time finite element methods a unified framework for numerical discretizations of odes abstract we present a unified framework for the numerical discretization of odes based on time finite element methods we relate time finite element methods to runge kutta methods with infinitely many stages by means of the corresponding numerical quadrature we establish the relation between time finite element methods and partitioned runge kutta methods we also provide order estimates and superconvergence of the corresponding numerical methods in use of the simplifying assumptions we apply time finite methods to hamiltonian systems and investigate the conservation of energy and symplectic structure for the resulting numerical discretizations for hamiltonian systems we also construct new classes of symplectic integrators by combining different time finite element methods and verify the results by performing some numerical experiments

a fuzzy and bipolar approach to preference modeling with application to need and desire fuzziness and bipolarity allow representing human knowledge taking into account the gradual and the dialectic properties of language focusing on the meaning of concepts under this cognitive and linguistic approach we explore preference relations examining their semantic decomposition through fuzzy preference structures and the specification of meaningful opposites in particular we introduce the preference aversion p a model which allows analyzing under an independent aggregation methodology the possible gains and losses like pros and cons towards a given set of alternatives as an attractive feature of this proposal we show that the p a model allows distinguishing between need and desire contrary to common preference models where both notions are indistinguishable

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

an axiomatic approach to fuzzy preference modelling abstract we propose an axiomatic approach to the definition of fuzzy strict preference p indifference i and incomparability j associated with a fuzzy preference relation r expression for p i and j are given via solving two systems of functional equations

coherent continuous systems and the generalized functional equation of associativity a theory of coherent continuous systems is outlined systems are to be decomposed hierarchically into modules for this purpose some results on the structure of continuous functions are adapted to generalize the classic 3 modules theorem of binary systems consideration of the modular structure of coherent continuous systems leads naturally to the generalized functional equation of associativity ggu v w hu hv w involving four unknown 2 place continuous functions the functional equation is solved under some regularity conditions motivated by the properties of coherent continuous systems the solutions obtained give rise to generalized notions of parallel and series system structures

discontinuous polynomial approximations in the theory of one step hybrid and multistep methods for nonlinear ordinary differential equations this paper studies the approximation of the solution of nonlinear ordinary differen tial equations by discontinuous piecewise polynomials of degree k and traces at the nodes of discretization a mesh dependent variational framework underlying this discontinuous ap proximation is derived several families of one step hybrid and multistep schemes are obtained it is shown that the convergence rate in the l2 norm is k 1 the nodal conver gence rate can go up to 2 k 2 depending on the particular scheme under consideration the mesh dependent variational framework introduced here is of special interest in the approxi mation of the solution of optimal control problems governed by differential equations 1 introduction the object of this paper is the study of discontinuous piecewise polynomial approximations to the solution of systems of nonlinear ordinary dif

another paraconsistent algebraic semantics for lukasiewicz pavelka logic abstract as recently proved in a previous work of turunen tsoukias and ozturk starting from an evidence pair a b on the real unit square and associated with a propositional statement we can construct evidence matrices expressed in terms of four values t f k u that respectively represent the logical valuations true false contradiction both true and false and unknown neither true nor false regarding the statement the components of the evidence pair a b are to be understood as evidence for and against respectively moreover the set of all evidence matrices can be equipped with an injective mv algebra structure thus the set of evidence matrices can play the role of truth values of a lukasiewicz pavelka fuzzy logic a rich and applicable mathematical foundation for fuzzy reasoning and in such a way that the obtained new logic is paraconsistent in this paper we show that a similar result can be also obtained when the evidence pair a b is given on the real unit triangle since the real unit triangle does not admit a natural mv structure we introduce some mathematical results to show how this shortcoming can be overcome and another injective mv algebra structure in the corresponding set of evidence matrices is obtained also we derive several formulas to explicitly calculate the evidence matrices for the operations associated to the usual connectives

