Torsional Rigidity of Minimal Submanifolds

We prove explicit upper bounds for the torsional rigidity ofextrinsic domains of minimal submanifolds P^m in ambient Riemannianmanifolds Nⁿ with a pole p. The upper bounds are given in termsof the torsional rigidities of corresponding Schwarz symmetrizationsof the domains in warped product model spaces. Our main resultsare obtained via previously established isoperimetric inequalities,which are here extended to hold for this more general settingbased on warped product comparison spaces. We also characterizethe geometry of those situations in which the upper bounds forthe torsional rigidity are actually attained and give conditionsunder which the geometric average of the stochastic mean exittime for Brownian motion at infinity is finite. 2000 MathematicsSubject Classification 53C42, 58J65, 35J25, 60J65.     

A p-Adic Quasi-Quadratic Time Point Counting Algorithm

In this article, we give an algorithm for the computation ofthe number of rational points on the Jacobian variety of a genericordinary hyperelliptic curve defined over a finite field of cardinality q with time complexity O(n^2+o(1))and space complexity O(n²), where n = log(q). In the lattercomplexity estimate the genus and the characteristic are assumedas fixed. Our algorithm forms a generalization of both, theAGM algorithm of J.-F. Mestre and the canonical lifting methodof T. Satoh. We canonically lift a certain arithmetic invariantof the Jacobian of the hyperelliptic curve in terms of thetaconstants. The theta null values are computed with respect toa semi-canonical theta structure of level 2p where >0 isan integer and . Theresults of this paper suggest a global positive answer to thequestion whether there exists a quasi-quadratic time algorithmfor the computation of the number of rational points on a genericordinary abelian variety defined over a finite field.     

Systems of cubic forms

We discuss the existence of rational and p-adic zeros of systemsof cubic forms. In particular, we prove that for p2 any systemof r cubic forms over Q_p in more than 125r³+705r²+210r variablesadmits a non-trivial p-adic zero, and that any system of r rationalcubic forms in more than O(r⁴ m⁶+r⁶ m⁵) variables admits a rationallinear space of zeros of dimension at least m.     

Canonical curves in P3

Let Hilb^6t–3(P³) be the Hilbert scheme of closed 1-dimensionalsubschemes of degree 6 and arithmetic genus 4 in P³. Let H bethe component of Hilb^6t–3(P³) whose generic point correspondsto a canonical curve, that is, a complete intersection of aquadric and a cubic surface in P³. Let F be the vector spaceof linear forms in the variables z₁, z₂, z₃, z₄. Denote by F_dthe vector space of homogeneous forms of degree d. Set X = (f₂,f₃)where f₂ P(F₂) is a quadric surface, and f₃ P(F₃/f₂ ·F) is a cubic modulo f₂. Wehave a rational map, : X ... Hdefined by (f₂,f₃) f₂ f₃. It fails to be regular along thelocus where f₂ and f₃ acquire a common linear component. Ourmain result gives an explicit resolution of the indeterminaciesof as well as of the singularities of H. 2000 Mathematical Subject Classification: 14C05, 14N05, 14N10,14N15.     

Double Hilbert transform along real-analytic surfaces in Rd+2

We give a necessary and sufficient condition for the doubleHilbert transform on ^d+2 to be bounded on L^p, 1 < p <. This generalizes a result of Carbery, Wainger and Wright [DukeMath. J. 101 (2000) 499–513] for d = 1.     

Two Interpolatory Cubic Splines

The third order and the uniform cubic spline are defined andshown to have O(h³) and O(h⁴) convergence respectively whenused for interpolation.     

Twistors and 3-symmetric spaces

We describe complex twistor spaces over inner 3-symmetric spacesG/H, such that H acts transitively on the fibre. As in the symmetriccase, the complex twistor spaces are flag manifolds G/K, whereK is the centralizer of a torus in G. Moreover, they carry analmost complex structure defined using the horizontal distributionof the normal connection on G/H that coincides with the complexstructure associated to a parabolic subgroup PG if it is integrable.Conversely, starting from a complex flag manifold G/P, thereexists a natural fibration with complex fibres on a 3-symmetricspace, called fibration of degree 3.     

Linear Systems and Quotients of Projective Space

A complete characterization of the categorical quotients of(P¹)ⁿ by the diagonal action of SL(2, C) with respect to anypolarization is given by M. Polito, in ‘SL(2, C)-quotientsde (P¹)ⁿ’, C. R. Acad. Sci. Paris Sér. I 321 (1995)1577–1582. In this paper, these categorical quotientsare obtained by certain linear systems on P^n–3, dependingon the given polarization. 2000 Mathematics Subject Classification14L24, 14L30     

On approximation theorems for controllability of non-linear parabolic problems

^** Email: anil{at}math.iitb.ac.in^*** Email: mcj{at}math.iitb.ac.in^**** Email: akp{at}math.iitb.ac.in In this paper, we consider the following control system governedby the non-linear parabolic differential equation of the form: [graphic: see PDF] where A is a linear operator with dense domain and f(t, y)is a non-linear function. We have proved that under Lipschitzcontinuity assumption on the non-linear function f(t, y), theset of admissible controls is non-empty. The optimal pair (u^*,y^*) is then obtained as the limit of the optimal pair sequence{(u_n^*, y_n^*)}, where u_n^* is a minimizer of the unconstrainedproblem involving a penalty function arising from the controllabilityconstraint and y_n^* is the solution of the parabolic non-linearsystem defined above. Subsequently, we give approximation theoremswhich guarantee the convergence of the numerical schemes tooptimal pair sequence. We also present numerical experimentwhich shows the applicability of our result.     

