A Perturbation Solution for a Two-dimensional Annular Electrochemical Machining Problem

Present address: Deparment of Applied Mathematics, University of Western Ontario, London, Canada. A two-dimensional electrochemical machining problem which consistsof an almost circular anode placed inside a circular cathodeis solved using a perturbation procedure. The smoothing of roughmetal parts is studied by solving the resulting perturbationequations numerically.     

Error estimates for the approximate solutions of a nonlinear hyperbolic equation given by finite volume schemes

This paper is devoted to the study of an error estimate of thefinite volume, approximation to the solution u L(R_N x R) ofthe equation u_t + div(Vf(u)) = 0, where v is a vector functiondepending on time and space. A 'h' error estimate for an initialvalue in BV(R^N) is shown for a large variety of finite volumemonotonous flux schemes, with an explicit or implicit time discretization.For this purpose, the error estimate is given for the generalsetting of approximate entropy solutions, where the error isexpressed in terms of measures in R^N and R^N x R. The study ofthe implicit schemes involves the study of the existence anduniqueness of the approximate solution. The cases where an 'h'error estimate can be achieved are also discussed.     

A stabilized finite-element method for the Stokes problem including element and edge residuals

^** Email: raraya{at}ing-mat.udec.cl^*** Corresponding author. Email: gbarrene{at}ing-mat.udec.cl^**** Email: valentin{at}lncc.br A new stabilized finite-element method is presented for theStokes problem. The method is of a Douglas–Wang type,and includes a positive jump term controlling the residual ofthe Cauchy stress tensor on the internal edges of the triangulation.A priori error estimates are obtained in the natural norms ofthe unknowns and an a posteriori error estimator is proposed,analysed and tested through numerical experiments.     

An LMI-based approach for robust stabilization of time delay systems containing saturating actuators

Email: elhaous_fati{at}yahoo.fr Corresponding author. Email: elh_tissir{at}yahoo.fr Received on September 8, 2005; Accepted on July 24, 2006 This paper deals with the problem of robust stabilization foruncertain systems with input saturation and time delay in thestate. The parameter uncertainties are time-varying and unknownbut are norm bounded. Sufficient conditions obtained via a linearmatrix inequality formulation are stated to guarantee the localstabilization. The method of synthesis consists in determiningsimultaneously a state feedback control law and an associateddomain of safe admissible states for which the stability ofthe closed-loop system is guaranteed when control saturationseffectively occur. Numerical examples are used to demonstratethe effectiveness of the proposed design technique.     

ON MINIMAL SURFACES WHOSE GAUSS MAP IS INVARIANT BY A HOLOMORPHIC FOLIATION

We study real minimal surfaces : M^{2 4} under the hypothesisthat the holomorphic Gauss map of the immersion is invariantby a holomorphic foliation with singularities. We give a sortof Huber–Osserman theorem regarding the algebraicity ofthe holomorphic Gauss map.     

The symplectic ideal and a double centraliser theorem

We interpret a result of Oehms as a statement about the symplecticideal. We use this result to prove a double centraliser theoremfor the symplectic group acting on , where V is the natural module for the symplectic group.This result was obtained in characteristic zero by Weyl. Furthermore,we use this to extend to arbitrary connected reductive groupsG with simply connected derived group the earlier result ofthe author that the algebra K[G] of infinitesimal invariantsin the algebra of regular functions on G is a unique factorisationdomain.     

A calibration algorithm for simulation-based pricing models

Don L. McLeish Department of Statistics and Actuarial Science, University of Waterloo, Waterloo,Ontario N2L 3G1, Canada Corresponding author. Email: mreesor{at}uwo.ca Email: dlmcleis{at}math.uwaterloo.ca Received on 20 February 2007. Accepted on 15 March 2007. Derivative pricing models require calibration to market conditionsin order to determine quantities such as hedging positions andthe prices of other instruments. For stochastic models and/orcomplex derivatives whose prices are not of an analytic form,prices must be computed via simulation and the calibration ismore difficult. A method to facilitate the calibration of simulation-basedpricing models is proposed. The algorithm uses a statisticallydesigned experiment to select the points at which simulationsare performed. The method is quite general as it is independentof the stochastic model for the underlying and allows for differentobjective functions that can incorporate information such asopen interest and volume. Furthermore, market prices from European-and/or American-style derivatives covering a range of strikeprices and maturities can be handled by this technique. Examplesshow the procedure is successful at calibrating well-known assetpricing models to both simulated and market data.     

