the factorization method for the numerical solution of two point boundary value problems for linear ode s the theoretical analysis and computational implementation of factorization based methods for the numerical solution of linear boundary value problems for ordinary differential equations are presented the methods are optimal with respect to certain clearly defined criteria numerical examples show the effectiveness of a general code based on the factorization method

a natural approach to the numerical integration of riccati differential equations this paper introduces a new class of methods which we call mobius schemes for the numerical solution of matrix riccati differential equations the approach is based on viewing the riccati equation in its natural geometric setting as a flow on the grassmannian of m dimensional subspaces of an n m dimensional vector space since the grassmannians are compact differentiable manifolds and the coefficients of the equation are assumed continuous there are no singularities or intrinsic instabilities in the associated flow the presence of singularities and numerical instabilities is an artifact of the coordinate system but since mobius schemes are based on the natural geometry they are able to deal with numerical instability and pass accurately through the singularities a number of examples are given to demonstrate these properties

the delivery man problem and cumulative matroids given a complete directed graph g v a the delivery man problem dmp consists of determining a hamiltonian circuit minimizing the sum of distances along the circuit from a given vertex v1 to every vertex of v including v1 itself there exists a number of applications of the dmp in the fields of distribution and machine scheduling the dmp is np hard the objective of this paper is to develop new theoretical results and an exact algorithm for the problem a new integer linear programming formulation is provided and results on the matroidal structure of a class of combinatorial problems are developed these are used to derive lower bounds for the dmp these bounds are embedded into an enumerative algorithm the largest problems solved to optimality with the proposed algorithm involve 60 vertices this compares favorably with previously published methods

the matrix sign function a survey of the matrix sign function is presented including some historical background definitions and properties approximation theory and computational methods and condition theory and estimation procedures applications to areas such as control theory eigendecompositions and roots of matrices are outlined and some new theoretical results are also given

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

multimodal admittance method in waveguides and singularity behavior at high frequencies this work presents a multimodal method for the propagation in a waveguide with varying height and its relation to trapped modes or quasi trapped modes the coupled mode equations are obtained by projecting the helmholtz equation on the local transverse modes to solve this problem we integrate the riccati equation governing the admittance matrix dirichlet to neumann operator for many propagating modes i e at high frequencies the numerical integration of the riccati equation shows that the rule is that this matrix has quasi singularities associated to quasi trapped modes

invariant imbedding the box scheme and an equivalence between them we show for general linear two point boundary value problems that the box scheme and invariant imbedding are equivalent in the sense that a specific algorithm for solving the difference equations is valid if and only if an appropriate imbedding is valid a key element in the equivalence is the role played by pivoting to solve the linear equations and switching to get a finite imbedding part of the equivalence proof shows that imbedding is the completion of the box scheme multiple imbeddings are introduced and a lower bound for the critical length problem is obtained

magnus integrators for solving linear quadratic differential games we consider magnus integrators to solve linear quadratic n player differential games these problems require to solve backward in time non autonomous matrix riccati differential equations which are coupled with the linear differential equations for the dynamic state of the game to be integrated forward in time we analyze different magnus integrators which can provide either analytical or numerical approximations to the equations they can be considered as time averaging methods and frequently are used as exponential integrators we show that they preserve some of the most relevant qualitative properties of the solution for the matrix riccati differential equations as well as for the remaining equations the analytical approximations allow us to study the problem in terms of the parameters involved some numerical examples are also considered which show that exponential methods are in general superior to standard methods

on the pollution effect in fe solutions of the 3d helmholtz equation we study convergence of the fem for a 3d problem of rigid scattering using separation of variables analytical results are presented for stability and error estimation for an h version galerkin fem with mesh refinement in radial direction we prove an upper bound that contains a pollution term similar to previous results in 1d and 2d this estimate is evaluated computationally using a 3d fem iem infinite elements implementation the tabulated and plotted errors show the theoretically predicted pollution effect in the fe error

