the algebraic entropy of classical mechanics we describe the lie algebra of classical mechanics modelled on the lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical poisson bracket it is a polynomially graded lie algebra a class we introduce we describe these lie algebras give an algorithm to calculate the dimensions cn of the homogeneous subspaces of the lie algebra of classical mechanics and determine the value of its entropy limn cn1 n it is 1 825 423 774 201 08 a fundamental constant associated with classical mechanics

physics of symplectic integrators perihelion advances and symplectic corrector algorithms symplectic integrators evolve dynamical systems according to modified hamiltonians whose error terms are also well defined hamiltonians the error of the algorithm is the sum of each error hamiltonian s perturbation on the exact solution when symplectic integrators are applied to the kepler problem these error terms cause the orbit to precess in this work by developing a general method of computing the perihelion advance via the laplace runge lenz vector even for nonseparable hamiltonians i show that the precession error in symplectic integrators can be computed analytically it is found that at leading order each paired error hamiltonians cause the orbit to precess oppositely by exactly the same amount after each period hence symplectic corrector or process integrators which have equal coefficients for these paired error terms will have their precession errors cancel at that order after each period with the use of correctable algorithms both the energy and precession error are of effective order n 2 where n is the nominal order of the algorithm thus the physics of symplectic integrators determines the optimal algorithm for integrating long time periodic motions

fuzzy stage characteristic preserving product life cycle modeling abstract stage concerns have been important in product life cycles plc such concerns are stage identification stage related strategies and here newly introduced stage modeling stage modeling is concerned with not only modeling but also aggregation of individual stages in an overall influencing manner it not only preserves the respective characteristics of the stages but also may be explored for the stage related strategy issue later to date this aspect of plc modeling has not yet been explored in this paper a fuzzy plc modeling capable of preserving the fuzzy individual characteristics of the stages is proposed the various concepts such as boundary identification fuzzy stages modeling and stages inter influence functions are introduced and discussed finally to illustrate and support the approach a numerical example and a comparison with classical data analytic procedures segmented polynomial regression and autoregressive integrated moving average is provided

reduction to least squares estimates in multiple fuzzy regression analysis in this paper we deal with the problem of least squares multiple regression with fuzzy data the regression coefficients are assumed to be real crisp a formula for solving the regression coefficients in one variable models is derived if each independent variable is effective i e its corresponding regression coefficient is nonzero the multiple regression problem can be replaced with a 0 1 programming problem its optimal solution is easily computed finally we also propose effective algorithms to compute the regression coefficients in a general case

general fuzzy least squares abstract a model for least squares fitting of fuzzy valued data is described a previous process obtained for triangular fuzzy numbers is generalized and improved to include all fuzzy numbers represented by single maxima piecewise continuous functions with compact support the new model is compared with the previous procedure and its applicability is demonstrated

fuzzy nonlinear regression with fuzzified radial basis function network a fuzzified radial basis function network frbfn is a kind of fuzzy neural network that is obtained by direct fuzzification of the well known neural model rbfn a frbfn contains fuzzy weights and can handle fuzzy in fuzzy out data this paper shows that a frbfn can also be interpreted as a kind of fuzzy expert system hence it owns the advantages of simple structure and clear physical meaning some metrics for fuzzy numbers have been extended to the metrics for n dimensional fuzzy vectors which are applicable to computations in frbfns the corresponding metric spaces for n dimensional fuzzy vectors are proved to be complete further frbfns are proved to be able to act as universal function approximators for any continuous fuzzy function defined on a compact set this paper applies the proposed frbfn to nonparametric fuzzy nonlinear regression problems for multidimensional lr type fuzzy data fuzzy nonlinear regression with frbfns can be formulated as a nonlinear mathematical programming problem two training algorithms are proposed to quickly solve the two types of problems under different criteria and constraint conditions namely the two stage and bp back propagation training algorithms simulation studies are carried out to verify the feasibility and demonstrate the advantages of the proposed approaches

