computations of multi fluid flows abstract full numerical simulations of three dimensional flows of two or more immiscible fluids of different densities and viscosities separated by a sharp interface with finite surface tension are discussed the method used is based on a finite difference approximation of the full navier stokes equations and explicit tracking of the interface between the fluids preliminary simulations of the rayleigh taylor instability and the motion of bubbles are shown

the entry exit theorem and relaxation oscillations in slow fast planar systems abstract the entry exit theorem for the phenomenon of delay of stability loss for certain types of slow fast planar systems plays a key role in establishing existence of limit cycles that exhibit relaxation oscillations the general existing proofs of this theorem depend on fenichel s geometric singular perturbation theory and blow up techniques in this work we give a short and elementary proof of the entry exit theorem based on a direct study of asymptotic formulas of the underlying solutions we employ this theorem to a broad class of slow fast planar systems to obtain existence global uniqueness and asymptotic orbital stability of relaxation oscillations the results are then applied to a diffusive predator prey model with holling type ii functional response to establish periodic traveling wave solutions furthermore we extend our work to another class of slow fast systems that can have multiple orbits exhibiting relaxation oscillations and subsequently apply the results to a two time scale holling tanner predator prey model with holling type iv functional response it is generally assumed in the literature that the non trivial equilibrium points exist uniquely in the interior of the domains bounded by the relaxation oscillations we do not make this assumption in this paper

a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

on the performance of linear programming heuristics applied on a quadratic transformation in the classification problem abstract we have applied lp heuristics to a quadratic transformation of the observation data in the classification problem four real world research data sets in the literature have been used to test the method these data sets include continuous integer and binary attributes for the observations to be classified in all four research data sets the proposed quadratic transformation method outperformed quadratic discriminant analysis

upwind differencing scheme for the time accurate incompressible navier stokes equations the two dimensional incompressible navier stokes equations are solved in a time accurate manner using the method of pseudocompres sibility using this method subiterations in pseudotime are required to satisfy the continuity equation at each time step an upwind differencing scheme based on flux difference splitting is used to compute the convective terms the upwind differencing is biased based on the sign of the local eigenvalue of the jacobian matrix of the convective fluxes both third order and fifth order differencing schemes are used on the convective fluxes throughout the grid s interior the equations are solved using an implicit line relaxation scheme this solution scheme is stable and is capable of running at large time steps in pseudo time leading to fast convergence for each physical time step a variety of computed results are presented to validate the present scheme results for the flow over an oscillating plate are compared with the exact analytic solution and good agreement is seen excellent comparison is obtained between the computed solution and the analytical results for inviscid channel flow with an oscillating back pressure flow solutions over a circular cylinder with vortex shedding are also presented finally the flow past an airfoil at 90 angle of attack is computed

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

multiple focus and hopf bifurcations in a predator prey system with nonmonotonic functional response in this paper we develop a criterion to calculate the multiplicity of a multiple focus for general predator prey systems as applications of this criterion we calculate the largest multiplicity of a multiple focus in a predator prey system with nonmonotonic functional response p x frac x ax 2 bx 1 studied by zhu campbell and wolkowicz siam j appl math 63 2002 pp 636 682 and prove that the degenerate hopf bifurcation is of codimension two furthermore we show that there exist parameter values for which this system has a unique positive hyperbolic stable equilibrium and exactly two limit cycles the inner one unstable and outer one stable numerical simulations for the existence of the two limit cycles bifurcated from the multiple focus are also given in support of the criterion

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

new lp based heuristics for the classification problem abstract three heuristic procedures for the two group discriminant problem are developed that minimize the total expected cost of misclassification in the training sample the approach is shown through a systematic experiment to generate near optimal solutions and to be efficient in terms of computational effort when compared to existing research

addressing the multigroup discriminant problem using multivariate statistics and mathematical programming in the paper we compare the performance of seven important multivariate and mathematical programming methods in the multigroup classification problem with simulated data our approach extends previous results from two to multiple groups and recognizes simultaneously different types of classification errors in a multivariate analysis of variance manova framework

bifurcation analysis of a predator prey system involving group defence a class of odes of generalized gause type modeling predator prey interaction is considered the prey are assumed to exhibit a phenomenon called group defence that is predation is decreased or even eliminated due to the ability of the prey to defend or disguise themselves as their numbers increase using the carrying capacity of the environment as the bifurcation parameter it is shown that the model undergoes a sequence of bifurcations that includes a homoclinic bifurcation as well as a hopf bifurcation conditions that hold even in the case of no group defence that ensure a subcritical hopf bifurcation and also the spontaneous appearance of a semistable periodic orbit that splits into a pair one stable and one unstable of periodic orbits are given ecological ramifications are considered unlike the classical model sufficient enrichment of the environment combined with group defence leads to extinction of the predator deterministically for almost all initial conditions providing strong support fo