automatic average case analysis of algorithms abstract many probabilistic properties of elementary discrete combinatorial structures of interest for the average case analysis of algorithms prove to be decidable this paper presents a general framework in which such decision procedures can be developed it is based on a combination of generating function techniques for counting and complex analysis techniques for asymptotic estimations we expose here the theory of exact analysis in terms of generating functions for four different domains the iterative recursive and unlabelled labelled data type domains we then present some major components of the associated asymptotic theory and exhibit a class of naturally arising functions that can be automatically analyzed a fair fragment of this theory is also incorporated into a system called lambda upsilon omega in this way using computer algebra one can produce automatically non trivial average case analyses of algorithms operating over a variety of decomposable combinatorial structures at a fundamental level this paper is part of a global attempt at understanding why so many elementary combinatorial problems tend to have elementary asymptotic solutions in several cases it proves possible to relate entire classes of elementary combinatorial problems whose structure is well defined with classes of elementary special functions and classes of asymptotic forms relative to counting probabilities or average case complexity

clustering in coagulation fragmentation processes random combinatorial structures and additive number systems asymptotic formulae and limiting laws we develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper in the case when the parametric function of the models is regularly varying with positive exponent the approach is based on khintchine s probabilistic method that grew out of the darwin fowler method in statistical physics our main result is the derivation of asymptotic formulae for the distribution of the largest and the smallest clusters components as the total size of a structure number of particles goes to infinity we discover that n1 l 1 is the threshold for the limiting distribution of the largest cluster as a byproduct of our study we prove the independence of the numbers of groups of fixed sizes as n this is in accordance with the general principle of asymptotic independence of sites in mean field models the latter principle is commonly accepted in statistical physics but not rigorously proved

asymptotics for logical limit laws when the growth of the components is in an rt class compton s method of proving monadic second order limit laws is based on analyzing the generating function of a class of finite structures for applications of his deeper results we previously relied on asymptotics obtained using cauchy s integral formula in this paper we develop elementary techniques based on a tauberian theorem of schur that significantly extend the classes of structures for which we know that compton s theory can be applied

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

construction of symplectic partitioned runge kutta methods with continuous stage hamiltonian systems as one of the most important class of dynamical systems are associated with a well known geometric structure called symplecticity symplectic numerical algorithms which preserve such a structure are therefore of interest in this article we study the construction of symplectic partitioned runge kutta methods with continuous stage this construction of symplectic methods mainly relies upon the expansion of orthogonal polynomials and the simplifying assumptions for partitioned runge kutta type methods by using suitable quadrature formulae it also provides a new and simple way to construct symplectic partitioned runge kutta methods in classical sense

hamilton s law of variable mass system and time finite element formulations for time varying structures based on the law summary traditional principles of mechanics are primarily conceived for constant mass systems which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems thus the numerical algorithms for time varying structures based on these principles are only suitable for this special case in this paper hamilton s law of variable mass system is derived based on meshchersky s fundamental equation and two classes of novel time finite element formulations for linear systems with arbitrary continuous time varying parameters are developed based on the previous law the formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated numerical examples demonstrate that compared with the algorithms for time varying structures that developed based on traditional principles of mechanics the proposed algorithms provide extended capabilities in both time varying mass problems that mass is gained or lost at any velocity such as rocket problem and moving mass problems such as vehicle bridge interaction problem besides the time varying stiffness and damping problems the proposed algorithms have a wider range of application in particular hamilton s law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time varying structures copyright 2014 john wiley sons ltd

