graph theoretical clustering based on limited neighbourhood sets abstract a method for clustering data according to a visual model of clusters is proposed the method uses either of two graphs which are defined according to relative distance and based on the gabriel graph and the relative neighbourhood graph respectively the method is locally sensitive hierarchic and based on the concept of limited neighbourhood sets clusters that are either disjoint or homogeneous and separable by sharp changes in point density may be detected

local discontinuous galerkin methods for nonlinear schrodinger equations in this paper we develop a local discontinuous galerkin method to solve the generalized nonlinear schrodinger equation and the coupled nonlinear schrodinger equation l 2 stability of the schemes are obtained for both of these nonlinear equations numerical examples are shown to demonstrate the accuracy and capability of these methods

multisymplectic geometry and multisymplectic preissman scheme for the kp equation the multisymplectic structure of the kp equation is obtained directly from the variational principal using the covariant de donder weyl hamilton function theories we reformulate the kp equation to the multisymplectic form which was proposed by bridges from the multisymplectic equation we can derive a multisymplectic numerical scheme of the kp equation which can be simplified to the multisymplectic 45 points scheme

backward error analysis for multi symplectic integration methods a useful method for understanding discretization error in the numerical solution of odes is to compare the system of odes with the modified equations obtained through backward error analysis and using symplectic integration for hamiltonian odes provides more incite into the modified equations in this paper the ideas of symplectic integration are extended to hamiltonian pdes and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations in particular local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution these results are also applied to the nonlinear wave equation

a conservative fourier pseudo spectral method for the nonlinear schrodinger equation a fourier pseudo spectral method that conserves mass and energy is developed for a two dimensional nonlinear schrodinger equation by establishing the equivalence between the semi norm in the fourier pseudo spectral method and that in the finite difference method we are able to extend the result in ref 56 to prove that the optimal rate of convergence of the new method is in the order of o n r 2 in the discrete l 2 norm without any restrictions on the grid ratio where n is the number of modes used in the spectral method and is the time step size a fast solver is then applied to the discrete nonlinear equation system to speed up the numerical computation for the high order method numerical examples are presented to show the efficiency and accuracy of the new method

multi symplectic integration of the camassa holm equation the camassa holm equation is rich in geometric structures it is completely integrable bi hamiltonian and it represents geodesics for a certain metric in the group of diffeomorphism here two new multi symplectic formulations for the camassa holm equation are presented and the associated local conservation laws are shown to correspond to certain well known hamiltonian functionals the multi symplectic discretisation of each formulation is exemplified by means of the euler box scheme numerical experiments show that the schemes have good conservative properties and one of them is designed to handle the conservative continuation of peakon antipeakon collisions

the midpoint scheme and variants for hamiltonian systems advantages and pitfalls the implicit midpoint scheme like higher order gauss collocation schemes is algebraically stable and symplectic and it preserves quadratic integral invariants it may appear particularly suitable for the numerical solution of highly oscillatory hamiltonian systems such as those arising in molecular dynamics or structural mechanics because there is no stability restriction when it is applied to a simple harmonic oscillator although it is well known that the midpoint scheme may also exhibit instabilities in various stiff situations one might still hope for good results when resonance type instabilities are avoided in this paper we investigate the suitability of the midpoint scheme for highly oscillatory frictionless mechanical systems where the step size k is much larger than the system s small parameter varepsilon in the case that the solution remains bounded as varepsilon rightarrow 0 we show that in general one must require that k 2 varepsilon be small enough or else even the errors in slowly varying quantities like the energy may grow undesirably especially when fast and slow modes are tightly coupled or worse the computation may yield misleading information in some cases this may already happen when k o varepsilon the same holds for higher order collocation at gaussian points the encountered restrictions on k are typically still better than the corresponding ones for explicit schemes

construction of multisymplectic schemes of any finite order for modified wave equations in this paper we analyze several symplectic discretizations of linear wave equations with periodic boundary conditions and show the discretizations are multisymplectic of arbitrary finite order for wave equations when the periodic boundary condition vanishes

symplectic energy momentum preserving variational integrators the purpose of this paper is to develop variational integrators for conservative mechanical systems that are symplectic and energy and momentum conserving to do this a space time view of variational integrators is employed and time step adaptation is used to impose the constraint of conservation of energy criteria for the solvability of the time steps and some numerical examples are given