The Baire Method for the Prescribed Singular Values Problem

The paper investigates the vectorial Dirichlet problem definedby S_j( u(x))=1,xnO; a.e., j=1,...,n, u(x)=\(x),\,x\in|O. end{cases}Here O is an open bounded subset of Rⁿ with boundary |O, and_j(A) (j=1,...,n) denote the singular values of the gradient u(x). The existence of solutions is established under one ofthe following assumptions: : O – Rⁿ is continuous on Oand locally contractive on O, or : |O – Rⁿ is contractiveon |O. This extends a result due to Dacorogna and Marcellini.The approach is based on the Baire category method developedearlier by the authors.     

Global smooth fibrations of R3 by oriented lines

A smooth fibration of ³ by oriented lines is given by a smoothunit vector field V on ³ all of whose integral curves are straightlines. Such a fibration is said to be nondegenerate if dV vanishesonly in the direction of V. Let be the space of oriented linesof ³ endowed with its canonical pseudo-Riemannian neutral metric.We characterize the nondegenerate smooth fibrations of ³ byoriented lines as the closed (in the relative topology) definiteconnected surfaces in . In particular, local conditions on imply the existence of a global fibration. Besides, for anysuch fibration the base space is diffeomorphic to the open discand the directions of the fibers form an open convex set ofthe two-sphere. We characterize as well, in a similar way, thesmooth (possibly degenerate) fibrations.     

The effects of an electric field on an autocatalytic ionic reaction in a system with high ionic strength

The effects of an applied electric field on an ionic autocatalyticreaction with a quadratic rate law are considered, where thereacting species, A⁺ and B⁺, are present in a system which alsoincludes non-reacting species C^- and D⁺. The conditions areestablished under which the general terms which describe theelectric field effects in the reaction-diffusion equations canbe simplified to those used in previous studies, where theseeffects are modelled by linear advection terms. The resultingequations are then studied in detail by first obtaining conditionsfor the existence of travelling waves of permanent form. Thisdiscussion shows that _B, the ratio of the diffusion coefficientsof B⁺ and A⁺, is a critical parameter, with different formsof behaviour arising for _B < 1 and _B > 1. This analysisis augmented by obtaining solutions valid for large times andlarge values of (the dirnensionless applied field). Numericalsolutions of initial-value problems are obtained for a rangeof values of and _B, guided by and interpreted through the analysispreviously obtained. These numerical integrations show the formationof reaction fronts, with the possibility of greatly increasedreaction rates caused by the applied electric field, as wellas propagating electrophoretic fronts in B⁺ being formed incases where a reaction front is also initiated. There is alsothe possibility of separate electrophoretic fronts in A⁺ andB⁺ being formed, which become increasingly separated as timeincreases with the reaction being completely inhibited.     

Sharp threshold of global existence for non-linear Gross-Pitaevskii equation in RN

^** Email: chenguanggan{at}hotmail.com This paper is concerned with the non-linear Gross–Pitaevskiiequation which describes the attractive Bose–Einsteincondensate under a magnetic trap. By an intricate variationalargument we derive out a sharp threshold of blowing up and globalexistence by applying the potential well argument and the concavitymethod. Furthermore, we answer the question: How small are theinitial data, the global solutions of the Cauchy problem ofthe equation exist for [graphic: see PDF]     

A Paley-Wiener theorem for Bergman spaces with application to invariant subspaces

An analogue of the Paley–Wiener theorem is developed forweighted Bergman spaces of analytic functions in the upper half-plane.The result is applied to show that the invariant subspaces ofthe shift operator on the standard Bergman space of the unitdisk can be identified with those of a convolution Volterraoperator on the space L²(⁺, (1/t)dt).     

Multiple criteria models for evaluation of competitive bids

Email: slliu{at}mail.east.net.cnAuthor for correspondence Email: mskklai{at}cityu.edu.hkEmail: sywang{at}iss02.iss.ac.cn In this paper, two general bidding systems are proposed to aidan owner to select one contractor based on multiple-attributedecision making models. They are named the general multiple-attributelower-bidder system and the general multiple-attribute average-bid system and are an extension (and refreshment) of the multiparameterbidding system and the average-bid method, respectively. Thepreference of the owner over the criteria (or attributes) relevantto construction work award is incorporated into the two newsystems if necessary and required by the owner. The values ofthe attributes (excluding the cost) required to be determinedby the owner in the multiparameter bidding system are not necessarilydetermined in the general multiple-attribute lower-bidder system.This reduces the burden to the owner. The two new bidding systemsare demonstrated via an example, and are compared with the multiparameterbidding system and the average-bid method. The comparisons showthat the preference of the owner in regard to the attributesrelevant to selection of contractors has a significant effecton bid evaluation. The two new systems preserve the form ofthe competitive bidding concept and can easily be applied topractice.     