a fully adaptive reaction diffusion integration scheme with applications to systems biology adaptive integration schemes for ode systems typically function by adjusting the time step size so as to keep the truncation error below some desired value for adaptive integration of pde systems involving coupled kinetic reaction and diffusion operations truncation error arises not only from the individual propagators but also from their method of coupling a common second order accurate method for coupling operators is strang s method of operator splitting we derive an expression for the truncation error resulting from strang splitting reaction and diffusion operators for an arbitrary number of spatial dimensions and demonstrate its use in adaptive time step algorithms in addition we present explanations of the second order implicit reaction and diffusion operators and their individual error calculations used in our implementation of the scheme finally using example simulations we discuss the use of this calculation for problems in systems biology

inferring operating rules for reservoir operations using fuzzy regression and anfis the methods of ordinary least squares regression olsr fuzzy regression fr and adaptive network based fuzzy inference system anfis are compared in inferring operating rules for a reservoir operations optimization problem dynamic programming dp is used as an example optimization tool to provide the input output data set to be used by olsr fr and anfis models the coefficients of an fr model are found by solving a linear programming lp problem the objective function of the lp is to minimize the total fuzziness of the fr model which is related to the width of fuzzy coefficients in the regression model before applying fr to the reservoir operations problem two fr formulations and interval regression ir are first examined in a simple tutorial example anfis is also used to derive the reservoir operating rules as fuzzy if then rules the olsr fr and anfis based rules are then simulated and compared based on their performance in simulation the methods are applied to a long term planning problem as well as to a medium term implicit stochastic optimization model the results indicate that fr is useful to derive operating rules for a long term planning model where imperfect and partial information is available anfis is beneficial in medium term implicit stochastic optimization as it is able to extract important features of the system from the generated input output set and represent those features as general operating rules

possibilistic linear systems and their application to the linear regression model abstract it is the purpose of this paper to illustrate possibilistic linear systems based on possibility measure and to formulate a linear regression analysis by possibilistic linear models linear regression by a possibilistic model is called possibilistic linear regression this is a new interpretation of fuzzy linear regression and also includes a new method by which interval analysis can be done in fuzzy numbers

portfolio selection based on fuzzy probabilities and possibility distributions abstract in this paper two kinds of portfolio selection models are proposed based on fuzzy probabilities and possibility distributions respectively rather than conventional probability distributions in markowitz s model since fuzzy probabilities and possibility distributions are obtained depending on possibility grades of security data offered by experts investment experts knowledge can be reflected a numerical example of a portfolio selection problem is given to illustrate our proposed approaches

further examination of fuzzy linear regression abstract fuzzy linear regression was originally introduced by tanaka et al 1982 sakawa and yano 1992 expanded fuzzy linear regression by introducing a method that allows fuzzy independent variables as well as a fuzzy dependent variable the purpose of this paper is to examine the strengths and weaknesses of this formulation suggest possible improvements and discuss fundamental questions about fuzzy linear regression

explicit lie poisson integration and the euler equations we give a wide class of lie poisson systems for which explicit lie poisson integrators preserving all casimirs can be constructed the integrators are extremely simple and have widespread applicability examples are the free rigid body a moment truncation and a new fast algorithm for the sine bracket truncation of the 2d euler equations

high order symplectic runge kutta nystro m methods a numerical method for ordinary differential equations is called symplectic if when applied to hamiltonian problems it preserves the symplectic structure in phase space thus reproducing the main qualitative property of solutions of hamiltonian systems the authors construct and test symplectic explicit runge kutta nystrom rkn methods of order 8 the outcome of the investigation is that existing high order symplectic rkn formulae require so many evaluations per step that they are much less efficient than conventional eighth order nonsymplectic variable step size integrators even for low accuracy however symplectic integration is of use in the study of qualitative features of the systems being integrated

composite methods for numerical solution of stiff systems of ode s this paper is concerned with methods which consist of sequential application of several different schemes for stiff ordinary differential systems with a predetermined ratio of step lengths we show that under some conditions such composite methods have higher order and better stability than the constituent schemes compositions of obrechkoff methods and of implicit runge kutta processes are investigated

integrators for lie poisson or dynamical systems abstract we reformulate the ge marsden 1988 lie poisson integration algorithm in terms of algebra variables alone and show how to implement it for regular quadratic lie algebras to arbitrary order and for arbitrary lie algebras to first order we also describe two generalizations of ruth s method 1983 that can be used to generate very fast algorithms