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

geometric integrators for the nonlinear schrodinger equation abstract recently an interesting new class of pde integrators multisymplectic schemes has been introduced for solving systems possessing a certain multisymplectic structure some of the characteristic features of the method are its local nature independent of boundary conditions and an equal treatment of spatial and temporal variables the nonlinear schrodinger equation nls has a multisymplectic formulation and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the nls in the numerical experiments the multisymplectic concatenated midpoint scheme a centered cell discretization is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff on the other hand an integrable hamiltonian semi discretization of nls from ablowitz and ladik al possesses a full set of global conservation laws and a noncanonical symplectic structure we generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the al type another approach examined in the paper is the introduction of transformations to reduce the al system to either 1 separable form or 2 canonical form and then apply standard schemes in the new coordinates all of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions the performance of the schemes as well as interrelations among various geometric features are discussed

open problems from the 12th international conference on fuzzy set theory and its applications eighteen open problems posed during fsta 2014 liptovsk jan slovakia are presented these problems concern fuzzy logics fuzzy partitions copulas triangular norms and related aggregation functions some open problems concerning effect and mv algebras are also included

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

galois connections and tense operators on q effect algebras for effect algebras the so called tense operators were already introduced by chajda and paseka they presented also a canonical construction of them using the notion of a time frame tense operators express the quantifiers it is always going to be the case that and it has always been the case that and hence enable us to express the dimension of time both in the logic of quantum mechanics and in the many valued logic a crucial problem concerning tense operators is their representation having an effect algebra with tense operators we can ask if there exists a time frame such that each of these operators can be obtained by the canonical construction to approximate physical real systems as best as possible we introduce the notion of a q effect algebra and we solve this problem for q tense operators on q representable q jauch piron q effect algebras

symplectic and multi symplectic methods for the nonlinear schrodinger equation the hamiltonian and the multi symplectic formulations of the nonlinear schrodinger equation are considered for the multi symplectic formulation a new six point difference scheme which is equivalent to the multi symplectic preissman integrator is derived numerical experiments are also reported c 2002 elsevier science ltd all rights reserved

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

stability and instability of solitary waves of the fifth order kdv equation a numerical framework abstract the spectral problem associated with the linearization about solitary waves of the generalized fifth order kdv equation is formulated in terms of the evans function a complex analytic function whose zeros correspond to eigenvalues a numerical framework based on a fast robust shooting algorithm on exterior algebra spaces is introduced the complete algorithm has several new features including a rigorous numerical algorithm for choosing starting values a new method for numerical analytic continuation of starting vectors the role of the grassmannian g 2 c 5 in choosing the numerical integrator and the role of the hodge star operator for relating 2 c 5 and 3 c 5 and deducing a range of numerically computable forms for the evans function the algorithm is illustrated by computing the stability and instability of solitary waves of the fifth order kdv equation with polynomial nonlinearity

verlet i r respa impulse is limited by nonlinear instabilities this paper shows that in molecular dynamics md when constant energy nve simulations of newton s equations of motion are attempted using the multiple time stepping mts integrator verlet i r respa impulse there are nonlinear instabilities when the longest step size is a third or possibly a fourth of the period s of the fastest motion s in the system this is demonstrated both through a thorough set of computer experiments and through the analysis of a nonlinear model problem the numerical experiments include not only the unconstrained dynamics simulation of a droplet of flexible water and a flexible protein but also the constrained dynamics simulation of a solvated protein representing a range of simulation protocols commonly in use by biomolecular modelers the observed and predicted instabilities match exactly previous work has identified and explained a linear instability for verlet i r respa impulse at around half the period of the fastest motion mandziuk and schlick discovered nonlinear resonances in single time stepping md integrators but unstable nonlinear resonances for mts integrators are reported here for the first time this paper also offers an explanation on the instability of mts constrained molecular dynamics simulations of explicitly solvated proteins more aggressive multiple step sizes are possible with mild langevin coupling or targeted langevin coupling and its combination with the mollified impulse method permits step sizes 3 to 4 times larger thanverlet i r respa impulse while still retaining some accuracy

monte carlo comparison of six hierarchical clustering methods on random data there is mounting evidence to suggest that the complete linkage method does the best clustering job among all hierarchical agglomerative techniques particularly with respect to misclassification in samples from known multivariate normal distributions however clustering methods are notorious for discovering clusters on random data sets also we compare six agglomerative hierarchical methods on univariate random data from uniform and standard normal distributions and find that the complete linkage method generally is best in not discovering false clusters the criterion is the ratio of number of within cluster distances to number of all distances at most equal to the maximum within cluster distance