Asymptotic analysis for a particle transport problem in a moving medium

In this paper we discuss the derivation of the diffusion theoryfor a linear particle transport in a moving gas. We performthe asymptotic analysis for a one-dimensional linear BGK modelproblem with a shifting Maxwellian which describes the timeevolution of the spatially dependent electron distribution functionin a moving weakly ionized host medium. The modified (compressed)Chapman-Enskog expansion procedure is applied to find the asymptoticsolution for small mean free path , showing that the differencebetween the exact and asymptotic solutions is of order ², uniformlyin time for arbitrary initial data.     

On a Comparison Principle and the Critical Fujita Exponents for Solutions of Semilinear Parabolic Inequalities

The paper examines blow-up phenomena for the inequality u_t–Lu–|u|^q–1u_t–L–||^q–1 (*) in the half-space x Rⁿ, n 1, where L is a linear second-order partial differential operatorin divergence form. The paper studies weak solutions of (*) that belong only locallyto the corresponding Sobolev spaces in the half-space x Rⁿ. It also requires no conditionsfor the behavior of solutions of (*) on the hyperplane t = 0. The existence of critical blow-up exponents is obtained forsolutions of (*) as a special case of a comparison principlefor the corresponding solutions of (*). For example, the well-knownFujita result is a consequence of the comparison principle. The approach developed in the paper is directly applicable tothe study of analogous problems involving nonlinear differentialoperators. Its elliptic analogue has been recently developedby the authors.     

Holomorphic vector fields transverse to polydiscs

In this article, we study holomorphic vector fields transverseto the boundary of a polydisc in ⁿ, n 3. We prove that, undera suitable hypothesis of transversality with the boundary ofthe polydisc, the foliation is the pull-back of a linear hyperbolicfoliation via a locally injective holomorphic map. This is then 3 version for one-dimensional foliations of a previous resultproved for n = 2 by Brunella and Sad and for codimension-onefoliations by Ito and Scárdua.     

Long Snakes in Powers of the Complete Graph with an Odd Number of Vertices

In [5] Abbott and Katchalski ask if there exists a constantc < 0 such that for every d 2 there is a snake (cycle withoutchords) of length at least c3^d in the product of d copies ofthe complete graph K₃. We show that the answer to the abovequestion is positive, and that in general for any odd integern there is a constant c_n such that for every d 2 there is asnake of length at least c_n n^d in the product of d copies ofthe complete graph K_n.     

ON THE LEAST n WITH {chi}(n) != 1

Suppose that a > 1. By using a method of Linnik employinghis Large Sieve one may derive the following result. The numberof primitive Dirichlet characters with conductor Q such that(n) = 1 for all n (log Q)^a is O(Q^2/a+). We improve the exponent2/a for a > 2 by using a refined version by Heath-Brown ofthe Halasz–Montgomery ‘Large Values method’.     

On optimal convergence rates for higher-order Navier-Stokes approximations. I. Error estimates for the spatial discretization

^** Email: bause{at}am.uni-erlangen.de Due to the increasing use of higher-order methods in computationalfluid dynamics, the question of optimal approximability of theNavier–Stokes equations under realistic assumptions onthe data has become important. It is well known that the regularitycustomarily hypothesized in the error analysis for parabolicproblems cannot be assumed for the Navier–Stokes equations,as it depends on non-local compatibility conditions for thedata at time t = 0, which cannot be verified in practice. Takinginto account this loss of regularity at t = 0, improved convergenceof the order (min{h^(5/2)–,h³/t^(1/4)+}), for any >0, is shown locally in time for the spatial discretization ofthe velocity field by (non-)conforming finite elements of third-orderapproximability properties. The error estimate itself is provedby energy methods, but it is based on sharp a priori estimatesfor the Navier–Stokes solution in fractional-order spacesthat are derived by semigroup methods and complex interpolationtheory and reflect the optimal regularity of the solution ast 0.     

Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

Wolfgang Hackbusch ^{In this paper, we discuss the application of hierarchical matrixtechniques to the solution of Helmholtz problems with largewave number in 2D. We consider the Brakhage–Werner integralformulation of the problem discretized by the Galerkin boundary-elementmethod. The dense n x n Galerkin matrix arising from this approachis represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats.A well-known multipole expansion is used to construct the 2-matrix. We present a new approach to dealingwith the numerical instability problems of this expansion: theparts of the matrix that can cause problems are approximatedin a stable way by an -matrix. Algebraic recompression methods are used to reducethe storage and the complexity of arithmetical operations ofthe -matrix.Further, an approximate LU decomposition of such a recompressed-matrix is aneffective preconditioner. We prove that the construction ofthe matrices as well as the matrix-vector product can be performedin almost linear time in the number of unknowns. Numerical experimentsfor scattering problems in 2D are presented, where the linearsystems are solved by a preconditioned iterative method.}