efficient matrix valued algorithms for solving stiff riccati differential equations in the time varying case a classical approach which has been widely used to compute the solution of the riccati matrix equation of for example size n n is to unroll the matrices into vectors and integrate the resulting system of n sup 2 vector differential equations directly if the system of vectorized differential equations is stiff the cost computation time and storage requirements of applying the popular backward differentiation formulas bdfs to the stiff equations will be very high for large n because a linear system of algebraic equations of size n sup 2 n sup 2 must be solved at each time step new matrix valued algorithms based on a matrix generalization of the bdfs are proposed for solving stiff riccatti differential equations the amount of work required to compute the solution per time step is only o n sup 3 flops by using the matrix valued algorithms whereas the classical approach requires o n sup 6 flops per time step

rcms right correction magnus series approach for oscillatory odes we consider rcms a method for integrating differential equations of the form y a a1 t y with highly oscillatory solution it is shown analytically and numerically that rcms can accurately integrate problems using stepsizes determined only by the characteristic scales of a1 t typically much larger than the solution wavelength in fact for a given t grid the error decays with or is independent of increasing solution oscillation rcms consists of two basic steps a transformation which we call the right correction and solution of the right correction equation using a magnus series with suitable methods of approximating the highly oscillatory integrals appearing therein rcms has high order of accuracy with little computational work moreover rcms respects evolution on a lie group we illustrate with application to the 1d schrodinger equation and to frenet serret equations the concept of right correction integral series schemes is suggested and right correction neumann schemes are discussed asymptotic analysis for a large class of odes is included which gives certain numerical integrators converging to exact asymptotic behaviour

fuzzy rough nearest neighbor algorithms in classification in this paper classification efficiency of the conventional k nearest neighbor algorithm is enhanced by exploiting fuzzy rough uncertainty the simplicity and nonparametric characteristics of the conventional k nearest neighbor algorithm remain intact in the proposed algorithm unlike the conventional one the proposed algorithm does not need to know the optimal value of k moreover the generated class confidence values which are interpreted in terms of fuzzy rough ownership values do not necessarily sum up to one consequently the proposed algorithm can distinguish between equal evidence and ignorance and thus the semantics of the class confidence values becomes richer it is shown that the proposed classifier generalizes the conventional and fuzzy knn algorithms the efficacy of the proposed approach is discussed on real data sets

stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures a stable approach for integrating the impedance matrix in cylindrical radial inhomogeneous structures is developed and studied a stroh like system using the time harmonic displacement traction state vector is used to derive the riccati matrix differential equation involving the impedance matrix it is found that the resulting equation is stiff and leads to exponential instabilities a stable scheme for integration is found in which a local expansion is performed by combining the matricant and impedance matrices this method offers a stable solution for fully anisotropic materials which was previously unavailable in the literature several approximation schemes for integrating the riccati equation in cylindrical coordinates are considered exponential magnus taylor series lagrange polynomials with numerical examples indicating that the exponential scheme performs best the impedance matrix is compared with solutions involving buchwald potentials in which the material is limited to piecewise constant transverse isotropy lastly a scattering example is considered and compared with the literature

a sextic formalism for three dimensional elastodynamics of cylindrically anisotropic radially inhomogeneous materials elasticity of arbitrary cylindrically anisotropic radially inhomogeneous media is considered the system of six first order ordinary differential equations describing time harmonic cylindrical waves propagating along the cylinder axis is obtained the matrix of the system coefficients has a specific algebraic symmetry underlying the structure of the formalism the matricant solution in the form of the peano expansion is discussed the wentzel kramers brillouin approximation is derived algebraic features of the solution are analysed and their physical content is elucidated the dispersion equation for waves in a hollow cylinder with free or clamped lateral surfaces is formulated in terms of the impedance admittance matrices their analytical properties are established which are useful for analysis of the resulting frequency spectrum the theory is extended to anisotropic radially inhomogeneous piezoelectric cylinders for which the octet generalization of the formalism is developed and studied

