a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

theory of markov systems with fuzzy states in this paper the theory of fuzzy logic is combined with the theory of markov systems and the concept of a markov system with fuzzy states is introduced for the first time the main objective is to develop a methodology of describing a markov population system with fuzzy states this is an effort to deal with the fact that in real applications one is often faced with fuzzy states this paper investigates the properties of a markov system with states that have imprecise boundaries a full description of a nonhomogeneous markov system with fuzzy states is outlined and the basic parameters of the system are provided more specifically the expected population structure of the system is found and given in a closed analytic form and the asymptotic behaviour of the system is provided under the more realistic assumption of a fuzzy set space

recovering exponential accuracy from collocation point values of smooth functions with end point singularities gibbs phenomenon is the particular manner how a global spectral approximation of a piecewise analytic function behaves at the jump discontinuity the truncated spectral series has large oscillations near the jump and the overshoot does not decay as the number of terms in the truncated series increases there is therefore no convergence in the maximum norm and convergence in smooth regions away from the discontinuity is also slow in gottlieb and shu 1995 5 a methodology is proposed to completely overcome this difficulty in the context of spectral collocation methods resulting in the recovery of exponential accuracy from collocation point values of a piecewise analytic function in this paper we extend this methodology to handle spectral collocation methods for functions which are analytic in the open interval but have singularities at end points with this extension we are able to obtain exponential accuracy from collocation point values of such functions similar to gottlieb and shu 1995 5 the proof is constructive and uses the gegenbauer polynomials c n l x the result implies that the gibbs phenomenon can be overcome for smooth functions with endpoint singularities

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

an averaging method for the fourier approximation to discontinuous functions both the truncated fourier integral and the truncated fourier series approximations for a discontinuous function bring about the inevitable oscillating error say the gibbs phenomenon most basic filtering methods for the gibbs phenomenon like the fejer averaging method and the lanczos averaging method have a disadvantage that the rise time is very slow even though the filtering effect is prominent away from the discontinuity in this paper we propose a new averaging method of polynomial type which improves the rise time of the existing method the present method can be regarded as a generalization of the traditional lanczos averaging method by several numerical examples we show the efficiency of the present method

jacobi approximations in certain hilbert spaces and their applications to singular differential equations jacobi approximations in certain hilbert spaces are investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using jacobi polynomials this method keeps the spectral accuracy some linear problems and a nonlinear logistic equation are considered the stabilities and the convergences of proposed schemes are proved strictly the main idea and techniques used in this paper are also applicable to other singular problems in multiple dimensional spaces

a basis deficiency allowing variation of the simplex method for linear programming h one of the most important and fundamental concepts in the simplex methodology basis is restricted to being a square matrix of the order exactly equal to the number of rows of the coefficient matrix such inflexibility might have been the source of too many zero steps taken by the simplex method in solving real world linear programming problems which are usually highly degen erate to dodge this difficulty we first generalize the basis to allow the deficient case characterized as one that has columns fewer than rows of the coefficient matrix variations of the primal and dual simplex procedures are then made and used to form a two phase method based on such a basis the number of whose columns varies dynamically in the solution process generally speaking the more degenerate a problem to be handled is the fewer columns the basis will have so thii renders the possibility of efficiently solving highly degenerate problems we also provide a valuable insight into the distinctive and favorable behavior of the proposed method by reporting our computational experiments 1998 elsevier science ltd all rights reserved

gegenbauer approximation in certain hilbert spaces and its applications to singular differential equations gegenbauer approximation in certain hilbert spaces is investigated several weighted inverse inequalities and poincare inequalities are obtained some approximation results are given singular differential equations are approximated by using gegenbauer polynomials this method keeps the spectral accuracy the main idea and techniques used in this paper are also applicable to some singular problems in multiple dimensional spaces

