Harmonic morphisms on heaven spaces

We prove that any (real or complex) analytic horizontally conformalsubmersion from a three-dimensional conformal manifold (M³,c_M) to a two-dimensional conformal manifold (N², c_N) can be,locally, ‘extended’ to a unique harmonic morphismfrom the (eaven)-space (H⁴, g) of (M³, c_N) to (N², c_N). Moreover,any positive harmonic morphism with two-dimensional fibres from(H⁴, g) is obtained in this way.     

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 Characterization of the Regularity of the Jacobian Determinant in the Framework of Bessel Potential Spaces on Domains

Let 2 m n. The paper gives necessary and sufficient conditionson the parameters s1, s2, ..., sm, p1, p2, ..., pm such thatthe Jacobian determinant extends to a bounded operator fromHs1p1 x Hs2p2 x ... x Hsmpm into S'. Here all spaces are definedon Rn or on domains Rn. In almost all cases the regularity ofthe Jacobian determinant is calculated exactly.     

On the Greatest Prime Divisor of the Sum of Two Squares of Primes

One of the most famous theorems in number theory states thatthere are infinitely many positive prime numbers (namely p =2 and the primes p 1 mod4) that can be represented in the formx²₁+x²₂, where x₁ and x₂ are positive integers. In a recentpaper, Fouvry and Iwaniec [2] have shown that this statementremains valid even if one of the variables, say x₂, is restrictedto prime values only. In the sequel, the letter p, possiblywith an index, is reserved to denote a positive prime number.As p²₁=p²₂ = p is even for p₁, p₂ > 2, it is reasonable toconjecture that the equation p²₁=p²₂ = 2p has an infinity ofsolutions. However, a proof of this statement currently seemsfar beyond reach. As an intermediate step in this direction,one may quantify the problem by asking what can be said aboutlower bounds for the greatest prime divisor, say P(N), of thenumbers p²₁=p²₂, where p₁, p₂ N, as a function of the realparameter N 1. The well-known Chebychev–Hooley methodcombined with the Barban–Davenport–Halberstam theoremalmost immediately leads to the bound P(N) N^1–, if N N_o(); here, denotes some arbitrarily small fixed positivereal number. The first estimate going beyond the exponent 1has been achieved recently by Dartyge [1, Théorème1], who showed that P(N) N^10/9–. Note that Dartyge'sproof provides the more general result that for any irreduciblebinary form f of degree d 2 with integer coefficients the greatestprime divisor of the numbers |f(p₁, p₂)|, p₁, p₂ N, exceedsN^_d–, where _d = 2 – 8/(d = 7). We in particular wantto point out that Dartyge does not make use of the specificfeatures provided by the form x²₁+x²₂. By taking advantage ofsome special properties of this binary form, we are able toimprove upon the exponent ₂ = 10/9 considerably.     

Finite Time Equipartition for Second-order Hyperbolic Systems

A sufficient condition for equipartition of energy for secondorder hyperbolic systems in three space variables is given.The condition states that the system should evolve in such away that the time derivative of a solution of the form (u₁,0)^T is connected with the space derivatives of a solution ofthe form (0, u₂)^T and the time derivative of (0, u₂)^T is connectedwith the space derivatives of (u₁, 0)^T.     

On the Discreteness and Convergence in n-Dimensional Mobius Groups

Throughout this paper, we adopt the same notations as in [1,6, 8] such as the Möbius group M(Rⁿ), the Clifford algebraC_n–1, the Clifford matrix group SL(2, _n), the Cliffordnorm of ||A||=(|a|²+|b|²+|c|²+|d|²) (1) and the Clifford metric of SL(2, _n) or of the Möbius groupM(Rⁿ) d(A₁,A₂)=||A₁–A₂||(|a₁–a₂|²+|b₁–b₂|²+|c₁–c₂|²+|d₁–d₂|²)(2) where |·| is the norm of a Clifford number and represents f_i M(), i = 1,2, and so on. In addition, we adopt some notions in [6, 12]:the elementary group, the uniformly bounded torsion, and soon. For example, the definition of the uniformly bounded torsionis as follows.     

