modulus less than unity for all v² >0. A complete characterizationof P-stable methods of fourth order corresponding to the rationalapproximation R₂₂(v²) is followed by several results for methodswhich have finite intervals of periodicity; in particular, weidentify methods which have order of dispersion 6 or 8 withlarge intervals of periodicity. There is also a detailed discussionof P-stable methods of sixth order corresponding to the rationalapproximation R₃₃(v²).     

The Stefan Problem with a Non-monotone Constitutive Relation

A finite difference approximation of the non-linear diffusionequation e_t = ((e))_xx' where is non-monotone, is shown to convergeto a measure valued solution. Some numerical results are described.     

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.     

On the non-existence of exceptional automorphisms on Shimura curves

We study the group of automorphisms of Shimura curves X₀(D,N) attached to an Eichler order of square-free level N in anindefinite rational quaternion algebra of discriminant D>1.We prove that, when the genus g of the curve is greater thanor equal to 2, Aut (X₀(D, N)) is a 2-elementary abelian groupwhich contains the group of Atkin–Lehner involutions W₀(D,N) as a subgroup of index 1 or 2. It is conjectured that Aut(X₀(D, N))=W₀(D, N) except for finitely many values of (D, N)and we provide criteria that allow us to show that this is indeedoften the case. Our methods are based on the theory of complexmultiplication of Shimura curves and the Cerednik–Drinfeldtheory on their rigid analytic uniformization at primes p| D.     

Optimal Approximation of the Electrostatic Capacity Matrix of Two Conducting Spheres by a Short-Distance Asymptotic Expansion

An integral representation for the electrostatic capacity matrixC=[c_ij]_i,j=1,2 of two conducting spheres of radii R₁, and R₂is obtained. A short-distance asymptotic expansion is then derivedand its approximation properties for fixed (surface) distancer between the spheres are investigated. An error function is defined for c_ij(r) and its nthorder asymptotic approximant it has the property following from the divergence of the expansion, and thereby shows thatthe optimal approximation of c_ij(r) is achieved by an approximantof finite order n = n_ij(r) depending possibly on r and the indicesi,j. The value gives the quality of approximation of c_ij by the asymptotic expansion for a givendistance r between the spheres. The point sets and are introduced in order to describe the distance ranges where c_ij can be approximatedwithin a given error >0 by an asymptotic approximant of given order n, or at least by theoptimal approximant, respectively. The optimal order n_ij(r)and the -approximation sets and D() are investigated numerically.     

Methods for the approximation of the matrix exponential in a Lie-algebraic setting

Discretization methods for ordinary differential equations basedon the use of matrix exponentials have been known for decades.This set of ideas has come off age and acquired greater interestrecently, within the context of geometric integration and discretizationmethods on manifolds based on the use of Lie-group actions. In the present paper we study the approximation of the matrixexponential in a particular context: given a Lie group G andits Lie algebra g, we seek approximants F(t B) of exp(t B) suchthat F(t B) G if B g. Having fixed a basis V₁, ..., V_d ofg, we write F(t B) as a composition of exponentials of the typeexp(_i (t) V_i), where _i for i = 1, 2, ..., d are scalar functions.In this manner it becomes possible to increase the order ofthe approximation without increasing the number of exponentialsto evaluate and multiply together. We study order conditionsand implementation details and conclude the paper with somenumerical experiments. Received 24 March 1999. Accepted 22 November 1999.