Diophantine analysis and torsion on elliptic curves

In a recent paper of Bennett and the author, it was shown thatthe elliptic curve defined by y² = x³ + Ax + B, where A andB are integers, has no rational points of finite order if Ais sufficiently large relative to B (at least if one assumesthe abc Conjecture of Masser and Oesterlé). In the presentarticle we show, perhaps surprisingly, that the rational torsionon the above curve is also quite restricted if B is sufficientlylarge relative to A. In particular, we demonstrate that forany > 0 there is a constant c such that if A and B are integerssatisfying |B| > c |A|⁶⁺, then the elliptic curve definedabove has no rational torsion points, other than a possiblepoint of order 2 (again making use of the abc Conjecture insome cases). We then extend this by proving similar resultsfor elliptic curves admitting non-trivial -isogenies, ellipticcurves written in other forms, and elliptic curves over certainnumber fields. Curiously, the results on isogenies lead to twounexpected irrationality measures for certain algebraic numbers.     

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in Rn

McMichael proved that the convolution with the (euclidean) arclengthmeasure supported on the curve t (t, t², ..., tⁿ), 0 < t< 1, maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if and only if 2n(n+1)/(n²+n+2) p 2. In proving this, a uniform estimate on damping oscillatoryintegrals with polynomial phase was crucial. In this paper,a remarkably simple proof of the same estimate on oscillatoryintegrals is presented. In addition, it is shown that the convolutionoperator with the affine arclength measure on any polynomialcurve in Rⁿ maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if p = 2n(n+1)/(n²+n+2).     

Sierpinski and non-Sierpinski curve Julia sets in families of rational maps

We discuss the dynamics as well as the structure of the parameterplane of certain families of rational maps with few criticalorbits. Our paradigm is the family R_t(z) = (1 + (4/27)z³/(1– z)), with dynamics governed by the behaviour of thepostcritical orbit (Rⁿ())_n. In particular, it is shown thatif escapes (that is, Rⁿ() tends to infinity), then the Juliaset of R is a Cantor set, or a Sierpiski curve, or a curve withone or else infinitely many cut-points; each of these casesactually occurs.     

Double Structures and Normal Bundle of Space Curves

It is shown that the general plane section of a double structureon an integral curve CP³ has a connected numerical character(that is, its Hilbert function is of decreasing type). The papergives applications, particularly to the stability of the normalbundle of space curves.     

A Classification of Horizontally Homothetic Submersions from Space Forms of Nonnegative Curvature

In this paper, it is proved that for n 2, any horizontallyhomothetic submersion : Rⁿ⁺¹ (Nⁿ, h) is a Riemannian submersionup to a homothety. It is also shown that if : Sⁿ⁺¹ (Nⁿ, h)is a horizontally homothetic submersion, then n = 2m, (Nⁿ, h)is isometric to CP^m and, up to a homothety, is a standard Hopffibration S^2m+1 CP^m. 2000 Mathematics Subject Classification53C20, 53C12.     

Convergence of a fully discrete approximation for advected mean curvature flows

Let (t) be a closed curve in R² which propagates in its normaldirection n with velocity V = --q.n-g, where is the mean curvatureof (t) and g and q are given represent, respectively, a forcingterm and a vector field. In this paper we prove that such flowscan be approximated by numerical solutions of advection Allen-Cahnequations. It is shown that the zero level set of the fullydiscrete solution using explicit time stepping converges evenpast singularities to the true interface provided that no fatteningoccurs and , h² O(⁴), where h and denote the mesh size andthe time step. For smooth flows an optimal O(²)-rate of convergenceis derived provided , h² O(⁵). The analysis is based on constructingfully discrete barriers via an explicit parabolic projectionand Lipschitz dependence of the viscosity solutions with respectto perturbations of data.     

The Alternative Dunford-Pettis Property, Conjugations and Real Forms of C*-Algebras

Let be a conjugation, alias a conjugate linear isometry oforder 2, on a complex Banach space X and let X be the real formof X of -fixed points. In contrast to the Dunford–Pettisproperty, the alternative Dunford–Pettis property neednot lift from X to X. If X is a C^*-algebra it is shown thatX has the alternative Dunford–Pettis property if and onlyif X does and an analogous result is shown when X is the dualspace of a C^*-algebra. One consequence is that both Dunford–Pettisproperties coincide on all real forms of C^*-algebras.     

Rank of Elliptic Curves over almost Separably Closed Fields

Let E be an elliptic curve over a finitely generated infinitefield K. Let K^s denote a separable closure of K, an elementof the Galois group G_K=Gal(K^s/K), and K^s() the invariant subfieldof K^s. If the characteristic of K is not 2 and belongs to asuitable open subgroup of G_K, then E(K^s()) has infinite rank.2000 Mathematics Subject Classification 11G05.     