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.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

The Natural Morphisms between Toeplitz Algebras on Discrete Groups

Let G be a discrete group and (G, G₊) be a quasi-ordered group.Set G₊(G₊)^–1 and G₁= (G₊\){e}. Let F^G₁(G) andF^G₊(G) be the corresponding Toeplitz algebras. In the paper,a necessary and sufficient condition for a representation ofF^G₊(G) to be faithful is given. It is proved that when G isabelian, there exists a natural C*-algebra morphism from F^G₁(G)to F^G₊(G). As an application, it is shown that when G = Z² andG₊ = Z₊ x Z, the K-groups K₀(F^G₁(G)) Z², K₁(F^G₁(G)) Z andall Fredholm operators in F^G₁(G) are of index zero.     

Linear Independence of Hecke Operators in the Homology of X0(N)

In Merel's recent proof [7] of the uniform boundedness conjecturefor the torsion of elliptic curves over number fields, a keystep is to show that for sufficiently large primes N, the Heckeoperators T₁, T₂, ..., T_D are linearly independent in theiractions on the cycle e from 0 to i in H₁(X₀(N) (C), Q). In particular,he shows independence when max(D⁸, 400D⁴) < N/(log N)⁴. Inthis paper we use analytic techniques to show that one can chooseD considerably larger than this, provided that N is large.     

On the Number of Singularities in Generic Deformations of Map Germs

Let f:Cⁿ, 0C^p, 0 be a K-finite map germ, and let i=(i₁, ...,i_k) be a Boardman symbol such that ⁱ has codimension n in thecorresponding jet space J^k(n, p). When its iterated successorshave codimension larger than n, the paper gives a list of situationsin which the number of ⁱ points that appear in a generic deformationof f can be computed algebraically by means of Jacobian idealsof f. This list can be summarised in the following way: f musthave rank n–i₁ and, in addition, in the case p=6, f mustbe a singularity of type ^i₁,i₂.     

On a Parabolic Symmetry of Finite Coxeter Groups

Let (W, S be a finite Coxeter system, and let JS. Any wW hasa unique factorization w = w^J w_J, where w_j belongs to the parabolicsubgroup W_J generated by J, and w^J is of minimal length in thecoset wW_J. It is shown here that w_I = w^J if and only if w^I =w_J, for all I, J S. Furthermore, a similar symmetry propertyin arbitrary (W_I, W_J-double cosets is conjectured, which linksthis result to the Solomon descent algebra of W. 2000 MathematicsSubject Classification 20F55.     

Exponential stability and maximal attractors for a one-dimensional nonlinear thermoviscoelasticity

This paper is concerned with the global existence, exponentialstability of solutions and associated nonlinear C₀-semigroupas well as the existence of maximal attractors in Hⁱ (i = 1,2, 4) for a nonlinear one-dimensional thermoviscoelasticitydescribing a kind of solid-like material. Some new ideas andmore delicated estimates are employed to prove the global existenceand exponential stability of solutions. The important featurefor the existence of maximal attractors in Hⁱ₊ (i = 1, 2, 4)is that the metric spaces H¹₊, H²₊ and H⁴₊ we work with arethree incomplete metric spaces, as can be seen from the physicalconstraints, i.e. > 0 and u > 0, with and u being absolutetemperature and deformation gradient (strain). For any positiveparameters ₁, ₂, ..., ₅ verifying some conditions, a sequenceof closed subspaces Hⁱ Hⁱ+ (i = 1, 2, 4) is found, and theexistence of maximal attractors in Hⁱ (i = 1, 2, 4) is established.     

On the Algorithmic Complexity of Minkowski's Reconstruction Theorem

In 1903 Minkowski showed that, given pairwise different unitvectors µ₁, ..., µ_m in Euclidean n-space Rⁿ whichspan Rⁿ, and positive reals µ₁, ..., µ_m such that^m_i=1µ_iµ_i = 0, there exists a polytope P in Rⁿ, uniqueup to translation, with outer unit facet normals µ₁, ...,µ_m and corresponding facet volumes µ₁, ..., µ_m.This paper deals with the computational complexity of the underlyingreconstruction problem, to determine a presentation of P asthe intersection of its facet halfspaces. After a natural reformulationthat reflects the fact that the binary Turing-machine modelof computation is employed, it is shown that this reconstructionproblem can be solved in polynomial time when the dimensionis fixed but is #P-hard when the dimension is part of the input. The problem of ‘Minkowski reconstruction’ has variousapplications in image processing, and the underlying data structureis relevant for other algorithmic questions in computationalconvexity.     