numerical integrators for the hybrid monte carlo method we construct numerical integrators for hamiltonian problems that may advantageously replace the standard verlet time stepper within hybrid monte carlo and related simulations past attempts have often aimed at boosting the order of accuracy of the integrator and or reducing the size of its error constants order and error constant are relevant concepts in the limit of vanishing step length we propose an alternative methodology based on the performance of the integrator when sampling from gaussian distributions with not necessarily small step lengths we construct new splitting formulae that require two three or four force evaluations per time step limited proof of concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard verlet method especially in problems with high dimensionality

fuzzy least squares linear regression analysis for fuzzy input output data a fuzzy regression model is used in evaluating the functional relationship between the dependent and independent variables in a fuzzy environment most fuzzy regression models are considered to be fuzzy outputs and parameters but non fuzzy crisp inputs in general there are two approaches in the analysis of fuzzy regression models linear programming based methods and fuzzy least squares methods in 1992 sakawa and yano considered fuzzy linear regression models with fuzzy outputs fuzzy parameters and also fuzzy inputs they formulated multiobjective programming methods for the model estimation along with a linear programming based approach in this paper two estimation methods along with a fuzzy least squares approach are proposed these proposed methods can be effectively used for the parameter estimation comparisons are also made between them

fuzzy least squares algorithms for interactive fuzzy linear regression models fuzzy regression analysis can be thought of as a fuzzy variation of classical regression analysis it has been widely studied and applied in diverse areas in general the analysis of fuzzy regression models can be roughly divided into two categories the first is based on tanaka s linear programming approach the second category is based on the fuzzy least squares approach in this paper new types of fuzzy least squares algorithms with a noise cluster for interactive fuzzy linear regression models are proposed these algorithms are robust for the estimation of fuzzy linear regression models especially when outliers are present numerical examples are given to detail the effectiveness of this approach

arbitrarily high order convected scheme solution of the vlasov poisson system abstract the convected scheme cs is a forward trajectory semi lagrangian method for solution of transport equations which has been most often applied to the kinetic description of plasmas and rarefied neutral gases in its simplest form the cs propagates the solution forward in time by advecting the so called moving cells along their characteristic trajectories and by remapping them on the mesh at the end of the time step the cs is conservative positivity preserving simple to implement and it is not subject to time step restriction to maintain stability recently guclu and hitchon 2012 1 a new methodology was introduced for reducing numerical diffusion based on a modified equation analysis the remapping error was compensated by applying small corrections to the final position of the moving cells prior to remapping while the spatial accuracy was increased from 2nd to 4th order the new scheme retained the important properties of the original method and was shown to be extremely simple and efficient for constant advection problems here the cs is applied to the solution of the vlasov poisson system which describes the evolution of the velocity distribution function of a collection of charged particles subject to reciprocal coulomb interactions the vlasov equation is split into two constant advection equations one in configuration space and one in velocity space and high order time accuracy is achieved by proper composition of the operators the splitting procedure enables us to use the constant advection solver which we extend to arbitrarily high order of accuracy in time and space a new improved procedure is given which makes the calculation of the corrections straightforward focusing on periodic domains we describe a spectrally accurate scheme based on the fast fourier transform the proposed implementation is strictly conservative and positivity preserving the ability to correctly reproduce the system dynamics as well as resolving small scale features in the solution is shown in classical 1d 1v test cases both in the linear and the non linear regimes

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

least squares model fitting to fuzzy vector data abstract we consider model fitting problems in which the mathematical model of an observed event is a system of equations involving model parameters and observables and in which the data are fuzzy vectors such problems naturally arise in applications when data are scarce and information is vague about distributions of variances that are contained in the observations then it may only be possible to obtain estimates of membership functions of the data a model can be fitted to such data by maximizing the membership values of the adjusted observations we achieve this by minimizing the sum of squares of the deviations of the membership values from one the optimization problem simplifies significantly if the membership functions of the fuzzy vectors belong to a class of conical functions which are defined in terms of an elliptic norm properties of these functions are discussed and a membership propagation formula derived which component wise obeys the extension principle the least squares model fitting to the described fuzzy vector data deviates in important aspects from ordinary least squares one obtains as a solution a fuzzy model parameter vector and a number of indicators which describe the quality of the fit viz the grade of compatibility between data and model their discord and dilators of model and data membership functions these new concepts are illustrated and discussed in the context of an example from a terminal ballistics problem