dominance based rough fuzzy set approach and its application to rule induction abstract theories of fuzzy sets and rough sets are related and complementary methodologies to handle uncertainty of vagueness and coarseness respectively marrying both leads to the hybrid notion of rough fuzzy sets in order to get a more accurate account of imperfect information in this paper our attention is paid to ordered fuzzy decision systems where condition criteria are preference ordered and decision classes are not only ordered but also fuzzy first the dominance based rough fuzzy approximations of an upward or downward cumulated fuzzy set are introduced in ordered fuzzy decision systems second lower and upper reducts relative to a certain cumulated fuzzy set are proposed to eliminate redundant criteria in the system then two approaches to attribute reduction are presented based on the discernibility matrix and the heuristic strategy respectively also decision rules are extracted directly from these approximations and some applicable and simplified decision rules are obtained according to requirements of decision makers finally a case study in bankruptcy risk analysis is used to illustrate the mechanism of the proposed methods

measuring fuzzy uncertainty first this paper reviews several well known measures of fuzziness for discrete fuzzy sets then new multiplicative and additive classes are defined we show that each class satisfies five well known axioms for fuzziness measures and demonstrate that several existing measures are relatives of these classes the multiplicative class is based on nonnegative monotone increasing concave functions the additive class requires only nonnegative concave functions some relationships between several existing and the new measures are established and some new properties are derived the relative merits and drawbacks of different measures for applications are discussed a weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced finally we comment on the construction of measures that may assess all of the uncertainties associated with a physical system

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

phase portrait of the matrix riccati equation the matrix riccati equation which arises from optimal control and filtering problems is a quadratic differential equation on the space of real symmetric n times n matrices it is closely related via compactification of the phase space to the differential equations on the grassmann manifold and the lagrange grassmann manifold whose flows are generated by the action of one parameter subgroups of the general linear group and of the symplectic group respectively we determine the complete phase portraits of the riccati equations on all three spaces the asymptotic behavior of every solution is described the phase portraits are characterized topologically as well as set theoretically although the riccati equation is not generally a morse smale vector field we are able to show that it possesses suitable generalizations of many of the important properties of morse smale vector fields in particular the riccati equation satisfies a generalized version of the morse inequalities for a morse smale dynamica

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

some aspects of using iv software to solve bvp via a riccati transformation we consider a numerical method for the solution of linear boundary value problems bvps with separated boundary conditions bcs for ordinary differential equations odes by using a particular linear transformation the problem is reformulated as three initial value problems ivps one of them being a riccati differential equation the nonlinearity of the riccati de makes it necessary to deal with the eventuality of modifying the structure of integration so called multiple embedding equations must thus be derived and a marching algorithm is incorporated a stability analysis of the three ivps indicates clearly for what class of problems the method is suited and why stiff iv methods should then frequently be used this leads us to consider the undesirable effect of superstability of implicit schemes which can occur when integrating the riccati de we interpret this occurrence in terms of the original bvp and the resulting analysis suggests some final considerations including the possibility of corrective measures

numerical integrators for quantum dynamics close to the adiabatic limit we study the numerical solution of singularly perturbed schro dinger equations with time dependent hamiltonian based on a reformulation of the equations we derive time reversible numerical integrators which can be used with step sizes that are substantially larger than with traditional integration schemes a complete error analysis is given for the adiabatic case to deal with avoided crossings of energy levels which lead to non adiabatic behaviour we propose an adaptive extension of the methods which resolves the sharp transients that appear in non adiabatic state transitions

the close relationships between methods for solving two point boundary value problems the well conditioning of a boundary value problem is shown to be related to bounding two quantities one involving the boundary conditions and the other involving the green s function in the case of a solution dichotomy they are related to known stability results these results easily explain why difficulties arise using superposition and reduced superposition it is shown how multiple shooting overcomes these difficulties by relating its matrix conditioning to the underlying boundary value problem some factorization methods for the multiple shooting method are considered these show the close relationships between multiple shooting the stabilized march and invariant imbedding and one factorization leads to an efficient new way to implement each of these three methods many of the results apply for collocation and finite difference methods too finally it is shown both theroetically and computationally when matrix compactification can lead to difficulty

comparison of the probabilistic approximate classification and the fuzzy set model abstract it is shown that the generalized notion probabilistic approximate classification of rough sets can be conveniently described by the concept of fuzzy sets a discussion of the proper choice of the definition for the membership function of the intersection union of fuzzy sets is also presented however from the point of view of the probabilistic approximation space it is argued that there does not exist a universal definition for the fuzzy intersection union operation