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

a classification model based on goal programming with non standard preference functions with application to the prediction of cinema going behaviour this paper presents a goal programming model that allows for the flexible handling of the two group classification problem the goal programming model is based around the concepts of non standard preference functions and penalty function modelling an extension to a generalised distance metric case is given the inclusion of multiple levels of classification based upon different levels of certainty is incorporated into the model the model is tested on a real life data set pertaining to cinema going attendance and conclusions are drawn both in the context of the methodology and of the application

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

a mathematical programming approach to clusterwise regression model and its extensions abstract the clusterwise regression model is used to perform cluster analysis within a regression framework while the traditional regression model assumes the regression coefficient to be identical for all subjects in the sample the clusterwise regression model allows to vary with subjects of different clusters since the cluster membership is unknown the estimation of the clusterwise regression is a tough combinatorial optimization problem in this research we propose a generalized clusterwise regression model which is formulated as a mathematical programming mp problem a nonlinear programming procedure with linear constraints is proposed to solve the combinatorial problem and to estimate the cluster membership and simultaneously moreover by integrating the cluster analysis with the discriminant analysis a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis the cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

artificial compressibility characteristics based schemes for variable density incompressible multispecies flows part ii multigrid implementation and numerical tests the paper presents an investigation of the accuracy and efficiency of artificial compressibility characteristics based cb schemes for variable density incompressible flows the cb schemes have been implemented in conjunction with a multigrid method for accelerating numerical convergence and a fourth order explicit runge kutta method for the integration of the governing equations in time the implementation of the cb schemes is obtained in conjunction with first second and third order interpolation formulas for calculating the variables at the cell faces of the computational volume the accuracy and efficiency of the schemes are examined against analytical and experimental results for diffusion broadening in two and three dimensional microfluidic channels a problem that has motivated the development of the present methods moreover unsteady inviscid simulations have been performed for variable density mixing layer the computations revealed that accuracy and efficiency depend on the cb scheme design the best multigrid convergence rates were exhibited by the conservative cb scheme which is obtained by the fully conservative formulation of the variable density incompressible equations

a hybrid genetic fuzzy neural network algorithm designed for classification problems involving several groups we propose a multigroup classification algorithm based on a hybrid genetic fuzzy neural net gfnn framework recent results on evolutionary computation and fuzzy neural network methodology are combined to effectively adapt the membership functions of the fuzzifier and the defuzzifier to the data set separate membership functions are defined for each dimension in the fuzzifier and for each fuzzy output group in the defuzzifier the signal inherent in the fuzzifier is aggregated by a suitable t norm and transmitted to the defuzzifier the defuzzifier aggregates the response i e the predicted group membership by a suitable conorm if misclassifications occur during training the membership functions of both the fuzzifier and the defuzzifier are adapted by a systematic robust procedure the algorithm is successfully tested with real economic data in total the gfnn performs as good as the best of the competing methods in our test the results suggest economically meaningful interpretations

global analysis in a predator prey system with nonmonotonic functional response a predator prey system with nonmonotonic functional response is considered global qualitative and bifurcation analyses are combined to determine the global dynamics of the model the bifurcation a

artificial compressibility characteristics based schemes for variable density incompressible multi species flows part i derivation of different formulations and constant density limit the paper presents various formulations of characteristics based schemes in the framework of the artificial compressibility method for variable density incompressible flows in contrast to constant density incompressible flows where the characteristics based variables reconstruction leads to a single formulation in the case of variable density flows three different schemes can be obtained henceforth labeled as transport conservative and hybrid schemes the conservative scheme results in pseudo compressibility terms in the multi species density reconstruction it is shown that in the limit of constant density the transport scheme becomes the original characteristics based scheme for incompressible flows but the conservative and hybrid schemes lead to a new characteristics based variant for constant density flows the characteristics based schemes are combined with second and third order interpolation for increasing the computational accuracy locally at the cell faces of the control volume numerical experiments for constant density flows reveal that all the characteristics based schemes result in the same flow solution but they exhibit different convergence behavior the multigrid implementation and numerical studies for variable density flows are presented in part ii of this study