Volume Estimate Via Total Curvature in Hyperbolic Spaces

Let D Hⁿ(–k²) be a convex compact subset of the hyperbolicspace Hⁿ(–k²) with non-empty interior and smooth boundary.It is shown that the volume of D can be estimated by the totalcurvature of D. More precisely, , where K denotes the Gauss–Kronecker curvature of D andVol(S^n–1) denotes the Euclidean volume of the sphere.2000 Mathematics Subject Classification 53C21.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

On The Curvature of the Real Milnor Fiber

Let C be a germ at O R² of a real analytic plane curve, andC^C its complexification; let C_t B be a fiber of a real smoothdeformation of C in the ball B = B(O,). The following inequalityis proved between the integrals of real curvature k of C_t andthose of Gaussian curvature K of : The sharpness of this inequality is proved in the case whereC is a real irreducible germ. Similar results are proved foran affine algebraic curve C R² of degree d. 2000 MathematicsSubject Classification 14H20, 14H50, 53A04.     

On the Centred Hausdorff Measure

Let v be a measure on a separable metric space. For t, q R,the centred Hausdorff measures µ^h with the gauge functionh(x, r) = r^t(vB(x, r))^q is studied. The dimension defined bythese measures plays an important role in the study of multifractals.It is shown that if v is a doubling measure, then µ^h isequivalent to the usual spherical measure, and thus they definethe same dimension. Moreover, it is shown that this is trueeven without the doubling condition, if q 1 and t 0 or ifq 0. An example in R² is also given to show the surprisingfact that the above assertion is not necessarily true if 0 <q < 1. Another interesting question, which has been askedseveral times about the centred Hausdorff measure, is whetherit is Borel regular. A positive answer is given, using the aboveequivalence for all gauge functions mentioned above.     

Asymptotic Interpolating Sequences in Uniform Algebras

T. Hosokawa, K. Izuchi and D. Zheng recently introduced theconcept of asymptotic interpolating sequences (of type 1) inthe unit disk for H(D). It is shown that these sequences coincidewith sequences that are interpolating for the algebra QA. Alsoa characterization is given of the interpolating sequences oftype 1 for H(D), and asymptotic interpolating sequences in thespectrum of H(D) are studied. The existence of asymptotic interpolatingsequences of type 1 for H() on arbitrary domains is verified.It is shown that any asymptotic interpolating sequence in auniform algebra eventually is interpolating.     

Differentiable Peak-Interpolation on Bounded Domains with Smooth Boundary

Let D be a bounded domain in Cⁿ with C² boundary, and, for 1 k , let A^k(D) be the algebra of functions holomorphic on Dand C^k on a neighbourhood of . It is shown that each peak-interpolation set for A^k(D) is afinite set.     

The Second Dual Algebra of the Measure Algebra of a Compact Group

It is shown that for every compact group G, L¹(G)^ is uniqueand minimal among all the closed subsets I of M(G)** such thatI is a proper (0, M(G)**) algebraic ideal, and such that I issolid with respect to absolute continuity; that is, n L¹(G)^whenever n M(G)** and n << µ L¹(G)^. 1991 MathematicsSubject Classification 43A20, 43A22.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

Finite element error bounds for a curve shrinking with prescribed normal contact to a fixed boundary

We approximate the evolution of a curve subject to motion bycurvature by linear finite elements. The curve evolves insidea given domain and meets orthogonally. We derive optimal boundsfor the error with respect to the L²- and H¹-norms and presentsome computed examples.     

Boundary Accessibility of a Domain Quasiconformally Equivalent to a Ball

A boundary point of a domain D in Rⁿ is said to be broadly accessibleif it ‘almost lies’ on the boundary of a round ballcontained in D. If f is a quasiconformal mapping of the unitball Bⁿ onto D, then it is shown that broadly accessible boundarypoints on D correspond under f to a set of full measure on Bⁿ.2000 Mathematics Subject Classification 30C65.     

A Simple Proof for a Spectral Factorization Theorem

It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H² (D) and (ii) Z^–1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary.     

Self-Adjoint Operators and Cones

Suppose that K is a cone in a real Hilbert space with K = {0},and that A: is a self-adjoint operator which maps K intoitself. If ||A|| is an eigenvalue of A, it is shown that ithas an eigenvector in the cone. As a corollary, it follows thatif ||A||ⁿ is an eigenvalue of Aⁿ, then ||A|| is an eigenvalueof A which has an eigenvector in K. The role of the support-boundaryof K in the simplicity of the principal eigenvalue ||A|| isinvestigated. If H is a separable Hilbert space, it is shownthat ||A|| (A); that is, the spectral radius of A lies in thespectrum of A. When A is compact, we obtain a very elementaryproof of the Krein-Rutman Theorem in the self-adjoint case withoutassuming that K = {0}.