development of a cost function for wastewater treatment systems with fuzzy regression abstract a fuzzy linear regression analysis is performed to study the cost function of wastewater that is developed by using a pool of data from 26 municipal wastewater treatment systems in taiwan the choice of a primary or more higher wastewater treatment process is intimately related to financial land availability effluent standards and managerial factors especially financial consideration plays an important role in the planning stage in addition cost estimation is difficult to precisely evaluate in an uncertain environment in the conventional regression model deviations between the observed values and the estimated values are supposed to be due to measurement errors here taking a different perspective these deviations are regarded as the fuzziness of the system s parameters thus these deviations are reflected in a linear function with fuzzy parameters using linear programming algorithm this fuzzy linear regression model might be very convenient and useful for finding a fuzzy structure in an evaluation system in this paper the details of the fuzzy linear regression concept and its applications to the cost function in an uncertain environment are shown and discussed

fuzzy strategic replacement analysis abstract this paper presents a fuzzy methodology for replacement of equipment issues such as fuzzy modeling of degradation parameters and determining fuzzy strategic replacement and economic lives are extensively discussed for the strategic purpose addible market and cost effects from the replacements against the counterpart i e the existing equipment cost and market obsolescence are modeled fuzzily and interactively in addition to the equipment deterioration both the standard fuzzy arithmetic and here re termed requisite constraint vertex fuzzy arithmetic or the vertex method are applied and investigated two numerical examples and the results are provided

practical symplectic partitioned runge kutta and runge kutta nystrom methods we present new symmetric fourth and sixth order symplectic partitioned runge kutta and runge kutta nystrom methods we studied compositions using several extra stages optimising the efficiency an effective error ef is defined and an extensive search is carried out using the extra parameters the new methods have smaller values of ef than other methods found in the literature when applied to several examples they perform up to two orders of magnitude better than previously known method which is in very good agreement with the values of ef

outliers detection and confidence interval modification in fuzzy regression abstract the existence of outliers in a set of experimental data can cause incorrect interpretation of the fuzzy linear regression results peters fuzzy sets and systems 63 1994 45 55 considered this problem for the nonfuzzy input and nonfuzzy output data type the present investigation focuses on nonfuzzy input and fuzzy output data type and proposes approaches to handle the outlier problem the main idea is to introduce a pre assigned k limiting value whose value must be determined based on the conditions of the current problem the advantages and problems of the proposed approaches are compared and discussed

portfolio selection based on upper and lower exponential possibility distributions in this paper two kinds of possibility distributions namely upper and lower possibility distributions are identified to reflect experts knowledge in portfolio selection problems portfolio selection models based on these two kinds of distributions are formulated by quadratic programming problems it can be said that a portfolio return based on the lower possibility distribution has smaller possibility spread than the one on the upper possibility distribution in addition a possibility risk can be defined as an interval given by the spreads of the portfolio returns from the upper and the lower possibility distributions to reflect the uncertainty in real investment problems a numerical example of a portfolio selection problem is given to illustrate our proposed approaches

composite integrators for bi hamiltonian systems abstract symmetric composition methods are applied to dynamical systems in bi hamiltonian form to lotka volterra equations to the completely integrable lorenz equation and to the periodic toda lattice the numerical results obtained show that the hamiltonians are preserved with high accuracy and the periodicity of the solutions are retained

numerical integration of lie poisson systems while preserving coadjoint orbits and energy in this paper we apply geometric integrators of the rkmk type to the problem of integrating lie poisson systems numerically by using the coadjoint action of the lie group g on the dual lie algebra mbox normalsize mathfrak g to advance the numerical flow we devise methods of arbitrary order that automatically stay on the coadjoint orbits first integrals known as casimirs are retained to machine accuracy by the numerical algorithm within the proposed class of methods we find integrators that also conserve the energy these schemes are implicit and of second order nonlinear iteration in the lie algebra and linear error growth of the global error are discussed numerical experiments with the rigid body and a finite dimensional truncation of the euler equations for a two dimensional 2d incompressible fluid are used to illustrate the properties of the algorithm