a numerical stable method for integration of the multicomponent species diffusion equations abstract a diagonally implicit method is shown to be an effective method for integrating the multicomponent species conservation equations the constitutive equation for multicomponent diffusion is recast into a form analogous to that for binary diffusion except that the diffusion coefficient is replaced with a matrix of effective multicomponent diffusion coefficients the resulting matrix has properties that allow the diagonal terms to be integrated implicitly and the off diagonal terms to be integrated explicitly numerical experiments show the integration procedure is stable for time steps much larger than the diffusion equation time step condition

constrained regression median for measuring possible salary discrimination abstract this study proposes a new analytical method for empirically examining possible salary discrimination against a protected e g female or minority group the proposed approach is referred to as constrained regression median crm the method can be used for regressing salary with job qualifications as dependent variables the result of crm is transformed into a 2 2 contingency table and the degree of salary discrimination is determined by the result of the contingency table the approach can be easily explained to and understood by lawyers judges and other individuals involved in a discrimination case the crm method is particularly useful and practical when a data set maintains the problems of outlier and or multicollinearity between job qualifications and gender or race

non conservative and conservative formulations of characteristics based numerical reconstructions for incompressible flows summary an investigation of characteristics based cb schemes for solving the incompressible navierstokes equations in conjunction with the artiflcial compressibility approach is presented both nonconservative and conservative characteristics based numerical reconstructions are derived and their accuracy and convergence properties are assessed analytically and numerically we demonstrate by means of eigenvalue analysis that there are difierences in the spectral characteristics of these formulations that result in difierent convergence properties numerical tests for two and threedimensional ows reveal that the two formulations provide similar accuracy but the non conservative formulation converges faster copyright c 2000 john wiley sons ltd

minimizing deviations from the group mean a new linear programming approach for the two group classification problem abstract this paper proposes a new linear programming approach to solve the two group classification problem in discriminant analysis this new approach is based on an idea from cluster analysis that objects within the same group should be more similar than objects between groups consequently it is desirable for the classification score of an object to be nearer to its mean classification score but further from the mean classification score of the other group this objective is accomplished by minimizing the total deviation of the classification scores of the objects from their group mean scores in a linear programming approach when applied to an actual managerial problem and simulated data the proposed linear programming approach performs well both in groups separation and group membership predictions of new objects moreover this new approach has an advantage of obtaining more stable classification function across different samples than most of the existing linear programming approaches

mixed integer programming approach of extended dea discriminant analysis abstract several unique features of data envelopment analysis dea are incorporated into discriminant analysis da such a combined use of the two approaches is referred to as dea da eur j operat res 1999 115 564 sueyoshi eur j operat res 2001 131 324 has recently extended dea da in a manner that it can deal with a negative value in data the new version is referred to as extended dea da an important feature of extended dea da is that it can estimate weights of a linear discriminant function by minimizing the total deviation of misclassified observations as an extension of his research this research proposes a new type of mixed integer programming formulations that provide these weight estimates by minimizing the total number of misclassified observations

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

a projection fem for variable density incompressible flows abstract this work describes a new finite element projection method for the computation of incompressible viscous flows of nonuniform density one original idea of the proposed method consists in factorizing the density variable partly outside and partly inside the time evolution operator in the momentum equation to prevent spatial discretization errors in the mass conservation to affect the kinetic energy balance of the fluid it is shown that unconditional stability in the incremental version of the projection method is possible provided two projections are performed per time step in particular a second order accurate bdf projection method is presented and its numerical performance is illustrated by test computations and comparisons

multiple bifurcation in a predator prey system with nonmonotonic predator response a model of a predator prey system showing group defence on the part of the prey is formulated and reduced to a three parameter family of quartic polynomial systems of equations mathematically this system contains the volterra lotka system and yields numerous kinds of bifurcation phenomena including a codimension two singularity of cusp type in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation biologically the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant so that the system exhibits an enrichment paradox and for certain choices of parameters coexistence of stable oscillation and a stable equilibrium

regular and slow fast codimension 4 saddle node bifurcations abstract using geometric singular perturbation theory including the family blow up as one of the main techniques we prove that the cyclicity i e maximum number of limit cycles in both regular and slow fast unfoldings of nilpotent saddle node singularity of codimension 4 is 2 the blow up technique enables us to use the well known results for slow fast codimension 1 and 2 hopf bifurcations slow fast bogdanov takens bifurcations and slow fast codimension 3 saddle and elliptic bifurcations