a logical approach to asymptotic combinatorics i first order properties we shall present a general framework for dealing with an extensive set of problems from asymptotic combinatorics this framework provides methods for determining the probability that a large finite structure ran domly chosen from a given class will have a given property our main concern is the asymptotic probability the limiting value as the size of the structure increases for example a common problem in elementary probability texts is to show that the asymptotic probabity that a per mutation will have no fixed point is l e we shall say nothing about the closely related problem of determining rates of convergence although the methods presented here may extend to such problems to develop a general approach we must fix a language for specifying properties of structures thus our approach is logical logic is the branch of mathematics that deals with problems of language in this paper we con sider properties expressible in the language of first order logic and speak of probabilities of first order sentences rather than properties in the sequel to this paper we consider properties expressible in the more general language of monadic second order logic clearly we must restrict the classes of structures we consider in order for questions about asymptotic probabilities to be meaningful and significant therefore we choose to consider only classes closed under disjoint unions and components see sect 1 for definitions this condition is general enough to include the classes of graphs permutations unary functions and many other important examples the main theorems of the paper are about classes satisfying this condition theorems 5 7 and 5 8 state that the existence of asymptotic probabilities for a special set of sentences called component bounded sentences defined in sect 5 is equivalent to a con dition on the growth of such a class theorem 5 9 gives necessary and suf ficient conditions for every first order sentence to have an asymptotic probability of 0 or 1 in nonfast growing classes in section 8 we present a long list of examples from the literature

valued preference structures abstract an approach to the axiomatics of valued preference modelling is suggested in the framework of this approach general definitions of strict preference indifference and incomparability relations associated with a valued preference relation are established via solving a system of functional equations any solution of this system satisfies most of the classical properties characterizations of important particular solutions are also proved

_ 1 free graphs asymptotic structure and a 0 1 law etude asymptotique de la structure des graphes etiquetes sans k l 1 etablissement de la l colorabilite et de la loi 0 1

a note on the continuous stage runge kutta nystrom formulation of hamiltonian boundary value methods hbvms abstract in recent years the class of energy conserving methods named hamiltonian boundary value methods hbvms has been devised for numerically solving hamiltonian problems in this short note we study their natural formulation as continuous stage runge kutta nystrom methods which allows a deeper insight in the methods

sufficient conditions for zero one laws we generalize a result of bateman and erdos concerning partitions thereby answering a question of compton from this result it follows that if k is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components and if it has the property that the number of indecomposables of size n is bounded above by a polynomial in n then k has a monadic second order 0 1 law moreover we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors and if it has the property that the number of indecomposables of size at most n is bounded above by a polynomial in log n then this class has a first order 0 1 law these results cover all known natural examples of classes of structures that have been proved to have a logical 0 1 law by compton s method of analyzing generating functions

analysis of optimal superconvergence of discontinuous galerkin method for linear hyperbolic equations in this paper we study the superconvergence of the error for the discontinuous galerkin dg finite element method for linear conservation laws when upwind fluxes are used we prove that if we apply piecewise k th degree polynomials the error between the dg solution and the exact solution is k 2 th order superconvergent at the downwind biased radau points with suitable initial discretization moreover we also prove the dg solution is k 2 th order superconvergent both for the cell averages and for the error to a particular projection of the exact solution the superconvergence result in this paper leads to a new a posteriori error estimate our analysis is valid for arbitrary regular meshes and for mathcal p k polynomials with arbitrary k geq1 and for both periodic boundary conditions and for initial boundary value problems we perform numerical experiments to demonstrate that the superconvergence rate proved in this paper is optimal

on time dynamics of coagulation fragmentation processes we establish a characterization of coagulation fragmentation processes such that the induced birth and death processes depicting the total number of groups at time t 0 are time homogeneous based on this we provide a characterization of mean field gibbs coagulation fragmentation models which extends the one derived by hendriks et al as a by product of our results the class of solvable models is widened and a question posed by n berestycki and pitman is answered under restriction to mean field models

independent process approximations for random combinatorial structures abstract many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables conditioned on the value of a weighted sum of the variables it is interesting to compare the combinatorial structure directly to the independent discrete process without renormalizing the quality of approximation can often be conveniently quantified in terms of total variation distance for functionals which observe part but not all of the combinatorial and independent processes among the examples are combinatorial assemblies e g permutations random mapping functions and partitions of a set multisets e g polynomials over a finite field mapping patterns and partitions of an integer and selections e g partitions of an integer into distinct parts and square free polynomials over finite fields we consider issues common to all the above examples including equalities and upper bounds for total variation distances existence of limiting processes heuristics for good approximations the relation to standard generating functions moment formulas and recursions for computing densities refinement to the process which counts the number of parts of each possible type the effect of further conditioning on events of moderate probability large deviation theory and nonuniform measures on combinatorial objects and the possibility of getting useful results by overpowering the conditioning