theoretical choice of the optimal threshold for possibilistic linear model with noisy input based on possibility concepts various possibilistic linear models plms have been proposed and their pivotal role in fuzzy modeling and associated applications has been established when adopting plms one has to adopt an appropriate threshold lambda value however choosing such a value is by no means trivial and is still an open theoretical issue in this paper we propose a solution by first extending the plm to its regularized version i e a regularized plm rplm such that its generalization capability can be enhanced the rplm is then formulated as a maximum a posteriori map framework which facilitates the determination of the theoretically optimal threshold value for the rplm with noisy input our mathematical derivations reveal the approximately inversely proportional relationship between the threshold and the standard deviation of gaussian noisy input this is also confirmed by the simulation results this finding is very helpful for the practical applications of both plms and rplms

fourth order symplectic integration abstract in this paper we present an explicit fourth order method for the integration of hamilton s equations this method preserves the property that the time evolution of such a system yields a canonical transformation from the initial conditions to the final state that is the integration step is an explicit symplectic map although the result is first derived for a specific type of hamiltonian it is shown to be quite general in particular the results can be applied to any lie group

fuzzy linear systems a general fuzzy linear system is investigated using the embedding approach conditions for the existence of a unique fuzzy solution to n n linear system are derived and a numerical procedure for calculating the solution is designed the applicability of the proposed model is illustrated with examples

fuzzy seasonality forecasting abstract in this paper a fuzzy forecasting technique for seasonality in the time series data is presented using the following procedure first with the fuzzy regression analysis the fuzzy trend of a time series is analyzed then the fuzzy seasonality is defined by realizing the membership grades of the seasons to the fuzzy regression model both making fuzzy forecast and crisp forecast are investigated seasonal fuzziness and trends are analyzed the method is applied to the sales forecasting problem of a food distribution company

possibilistic linear regression analysis for fuzzy data abstract fuzzy data given by expert knowledge can be regarded as a possibility distribution by which possibilistic linear systems are defined recently it has become important to deal with fuzzy data in connection with expert knowledge three formulations of possibilistic linear regression analysis are proposed here to deal with fuzzy data since our formulations can be reduced to linear programming problems the merit of our formulations is to be able to obtain easily fuzzy parameters in possibilistic linear models and to add other constraint conditions which might be obtained from expert knowledge of fuzzy parameters this approach can be regarded as a fuzzy interval analysis in a fuzzy environment

combining metamodels with rational function representations of discretization error for uncertainty quantification abstract techniques for producing metamodels for the efficient monte carlo simulation of high consequence systems are presented the bias of f e m mesh discretization errors is eliminated or minimized by extrapolation using rational functions rather than the power series representation of richardson extrapolation examples including estimation of the vibrational frequency of a one dimensional bar show that the rational function model gives more accurate estimates using fewer terms than richardson extrapolation an important consideration for computational reliability assessment of high consequence systems where small biases in solutions can significantly affect the accuracy of small magnitude probability estimates rational function representation of discretization error enable the user to accurately extrapolate to the continuum from numerical experiments performed outside the asymptotic region of the usual power series allowing use of coarser meshes in the numerical experiments resulting in significant savings

method for treating discretization error in nondeterministic analysis a response surface methodology based technique is presented for treating discretization error in non deterministic analysis the response surface or metamodel is estimated from computer experiments which vary both uncertain physical parameters and the fidelity of the computational mesh the resultant metamodel is then used to propagate the variabilities in the continuous input parameters while the mesh size is taken to zero its asymptotic limit with respect to mesh size the metamodel is equivalent to richardson extrapolation in which solutions on coarser and finer meshes are used to estimate discretization error the method is demonstrated on a one dimensional prismatic bar in which uncertainty in the third vibration frequency is estimated by propagating variations in material modulus density and bar length the results demonstrate the efficiency of the method for combining non deterministic analysis with error estimation to obtain estimates of total simulation uncertainty the results also show the relative sensitivity of failure estimates to solution bias errors in a reliability analysis particularly when the physical variability of the system is low