double quantitative rough fuzzy set based decisions as two important expanded quantification rough set models the probabilistic rough set prs model and the graded rough set grs model are used to measure relative quantitative information and absolute quantitative information between the equivalence classes and a basic concept respectively the decision theoretic rough set dtrs model is a special case of prs model which mainly utilizes the conditional probability to express relative quantification since the fuzzy concept is more general than classical concept in real life how to make decision for a fuzzy concept using relative and absolute quantitative information is becoming a hot topic in this paper a couple of double quantitative decision theoretic rough fuzzy set dq dtrfs models based on logical conjunction and logical disjunction operation are proposed furthermore we discuss decision rules and the inner relationship between these two models then an experiment in the medical diagnosis is studied to support the theories finally to apply our methods to solve a pattern recognition problem in big data experiments on data sets downloaded from uci are conducted to test the proposed models in addition we also offer a comparative analysis using two non rough set based methods from the results obtained one finds that the proposed method is efficient for dealing with practical issues

multimodal method and conformal mapping for the scattering by a rough surface the scattering of a wave from a periodic rough surface in two dimensions is considered the proposed method is based on the use of a conformal mapping and of the multimodal admittance method the use of the conformal mappings induces a spatially varying refractive index and transforms the boundary condition on the rough surface into a boundary condition on a flat surface then the multimodal admittance method reduces this problem to a riccati equation for the modal admittance matrix which is solved numerically with a magnus mobius scheme the method is shown to converge exponentially with the number of fourier modes the method also allows to find geometries having trapped modes or quasi trapped modes at a given frequency by looking at singularities or quasi singularities of this riccati equation besides a simple perturbation expansion based on a small roughness approximation is developed

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

superposition laws for solutions of differential matrix riccati equations arising in control theory representation formulas are given for the general solution of the n times n matrix riccati equation dot w a wb cw wdw using n known solutions with n 1 5 n representations the 5 representation is a superposition formula in that it expresses the general solution explicitly as a function of five particular solutions and n2arbitrary constants n geq 2 using no further information the representation formulas can be used in numerical calculations the 4 and 5 representations are specially useful when a solution w t has a singularity for some finite t t 0 they also clarify the properties of the solution space the matrix elements of w t are meromorphic functions of t having simple poles as the only possible singularities the relation between the representation formulas and previously known results is discussed

a threshold accepting heuristic with intense local search for the solution of special instances of the traveling salesman problem in real life scheduling variations of the standard traveling salesman problem are very often encountered the aim of this work is to present a new heuristic method for solving three such special instances with a common approach the proposed algorithm uses a variant of the threshold accepting method enhanced with intense local search while the candidate solutions are produced through an insertion heuristic scheme the main characteristic of the algorithm is that it does not require modifications and parameter tuning in order to cope with the three different problems computational results on a variety of real life and artificial problems are presented at the end of this work and prove the efficiency and the ascendancy of the proposed method over other algorithms found in the literature

coarse grid computation of the one way propagation of coupled modes in a varying cross section waveguide a one way approximation is investigated for the computation of wave propagation in varying cross section waveguides the proposed method derives as a basic approximation of the extensively studied multimodal admittance method when integrated with a magnus scheme this matrix one way equation exhibits an unexpected behavior as the deviation from the exact solution is minimum when only two discretization points per wavelength are taken this peculiar property makes this method efficient to compute the wave propagation for a large variety of geometries beyond the initially stated framework of weakly non uniform waveguides

the riccati method for the helmholtz equation the operator riccati equation for the dirichlet to neumann map is derived from the exact operator factorization of the two dimensional variable coefficient helmholtz equation numerical schemes are developed for the operator riccati equation and its variant using a local eigenfunction expansion this leads to a practical computational method for acoustic wave propagation over large range distances since the boundary value problem of the helmholtz equation is reduced to initial value problems that are solved by marching in the range the efficiency and accuracy of the method is demonstrated by numerical experiments including the plane parallel range dependent waveguide benchmark problem proposed by the acoustical society of america

one way large range step methods for helmholtz waveguides a useful approach for long range computation of the helmholtz equation in a waveguide is to re formulate it as the operator differential riccati equation for the dirichlet to neumann map for waveguides with slow range dependence the piecewise range independent approximation is used to derive a second order range stepping method for this one way re formulation the range marching formulas are exact for each range independent piece and a large range step is possible if the range dependence is gradual based on a fourth order conservative exponential method for linear evolution equations a fourth order method that admits even larger range steps is developed for the one way re formulation numerical examples are used to demonstrate the improved accuracy of the fourth order method