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.     

On the L-Function of the Curves y2 = x5 + A

Let C_A/Q be the curve y² = x⁵ + A, and let L(s, J_A) denote theL-series of its Jacobian. Under the assumption that the signin the functional equation for L(s, J_A) is +1, the criticalvalue L(1, J_A) is evaluated in terms of the value of a thetaseries for depending on Aat a complex multiplication point coming from Q(₅).     

On the Local and Superlinear Convergence of Quasi-Newton Methods

This paper presents a local convergence analysis for severalwell-known quasi-Newton methods when used, without line searches,in an iteration of the form to solve for x^* such that Fx^* = 0. The basic idea behind theproofs is that under certain reasonable conditions on x_o, Fand x_o, the errors in the sequence of approximations {H_k} toF'(x^*)^–1 can be shown to be of bounded deterioration inthat these errors, while not ensured to decrease, can increaseonly in a controlled way. Despite the fact that H_k is not shownto approach F'(x^*)^–1, the methods considered, includingthose based on the single-rank Broyden and double-rank Davidon-Fletcher-Powellformulae, generate locally Q-superlinearly convergent sequences{x_k}.     

Symmetric Groups as Products of Abelian Subgroups

A proof is given that the full symmetric group over any infiniteset is the product of finitely many Abelian subgroups. In fact,289 subgroups suffice. Sharp bounds are also obtained on theminimal number k, such that the finite symmetric group S_n isthe product of k Abelian subgroups. Using this, S_n is provedto be the product of 72n^1/2(log n)^3/2 cyclic subgroups. 2000Mathematics Subject Classification 20B30, 20D40.     

A Krylov subspace algorithm for multiquadric interpolation in many dimensions

We consider the version of multiquadric interpolation wherethe interpolation conditions are the equations s(x_i) = f_i, i= 1,2,..., n, and where the interpolant has the form s(x) =_j=1ⁿ _j (||x – x_j ||² + c²)^1/2 + x R^d, subject to theconstraint _j=1ⁿ _j = 0. The points x_i R^d, the right-hand sidesf_i, i = 1,2,...,n, and the constant c are data. The equationsand the constraint define the parameters _j, j = 1,2,...,n, and. The resultant approximation s f is useful in many applications,but the calculation of the parameters by direct methods requiresO (n³) operations, and n may be large. Therefore iterative proceduresfor this calculation have been studied at Cambridge since 1993,the main task of each iteration being the computation of s(x_i),i = 1,2,...,n, for trial values of the required parameters.These procedures are based on approximations to Lagrange functions,and often they perform very well. For example, ten iterationsusually provide enough accuracy in the case d = 2 and c = 0,for general positions of the data points, but the efficiencydeteriorates if d and c are increased. Convergence can be guaranteedby the inclusion of a Krylov subspace technique that employsthe native semi-norm of multiquadric functions. An algorithmof this kind is specified, its convergence is proved, and carefulattention is given to the choice of the operator that definesthe Krylov subspace, which is analogous to pre-conditioningin the conjugate gradient method. Finally, some numerical resultsare presented and discussed, for values of d and n from theintervals [2,40] and [200,10 000], respectively.     

A Strong Law for the Largest Nearest-Neighbour Link between Random Points

Suppose that X₁, X₂, X₃, ... are independent random points inR^d with common density f, having compact support with smoothboundary , with f| continuous. Let Rⁿ_{i, k} denote the distancefrom X_i to its kth nearest neighbour amongst the first n points,and let M_{n, k} = max_in Rⁿ_{i, k}. Let denote the volume of theunit ball. Then as n , , almost surely If instead the points lie in a compact smooth d-dimensionalRiemannian manifold K, then nM^d_{n, k}/log n (min_Kf)^–1,almost surely.