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis

a posteriori error estimation for discontinuous galerkin solutions of hyperbolic problems abstract we analyze the spatial discretization errors associated with solutions of one dimensional hyperbolic conservation laws by discontinuous galerkin methods dgms in space we show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a radau polynomial of degree p 1 on each element we also prove that the local and global discretization errors are o x2 p 1 and o x2p 1 at the downwind point of each element this strong superconvergence enables us to show that local and global discretization errors converge as o xp 2 at the remaining roots of radau polynomial of degree p 1 on each element convergence of local and global discretization errors to the radau polynomial of degree p 1 also holds for smooth solutions as p these results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth

superconvergence of a discontinuous finite element method for a nonlinear ordinary differential equation abstract in this paper n degree discontinuous finite element method with interpolated coefficients for an initial value problem of nonlinear ordinary differential equation is introduced and analyzed by using the finite element projection for an auxiliary linear problem as comparison function an optimal superconvergence u u o h n 2 n 2 at n 1 order characteristic points in each element respectively is proved finally the theoretic results are tested by a numerical example

an adaptive discontinuous galerkin method for very stiff systems of ordinary differential equations abstract we present a discontinuous galerkin dg method for the numerical solution of stiff systems of ordinary differential equations odes we use a standard dg variational formulation with polynomials of degree k in each time interval we show that the method is a stable for every k we then introduce a hierarchical legendre finite element basis and we show that a whole family of approximations can be obtained simply by truncating the last p degrees of freedom from the computed solution we show that these approximations converge to order k 1 p in l2 norm and to order k 1 2 p in supremum norm we then show how this can be used to control the error and the time step length we present numerical examples of solutions on very stiff problems and on stiff problems with very long time integration where the time step length can vary on many orders of magnitude

a posteriori analysis of a multirate numerical method for ordinary differential equations in this paper we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model the main purpose of this paper is to present a hybrid a priori a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations the use of only a finite number of iterations in the iterative solution of the discrete equations the numerical error arising in the solution of each component and the effects on stability arising from the multirate solution the hybrid estimate has the form of a computable a posteriori leading order expression and a provably higher order a priori expression we support this estimate by an a priori convergence analysis we present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate

inherently energy conserving time finite elements for classical mechanics in this paper we develop a finite element method for the temporal discretization of the equations of motion the continuous galerkin method is based upon a weighted residual statement of hamilton s canonical equations we show that the proposed finite element formulation is energy conserving in a natural sense a family of implicit one step algorithms is generated by specifying the polynomial approximation in conjunction with the quadrature formula used for the evaluation of time integrals the numerical implementation of linear quadratic and cubic time finite elements is treated in detail for the model problem of a circular pendulum in addition to that concerning dynamical systems with several degrees of freedom we address the design of nonstandard quadrature rules which retain the energy conservation property our numerical investigations assess the effect of numerical quadrature in time on the accuracy and energy conservation property of the time stepping schemes

some properties of choice functions based on valued binary relations abstract choice functions based on t norms of valued binary relations are introduced strict preference is also specified with the use of a t norm properties of the choice functions are investigated and rationality conditions are studied some classical particular cases are presented

error estimates for discontinuous galerkin approximations of implicit parabolic equations we analyze the classical discontinuous galerkin method for implicit parabolic equations symmetric error estimates for schemes of arbitrary order are presented the ideas developed allow certain assumptions frequently required in previous work to be relaxed for example different discrete spaces are allowed at each time step and the spatial operator is not required to be self adjoint or independent of time error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate these projections are local and enjoy optimal approximation properties when the solution is sufficiently regular

complexity of the first order theory of almost all finite structures a first order sentence of a relational type l is true almost everywhere if the proportion of its models among the structures of type l and cardinality m tends to 1 when m tends to it is shown that th l the set of sentences of type l true almost everywhere is complete in pspace further various upper and lower bounds of the complexity of this theory are obtained for example if the arity of the relation symbols of l is d 2 and if pr th l is the set of prenex sentences of th l then pr th l dspace n log n d and pr th l ntime o n log n d if r is a binary relation symbol and l r th l is the theory of almost all graphs then pr th l nspace o n log n these results are optimal modulo open problems in complexity such as ntime t dspace t and nspace s dspace s2

discontinuous galerkin methods for hamiltonian odes and pdes in this article we present a unified framework of discontinuous galerkin dg discretizations for hamiltonian odes and pdes we show that with appropriate numerical fluxes the numerical algorithms deduced from dg discretizations can be combined with the symplectic methods in time to derive the multi symplectic prk schemes the resulting numerical discretizations are applied to the linear and nonlinear schrodinger equations some conservative properties of the numerical schemes are investigated and confirmed in the numerical experiments

fuzzy rationality measures fuzzy preference relations formalize intensity of individual preferences over fixed sets of alternatives it is therefore natural to extend to fuzzy preferences the notion of rationality or consistency with this goal in mind we address in this paper the problem of giving an axiomatic basis for defining the concept of fuzzy rationality specifically we establish a collection of conditions that any fuzzy rationality measure must satisfy some examples of fuzzy rationality measures are then given and analyzed

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

connectives and quantifiers in fuzzy sets abstract we discuss some of the issues involved in the selection of appropriate operators for implementing the union and intersection of fuzzy subsets we discuss the extension principles used in extending operators to fuzzy subsets we provide a structure for aggregating linguistic variables in the theory of approximate reasoning we look at mean operators on fuzzy subsets as well as the inclusion of importances in aggregation we discuss non monotone set operators we introduce the concept of linguistic quantifiers and look at some applications of these structures we are particularly concerned with their application to linguistic summarizers syllogistic reasoning and multi criteria decision making

strict preference relations based on weak t norms abstract in this paper we introduce and investigate weak t norms and related concepts in order to give a general representation of valued strict preference relations associated with a given valued preference relation all the well known forms of valued strict preference relations are particular cases of our results moreover when some types of rationality assumptions are made our unified approach seems to be the only possible one

equivalence and compositions of fuzzy rationality measures an axiomatic basis for fuzzy rationality measures has already been introduced by the authors in a previous paper 5 formalizing the fact that there exist degrees of consistency when preferences over a fixed set of alternatives are expressed in terms of fuzzy binary preference relations this paper deals with some practical consequences on the one hand similarities and compositions of fuzzy rationality measures are considered showing natural ways of deriving new measures on the other if basic stability properties are introduced in order to assure that small intensity measurement errors never lead to big changes in the associate rationality value it is shown that crisp i e binary rationality measures present serious difficulties when applied to fuzzy preference relations

discontinuous galerkin methods of the non selfadjoint steklov eigenvalue problem in inverse scattering abstract in this paper we apply discontinuous galerkin methods to the non selfadjoint steklov eigenvalue problem arising in inverse scattering the variational formulation of the problem is non selfadjoint and does not satisfy h1 elliptic condition by using the spectral approximation theory of compact operators we prove the spectral approximation and optimal convergence order for the eigenvalues finally some numerical experiments are reported to show that the proposed numerical schemes are efficient for real and complex steklov eigenvalues

size of the largest cluster under zero range invariant measures we study the finite zero range process with occupancy dependent rate function g under the invariant measure which can be written explicitly in terms of g particles are distributed over sites and we regard all particles at a fixed site as a cluster in the density one case that is equal numbers of particles and sites we determine asymptotically the size of the largest cluster as the number of particles tends to infinity and determine its dependence on the rate function

symmetric integrators based on continuous stage runge kutta nystrom methods for reversible systems abstract in this paper we study symmetric integrators for solving second order ordinary differential equations on the basis of the notion of continuous stage runge kutta nystrom methods the construction of such methods heavily relies on the legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions new families of symmetric integrators as illustrative examples are presented for comparing the numerical behaviors of the presented methods some numerical experiments are also reported

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

on a family of connectives for fuzzy sets abstract in this paper we find the general solution of the functional equation s t x y t x n y x where s is a t conorm t is a t norm and n is a strong negation on the unit interval in particular the result yields a family of connectives for fuzzy sets

enabling local time stepping in the parallel implicit solution of reaction diffusion equations via space time finite elements on shallow tree meshes for many applications local time stepping offers an interesting and worthwhile alternative to the by now well established global time step control in fact local time stepping can allow for a highly detailed resolution of localized features of the solution with strongly reduced computational cost when compared to global time step control however local time stepping is not applicable in a straight forward manner in the context of fully implicit time discretizations here we present a method for the efficient parallel adaptive solution of non linear partial differential equations in particular reaction diffusion equations using spatially adapted time step sizes in the context of a fully implicit solution strategy our proposed method uses a discontinuous galerkin method in time approach within a full space time approach moreover it is designed from scratch for efficient parallel computation we employ shallow tree based mesh data structures in order to ensure a low memory footprint of the adaptive meshes by solving the time dependent partial differential equation on a d 1 dimensional non conforming mesh space time adaptivity is naturally achieved in combination with a discontinuous galerkin method in time the size of the arising systems can be precisely controlled we additionally introduce and discuss a stabilization scheme for space time mortar element methods that also has a highly positive impact on the efficiency of preconditioning techniques for the arising systems of equations we present results from extensive numerical experiments that address the question of convergence and efficiency linear and non linear solver performance parallel scalability up to 2048 cores as well as accuracy for the linear heat equation and a real world non linear reaction diffusion equation from the field of computational electrocardiology

superconvergence of discontinuous galerkin method for scalar nonlinear hyperbolic equations in this paper we study the superconvergence behavior of the semi discrete discontinuous galerkin dg method for scalar nonlinear hyperbolic equations in one spatial dimension superconvergence results for problems with fixed and alternating wind directions are established on the one hand we prove that if the wind direction is fixed i e the derivative of the flux function is bounded away from zero both the cell average error and numerical flux error at cell interfaces converge at a rate of 2k 1 when upwind fluxes and piecewise polynomials of degree k are used moreover we also prove that the function value approximation of the dg solution is superconvergent at interior right radau points and the derivative value approximation is superconvergent at interior left radau points with an order of k 2 and k 1 respectively as a byproduct we show a k 2 th order superconvergence of the dg solution towards the gauss radau projection of the exact solution on the other hand superconvergence resu

high order symplectic integrators based on continuous stage runge kutta nystrom methods abstract on the basis of the previous work by tang and zhang 37 in this paper we present a more effective way to construct high order symplectic integrators for solving second order hamiltonian equations instead of analyzing order conditions step by step as shown in the previous work the new technique of this paper is using legendre expansions to deal with the simplifying assumptions for order conditions with the new technique high order symplectic integrators can be conveniently devised by truncating an orthogonal series

exponential collocation methods for conservative or dissipative systems abstract in this paper we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods it is shown that these novel methods can be of arbitrarily high order and exactly or approximately preserve first integrals or lyapunov functions we also consider order estimates of the new methods furthermore we explore and discuss the application of our methods in important stiff gradient systems and it turns out that our methods are unconditionally energy diminishing and strongly damped even for very stiff gradient systems practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear schrodinger equation as a byproduct of this paper arbitrary order trigonometric rkn collocation methods are also presented and analysed for second order highly oscillatory general systems the paper is accompanied by numerical results that demonstrate the great potential of this work

existence of gelling solutions for coagulation fragmentation equations we study the smoluchowski coagulation fragmentation equation which is an infinite set of non linear ordinary differential equations describing the evolution of a mono disperse system of particles in a well stirred solution approximating the solutions of the smoluchowski equations by a sequence of finite markov chains we investigate the qualitative behavior of the solutions we determine a device on the finite chains which can detect the gelation phenomena the density dropping phenomena it shows how the gelation phenomena are reflected on the sequence of finite markov chains using this device we determine various types of gelation kernels and get the bounds of gelation times

time discretization of parabolic problems by the hp version of the discontinuous galerkin finite element method the discontinuous galerkin finite element method dgfem for the time discretization of parabolic problems is analyzed in the context of the hp version of the galerkin method error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp dgfem gives spectral convergence in problems with smooth time dependence in conjunction with geometric time partitions it is proved that the hp dgfem results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms for the h version dgfem algebraically graded time partitions are determined that give the optimal algebraic convergence rates a fully discrete hp scheme is discussed exemplarily for the heat equation the use of certain mesh design principles for the spatial discretizations yields exponential rates of convergence in time and space numerical examples confirm the theoretical results

the lack of continuity and the role of infinite and infinitesimal in numerical methods for odes the case of symplecticity when numerically integrating canonical hamiltonian systems the long term conservation of some of its invariants for example the hamiltonian function itself assumes a central role the classical approach to this problem has led to the definition of symplectic methods among which we mention gauss legendre collocation formulae indeed in the continuous setting energy conservation is derived from symplecticity via an infinite number of infinitesimal contact transformations however this infinite process cannot be directly transferred to the discrete setting by following a different approach in this paper we describe a sequence of methods sharing the same essential spectrum and then the same essential properties which are energy preserving starting from a certain element of the sequence on i e after a finite number of steps

a linearly implicit energy preserving exponential integrator for the nonlinear klein gordon equation abstract in this paper we generalize the exponential energy preserving integrator proposed in the recent paper siam j sci comput 38 2016 a1876 a1895 for conservative systems which now becomes linearly implicit by further utilizing the idea of the scalar auxiliary variable approach comparing with the original exponential energy preserving integrator which usually leads to a nonlinear algebraic system our new method only involves a linear system with a constant coefficient matrix taking the nonlinear klein gordon equation and the nonlinear schrodinger equation for examples we derive the concrete energy preserving schemes and demonstrate their high efficiency through numerical experiments

the time finite element as a robust general scheme for solving nonlinear dynamic equations including chaotic systems schemes that can be proven to be unconditionally stable in the linear context can yield unstable solutions when used to solve nonlinear dynamical problems hence the formulation of numerical strategies for nonlinear dynamical problems can be particularly challenging in this work we show that time finite element methods because of their inherent energy momentum conserving property in the case of linear and nonlinear elastodynamics provide a robust time stepping method for nonlinear dynamic equations including chaotic systems we also show that most of the existing schemes that are known to be robust for parabolic or hyperbolic problems can be derived within the time finite element framework thus the time finite element provides a unification of time stepping schemes used in diverse disciplines we demonstrate the robust performance of the time finite element method on several challenging examples from the literature where the solution behavior is known to be chaotic

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

symplecticity preserving continuous stage runge kutta nystrom methods abstract in this paper we develop continuous stage runge kutta nystrom csrkn methods for numerical integration of second order ordinary differential equations odes written in the form q f t q numerous odes in such form can be reduced to first order odes with the separable form of hamiltonian systems and symplecticity preserving discretizations of these systems are of interest for the sake of designing symplectic csrkn methods we explore the sufficient conditions for symplecticity and we show a simple way to derive symplectic rkn type integrators by using legendre polynomial expansion numerical results show the efficiency of the presented methods

on fuzzy strict preference indifference and incomparability relations abstract the existence of fuzzy models for strict preference indifference and incomparability satisfying all classical properties is established a particular class of such models is completely described