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

a classification of formulations for the time dependent traveling salesman problem abstract the time dependent traveling salesman problem tdtsp is a generalization of the classical traveling salesman problem where the cost of any given arc is dependent of its position in the tour the tdtsp can model several real world applications e g one machine sequencing in this paper we present a classification of formulations for the tdtsp this framework includes both new and old formulations the new formulation presented in this paper is derived from a quadratic assignment model for the tdtsp in a first step lawler s transformation procedure is used to derive an equivalent linearized version of the quadratic model in a second step a stronger formulation is obtained by tightening some constraints of the previous formulation it is shown that in terms of linear relaxations the latter formulation is either equivalent or better than other formulations already known from the literature finally we compare these formulations with other well known formulations for the classical traveling salesman problem

a domain decomposition method for the helmholtz equation in a multilayer domain the two dimensional helmholtz equation for problems where the physical domain consists of layers with different material properties is studied an efficient preconditioner for iterative solution of the problem is constructed the problem is discretized with fourth order accurate finite difference operators for the construction of the radiation boundary conditions a fourth order finite element method also is used the large sparse complex indefinite and ill conditioned system of equations that arises is solved with preconditioned restarted gmres a domain decomposition method is used in which the preconditioning is based on the schur complement algorithm with fast poisson type preconditioners for the subdomains the memory requirements for the preconditioner are nearly linear in the number of unknowns the arithmetic complexity for each iteration is low whereas the construction of the preconditioner is a bit more expensive electromagnetic wave propagation in a three layered waveguide is used as a model problem numerical experiments show that convergence is achieved in a few iterations compared with banded gaussian elimination which is a standard solution method the iterative method shows significant gain in both memory requirements and arithmetic complexity furthermore the relative gain grows when the problem size increases

fuzzy rough approximations for set valued data rough set theory is one of important tools of soft computing and rough approximations are the essential elements in rough set models however the existing fuzzy rough set model for set valued data which is directly constructed based on a kind of similarity relation fail to explicitly define fuzzy rough approximations to solve this issue in this paper we propose two types of fuzzy rough approximations and define two corresponding relative positive region reducts furthermore two discernibility matrices and two discernibility functions are introduced to acquire these new proposed reducts and the relationships among the new reducts and the existing reducts are also be provided theoretical analyses demonstrate that the new types of reducts have less redundancy and are more diverse no lower number of reducts than those obtained by means of the existing matrices and experimental results illustrate the new reducts found by our methods outperform those obtained by existing method

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

traveling salesman problem with clustering in the original traveling salesman problem the traveling salesman has the task to find the shortest closed tour through a proposed set of nodes touching each node exactly once and returning to the initial node at the end for the sake of the tour length to be minimized nodes close to each other might not be visited one after the other but separated in the tour however for some practical applications it is useful to group nodes to clusters such that all nodes of a cluster are visited contiguously here we present an approach which leads to an automatic clustering with a clustering parameter governing the sizes of the clusters

incomplete factorization based preconditionings for solving the helmholtz equation preconditioning techniques based on incomplete factorization of matrices are investigated to solve highly indefinite complex symmetric linear systems a novel preconditioning is introduced the real part of the matrix is made positive definite or less indefinite by adding properly defined perturbations to the diagonal entries while the imaginary part is unaltered the resulting preconditioning matrix which is obtained by applying standard methods to the perturbed complex matrix turns out to perform significantly better than classical incomplete factorization schemes for realistic values of the gmres restart parameter spectacular reduction of iteration counts is observed a theoretical spectral analysis is provided in which the spectrum of the preconditioner applied to indefinite matrix is related to the spectrum of the same preconditioner applied to a stieltjes matrix extracted from the indefinite matrix results of numerical experiments are reported which display the efficiency of the new preconditioning copyright 2001 john wiley sons ltd