interval regression analysis by quadratic programming approach when we use linear programming in possibilistic regression analysis some coefficients tend to become crisp because of the characteristic of linear programming on the other hand a quadratic programming approach gives more diverse spread coefficients than a linear programming one therefore to overcome the crisp characteristic of linear programming we propose interval regression analysis based on a quadratic programming approach another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression by changing the weights of the quadratic function we can analyze the given data from different viewpoints for data with crisp inputs and interval outputs the possibility and necessity models can be considered therefore the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed even though there always exist possibility estimation models the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model thus we consider polynomials as regression models since any curve can be represented by the polynomial approximation using polynomials we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models furthermore from the obtained possibility and necessity regression models a trapezoidal fuzzy output can be constructed

fuzzy sets as a basis for a theory of possibility the theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable more specifically if f is a fuzzy subset of a universe of discourse u u which is characterized by its membership function f then a proposition of the form x is f where x is a variable taking values in u induces a possibility distribution tx which equates the possibility of x taking the value u to f u the compatibility of u with f in this way x becomes a fuzzy variable which is associated with the possibility distribution tx in much the same way as a random variable is associated with a probability distribution in general a variable may be associated both with a possibility distribution and a probability distribution with the weak connection between the two expressed as the possibility probability consistency principle a thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is in the main possibilistic rather than probabilistic in nature thus by employing the concept of a possibility distribution a proposition p in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p several types of conditional translation rules are discussed and in particular a translation rule for propositions of the form x is f is possible where is a number in the interval 0 1 is formulated and illustrated by examples

fuzzy data analysis by possibilistic linear models abstract since fuzzy data can be regarded as distribution of possibility fuzzy data analysis by possibilistic linear models is proposed in this paper possibilistic linear systems are defined by the extension principle fuzzy parameter estimations are discussed in possibilistic linear systems and possibilistic linear models are employed for fuzzy data analysis with non fuzzy inputs and fuzzy outputs defined by fuzzy numbers the estimated possibilistic linear system can be obtained by solving a linear programming problem this approach can be regarded as fuzzy interval analysis

preservation of stability properties near fixed points of linear hamiltonian systems by symplectic integrators abstract based on reasonable testing model problems we study the preservation by symplectic runge kutta method srk and symplectic partitioned runge kutta method sprk of structures for fixed points of linear hamiltonian systems the structure preservation region provides a practical criterion for choosing step size in symplectic computation examples are given to justify the investigation

identification of possibilistic linear systems by quadratic membership functions of fuzzy parameters abstract we have already discussed several models of the possibilities linear regression analysis under the assumption that probabilitioes parameters in the models are non interactive i e the possibilistic distribution of parameters is defined by minimum operators in this paper quadratic membership functions as defined by a celmi s are considered to propose an identification method of interactive fuzzy parameters in possibilistic linear systems our method can be reduced to linear programming so that it is very easy to obtain the posssibilistic distribution of parameters

fuzzy regression methods a comparative assessment abstract the fundamental differences between fuzzy regression and ordinary regression are identified in this paper fuzzy regression can be used to fit fuzzy data and crisp data into a regression model whereas ordinary regression can only fit crisp data through a comprehensive literature review three approaches of fuzzy regression are summarized the first approach of fuzzy regression is based on minimizing fuzziness as an optimal criterion the second approach uses least squares of errors as a fitting criterion and two methods are summarized in this paper the third approach can be described as an interval regression analysis for each fuzzy regression method numerical examples and graphical presentations are used to evaluate their characteristic and differences with ordinary least squares regression based on the comparative assessment the fundamental differences between ordinary least squares regression and conventional fuzzy regression are concluded that is ordinary least squares regression modeling data with randomness type of uncertainty and conventional fuzzy regression modeling data with fuzziness type of uncertainty in order to integrate both randomness and fuzziness types of uncertainty into one regression model the concept of hybrid fuzzy least squares regression analysis is proposed in this paper and the details of its method are derived in the accompanying paper

explicit unitary schemes to solve quantum operator equations of motion to solve operator equations of motion in quantum mechanics and in a quantum scalar field theory we propose an explicit unitary scheme with arbitrary high order of accuracy

higher order hybrid monte carlo algorithms we present a simple recursive iteration of the leapfrog discretization of newton s equations which leads to a removal of the finite step size error to any desired order this is done in a manner that preserves phase space areas and reversibility as required for use in the hybrid monte carlo method for simulating fermionic fields the resulting asymptotic volume dependence is exp ln ital v sup 1 2 we test the scheme on the 2 1 dimensional hubbard model