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

local discontinuous galerkin methods for two classes of two dimensional nonlinear wave equations in this paper we develop analyze and test local discontinuous galerkin dg methods to solve two classes of two dimensional nonlinear wave equations formulated by the kadomtsev petviashvili kp equation and the zakharov kuznetsov zk equation our proposed scheme for the kadomtsev petviashvili equation satisfies the constraint from the pde which contains a non local operator and at the same time has the local property of the discontinuous galerkin methods the scheme for the zakharov kuznetsov equation extends the previous work on local discontinuous galerkin method solving one dimensional nonlinear wave equations to the two dimensional case l2 stability of the schemes is proved for both of these two nonlinear equations numerical examples are shown to illustrate the accuracy and capability of the methods

discrete gradient methods for solving odes numerically while preserving a first integral we show that all direct methods for preserving a first integral during the numerical integration of an ordinary differential equation fit into the unified framework of discrete gradient methods using this framework we construct several new integral preserving schemes

a new class of energy preserving numerical integration methods the first ever energy preserving b series numerical integration method for ordinary differential equations is presented and applied to several hamiltonian systems related novel lie algebraic results are also discussed

a simple framework for the derivation and analysis of effective one step methods for odes in this paper we provide a simple framework to derive and analyse a class of one step methods that may be conceived as a generalization of the class of gauss methods the framework consists in coupling two simple tools firstly a local fourier expansion of the continuous problem is truncated after a finite number of terms and secondly the coefficients of the expansion are computed by a suitable quadrature formula different choices of the basis lead to different classes of methods even though we shall here consider only the case of an orthonormal polynomial basis from which a large subclass of runge kutta methods can be derived the obtained results are then applied to prove in a simplified way the order and stability properties of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see 2 and references therein a few numerical tests are also included in order to confirm the effectiveness of the methods resulting from our analysis

probabilities of first order sentences about unary functions soit f un entier positif fixe et une phrase dans le calcul de predicats d ordre 1 des fonctions unaires pour des entiers positifs n une n structure est un modele d univers 0 1 n 1 et de f fonctions unaires et n est le rapport du nombre de n structures satisfaisant a n nf le nombre de n structures on montre que lim n n exite pour tout tel et sa valeur est donnee par une expression constituee de constantes entieres et des operateurs et exp x

construction of runge kutta type methods for solving ordinary differential equations in this paper we study runge kutta methods with continuous stage introduced by butcher in 1987 by setting the coefficients of this family of methods and choosing the appropriate numerical integration we derive new classes of runge kutta methods furthermore we extend the w transformation technique by permitting w to be a non square matrix this allows us to construct more high order implicit runge kutta methods with some geometric properties specially we provide runge kutta methods with continuous stage which are conjugate symplectic symmetric or energy preserving for solving hamiltonian systems we also present some numerical experiments to verify our results

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

a general model for deriving preference structures from data in this paper we comment upon the integrated model for valued preferences introduced by fodor ovchinnikov and roubens in particular while on one hand we revise basic assumptions and point out their intuitive meaning on the other hand we propose an alternative mathematical justification of such a model which allows not only a better understanding of the obtained results but also a functional characterization of the whole family of solutions

a note on the efficient implementation of hamiltonian bvms we discuss the efficient implementation of hamiltonian bvms hbvms a recently introduced class of energy preserving methods for canonical hamiltonian systems see brugnano et al 8 and references therein also sketching their blended formulation we also discuss the case of separable problems for which the structure of the problem can be exploited to gain efficiency

superconvergence of discontinuous galerkin solutions for delay differential equations of pantograph type this paper is concerned with the superconvergence of the discontinuous galerkin solutions for delay differential equations with proportional delays vanishing at t 0 two types of superconvergence are analyzed here the first is based on interpolation postprocessing to improve the global convergence order by finding the superconvergence points of discontinuous galerkin solutions the second type follows from the integral iteration which just requires a local integration procedure applied to the discontinuous galerkin solution thus increasing the order of convergence the theoretical results are illustrated by a broad range of numerical examples