a numerical method for the incompressible navier stokes equations based on an approximate projection in this method we present a fractional step discretization of the time dependent incompressible navier stokes equations the method is based on a projection formulation in which we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields our treatment of the diffusion convection step uses a specialized second order upwind method for differencing the nonlinear convective terms that provides a robust treatment of these terms at a high reynolds number in contrast to conventional projection type discretizations that impose a discrete form of the divergence free constraint we only approximately impose the constraint i e the velocity field we compute is not exactly divergence free the approximate projection is computed using a conventional discretization of the laplacian and the resulting linear system is solved using conventional multigrid methods numerical examples are presented to validate the second order convergence of the method for euler finite reynolds number and stokes flow a second example illustrating the behavior of the algorithm on an unstable shear layer is also presented

evaluating the maximize minimum distance formulation of the linear discriminant problem abstract the maximize minimum distance mmd linear programming model for the two group discriminant problem has been noted to produce occasionally a trivial identically zero discriminant function one which classifies all observations into a single category in tests against other methods both parametric and nonparametric mmd has fared poorly in this paper we attribute the propensity of the mmd model to produce trivial solutions to a specific aspect of its formulation this same facet may also cause unnessarily high misclassification rates even when a nontrivial function is found we note a simple revision of a model which ensures an acceptable solution in those instancesin which the calibration samples can be classified with 100 accuracy by a single function this rises the question of whether the inferior performance of mmd in previous studies was due to inherent limitations in mmd or to the particular formulation used

simple but powerful goal programming models for discriminant problems abstract conventional statistical analysis includes the capacity to systematically assign individuals to groups we suggest alternative assignment procedures utilizing a set of interrelated goal programming formulations we further demonstrate via simple illustration the potential of these procedures to play a significant part in addressing the discriminant problem and indicate fundamental ideas that lay the foundation for other more sophisticated approaches

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

multiple canard cycles in generalized lienard equations the paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields the results deal with any number of parameters proofs are based on the techniques introduced in canard cycles and center manifolds f dumortier and r roussarie 1996 mem amer math soc 121 the presentation is limited to generalized lienard equations ex x c x x c 0

second order mathematical programming formulations for discriminant analysis this paper introduces a nonparametric formulation based on mathematical programming mp for solving the classification problem in discriminant analysis which differs from previously proposed mp based models in that even though the final discriminant function is linear in terms of the parameters to be estimated the formulation is quadratic in terms of the predictor attribute variables including second order i e quadratic and cross product terms of the attribute variables in the model is similar in concept to the usual treatment of multiple predictor variables in statistical methods such as fisher s linear discriminant analysis and allows an analysis of how including nonlinear terms and interaction effects affect the predictive ability of the estimated classification function using simulation experiments involving data conditions for which nonlinear classifiers are appropriate the classificatory performance of this class of second order mp models is compared with that of existing statistical linear and quadratic and first order mp based formulations the results of these experiments show that the proposed formulation appears to be a very attractive alternative to previously introduced linear and quadratic statistical and linear mp based classification methods

unsteady solution of incompressible navier stokes equations abstract a new time accurate finite difference numerical method for solving incompressible navier stokes equations is presented with primitive variables as the unknowns the numerical scheme is a crank nicolson implicit treatment of all terms of the equations with central differencing for space derivatives the pressure and the nonlinearities in convection terms are not lagged to obtain the incompressible solution at the advanced time level a continuous auxiliary system is introduced in pseudo time with artificial compressibility time accurate solutions are presented for two dimensional fluid flows in a square cavity with an impulsively starting lid and with an oscillating lid

on detecting nonlinear patterns in discriminant problems we propose a two stage model for detecting nonlinear patterns in discriminant problems and for solving the problem the model deploys a linear programming based discriminator lpbd in stage one for treating linear patterns and a probabilistic neural network pnn in stage two for handling nonlinear patterns the lpbd in stage one divides the decision space into a clear zone where observations are almost linearly separable and a gray zone where nonlinear patterns are more likely to occur the pnn in stage two analyzes the gray zone and determines whether a significant nonlinear patterns exist in the observations our goal is to avoid using a nonlinear model unless the pnn strongly suggests so to maintain good interpretability and avoid overfitting our computational study demonstrates the effectiveness of the two stage model in both classification accuracy and computational efficiency

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

a three dimensional incompressible navier stokes flow solver using primitive variables an implicit finite difference computer code has been developed to solve the incompressible navier stokes equations in a three dimensional curvilinear coordinate system the pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation the solution procedure employs an implicit approximate factorization scheme the reynolds stresses which are uncoupled from the implicit scheme are lagged by one time step to facilitate implementing various levels of the turbulence model test problems for external and internal flows are computer and the results are compared with existing experimental data the application of this technique for general three dimensional problems is then demonstrated

