Linear Dependence Relations Connecting Equal Interval Nth Degree Splines and Their Derivatives

Present address: The Polytechnic of the South Bank, Borough Road, London, S.E.1 England. The linear dependence of the values of Nth degree spline andits pth derivatives at N successive equally spaced knots isshown and the constants associated with this linear dependencecalculated. A recurrence relation which enables the constants to be foundfor any N is also given. The results are applied to equal intervalinterpolation.     

Qualitative dynamics, root-loci and asymptotic structure of universal adaptive stabilizers

A problem of universal adaptive stabilization in Rⁿ is approachedin a qualitative manner to relate the form of the trajectoriesin the extended state space Rⁿ⁺¹ to the root locus of the associatedfixed-parameter linear system. Relationships are derived betweenthe values of the limit gain and the initial conditions. Numericalstudies are used to illustrate and support these ideas by computationof generic trajectories and attempted computation of certainnon-generic possibilities. The implications of the study formore general dynamic situations are outlined. These authors are also with the Department of Mathematics,University of Exeter, EX4 4QE.     

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.     

Optimal boundary feedback stabilization of a string with moving boundary

Email: gugat{at}am.uni-erlangen.de Received on April 30, 2006; We consider a finite string that is fixed at one end and subjectto a feedback control at the other end which is allowed to move.We show that the behaviour is similar to the situation whereboth ends are fixed: As long as the movement is not too fast,the energy decays exponentially and for a certain parameterin the feedback law it vanishes in finite time. We considermovements of the boundary that are continuously differentiablewith a derivative whose absolute value is smaller than the wavespeed. We solve a problem of worst-case optimal feedback control,where the parameter in the feedback law is chosen such thatthe worst-case L^p-norm of the space derivative at the fixedend of the string is minimized (p [1, )). We consider the worstcase both with respect to the initial conditions and with respectto the boundary movement. It turns out that the parameter forwhich the energy vanishes in finite time is optimal in thissense for all p.     

Computation and parametrization of periodic and connecting orbits

Email: g.moore{at}ma.ic.ac.uk or na.gmoore New algorithms for the computation of periodic and connectingorbits of autonomous differential equations are developed. Previousmethods have parametrized these curves by the independent timevariable, but our procedure is based on the canonical arclengthparametrization.     

The Convergence of Operator Approximations at Turning Points

Present address: Department of Mathematics, University of Reading, Reading RG6 2AX. We consider the convergence of solution curves of approximationsto parameter-dependent operator equations of the form G(, x)= 0. Provided G_x(, x) remains non-singular this problem is cateredfor by a simple extension to standard theory. In this paper,however, attention is concentrated on solution curves throughcertain singular points (⁰, x⁰), and the main result is thatconvergence depends on consistency and stability results forthe linear eigenvalue problem G_x(⁰, x⁰)0 = 0.     

Realistic a Priori and a Posteriori Error Bounds for Computed Eigenvalues

Permanent address: Department of Engineering Mathematics, Cairo University, Giza, Egypt. A priori and a posteriori error bounds are given for the computedeigenpair (, ) of the eigenvalue problem Ax = x, which are shownto be more realistic than some of the available ones. A simplemethod is also presented for computing the backward error. Finallya scaling procedure is explained for reducing the residual error.     

Decentralized dynamic pole assignment with low-order compensators

N. Karcanias Control Engineering Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square London EC1V OHB, UK Email: n.karcanias{at}city.ac.uk Received on June 14, 2006; Accepted on October 2, 2006 The problem of arbitrary pole placement via dynamic decentralizedoutput feedback is studied for minimal systems described bya proper transfer function matrix P(s) R^{m x p}(s) (m = m_i andp = p_i), with McMillan degree n. The family of controllersto be used includes those decentralized controllers with channelswhose ith channel has maximum observability index at most d_i.The method presented here is based on asymptotic linearizationaround a decentralized degenerate compensator of the pole placementmap related to the problem. It is shown that the method worksgenerically when m⁺p > n, where m⁺ = min{d_i(p_i + m_i –1) + m_i}, i = 1, ..., , and the smallest d_i of the compensatorof the ith channel is the integral part of (n – pm_i)/p(p_i+ m_i – 1).     

A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes

This paper considers a finite-element approximation of a Poissonequation in a region with a curved boundary on which a Neumanncondition is prescribed. Piecewise linear and bilinear elementsare used on unfitted meshes with the region of integration beingreplaced by a polygonal approximation. It is shown, despitethe variational crimes, that the rate of convergence is stillorder (h) in the H¹ norm. Numerical examples show that the methodis easy to implement and that the predicted rate of convergenceis obtained. Supported by SERC postdoctoral fellowship RF/5830.     

Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations

An elliptic boundary-value problem on a domain with prescribedDirichlet data on _I is approximated using a finite-elementspace of approximation power h^K in the L² norm. It is shownthat the total flux across _I can be approximated with an errorof O(h^K) when is a curved domain in Rⁿ (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h^2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on _I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.     

Characterization of Distributions through Maximization of {alpha}-Entropies

This paper provides characterizations of the Pareto distributionof the second kind, the Pearson type-VII distribution and thet-distribution. These characterizations have been obtained throughmaximization of -entropies. As 1, characterizations of the exponential,the Laplace, and the normal distributions are obtained. On sabbatical leave from the Department of Statistics, YarmoukUniversity, Irbid, Jordan.     

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.     

New perturbation analyses for the Cholesky factorization

We present new perturbation analyses for the Cholesky factorizationA = R^T R of a symmetric positive definite matrix A. The analysesmore accurately reflect the sensitivity of the problem thanprevious normwise results. The condition numbers here are alteredby any symmetric pivoting used in PAP^T = R^TR, and both numericalresults and an analysis show that the standard method of pivotingis optimal in that it usually leads to a condition number veryclose to its lower limit for any given A. It follows that thecomputed R will probably have greatest accuracy when we usethe standard symmetric pivoting strategy. Initially we give a thorogh analysis to obtain both first-orderand strict normwise perturbation bounds which are as tight aspossible, leading to a definition of an optimal condition numberfor the problem. Then we use this approach to obtain reasonablyclear first-order and strict componentwise perturbation bounds. We complete the work by giving a much simpler normwise analysiswhich provides a somewhat weaker bound, but which allows usto estimate the condition of the problem quite well with anefficient computation. This simpler analysis also shows whythe factorization is often less sensitive than we previouslythought, and adds further insight into why pivoting usuallygives such good results. We derive a useful upper bound on thecondition of the problem when we use pivoting. This research was supported by the Natural Sciences and EngineeringResearch Ciuncil of Canada Grant OGP0009236. This research was supported in part by the US National ScienceFoundation under grant CCR 95503126.     

Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. II: convection-diffusion-type problems

e-mail address: stynes{at}bureau.ucc.ie We consider singularly perturbed high-order elliptic two-pointboundary value problems of convection-diffusion type. Undersuitable hypotheses, the coercivity of the associated bilinearform is proved and a representation result for the solutionsof such problems is given. A family of Galerkin finite-elementmethods based on piecewise polynomial test/trial functions ona Shishkin mesh is constructed and proved to be convergent,uniformly in the perturbation parameter, in energy and Wnorms. Numerical results are presentedfor a second-order problem and fourth-order problems.     

On the Structure of the Moving Finite-element Equations

Address from 1st April 1985, School of Mathematics, Universityof Bristol, University Walk, Bristol BS8 1TW. The morning finite-element method for evolutionary partial differentialequations leads to a coupled non-linear system of ordinary differentialequations in time, with a coefficien matrix A, say, for thetime derivaties, We show for linear elements in any number ofdimensions, A can be written in the form M^TCM, where the matrixC depends solely on the mesh geometry and the matrix M on thegradient of the section, As a simple consequence we show thatA is singular only in the cases (i) element degeneracy () and (ii) collinearity of nodes (M not out of fullrank). We give constructions for the inversion of A in all cases. In one dimension, if A is non-singular, it has a simple explicitinverse. If A is singular we replace it by reduced matrix A^*.It can be shown that every case the spectral radius of the Jacobiiteration matrix ia ?and that A or A^* can be efficiently invertedby conjugate gradient methods. Finally, we discuss the applicability of these arguments tosystem of equations in any number of dimensions.     

Estimation of nonlinear systems without a priori structural information and its application in Hammerstein model identification with dynamic nonlinearities

Email: er-wei-bai{at}uiowa.edu Received on July 30, 2005; Accepted on July 23, 2006 In the paper, we discuss identification of a nonlinear systemwithout structural information and propose two methods, thekernel method and the orthonormal basis method. The convergenceresults are established for both methods without a priori structuralinformation. We then apply the results to identification ofHammerstein models with an unknown dynamic nonlinearity. Itis also shown that identification of the linear part in Hammersteinmodels is possible with no knowledge of the dynamic nonlinearity.     

Shape-preserving interpolation in R3

Received on 7 September 1993. Revised on 20 February 1996. In this paper we develop and test a simple automatic algorithmfor constructing curvature- and torsion-continuous interpolantsin R³ which are shape-preserving in a sense that takes intoaccount the convexity, torsion, coplanarity and collinearityinformation contained in the polygonal line connecting the interpolationpoints. This algorithm exploits the asymptotic properties ofa family of C² polynomial splines of non-uniform degree, whichtend to the above-mentioned polygonal line, as the segment degreestend to infinity. The performance of the algorithm is testedfor a three-dimensional data set, containing coplanar and collineargroups of points as well. E-mail address: kaklis{at}deslab.ntua.gr Tel.: +30-1-7721419Telefax: +30-1-7721408 E-mail address: menelaos{at}sccm.stanford.edu     

The multidimensional positive and nearly conservative Lagrange-Galerkin method. Part II: the use of C1 elements

In this paper we consider a monotonic and conservative variationon the direct Lagrange-Galerkin method due to Priestley, andaddress certain problems with both the original Lagrange-Galerkinmethod and this new method which disappear with the use of higher-orderC¹ basis functions. Here we concentrate on the use of Hermite-cubics,demonstrating their superior stability in the Lagrange- Galerkinmethod, and the adaptations of the recovery techniques neededto ensure monotonicity and near conservation. Current address: Schlumberger GeoQuest, Abingdon Business Park,Oxon OX14 10 UK.     

Necessary conditions for infinite-horizon Volterra optimal control problems with time delay

Email: csfvega{at}dm.uba.ar Received on August 17, 2006; Accepted on September 8, 2007 Necessary conditions are proved for optimal control problemsinvolving an infinite horizon and terminal conditions at infinitywhose states are governed by Volterra integral equations withnon-linear time delay.