cylindrical quasi solitons of the zakharov kuznetsov equation abstract evolutions and interactions of two dimensional solitary waves of the zakharov kuznetsov equation are investigated numerically formations of cylindrical bell shaped pulses are observed in the initial value problems a single bell shaped pulse propagates stably without any deformation like a soliton two similar pulses exchange their amplitudes without merging and two dissimilar ones undergo overtaking collision after a collision of two pulses the strong pulse becomes somewhat stronger and the weak one becomes weaker with radiation of ripples so that the collision process is slightly inelastic generated ripples are very small for center to center collision of similar pulses the cylindrically symmetric pulses of the zakharov kuznetov equation are thus found to behave approximately soliton like some properties of collision process are interpreted in terms of the conservation laws

multi symplectic runge kutta nystrom methods for nonlinear schrodinger equations with variable coefficients in this paper we consider runge kutta nystrom rkn methods applied to nonlinear schrodinger equations with variable coefficients nlsevc concatenating symplectic nystrom methods in spatial direction and symplectic runge kutta methods in temporal direction for nlsevc leads to multi symplectic integrators i e to numerical methods which preserve the multi symplectic conservation law mscl we present the corresponding discrete version of mscl it is shown that the multi symplectic rkn methods preserve not only the global symplectic structure in time but also local and global discrete charge conservation laws under periodic boundary conditions we present a 4 order multi symplectic rkn method and use it in numerical simulation of quasi periodically solitary waves for nlsevc and we compare the multi symplectic rkn method with a non multi symplectic rkn method on the errors of numerical solutions the numerical errors of discrete energy discrete momentum and discrete charge the precise conservation of discrete charge under the multi symplectic rkn discretizations is attested numerically some numerical superiorities of the multi symplectic rkn methods are revealed

distributivity and associativity in effect algebras we prove large associativity of the partial sum in effect algebras and present an overview of distributivity like properties of partial operations and in effect algebras with respect to possibly infinite suprema and infima and vice versa generalizing several previous results

optimal error estimate of a linear fourier pseudo spectral scheme for two dimensional klein gordon schrodinger equations abstract the focus of this paper is on the optimal error bounds of a fourier pseudo spectral conservative scheme for solving the 2 dimensional nonlinear klein gordon schrodinger equations the proposed fourier pseudo spectral scheme not only conserves the mass and energy in the discrete level but also is efficient in practical computation because only two linear systems need to be solved at each time step based on the equivalence between the semi norm derived by the fourier pseudo spectral method and that by the finite difference method the pseudo spectral solution of the proposed scheme is proved strictly to be bounded and convergent with the order of o n r 2 in the discrete l 2 norm where n is the number of nodes and is the time step size some numerical experiments are carried out to validate the theoretical analysis

multisymplectic geometry and multisymplectic preissmann scheme for the kdv equation the multisymplectic structure of the kdv equation is presented directly from the variational principle from the numerical view point we give a multisymplectic twelve points scheme which is equivalent to the multisymplectic preissmann scheme finally we test the twelve points scheme on solitary waves over long time intervals

numerical methods for hamiltonian pdes the paper provides an introduction and survey of conservative discretization methods for hamiltonian partial differential equations the emphasis is on variational symplectic and multi symplectic methods the derivation of methods as well as some of their fundamental geometric properties are discussed basic principles are illustrated by means of examples from wave and fluid dynamics

classification by restricted random walks we define a random walk in a data set of a metric space in order that the random walk depends on the pattern of the data restrictions are imposed during its generation since such a restricted random walk investigates only a local subset of the data a series of random walks has to be realized for describing the entire data set an agglomerative graph related classification method is introduced whose hierarchy is based on these restricted random walks it is demonstrated on various examples that this new technique is able to detect efficiently clusters of different shapes without specifying the number of groups in advance

multisymplectic conservation laws for differential and differential difference equations many well known partial differential equations can be written as multisymplectic systems such systems have a structural conservation law from which scalar conservation laws can be derived these c

on multisymplecticity of sinc gauss legendre collocation discretizations for some hamiltonian pdes abstract in this paper a new multisymplectic scheme based on sinc collocation discretization in space and gauss legendre collocation discretization in time for some hamiltonian pdes is developed the scheme preserves the multisymplectic geometry structure exactly by satisfying the discrete multisymplectic conservation law moreover the discrete energy and momentum conservative properties of the multisymplectic integrator are also discussed in order to testify the superiority of the multisymplectic method it is applied to nonlinear dirac equation numerical experiments are given to illustrate the accuracy of the approximation and the conserved quantities