High-accuracy P-stable Methods for y' = f(t, y)

We obtain a one-parameter family of sixth-order P-stable methodsfor the numerical integration of periodic or near-periodic differentialequations that are defined by initial-value problems of theform: y^" = f(t, y), y(t₀)= y₀, y^'(t₀)= y^’₀. Our P-stablemethods are symmetric and involve three function evaluationsper step (periteration, in case f(t, y) is non-linear in y).For non-linear problems, starting values for the solution ofthe implicit equations by modified Newton's method are suggestedand illustrated by an example.     

The Banach Space B(l2) is Primary

We prove that if A is an injective operator system on l² andP is a completely bounded projection on A then either PA or(I–P)A is completely boundedly isomorphic to A. We alsoprove that if B(l²) is linearly homeomorphic to X Y then eitherX or Y is linearly homeomorphic to B(l²). Current address: Merton College, Oxford 0X1 4JD     

Isomorphisms between Semisimple Weighted Measure Algebras

It is shown that, if is an isomorphism between semisimple weightedmeasure algebras M(w₁) and M(w₂), then maps L¹(R⁺, w₁) ontoL¹(R⁺, w₂). This is used to describe all the automorphisms ofM(R⁺, w). A necessary and sufficient condition is given forM(w₁) and M(w₂) to be isomorphic.     

The density of rational points on non-singular hypersurfaces, II

For any integers d,n 2, let X Pⁿ be a non-singular hypersurfaceof degree d that is defined over the rational numbers. The mainresult in this paper is a proof that the number of rationalpoints on X which have height at most B is O(B^{n – 1} +), for any > 0. The implied constant in this estimate dependsat most upon d, and n. 2000 Mathematics Subject Classification11D45 (primary), 11G35, 14G05 (secondary).     

Finely Harmonic Functions need not be Quasi-Analytic

If a function f is harmonic on a connected open set R^d andis constant in a neighbourhood of one point in then it is identicallyconstant. We give an example of a non-constant finely harmonicfunction defined on E={zC||z|<2}\p which is identically zeroon |z|1. The exceptional set P is of capacity zero so E is finelyopen and finely connected. This example therefore shows thatfinely harmonic functions are not globally determined by theirlocal behaviour.     

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P²ⁿ⁺¹ be a two-cell complex which is formed by attachinga (2n + 1)-cell to a 2m-sphere by a suspension map. We constructa universal space U for P²ⁿ⁺¹ in the category of homotopy associative,homotopy commutative H-spaces. By universal, we mean that Uis homotopy associative, homotopy commutative and has the propertythat any map f: P²ⁿ⁺¹ Y to a homotopy associative, homotopycommutative H-space Y extends to a uniquely determined H-map: U Y. We then prove upper and lowerbounds of the H-homotopy exponent of U. In the case of a modp^r, Moore space U is the homotopy fibre S²ⁿ⁺¹{p^r} of the p^r-powermap on S²ⁿ⁺¹, and we reproduce Neisendorfer's result that S²ⁿ⁺¹{p^r}is homotopy associative, homotopy commutative and that the p^r-powermap on S²ⁿ⁺¹{p^r} is null homotopic.     

Free Lie Algebras and Adams Operations

Let G be a group and K a field. For any finite-dimensional KG-moduleV and any positive integer n, let L^n(V) denote the nth homogeneouscomponent of the free Lie K-algebra generated by (a basis of)V. Then L^n(V) can be considered as a KG-module, called the nthLie power of V. The paper is concerned with identifying thismodule up to isomorphism. A simple formula is obtained whichexpresses L^n(V) in terms of certain linear functions on theGreen ring. When n is not divisible by the characteristic ofK these linear functions are Adams operations. Some resultsare also obtained which clarify the relationship between Adamsoperations defined by means of exterior powers and symmetricpowers and operations introduced by Benson. Some of these resultsare put into a more general setting in an appendix by StephenDonkin.     

A Geometric Characterization of the Perfect Ree-Tits Generalized Octagons

Given a generalized octagon which satisfies certain geometricconditions, we show that it must be isomorphic to the Ree–Titsoctagon arisingnaturally from a Ree group of type ²F₄ over aperfect field of characteristic 2, in which the Frobenius automorphismhas a square root. 1991 Mathematics Subject Classification:51E12.     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K overa number field K satisfying the property that there exist P₁,P₂ K[t] such that the twists and are of positiverank over K(t). As a consequence of this result on twists, weshow that for those elliptic curves E/K, and for each , the rank of E over the fixed field (K^ab) under is infinite, where K^ab is the maximal abelian extension ofK.