a rst a generalization of rough set theory the paper presents a transition from the crisp rough set theory to a fuzzy one called alpha rough set theory or in short a rst all basic concepts or rough set theory are extended i e information system indiscernibility dependency reduction core definability approximations and boundary the resulted theory takes into account fuzzy data and allows the approximation of fuzzy concepts besides the control of knowledge granularity is natural in a rst which is based on a parameterized indiscernibility relation a rst is developed to recognize non deterministic relationships using notions as a dependency a reduct and so forth on the other hand we introduce a notion of relative dependency as an alternative of the absolute definibility presented in rough set theory the extension a rst leads naturally to the new concept of alpha rough sets which represents sets with fuzzy non empty boundaries 2000 elsevier science inc all rights reserved

on magnus integrators for time dependent schrodinger equations numerical methods based on the magnus expansion are an efficient class of integrators for schrodinger equations with time dependent hamiltonian though their derivation assumes an unreasonably small time step size as would be required for a standard explicit integrator the methods perform well even for much larger step sizes this favorable behavior is explained and optimal order error bounds are derived that require no or only mild restrictions of the step size in contrast to standard integrators the error does not depend on higher time derivatives of the solution which is in general highly oscillatory

rough sets theory for multicriteria decision analysis the original rough set approach proved to be very useful in dealing with inconsistency problems following from information granulation it operates on a data table composed of a set u of objects actions described by a set q of attributes its basic notions are indiscernibility relation on u lower and upper approximation of either a subset or a partition of u dependence and reduction of attributes from q and decision rules derived from lower approximations and boundaries of subsets identified with decision classes the original rough set idea is failing however when preference orders of attribute domains criteria are to be taken into account precisely it cannot handle inconsistencies following from violation of the dominance principle this inconsistency is characteristic for preferential information used in multicriteria decision analysis mcda problems like sorting choice or ranking in order to deal with this kind of inconsistency a number of methodological changes to the original rough sets theory is necessary the main change is the substitution of the indiscernibility relation by a dominance relation which permits approximation of ordered sets in multicriteria sorting to approximate preference relations in multicriteria choice and ranking problems another change is necessary substitution of the data table by a pairwise comparison table where each row corresponds to a pair of objects described by binary relations on particular criteria in all those mcda problems the new rough set approach ends with a set of decision rules playing the role of a comprehensive preference model it is more general than the classical functional or relational model and it is more understandable for the users because of its natural syntax in order to workout a recommendation in one of the mcda problems we propose exploitation procedures of the set of decision rules finally some other recently obtained results are given rough approximations by means of similarity relations rough set handling of missing data comparison of the rough set model with sugeno and choquet integrals and results on equivalence of a decision rule preference model and a conjoint measurement model which is neither additive nor transitive

stable parallel elimination for boundary value odes a parallelizable and vectorizable algorithm for solving linear algebraic systems arising from two point boundary value odes is described the method is equivalent to gaussian elimination with row partial pivoting applied to a certain column reordered version of the usual almost block diagonal coefficient matrix we present analytical and numerical evidence to show that the algorithm is stable in most circumstances results from implementation on a shared memory multiprocessor and a vector processor are given the approach can be extended to handle problems with multipoint and integral conditions or with algebraic parameters

decision theoretic rough fuzzy set model and application this article investigates the rough approximation of a fuzzy concept on a probabilistic approximation space we propose the probabilistic rough fuzzy set by defining the conditional probability of a fuzzy event then we establish the model of probabilistic rough fuzzy set and discuss several properties in detail furthermore three generalizations of probabilistic rough fuzzy set namely 0 5 probabilistic rough fuzzy set variable precision probabilistic rough fuzzy set and bayesian rough fuzzy set are reported in order to give a systematic method of selecting parameters for the probabilistic rough fuzzy set we propose a decision theoretic rough fuzzy set that is we formulate a non parametric definition of the probabilistic rough fuzzy set moreover we illustrate the motivation and verify the validity of the decision theoretic rough fuzzy set by using a credit card applicant decision making problem furthermore the interrelationship between the decision theoretic rough fuzzy set and the probabilistic rough fuzzy set is explained the main contribution of this paper is twofold one is to extend the probabilistic rough set to fuzzy environment i e the probabilistic rough fuzzy set model another is to present an approach to select parameters needed in probabilistic rough fuzzy set modeling by using the process of decision making under conditions of risk