canonical transforms and the efficient integration of quantum mechanical wave equations the integration of time dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry the wave function s energy spectrum as well as its momentum spectrum impose fundamental limits on the performance of numerical algorithms for the solution of wave equations we demonstrate how canonical transforms may be applied to negotiate these limitations and to increase the performance of numerical algorithms by up to several orders of magnitude our approach includes the so called kramers henneberger transform as a special case and puts forward modifications toward an improved numerical efficiency

an orderwise polynomial regression procedure for fuzzy data a procedure for polynomial fit with fuzzy data is presented as in real case studies there is often dependence between modes and spreads we propose a regression polynomial model capable to take into account possible interactions between the estimated spreads and modes this method finds applications in several fields such as reliability quality control psychometrics marketing image processing etc an application to a software reliability problem is also presented

high order staggered schemes for compressible hydrodynamics weak consistency and numerical validation abstract staggered grids schemes formulated in internal energy are commonly used for cfd applications in industrial context here we prove the consistency of a class of high order lagrange remap staggered schemes for solving the euler equations in 1d and 2d on cartesian grids the main result of the paper is that using an a posteriori internal energy corrector the lagrangian schemes are proved to be conservative in mass momentum and total energy and to be weakly consistent with the 1d lagrangian formulation of the euler equations extension in 2d is done using directional splitting methods and face staggering numerical examples in both 1d and 2d illustrate the accuracy the convergence and the robustness of the schemes

symbolic calculation in development of algorithms split step methods for the gross pitaevskii equation we employ symbolic calculation to perform a systematic study of the accuracy of split step fourier transform methods for the time dependent gross pitaevskii equation gpe provided the most recent approximation for the wave function is always used in the nonlinear atom atom interaction energy every split step algorithm we have tried has the same order time stepping error for the nonlinear gpe and for the linear schroedinger equation

high order split step exponential methods for solving coupled nonlinear schrodinger equations numerical integration schemes for coupled time dependent nonlinear schrodinger equations are examined using exponential splitting step methods exponentiation of the nonlinear potential term is reduced to the exponential of a kinetic energy term which can be calculated by fast fourier transforms high order iteration schemes involving a minimum number of product operators are shown to yield highly accurate amplitude and phase these new splitting methods are shown to be highly efficient both with respect to accuracy and integration time

on the numerical integration of ordinary differential equations by symmetric composition methods differential equations of the form dot x x a b are considered where the vector fields a and b can be integrated exactly enabling numerical integration of x by composition of the flows of a and b various symmetric compositions are investigated for order complexity and reversibility free lie algebra theory gives simple formulae for the number of determining equations for a method to have a particular order a new more accurate way of applying the methods thus obtained to compositions of an arbitrary first order integrator is described and tested the determining equations are explored and new methods up to 100 times more accurate at constant work than those previously known are given

evaluating explicit functional relationships from fuzzy observations starting from ideas due to zadeh and sugeno we suggest two methods to transfer information given by fuzzy observations to fuzzy sets on the parameter region of a given explicit functional relationship expected cardinality and fuzzy expectation moreover we mention their use in soft modelling when several functional relationships are to be compared for the same fuzzy observations the procedures are illustrated by an example

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

dependency between degree of fit and input noise in fuzzy linear regression using non symmetric fuzzy triangular coefficients fuzzy linear regression flr model can be thought of as a fuzzy variation of classical linear regression model it has been widely studied and applied in diverse fields when noise exists in data it is a very meaningful topic to reveal the dependency between the parameter h i e the threshold value used to measure degree of fit in flr model and the input noise in this paper the flr model is first extended to its regularized version i e regularized fuzzy linear regression rflr model so as to enhance its generalization capability then rflr model is explained as the corresponding equivalent maximum a posteriori map problem finally the general dependency relationship that the parameter h with noisy input should follow is derived particular attention is paid to the regression model using non symmetric fuzzy triangular coefficients it turns out that with the existence of typical gaussian noisy input the parameter h is inversely proportional to the input noise our experimental results here also confirm this theoretical claim obviously this theoretical result will be helpful to make a good choice for the parameter h and to apply flr techniques effectively in practical applications