the perturbed nonhomogeneous markov system abstract in this paper we introduce for the first time the concept of a perturbed nonhomogeneous markov system p nhms the expected population structure is found and its asymptotic behavior is provided under more realistic assumptions than previous studies by relaxing the assumption that the imbedded nonhomogeneous markov chain of a nhms is converging to a homogeneous markov chain also we study the sensitivity of the limiting expected structure on perturbations of the limiting input probabilities for a nhms moreover the asymptotic behavior of the variances and covariances of the class sizes of a p nhms is considered and found in an elegant closed analytic form

multi phase dynamic constraint aggregation for set partitioning type problems dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems this speed up is mostly due to the use at each iteration of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model this aggregation is updated when needed to ensure the exactness of the overall approach in this paper we propose a new version of this method called the multi phase dynamic constraint aggregation method which essentially adds to the original method a partial pricing strategy that involves multiple phases this strategy helps keeping the size of the aggregated problem as small as possible yielding a faster average computation time per iteration and fewer iterations we also establish theoretical results that provide some insights explaining the success of the proposed method tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2 000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4 5

an investigation of resolution of 2 variate gibbs phenomenon abstract in this paper we consider an implementation approach of a method for the resolution of the gibbs phenomenon for f x y which has been discussed extensively by jung and shizgal j h jung b d shizgal generalization of the inverse polynomial reconstruction method in the resolution of the gibbs phenomenon j comput appl math 172 2004 131 151 j h jung b d shizgal inverse polynomial reconstruction of two dimensional fourier images j sci comput 25 3 2005 367 399 b d shizgal j h jung towards the resolution of the gibbs phenomenoa j comput appl math 161 2003 41 65 here we go through some details of applying that technique and also we try to work out its application on some time dependent partial differential equations numerical experiments confirm the efficiency of the method

gibbs phenomenon for fourier partial sums on zp abstract it is of interest to know whether the gibbs phenomenon occurs on a local field a p adic heaviside function on the group of p adic integers is defined as an analogy of real variable heaviside function and it is also shown that there exists the gibbs phenomenon with an undershoot of at least 1 p 1 as an approximate level as a consequence of the theorem we see that the fourier partial sum for a heaviside function does not converge at the point of discontinuity

asymptotic behavior of nonhomogeneous semi markov systems abstract we study the asymptotic behavior of a nonhomogeneous semi markov system population in discrete time after a series of definitions lemmas and theorems we firstly establish the conditions under which the ergodic behavior of a nonhomogeneous semi markov chain exists and then find the limit of the basic matrix of the chain q n s in closed form finally the existence of the asymptotic population structure of the nonhomogeneous semi markov system is studied and the limit is provided in closed analytic form

asymptotic behaviour of the survival probabilities in an inhomogeneous semi markov model for the migration process in credit risk abstract we start with the stochastic foundation of the general discrete time market of defaultable bonds we prove that the above market is viable if and only if there exists an equivalent martingale measure from which we construct the forward probability measure and under which the discounted default free bond price is a martingale assuming that the migration process of defaultable bonds evolves as an inhomogeneous semi markov process we study the asymptotic behaviour of survival probabilities we provide a method of estimating real world transition probability sequences for the semi markov process and statistics for testing their homogeneity over time

nonstationary cyclic behavior in markov systems abstract we introduce the concept of the nonstationary cyclic markov system as a natural extension of the cyclic systems found in recent literature we find the sets of d cyclic asymptotically attainable structures and establish the relationships that exist between these structures for a nonhomogeneous markov system in a deterministic or stochastic environment the inherent connection between nonstationary cyclicity and stochastic environment is examined also we investigate the conditions under which there exists a geometric convergence to the equilibrium

on the gibbs phenomenon iii recovering exponential accuracy in a sub interval from a spectral partial sum of a piecewise analytic function we continue the investigation of overcoming the gibbs phenomenon i e obtaining exponential accuracy at all points including at the discontinuities themselves from the knowledge of a spectral partial sum of a discontinuous but piecewise analytic function we show that if we are given the first n expansion coefficients of an l 2 function f x in terms of either the trigonometric polynomials or the legendre polynomials we can construct an exponentially convergent approximation to the point values of f x in any sub interval in which it is analytic