the time dependent traveling salesman problem abstract a problem often considered in operations research and computational physics is the traveling salesman problem in which a traveling salesperson has to find the shortest closed tour between a given set of cities touching each city exactly once the distances between the single nodes are known to the traveling salesperson an extension of this problem is the time dependent traveling salesman problem in which these distances vary in time i will show how this more complex problem is treated with physical optimization algorithms like simulated annealing i will present results for the problem of the 127 beergardens in the area of augsburg in which i define a traffic zone in which traffic jams occur in the afternoon

accurate numerical integration of stiff differential riccati equations abstract in this paper accurate numerical integration methods for four times continuously differentiable coefficient stiff riccati matrix differential equations are proposed first a discrete numerical solution is constructed at a net of points of the predetermined existence interval using implicit one step matrix methods finally using linear b spline matrix functions a continuous numerical solution whose error is uniformly upper bounded by a prefixed error in the existence domain is constructed

two new operators in rough set theory with applications to fuzzy sets in this paper two new operators are introduced for the rough set theory using them two inequalities well known in the rough set theory can now be modified to become equalities with this change no information will be lost in the new expressions hence many properties in rough set theory can be improved and in particular the union the intersection and the complement operations can be redefined based on the two equalities furthermore the collection of rough sets of an approximation space forms a boolean algebra under these new operators finally roughness properties of fuzzy sets are analyzed using the new operations

a fourth order magnus scheme for helmholtz equation for wave propagation in a slowly varying waveguide it is necessary to solve the helmholtz equation in a domain that is much larger than the typical wavelength standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications a popular method is to approximate the waveguide by segments that are uniform in the propagation direction and use separation of variables in each segment for a slowly varying waveguide it is possible that the length of such a segment is much larger than the typical wavelength to reduce memory requirements it is advantageous to reformulate the boundary value problem of the helmholtz equation as an initial value problem using a pair of operators such an operator marching scheme can also be solved with the piecewise uniform approximation of the waveguide this is related to the second order midpoint exponential method for a system of linear odes in this paper we develop a fourth order operator marching scheme for the helmholtz equation using a fourth order magnus method

axiomatic approaches to rough approximation operators on complete completely distributive lattices we proposed a pair of rough approximation operators on a complete completely distributive lattice ccd lattice for short in 2015 in this paper we further discuss its properties and study the axiomatic approaches to the rough approximation operators through these axioms fuzzy rough approximation operators can be seen as special cases of rough approximation operators on a ccd lattice we also discuss the axiomatic approaches to generalized rough sets on ccd lattices

rough fuzzy functions in classification this paper generalizes the concept of rough membership functions in pattern classification tasks to rough fuzzy membership functions and rough fuzzy ownership functions unlike the rough membership value of a pattern which is sensitive only towards the rough uncertainty associated with the pattern the rough fuzzy membership or ownership value of the pattern signifies the rough uncertainty as well as the fuzzy uncertainty associated with the pattern in this paper various set theoretic properties of the rough fuzzy functions are exploited to characterize the concept of rough fuzzy sets these properties are also used to measure the rough fuzzy uncertainty associated with the given output class finally a few possible applications of the rough fuzzy functions are mentioned

staircase matrices and systems difference equations spline approximations and multiperiod linear programs all give rise to linear equation systems that have a characteristic staircase structure a staircase system s variables can be partitioned into a natural sequence of periods in such a way that every equation involves at most the variables of two adjacent periods this paper surveys the numerous properties of staircase systems and of the staircase matrices that comprise their coefficients particular attention is given to reduced and sparse staircase forms and to the varieties of gaussian elimination for solving staircase systems

decoupling and stability of algorithms for boundary value problems the ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals it is shown that most methods like multiple shooting collocation invariant imbedding and difference methods derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later for instance when solving a linear system this decoupling is related to the dichotomy of the ordinary differential equations in fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed how this stability is related to the use of the boundary conditions is also investigated

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough