Implicit algorithms and their linearization for the transient incompressible Navier-Stokes equations

Email: na.asmith2{at}na-net.ornl.gov Email: na.silvester{at}na-net.ornl.gov The issue of appropriate time discretization methods for theunsteady incompressible Navier-Stokes equations is consideredfrom a practical perspective. Conventional implicit time-steppingalgorithms are not feasible for long-time simulations sincethey inherit the quadratic nonlinearity of the steady-stateequations. As a result, two new linearized versions of the ‘pure’algorithms are analyzed herein. These have similar stabilityproperties and comparable accuracy to the underlying nonlinearmethods.     

Modelling the growth of soil-borne fungi in response to carbon and nitrogen

Email: Angelique.Lamour{at}medew.fyto.wau.nl Growth of soil-borne fungi is poorly described and understood,largely because non-destructive observations on hyphae in soilare difficult to make. Mathematical modelling can help in theunderstanding of fungal growth. Except for a model by Paustian& Schnürer (1987a), fungal growth models do not considercarbon and nitrogen contents of the supplied substrate, althoughthese nutrients have considerable effects on hyphal extensionin soil. We introduce a fungal growth model in relation to soilorganic matter decomposition dealing with the detailed dynamicsof carbon and nitrogen. Substrate with a certain carbon: nitrogenratio is supplied at a constant rate, broken down and then takenup by fungal mycelium. The nutrients are first stored internallyin metabolic pools and then incorporated into structural fungalbiomass. Standard mathematical procedures were used to obtainoverall-steady states of the variables (implicitly from a cubicequation) and the conditions for existence. Numerical computationsfor a wide range of parameter combinations show that at mostone solution for the steady state is biologically meaningful,specified by the conditions for existence. These conditionsspecify a constraint, namely that the ‘energy’ (interms of carbon) invested in breakdown of substrate should beless than the ‘energy’ resulting from breakdownof substrate, leading to a positive carbon balance. The biologicalinterpretation of the conditions for existence is that for growththe ‘energy’ necessary for production of structuralfungal biomass and for maintenance should be less than the mentionedpositive carbon balance in the situation where all substrateis colonized. In summary, the analysis of this complicated fungalgrowth model gave results with a clear biological interpretation.     

Statistical solution to the capacity problem in direct-sequence code-division multiple-access communication systems

Tohru Kohda Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812-8581, Japan Minoru Tabata Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan Email: eshima{at}med.oita-u.ac.jp Received on September 21, 2005; Accepted on May 2, 2006 Capacity in the direct-sequence code-division multiple-access(DS/CDMA) communication system was considered according to thecode acquisition performance with the conventional serial-searchmethod, because code acquisition needs a difficult operation.Since capacity in DS/CDMA systems is defined by the maximumnumber of users that can simultaneously transmit their signalswith the same carrier frequencies, assuring a larger capacityis very important in DS/CDMA systems with respect to the economyof frequency source. This paper reconsiders the capacity problemthrough a statistical approach to code acquisition. First, aDS/CDMA system model is reviewed. Second, properties of thecounting method for code acquisition are discussed. It is provedthat the method can acquire the target user's code under anynumber of interference users that simultaneously transmit theirsignals, and that the method can guarantee a considerable precisionin code acquisition. Third, an observation time necessary forcode acquisition is given from a statistical discussion, andthe original counting method is modified into a simpler one.It is concluded that the capacity of the DS/CDMA communicationsystem can be set by the ‘bit error-rate-based capacityor signal-to-noise ratio-based capacity’, i.e. ‘post-acquisition-basedcriterion’, rather than the ‘acquisition-based capacity’.     

Convergence estimates for the numerical approximation of homoclinic solutions

Received on 21 November 1995. Revised on 12 July 1996. This article is concerned with the numerical computation ofhomoclinic solutions converging to a hyperbolic or semi-hyperbolicequilibrium of a system u = f(u, µ). The approximationis done by replacing the original problem with a boundary valueproblem on a finite interval and introducing an additional phasecondition to make the solution unique. Numerical experimentshave indicated that the parameter µ is much better approximatedthan the homoclinic solution. This was proved in Schecter (1995IMA J. Numer Anal. 15, 23–60) for phase conditions satisfyingan additional ‘niceness’ assumption, which is unfortunatelynot satisfied for the phase condition most commonly used innumerical experiments and which actually suggested the super-convergenceresult. Here, this result is proved for arbitrary phase conditions.Moreover, it is shown that it suffices to approximate the originalboundary value problem to first order when considering semi-hyperbolicequilibria, extending a result of Schecter (1993 SIAM J. NumerAnal. 30, 1155–78). Permanent address: WIAS, MohrenstraBe 39, 10117 Berlin, Germany     

The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Motivated in part by the first author's work [23] on the Weyl-Berryconjecture for the vibrations of ‘fractal drums’(that is, ‘drums with fractal boundary’), M. L.Lapidus and C. Pomerance [31] have studied a direct spectralproblem for the vibrations of ‘fractal strings’(that is, one-dimensional ‘fractal drums’) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the ‘critical interval’0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in the‘midfractal’ case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the ‘partialRiemann hypothesis’ for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis.     

Predicting stock returns and assessing prediction performance

Email: r.d.baker{at}salford.ac.uk Received on 1 November 2006. Accepted on 15 March 2007. We use regression methods to predict the expected monthly returnon stocks and the covariance matrix of returns, the predictorvariables being a company's ‘fundamentals’, suchas dividend yield and the history of previous returns. Predictionsare evaluated out of sample for shares traded on the LondonStock Exchange from 1976 to 2005. We explore and evaluate manymodelling and inferential approaches, including the use of weightedregression, discounted regression, shrinkage of regression coefficientsand the transformation to normality of predictor variables.We also investigate alternative covariance matrix models, suchas a two-index model and a shrinkage model. Using suitable statisticsto enable the out-of-sample performance of competing methodologiesto be compared is crucial, and we develop some new statisticsand a graphical aid for this purpose. What is original in thispaper is an evaluation of many modelling and inferential proceduresfor which conflicting claims have been made in the literatureand the development of new measures of portfolio performance.     

An Algorithm with a Point-Linearized Input-Output Mapping for Integrated System Optimization and Parameter Estimation

An improved algorithm for centralized integrated system optimizationand parameter estimation based on a point-linearized I-O mappingand Newton step in updating is described. Two convergence theoremsunder different assumptions are investigated. Optimality ofthe method and the relationship between the algorithmic fixed-pointset and the optimum solution set of the system are presented.Simulation reveals that, because of the adoption of point-linearizedI-O mapping and a Newton step in updating, the proposed techniqueenjoys two benefits of easier computation and a higher convergencerate. On study leave from the Anhui Institute of Machinery & Electronics,Wuhu, Anhui Province, China On study leave from the Institute of Systems Engineering, Xi’anJiaotong University, Xi’an, China     

Maximin investment problems for discounted and total wealth

Email: nikolaidokuchaev{at}trentu.ca Received on 27 March 2006. Accepted on 14 September 2007. We study an optimal investment problem for a continuous-timeincomplete market model such that the risk-free rate, the appreciationrates and the volatility of the stocks are all random; theyare not necessarily adapted to the driving Brownian motion,and their distributions are unknown, but they are supposed tobe currently observable. The optimal investment problem is statedin ‘maximin’ setting which leads to maximizationof the minimum of expected utility over all distributions ofparameters. We found that the presence of the non-discountedwealth in the performance criterion (in addition to the discountedwealth) implies an additional condition for the saddle pointof the maximin problem: the saddle point must include the minimumof the possible risk-free return. This is different from thecase when the utility depends on the discounted wealth only.Using this result, the maximin problem is reduced to a linearparabolic equation and minimization over two scalar parameters.It is an important development of the results obtained in Dokuchaev(2002, Dynamic Portfolio Strategies: Quantitative Methods andEmpirical Rules for Incomplete Information. Boston: Kluwer;2006, IMA J. Manage. Math., 17, 257–276).     

Reduced-basis output bound methods for parabolic problems

^** Email: rovas{at}uiuc.edu^*** Email: luc_machiels{at}mckinsey.com^**** Corresponding author. Email: maday{at}ann.jussieu.fr In this paper, we extend reduced-basis output bound methodsdeveloped earlier for elliptic problems, to problems describedby ‘parameterized parabolic’ partial differentialequations. The essential new ingredient and the novelty of thispaper consist in the presence of time in the formulation andsolution of the problem. First, without assuming a time discretization,a reduced-basis procedure is presented to ‘efficiently’compute accurate approximations to the solution of the parabolicproblem and ‘relevant’ outputs of interest. In addition,we develop an error estimation procedure to ‘a posteriorivalidate’ the accuracy of our output predictions. Second,using the discontinuous Galerkin method for the temporal discretization,the reduced-basis method and the output bound procedure areanalysed for the semi-discrete case. In both cases the reduced-basisis constructed by taking ‘snapshots’ of the solutionboth in time and in the parameters: in that sense the methodis close to Proper Orthogonal Decomposition (POD).     

An inventory model for stock with advertising sensitive demand

K. S. Chaudhuri Department of Mathematics, Jadavpur University, Kolkata-700 032, West Bengal, India Corresponding author. Email: shib_sankar{at}yahoo.com Email: k_s_chaudhuri{at}yahoo.com The effect of advertising media and sales effort on future demandof merchandize is considered. In an oligopolistic marketingsystem, it is quite natural to boost sales by using advertisingand sales effort to earn more money from a business sector.Determination of demand and costs due to advertising and saleseffort is quite difficult. The approach in this paper is toconcentrate on investment for the purpose of advertising andincreasing sales effort in order to maximize profit. The paperextends an inventory model over finite and infinite time horizonswhere demand is influenced by the level of stock, advertisingmedia and sales effort. The associated profit maximization problemis solved by the ‘Euler–Lagrange method’.The model is illustrated with a numerical example.     

On the independence of expansions of algebraic numbers in an integer base

Let b 2 be an integer. According to a conjecture of ÉmileBorel, the b-adic expansion of any irrational algebraic numberbehaves in some respects ‘like a random sequence’.We give a contribution to the following related problem: let and ' be irrational algebraic numbers, then prove that theirb-adic expansions either have the same tail, or behave in somerespects ‘like independent random sequences’.     

Necessary and Sufficient Tauberian Conditions, Under Which Convergence Follows From Summability (C, 1)

Let (s_k: k = 0, 1, ...) be a sequence of real numbers whichis summable (C, 1) to a finite limit. We prove that (s_k) isconvergent if and only if the following two conditions are satisfied: where _n denotes the integer part of the productn. Both conditions are clearly satisfied if (s_k) is slowly decreasingin the sense of R. Schmidt and G. H. Hardy. The symmetric counterparts of the conditions above are thosewhen ‘limsup’ and ‘liminf’ are interchangedon the left-hand sides, while the inequality sign ‘ ’is changed for the opposite ‘ ’ in them. Next, let (s_k) be a sequence of complex numbers which is summable(C, 1) to a finite limit. We prove that (s_k) is convergent ifand only if one of the following conditions is satisfied: We also prove a general Tauberian theorem forsequences in ordered linear spaces.     

Two-dimensional nonlinear advection-diffusion in a model of surfactant spreading on a thin liquid film

The spreading of a localized monolayer of dilute, insoluble surfactant, discharged from a point source that moves at constant speed over a thin liquid film coating a planar substrate, is described according to lubrication theory by a pair of coupled nonlinear evolution equations for the monolayer concentration and the film depth h. Numerical and asymptotic techniquesare here used to show that the extent and structure of sucha spreading asymmetric monolayer can be well approximated bya single nonlinear advection–diffusion equation involving alone. At large times the solution is composed of three, spatiallydistinct, asymptotic regions: (i) a quasi-steady ‘nose’region (containing the source), in which there is a dominantbalance between two-dimensional nonlinear diffusion and advection;(ii) an ‘advective’ region, in which longitudinaladvection balances transverse diffusion; and (iii) a ‘tail’region, in which unsteady diffusion is dominant. In each region,local similarity solutions are obtained either exactly (inthe advective region) or approximately (elsewhere) by rescalingnumerical solutions of the initial-value problem. If the sourceconcentration decreases with time, it is demonstrated that the monolayer’s width is greatest in the tail region, whereasfor a source of increasing concentration the monolayer is widestin the advective region. For the simpler one-dimensional problemof a monolayer spreading from a line source, the same balanceshold but with transverse diffusion eliminated; here self-similarsolutions are found in all three regions that agree closelywith numerical solutions of the initial-value problem. Received 7 October, 1998. Revised 11 April, 2000. ⁺ antoine@mip.ups-tlse.fr Present address: Division of Theoretical Mechanics, Schoolof Mathematical Sciences, University of Nottingham , UniversityPark, Nottingham NG7 2RD, UK. Oliver.Jensen@nottingham.ac.uk.     

A queueing model of activities in an intensive care unit

^** Email: griffiths{at}cardiff.ac.uk Activities in an intensive care unit (ICU) at a major teachinghospital are modelled by means of a queue-theoretic approach.Using data supplied by the ICU relating to the admissions process,the bed availability and the length of stay of patients, itwas possible to fit theoretical distributions to the observed‘arrival’ and ‘service’ distributions.Queueing equations relevant to a multi-channel system havingrandom arrivals and hyper-exponential service times for eachchannel are set up, and solved iteratively. Results obtainedmatch well with observations, and the model is then utilisedto investigate several ‘what if? ’ scenarios. Referenceis made to a simulation model developed in conjunction withthe queueing model.     

Control consolidation with a threshold: an algorithm

Email: ariane.chapelle{at}ulb.ac.be Corresponding author. Email: aszafarz{at}ulb.ac.be Received on 14 October 2005. Accepted on 21 November 2006. Besides multiple voting rights, board representation and shareholders’monitoring, control tunnelling over firms can be reached throughpyramids, cross-ownership and other complex features. This phenomenonis frequent in Europe and Asia. However, the theoretical literaturehas not yet converged towards a well-defined and robust measurementof integrated control that takes into account the thresholdfor control as applied in practice. Based on graph theory, thispaper aims at filling this gap and proposes a new algorithmfor evaluating the control tunnelling exerted by the firms’ultimate shareholders. Then, the paper discusses the variousforms of control existing next to voting shares, like multiplevoting rights, board representation and active monitoring, beforesuggesting ways to include them into the modelling of control.     

Timeout control scheme for overloaded M/E2/1 queue

A timeout scheme is considered for controlling an infinite ‘firstcome, first served’ overloaded single-server queue. Inthe overload situation, a customer-rejection mechanism is usedfor timing out ‘older’ customers in the queue, i.e.excluding those who have waited longer than a certain time.Applying ‘level-crossing analysis’ to an M/E₂/1queue, exact analytic expressions of performance such as thedensity and distribution functions of waiting time of the customerswho get served, the mean delay of customers, successful throughput,and ‘goodput’ are determined for this queue.     

Stability in the Sense of Bounded Average Power

An R^m-valued sequence (x_k): = (x_k : k = 1, 2, ...), e.g. generatedrecursively by x_k = f_k (x_k–_k, U_k), is called ‘averagepth power bounded’ if (1/K) is bounded uniformly in K= 1, 2,.... (The case p = 2 may correspond to ‘power’in the physical sense.) This is a notion of stability. Givenestimates of the form: f_k (x, u) < a x + ¶ _k conditionsare obtained on the coefficient sequence (a_k) and the inputestimates e_k:=¶_k (u_k) which ensure this form of stabilityfor the output (x_k). In particular, a condition (utilized inan application to adaptive control) is obtained which imposes(i) a bound b on (a_k) and a ‘sparsity measure’ m(K) on #{kK: a_k>} as K ( >1) (ii) average pth power boundednesson (e_k), and (iii) a growth condition on (e_k) related to b andm (•). This condition is sharp.     

Bayesian networks applied to credit scoring

Email: kchang{at}gmu.eduEmail: RobertFung{at}Fairlsaac.comEmail: alan.lucas{at}hotmail.com^¶Email: BobOliver{at}Fairlsaac.com^||Email: NShikaloff{at}Fairlsaac.com The objectives of this paper are to apply the theory and numericalalgorithms of Bayesian networks to risk scoring, and comparethe results with traditional methods for computing scores andposterior predictions of performance variables. Model identification,inference, and prediction of random variables using Bayesiannetworks have been successfully applied in a number of areas,including medical diagnosis, equipment failure, informationretrieval, rare-event prediction, and pattern recognition. Theability to graphically represent conditional dependencies andindependencies among random variables may also be useful incredit scoring. Although several papers have already appearedin the literature which use graphical models for model identification,as far as we know there have been no explicit experimental resultsthat compare a traditionally computed risk score with predictionsbased on Bayesian learning algorithms. In this paper, we examine a database of credit-card applicantsand attempt to ‘learn’ the graphical structure ofthe characteristics or variables that make up the database.We identify representative Bayesian networks in a developmentsample as well as the associated Markov blankets and cliquestructures within the Markov blanket. Once we obtain the structureof the underlying conditional independencies, we are able toestimate the probabilities of each node conditional on its directpredecessor node(s). We then calculate the posterior probabilitiesand scores of a performance variable for the development sample.Finally, we calculate the receiver operating characteristic(ROC) curves and relative profitability of scorecards basedon these identifications. The results of the different modelsand methods are compared with both development and validationsamples. Finally, we report on a statistical entropy calculationthat measures the degree to which cliques identified in theBayesian network are independent of one another.     

On the Existence of Elements of Non-Nilpotent Finite Closed Descent in Commutative Radical FreChet Algebras

It is established that in a commutative radical Fréchetalgebra, elements of non-nilpotent finite closed descent existif a locally non-nilpotent element of locally finite closeddescent exists. Thus if C[[X]] can be embedded into the unitizationof the algebra in such a way that X is mapped to an elementwhich is locally non-nilpotent, then it is possible to embedthe ‘structurally rich’ algebra C₁.     

Barrelled Countable Enlargements and the Bounding Cardinal

Replacing c with the bounding cardinal improves two standardBCE results of Robertson, Tweddle and Yeomans, and is optimalfor the BCE codimension/inheritance result of Bonet and PérezCarreras. Indeed, is the smallest infinite-dimensionality formetrizable barrelled spaces, and is the largest cardinal suchthat every subspace of codimension less than in a metrizablebarrelled space is itself barrelled. We thus reconfirm as oneof two ‘optimal cardinals for metrizable barrelled spaces’.Esthetically pleasing, these properties of immediately solvethe normable BCE problem without extra-ZFC axiomatic assumptions,reduce the separable quotient problem to Banach spaces E with density character of E c, and are part of the solution toone version of the metrizable BCE problem found in our sequel,devoted to the other ‘optimal cardinal’.