symplectic integration of hamiltonian wave equations the numerical integration of a wide class of hamiltonian partial differential equations by standard symplectic schemes is discussed with a consistent hamiltonian approach we discretize the hamiltonian and the poisson structure separately then form the the resulting ode s the stability accuracy and dispersion of different explicit splitting methods are analyzed and we give the circumstances under which the best results can be obtained in particular when the hamiltonian can be split into linear and nonlinear terms many different treatments and examples are compared

exponential possibility regression analysis by identification method of possibilistic coefficients abstract this paper introduces exponential possibility regression analysis by the proposed identification method to identify the exponential possibility distribution first we solve all possible different sets of the input output linear systems from the given data and obtain the corresponding coefficient vectors then we select minimum number of coefficient vectors which represent all the given data using the selected coefficient vectors we apply the identification method to determine a possibility distribution of coefficients in the fuzzy linear system as a criterion to select minimum coefficient vectors we use the inner product by the proposed approach the smaller area of possibility distribution can be obtained this is the main characteristic of our proposed regression comparing with the other regression analyses

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

composition methods for differential equations with processing we construct numerical integrators for differential equations up to order 12 obtained by composition of basic integrators the following cases are considered i composition for a system separable in two solvable parts ii composition using as basic methods a first order integrator and its adjoint iii composition using second order symmetric methods and iv composition using fourth order symmetric methods each scheme is implemented with a processor or corrector to improve their efficiency and this can be done virtually cost free

higher order splitting algorithms for solving the nonlinear schrodinger equation and their instabilities since the kinetic and potential energy terms of the real time nonlinear schroedinger equation can each be solved exactly the entire equation can be solved to any order via splitting algorithms we verified the fourth order convergence of some well known algorithms by solving the gross pitaevskii equation numerically all such splitting algorithms suffer from a latent numerical instability even when the total energy is very well conserved a detail error analysis reveals that the noise or elementary excitations of the nonlinear schroedinger equation obeys the bogoliubov spectrum and the instability is due to the exponential growth of high wave number noises caused by the splitting process for a continuum wave function this instability is unavoidable no matter how small the time step for a discrete wave function the instability can be avoided only for delta tk sub max sup 2 or approx 2 pi where k sub max pi delta x

any order propagation of the nonlinear schrodinger equation we derive an exact propagation scheme for nonlinear schrodinger equations this scheme is entirely analogous to the propagation of linear schrodinger equations we accomplish this by defining a special operator whose algebraic properties ensure the correct propagation as applications we provide a simple proof of a recent conjecture regarding higher order integrators for the gross pitaevskii equation extend it to multicomponent equations and to a new class of integrators

the normalized interval regression model with outlier detection and its real world application to house pricing problems tanaka and his colleagues initially proposed fuzzy linear regression models in 1982 from then on fuzzy regression analysis has been widely studied by many researchers since tanaka s model uses the inclusion relationship between the given data and the estimated data it was pointed out that tanaka s model is sensitive to outliers how to deal with the outlier problem has been a very important issue and attracted much attention from fuzzy regression researchers given crisp multiple independent variables and single interval dependent variable we introduce the normalized upper and lower interval regression models and propose two outlier detection approaches for them the dual problem of the normalized upper interval regression model is used to find out potential outliers for the normalized lower interval regression model the existence of the outliers makes it infeasible the relaxation approach is proposed to find out the outliers by deleting the outliers we can find out the possibilistic functional relationship of the major data a case study involving a house pricing problem is analyzed in detail household size loan ratio and annual household income are identified as the independent variables and acceptable purchase price is the interval dependent variable with the data collected from the survey in shanghai the normalized upper and lower interval regression models are built with deleting the outliers the proposed models provide managerial insights into the real estate market and important policy implications in regulating urban land development

exponential possibility regression analysis abstract this paper aims at giving the application of possibility theory based on exponential distributions to regression analysis a general outline of the possibility theory is illustrated by exponential possibility distributions it is shown that possibility analysis is well corresponding to statistical analysis using exponential possibility analysis we propose possibility regression analysis which is suitable for rough phenomena arising from social and economic systems the spread of the given data can be directly transferred into the exponential possibility distribution of coefficients in the regression model by the proposed method