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

optimization approaches to supervised classification abstract the supervised classification problem one of the oldest and most recurrent problems in applied data analysis has always been analyzed from many different perspectives when the emphasis is placed on its overall goal of developing classification rules with minimal classification cost supervised classification can be understood as an optimization problem on the other hand when the focus is in modeling the uncertainty involved in the classification of future unknown entities it can be formulated as a statistical problem other perspectives that pay particular attention to pattern recognition and machine learning aspects of supervised classification have also a long history that has lead to influential insights and different methodologies in this review two approaches to supervised classification strongly related to optimization theory will be discussed and compared in particular we will review methodologies based on mathematical programming models that optimize observable criteria linked to the true objective of misclassification error or cost minimization and approaches derived from the minimization of known bounds on the true misclassification error the former approach is known as the mathematical programming approach to supervised classification while the latter is in the origin of the well known classification support vector machines throughout the review two group as well as general multi group problems will be considered and the review will conclude with a discussion of the most promising research directions in this area

a second order projection method for variable density flows this paper describes a second order projection method for variabledensity incompressible flows the method is suitable for both finite amplitude density variations and for fluids that are modeled using a boussinesq approximation it is based on a second order fractional step scheme in which diffusion convection terms are advanced without enforcing the incompressibility condition and the resulting intermediate velocity field is then projected onto the space of discretely divergencefree vector fields the nonlinear convection terms are treated using a godunov type procedure that is second order for smooth flow and remain stable and non oscillatory for nonsmooth flows with low fluid viscosities the method is described for finite amplitude density variation and the simplifications for a boussinesq approximation are sketched numerical results are presented that validate the convergence properties of the method and demonstrate its performance on more realistic problems

a comparison of standard and two stage mathematical programming discriminant analysis methods abstract classification models can be developed using standard or two stage mathematical programming mp discriminant analysis methods in standard mp discriminant analysis methods discriminant functions are generated by solving a single mp model in two stage mp methods observations that are difficult to classify are identified in the first stage with greater emphasis being given to these observations in the second stage mp model for discriminant function generation in this paper two two stage methods are described and compared with two standard mp models the model for minimisation of the sum of deviations and the model for maximisation of classification accuracy the performance of these mp discriminant analysis methods and fisher s linear discriminant analysis a parametric statistical technique is then evaluated on a published data set and on a number of simulated data sets

the two group discriminant problem with equal group mean vectors an experimental evaluation of six linear nonlinear programming formulations abstract various parametric and nonparametric approaches to multiple discriminant analysis attempt to discriminate among or classify entities e g loan applicants customers employees businesses based on several of their distinguishing characteristics called discriminant variables statistical parametric procedures require that the mean vectors of discriminant variables for the populations of entities be different across groups this requirement may not always be met in practical settings this paper reports on a preliminary monte carlo simulation experiment which compares the performance of six 1 p norm distance models including two linear and four nonlinear formulations as well as two statistical procedures to address the discriminant problem under equal mean vectors the experimental data were generated from multivariate normal or nonnormal populations with equal or unequal dispersion matrices and with or without outliers the results indicate that when population mean vectors are equal the most significant characteristics which affects the performance of all of the methods is the similarity or dissimilarity of dispersion matrices the departure from normality and the presence of outliers and some of the interactions between these three factors are also statistically significant

canard phenomenon for an sis epidemic model with nonlinear incidence abstract by using the singular perturbation theory especially on canard cycles the canard phenomenon for an sis epidemic model with nonlinear incidence is investigated the phenomenon suggests the existence of disease outbreak in the infection transmission it is proved that the cyclicity of any possible slow fast cycle is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the slow fast cycle by small perturbations we also indicate the regions in parameter space where the corresponding slow fast cycle has cyclicity at most one or at most two

linear goal programming in estimation of classification probability classification of objects into groups is a common practice in everyday life normally the scores of some pertinent criteria about the objects are available and form the most important factor in the classification decision discriminant analysis 4 logistic regression 1 3 and linear goal programming 2 5 are alternative techniques used to determine the criterion weights classification with these techniques is based upon the weighted sum of the criterion scores of each object the classification itself ignores more discriminating information about objects within groups for example dividing companies into bankrupt and non bankrupt groups ignores the fact that the non bankrupt group contains companies of varying qualities thus it is more appropriate to determine the group membership probabilities for each object instead of just the pure classification into one of the groups these probabilities can readily be used to discriminate objects within the same group

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

bifurcations in a predator prey system of leslie type with generalized holling type iii functional response abstract we consider a predator prey system of leslie type with generalized holling type iii functional response p x m x 2 a x 2 b x 1 by allowing b to be negative b 2 a p x is monotonic for b 0 and nonmonotonic for b 0 when x 0 the model has two non hyperbolic positive equilibria one is a multiple focus of multiplicity one and the other is a cusp of codimension 2 for some values of parameters and a degenerate bogdanov takens singularity focus or center case of codimension 3 for other values of parameters when there exist a multiple focus of multiplicity one and a cusp of codimension 2 we show that the model exhibits subcritical hopf bifurcation and bogdanov takens bifurcation simultaneously in the corresponding small neighborhoods of the two degenerate equilibria respectively different phase portraits of the model are obtained by computer numerical simulations which demonstrate that the model can have i a stable limit cycle enclosing two non hyperbolic positive equilibria ii a stable limit cycle enclosing an unstable homoclinic loop iii two limit cycles enclosing a hyperbolic positive equilibrium iv one stable limit cycle enclosing three hyperbolic positive equilibria or v the coexistence of three stable states two stable equilibria and a stable limit cycle when the model has a bogdanov takens singularity of codimension 3 we prove that the model exhibits degenerate focus type bogdanov takens bifurcation of codimension 3 these results not only demonstrate that the dynamics of this model when b 2 a are much more complex and far richer than the case when b 0 but also provide new bifurcation phenomena for predator prey systems

canards heteroclinic and homoclinic orbits for a slow fast predator prey model of generalized holling type iii abstract for a classical ratio dependent predator prey model with the generalized holling type iii functional response it was previously investigated in 20 by hsu and huang for global stability of an equilibrium and in 21 by huang ruan and song for subcritical hopf and bogdanov takens bifurcations here in this model when prey reproduces much faster than predator by using geometric singular perturbation theory we achieve much richer new dynamical phenomena than the existing ones such as the existence of canard cycles canard explosion and relaxation oscillations heteroclinic and homoclinic orbits cyclicity of slow fast cycles and the coexistence of the hopf cycle and the relaxation oscillation on global stability of the equilibrium we also provide less restricted conditions than the existing ones

mathematical programming based heuristics for improving lp generated classifiers for the multiclass supervised classification problem abstract mathematical programming is used as a nonparametric approach to supervised classification however mathematical programming formulations that minimize the number of misclassifications on the design dataset suffer from computational difficulties we present mathematical programming based heuristics for finding classifiers with a small number of misclassifications on the design dataset with multiple classes the basic idea is to improve an lp generated classifier with respect to the number of misclassifications on the design dataset the heuristics are evaluated computationally on both simulated and real world datasets

a comparison of a robust mixed integer approach to existing methods for establishing classification rules for the discriminant problem abstract a nonparametric mixed integer programming formulation to solve the classification problem in linear discriminant analysis is proposed the classification performance of this formulation is compared to the msd linear programming approach and two commonly used statistical methods fisher s linear discriminant function and the quadratic discriminant function using real data with highly nonnormal distributions the mixed integer formulation is shown to outperform any of the other three approaches to study the performance of the mixed integer formulation systematically a monte carlo simulation experiment is conducted sampling from several different distributions the results show that the mixed integer formulation outperforms the other three approaches on the training samples except when the variance covariances are heterogeneous in which case the quadratic function classifies better when holdout samples are used to evaluate the relative performance the mixed integer approach classifies well when the data are highly discrete and the variance covariances are homogeneous but does not fare as well under other data conditions therefore this study suggests that under certain conditions the mixed integer approach is an attractive alternative to establish classification methods

a combinatorial approach to the classification problem we study the two group classification problem which involves classifying an observation into one of two groups based on its attributes the classification rule is a hyperplane which misclassifies the fewest number of observations in the training sample exact and heuristic algorithms for solving the problem are presented computational results confirm the efficiency of this approach

dea discriminant analysis methodological comparison among eight discriminant analysis approaches abstract discriminant analysis da is a classification method that can predict the group membership of a newly sampled observation recently a new type of non parametric da approach is proposed to provide a set of weights of a discriminant function consequently yielding an evaluation score for the determination of group membership the non parametric da is referred to as data envelopment analysis discriminant analysis dea da because it maintains its discriminant capabilities by incorporating the non parametric feature of dea into da in this study a use of the mixed integer approach of dea da is compared with other da methods it is confirmed that it performs at least as well as the other well known da methods the proposed approach is further reformulated in a manner that it can deal with classification of more than two groups

efficient and non reflecting far field boundary conditions for incompressible flow calculations abstract traditionally the artificial compressibility ac method of chorin is used for simulation of low speed and incompressible flows in this method the difficulty of continuity and momentum equations decoupling is removed by adding an artificial time derivative of pressure to the continuity equation for the first time a fully two dimensional upwind scheme was presented for ac equations by using characteristic structure of equations by the authors in this paper a new remote boundary calculation method is presented by using the idea of characteristics for these equations instead of simple boundary conditions which usually are employed for incompressible flows the flow quantities at the far field boundaries are evaluated by compatibility relations of characteristic equations this is implemented by assuming a row of ghost cells outside of the far field boundary and using the flux calculation method based on characteristics similar to the inside of computational domain the idea in conjunction with multidimensional characteristic based scheme was tested for incompressible flow around circular cylinder in comparison with conventional far field boundary condition and showed good improvements in the terms of accuracy and convergence speed

an adaptive statistical method for the discriminant problem abstract this paper proposes an adaptive statistical method for the discriminant problem the method selects fisher s linear or smith s quadratic discriminant function or the nearest neighbor method for use on the holdout sample depending upon which method minimizes the sum of overall accuracy and balance on the estimation sample a simulation study examines the two group discriminant problem with variables generated from both bivariate normal and nonnormal distributions the resulting misclassification rates indicate that the adaptive method is an effective alternative to existing statistical and linear programming methods

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

a piecewise linear programming approach to the two group discriminant problem an adaptation to fisher s linear discriminant function model abstract most existing linear programming lp models have optimization objectives that are very different from fisher s linear discriminant function fldf an lp technique that adapts to fldf to solve the two group classification problem is desirable as fldf is one of the most popular classification rules therefore this paper introduces a piecewise linear programming plp p approach that has an optimization objective very similar to that of fldf to solve the two group classification problem in discriminant analysis moreover the paper compares the classificatory performance between fldf and the new plp p model and shows that the results from both approaches are as good as each other when applied to three published data sets however the new plp p is more flexible than fldf in terms of adding different types of constraints and weighting individual observations the results of a simulation experiment confirm the value of our proposed approach

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

bifurcation analysis of a predator prey system with nonmonotonic functional response we consider a predator prey system with nonmonotonic functional response p x frac mx ax 2 bx 1 by allowing b to be negative b 2 sqrt a p x is concave up for small values of x 0 a

singular perturbations of the holling i predator prey system with a focus abstract we study the occurrence of limit cyles in a model for two species predator prey interaction referred to as holling i this model is based on the phenomenological fitting of the so called functional response function to observed data in nature by a continuous piecewise differentiable function of the original models presented by holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet we extend the work of liu who showed that for certain values of parameters of the holling i system at least two nested limit cycles will occur surrounding a focus his case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities i e a singular perturbation problem with slow fast solutions we prove that exactly two hyperbolic limit cycles occur after perturbation

a nonlinear multigrid method for the three dimensional incompressible navier stokes equations a nonlinear multigrid method is developed for solving the three dimensional navier stokes equations in conjunction with the artificial compressibility formulation the method is based on the full multigrid fmg full approximation storage fas algorithm and is realized via an unsteady type procedure according to which the equations are not solved exactly on the coarsest grid but some pseudo time iterations are performed on the finer grids and some on the coarsest grid the multigrid method is implemented in conjunction with a third order upwind characteristics based scheme for the discretization of the convection terms and the fourth order runge kutta scheme for time integration the performance of the method is investigated for three dimensional flows in straight and curved channels as well as flow in a cubic cavity the multigrid acceleration is assessed in contrast to the single grid and mesh sequencing algorithms the effects of various multigrid components on the convergence acceleration such as prolongation operators as well as pre and postrelaxation iterations are also investigated