rate of convergence asymptotically attainable structures and sensitivity in non homogeneous markov systems with fuzzy states this paper is concerned with the rate of convergence the asymptotically attainable structures and the sensitivity of a non homogeneous markov system nhms in short with fuzzy states more specifically it is proved that under some realistic assumptions easily met in practice the convergence rate of the relative population structure to its limit is geometric furthermore the asymptotically attainable structures in a nhms with fuzzy states are given more precisely the problem of finding which states are possible as limiting ones in a nhms with fuzzy states is studied provided that the limit of the sequence of the input probability vectors is controlled moreover the sensitivity of the limiting expected population structure on perturbations of the limiting input probabilities for the system is also studied two applications in cognitive science and in manpower planning are given where the sensitivity of two nhms with three states is studied provided that the limiting input probability vector is perturbed

semi markov migration process in a stochastic market in credit risk abstract in the present the idea of stochastic market environment comes into play to express the changes in general economy which affects any industry in small or great amounts of turbulence we model the evolution of the market among its possible states as an f inhomogeneous semi markov process this idea leads us to modeling the migration process of defaultable bonds as different f inhomogeneous semi markov process the survival probabilities of a defaultable bond in every credit grade are found the asymptotic behaviour of the survival probabilities is established under certain conditions also it is proved under what conditions the convergence is geometrically fast the stochastic foundation of the general stochastic discrete time market is provided by proving that the market is viable if and only if there exists an equivalent martingale measure from which we construct the forward probability measure and under which the discounted default free bond price process for all possible states of the market is a martingale the term structure of credit spread and the change of real world probability measure to forward probability measure are studied in the form of a theorem it is proved that under certain conditions changing the real probability measure to a forward probability measure does not affect the inhomogeneous semi markov process modeling the migration of defaultable bonds that is it is proved that it only changes the basic parameters and we provide a relation among the transition probabilities under the two measures finally parameter estimation and calibration of the inhomogeneous semi markov chain in stochastic environment is being provided

fuzzy semi markov migration process in credit risk we explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory we prove that a fuzzy market is viable if and only if an equivalent martingale measure exists from which we construct the forward probability measure and under which the discounted price of a default free bond is a martingale we model the evolution of credit migration of a defaultable bond as an inhomogeneous semi markov process with fuzzy states we study the effects of changing the real probability measure to a forward probability measure in addition we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default finally we discuss parameter estimation and calibration of the inhomogeneous markov chain with fuzzy states

lagrange interpolation for continuous piecewise smooth functions this note is devoted to lagrange interpolation for continuous piecewise smooth functions a new family of interpolatory functions with explicit approximation error bounds is obtained we apply the theory to the classical lagrange interpolation

accuracy of high order and spectral methods for hyperbolic conservation laws with discontinuous solutions higher order and spectral methods have been used with success for elliptic and parabolic initial and boundary value problems with smooth solutions on the other hand higher order methods have been applied to hyperbolic problems with less success as higher order approx imations of discontinuous solutions suffer from the gibbs phenomenon we extend past work and show that spectral methods yield spectral convergence of moments even when applied to problems with discontinuous solutions besides spectral fourier methods for periodic domains we also prove high order convergence for adjoint consistent non periodic numerical methods exemplified by the discontinuous galerkin finite element method

periodicity of asymptotically attainable structures in nonhomogeneous markov systems we find the sets of d periodic asymptotically attainable structures and we establish the periodicities that exist between these structures for a nonhomogeneous markov system in the case where the imbedded nonhomogeneous markov chain is periodic with period d also it is proved that under certain conditions each converging subsequence of the sequence of relative structures has a geometric rate of convergence

stability analysis and stabilization of discrete time non homogeneous semi markov jump linear systems a polytopic approach abstract this paper addresses the issues of stability analysis and stabilizing controller design for two classes of discrete time non homogeneous semi markov jump linear systems s mjlss the first class is concerned with the sojourn time probability mass functions that are independent of jump instants while the other considers the existence of probability mass functions of sojourn time depending on jump instants new techniques are developed respectively for both classes of systems by transforming their corresponding time varying semi markov kernels into more tractable forms based on the stochastic stability allowing for bounded sojourn time the derived stability and stabilization criteria containing the probability distribution information of random sojourn time can be numerically tested in order to reduce the possible conservatism of the obtained results a novel class of polytopic quadratic lyapunov function is constructed which depends on both the current mode and the elapsed time in the current mode finally the proposed control strategies are applied to an illustrative application of automotive electronic throttle valve to show the effectiveness of the theoretical results the results also reveal the significance of considering the inhomogeneity of s mjlss and the importance of our constructed lyapunov function

on exponential convergence of gegenbauer interpolation and spectral differentiation this paper is devoted to a rigorous analysis of exponential convergence of polynomial interpolation and spectral differentiation based on the gegenbauer gauss and gegenbauer gauss lobatto points when the underlying function is analytic on and within an ellipse sharp error estimates in the maximum norm are derived

on the solution of highly degenerate linear programs the solution of scheduling problems often gives rise to highly degenerate linear programmes which cause significant computational difficulties for the revised simplex method wolfe s highly effective ad hoc method for overcoming the cycling or stalling problems associated with degeneracy is described here it is given a geometric interpretation in terms of finding a direction of recession for a reduced problem which is fundamental to a full understanding of the procedure an example of an aircrew scheduling problem is used to illustrate the effectiveness of the method

resolving degeneracy in combinatorial linear programs steepest edge steepest ascent and parametric ascent while variants of the steepest edge pivot rule are commonly used in linear programming codes they are not known to have the theoretically attractive property of avoiding an infinite sequence of pivots at points of degeneracy an example is presented demonstrating that the steepest edge pivot rule can fail to terminate finitely it is then shown that a natural extension of the steepest edge pivot rule based on steepest ascent is guaranteed to determine a direction of ascent or a proof that no such direction exists after a finite number of pivots although without modification the extension may not terminate with an ascent direction corresponding to an edge finally it is demonstrated that a computationally more efficient pivot rule proposed by magnanti and orlin has similar theoretical properties to steepest ascent with probability 1independent of the linear program being solved unlike alternative methods such as primal lexicographic rules and bland s rule the algorithms described here have the advantage that they choose the pivot element without explicit knowledge of the set of all active constraints at a point of degeneracy thus making them attractive in combinatorial settings where the linear program is represented implicitly

a stability barrier for reconstructions from fourier samples we prove that any stable method for resolving the gibbs phenomenon that is recovering high order accuracy from the first m fourier coefficients of an analytic and nonperiodic function can converge at best root exponentially fast in m any method with faster convergence must also be unstable and in particular exponential convergence implies exponential ill conditioning this result is analogous to a recent theorem of platte trefethen and kuijlaars concerning recovery from pointwise function values on an equispaced m grid the main step in our proof is an estimate for the maximal behavior of a polynomial of degree n with bounded m term fourier series this estimate is related to a conjecture of hrycak and grochenig in the final part of the paper we discuss the implications of our main theorem to polynomial based interpolation and least squares approaches for overcoming the gibbs phenomenon

evolution of a time dependent markov model for training and recruitment decisions in manpower planning the current paper deals with the asymptotic behavior of a generalized time dependent markov manpower planning model this model on the one hand takes into account the need of the employees for eternal training and on the other hand its workforce demand at each time period is satisfied through internal mobility and two streams of recruitment the first originating from the outside environment and the second from a supplementary auxiliary system for the model under discussion we provide two sets of conditions concerning the existence and the specification of its expected limiting structure a numerical application illustrating the present results in a typical manpower system follows

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

on the integer properties of scheduling set partitioning models abstract many scheduling problems arising in the transportation industry can be posed as massive set partitioning zero one integer programmes for reasons of computational complexity it is generally unrealistic to attempt to solve the model in this form using conventional integer linear programming by the imposition of additional structure derived from the real world problem but not already implicit in the mathematical model it is possible to significantly reduce the effects of computational complexity and provide an effective method of obtaining good feasible solutions in this paper recent results in graph theory concerning natural integer properties of set partitioning integer programmes are discussed these results motivate the development of further implicit constraints which simultaneously reduce the dimensionality and increase the proportion of integer basic feasible solutions of the set partitioning linear programme

asymptotically attainable structures in nonhomogeneous markov systems in this paper we provide and study a new aspect in the classical problem of the asymptotic behavior of a nonhomogeneous markov system nhms the set of asymptotically attainable structures is determined as the convex hull of k vertices which are given also the rate of convergence to asymptotically attainable structures is studied and conditions are provided for the convergence to be geometrically fast finally an illustration of the results with data from manpower systems is provided

a new degeneracy method and steepest edge based conditioning for lp a new recursive method for resolving degeneracy in simplex like methods for linear programming lp is described the method provides a guarantee of termination even in the presence of round off errors and is readily implemented in contrast to a previous method of the author this method works throughout in the primal space one consequence is that the steepest edge criterion can be used on all iterations and at all levels of recursion it is also shown that the associated steepest edge coefficients provide information from which the expected condition of the current lp basis can be calculated cheaply this provides a more accurate indication of the actual condition of a system than is obtained from norm based condition numbers this idea also enables the condition of null space matrices to be estimated

trouble with gegenbauer reconstruction for defeating gibbs phenomenon runge phenomenon in the diagonal limit of gegenbauer polynomial approximations to defeat gibbs phenomenon in fourier and chebyshev series gottlieb et al d gottlieb c w shu a solomonoff h vandeven on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function j comput appl math 43 1992 81 98 developed a gegenbauer reconstruction the partial sums of the fourier or other spectral series are reexpanded as a series of gegenbauer polynomials c n m x recovering spectral accuracy even in the presence of shock waves or other discontinuities to achieve a rate of convergence which is exponential in n however gegenbauer reconstruction requires increasing the order m of the polynomials linearly with the truncation n of the series m bn for some constant b 0 when the order m is fixed it is well known that the gegenbauer series converges as n everywhere on x 1 1 if f x the function being expanded is analytic on the interval but what happens in the diagonal limit in which m n tend to infinity simultaneously we show that singularities of f x off the real axis can destroy convergence of this diagonal approximation process in the sense that the error diverges for subintervals of x 1 1 gegenbauer reconstruction must therefore be constrained to use a sufficiently small ratio of order m to truncation n this off axis singularity constraint is likely to impair the effectiveness of the reconstruction in some applications

on the limiting behaiour of a nonhomogeneour markov chain model in manpower systems summary we study the limiting structure of an expanding manpower system where the nonhomogeneous markov chain model proposed by young vassiliou 1974 is applicable the limiting structure and the relative limiting structure exist under certain conditions which are easily met in practical situations finally the problem of weak ergodicity in the above manpower systems is studied necessary and sufficient conditions are provided for weak ergodicity

discontinuous piecewise linear optimization a theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known a penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions it is straightforwardly extendable using standard nonlinear programming techniques tononlinear discontinuous piecewise differentiable functions the descent algorithm which is elaborated uses active set and projected gradient approaches it is a generalization of the ideas used by conn to deal with nonsmoothness in thel1 exact penalty function and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part the constrained case is reduced to the unconstrained minimization of a piecewise linear l1 exact penalty function we also discuss how the algorithm is modified when it encounters degenerate points preliminary numerical results are presented the algorithm is applied to discontinuous optimization problems from models in industrial engineering 1998 the mathematical programming society inc published by elsevier science b v