short note on the characteristics based acm for incompressible flows in this paper the revised characteristics based cb method for incompressible flows recently derived by neofytou p neofytou revision of the characteristic based scheme for incompressible flows j comput phys 222 2007 475 484 has been further investigated we have derived all the formulas for pressure and velocities from this revised cb method which is based on the artificial compressibility method acm a j chorin a numerical solution for solving incompressible viscous flow problems j comput phys 2 1967 12 then we analyze the formulations of the original cb method d drikakis p a govatsos d e papatonis a characteristic based method for incompressible flows int j numer meth fluids 19 1994 667 685 e shapiro d drikakis non conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows int j numer meth eng 66 2006 1466 1482 d drikakis p k smolarkiewicz on spurious vortical structures j comput phys 172 2001 309 325 f mallinger d drikakis instability in three dimensional unsteady stenotic flows int j heat fluid flow 23 2002 657 663 e shapiro d drikakis artificial compressibility characteristics based schemes for variable density incompressible multi species flows parts i derivation of different formulations and constant density limit j comput phys 210 2005 584 607 y zhao b zhang a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids comput meth appl mech eng 190 5 7 2000 733 756 to investigate their consistency with the governing flow equations after convergence has been achieved furthermore we have implemented both formulations in an unstructured grid finite volume solver y zhao b zhang a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids comput meth appl mech eng 190 5 7 2000 733 756 detailed numerical experiments show that both methods give almost identical solutions and convergence rates both can generate solutions which agree well with published results and experimental measurements we thus conclude that both methods being upwind schemes designed for the acm have the same performances in terms of accuracy and convergence speed even though the revised method is more complex with less stringent assumptions made while the original cb method is simpler due to the use of extra simplifying assumptions

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

credit risk evaluation using multi criteria optimization classifier with kernel fuzzification and penalty factors abstract with the fast development of financial products and services bank s credit departments collected large amounts of data which risk analysts use to build appropriate credit scoring models to evaluate an applicant s credit risk accurately one of these models is the multi criteria optimization classifier mcoc by finding a trade off between overlapping of different classes and total distance from input points to the decision boundary mcoc can derive a decision function from distinct classes of training data and subsequently use this function to predict the class label of an unseen sample in many real world applications however owing to noise outliers class imbalance nonlinearly separable problems and other uncertainties in data classification quality degenerates rapidly when using mcoc in this paper we propose a novel multi criteria optimization classifier based on kernel fuzzification and penalty factors kfp mcoc firstly a kernel function is used to map input points into a high dimensional feature space then an appropriate fuzzy membership function is introduced to mcoc and associated with each data point in the feature space and the unequal penalty factors are added to the input points of imbalanced classes thus the effects of the aforementioned problems are reduced our experimental results of credit risk evaluation and their comparison with mcoc support vector machines svm and fuzzy svm show that kfp mcoc can enhance the separation of different applicants the efficiency of credit risk scoring and the generalization of predicting the credit rank of a new credit applicant

a fuzzy neural network algorithm for multigroup classification we propose a multigroup classification algorithm based on a hybrid fuzzy neural net framework a key feature of the approach is the adaptation of membership functions to new data in this way learning is reflected in the shape of the membership functions by defining separate membership functions for each fuzzy output class we allow dynamic adjustment of the functions during training the algorithm is successfully tested with real economic data the results suggest economically meaningful interpretations

an experimental comparison of several approaches to the discriminant problem with both qualitative and quantitative variables abstract this paper reports the results of an experimental comparison between a linear programming approach and the well known statistical procedure by fisher for solving discriminant analysis problems this work represents an extension of the work of bajgier and hill as we incorporate qualitative variables into our design as well as enlarge the estimation sample from 30 to 90 our results indicate that both methods are enhanced by the inclusion of the qualitative variables but that the fisher approach seems preferable lastly we discuss the effect of the different experimental factors on the relative performance of the two methods

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

improper linear discriminant classifiers abstract mathematical programming formulations for determining linear discriminant classifiers have received considerable interest in recent years although their performance rivals classical methods and requires fewer assumptions many of the formulations exhibit undesirable properties in this paper we draw attention to two additional problems that the mathematical programming formulations face a solution for each type of problem is presented

multidimensional upwinding for incompressible flows based on characteristics in this paper the multidimensional characteristic based upwind scheme mcb which has been recently introduced by the authors is applied to two another benchmark problems namely flow in a channel with a backward facing step and two dimensional steady and unsteady flows past a circular cylinder extension of mcb scheme for calculating convective fluxes on non cartesian grids is presented here for the flow over backward facing step obtained results were compared against well known experimental data and the results show high accuracy of mcb scheme and faster convergence rate with respect to conventional cb scheme in the case of flow over circular cylinder the flow at steady and transient regimes is investigated by mcb scheme again the results obtained by mcb are compared to the other results in the literature and show good agreement with them also rapid convergence rate of mcb was observed in this case too it is concluded that the genuinely multidimensional characteristic based mcb scheme has been introduced earlier by the authors is a robust and powerful scheme for modeling incompressible viscous flows for achieving the high accuracy and remarkable advantage in convergence rate with respect to conventional characteristic based schemes