multi symplectic runge kutta methods for nonlinear dirac equations in this paper we consider the multi symplectic runge kutta msrk methods applied to the nonlinear dirac equation in relativistic quantum physics based on a discovery of the multi symplecticity of the equation in particular the conservation of energy momentum and charge under msrk discretizations is investigated by means of numerical experiments and numerical comparisons with non msrk methods numerical experiments presented reveal that msrk methods applied to the nonlinear dirac equation preserve exactly conservation laws of charge and momentum and conserve the energy conservation in the corresponding numerical accuracy to the method utilized it is verified numerically that msrk methods are stable and convergent with respect to the conservation laws of energy momentum and charge and msrk methods preserve not only the inner geometric structure of the equation but also some crucial conservative properties in quantum physics a remarkable advantage of msrk methods applied to the nonlinear dirac equation is the precise preservation of charge conservation law

backward error analysis for numerical integrators backward error analysis has become an important tool for understanding the long time behavior of numerical integration methods this is true in particular for the integration of hamiltonian systems where backward error analysis can be used to show that a symplectic method will conserve energy over exponentially long periods of time such results are typically based on two aspects of backward error analysis i it can be shown that the modified vector fields have some qualitative properties which they share with the given problem and ii an estimate is given for the difference between the best interpolating vector field and the numerical method these aspects have been investigated recently for example by benettin and giorgilli in j statist phys 74 1994 pp 1117 1143 by hairer in ann numer math 1 1994 pp 107 132 and by hairer and lubich in numer math 76 1997 pp 441 462 in this paper we aim at providing a unifying framework and a simplification of the existing results and corresponding proofs our approach to backward error analysis is based on a simple recursive definition of the modified vector fields that does not require explicit taylor series expansion of the numerical method and the corresponding flow maps as in the above cited works as an application we discuss the long time integration of chaotic hamiltonian systems and the approximation of time averages along numerically computed trajectories

homogeneous orthocomplete effect algebras are covered by mv algebras the aim of our paper is twofold first we thoroughly study the sets of meager and hypermeager elements second we study a common generalization of orthocomplete and lattice effect algebras we show that every block of an archimedean homogeneous effect algebra satisfying this generalization is lattice ordered hence such effect algebras can be covered by ranges of observables as a corollary this yields that every block of a homogeneous orthocomplete effect algebra is lattice ordered therefore finite homogeneous effect algebras are covered by mv algebras

continuous lattice effect algebras admitting order continuous states we prove that a necessary and sufficient condition for the existence of a faithful o continuous state on a complete modular atomic effect algebra e is the separability of e moreover we generalize the famous kaplansky theorem about order continuity of complemented complete modular lattices onto complete modular atomic effect algebras some other statements about the algebraic structure of modular effect algebras are shown we prove that every chain and every block in an irreducible complete modular atomic effect algebra is finite moreover every complete atomic modular effect algebra is compactly generated by finite elements

the multi symplectic fourier pseudospectral method for solving two dimensional hamiltonian pdes in this paper the multi symplectic fourier pseudospectral msfp method is generalized to solve two dimensional hamiltonian pdes with periodic boundary conditions using the fourier pseudospectral method in the space of the two dimensional hamiltonian pde 2d hpde the semi discrete system obtained is proved to have semi discrete multi symplectic conservation laws and a global symplecticity conservation law then the implicit midpoint rule is employed for time integration to obtain the msfp method for the 2d hpde the fully discrete multi symplectic conservation laws are also obtained in addition the proposed method is applied to solve the zakharov kuznetsov zk equation and the kadomtsev petviashvili kp equation numerical experiments on soliton solutions of the zk equation and the kp equation show the high accuracy and effectiveness of the proposed method

general formulation and evaluation of agglomerative clustering methods with metric and non metric distances abstract agglomerative clustering methods with stopping criteria are generalized clustering related concepts are rigorously formulated with special consideration on metricity of object space a new definition of combinatoriality is given and a stronger proposition of monotonicity is proven specializations of the general method are applied to non attributive non metric and attributive pseudometric representations of biosequences the furthest neighbor method is shown suitable for non metric use in metric object space four inter clusteral distance functions including a new truly context sensitive method are compared using a method independent goodness criterion for biosequence clustering the new method overcomes the upgma upgmc and furthest neighbor methods

on tense mv algebras the main aim of this article is to study tense mv algebras which are just mv algebras with new unary operations g and h which express a universal time quantifiers tense mv algebras were introduced by d diaconescu and g georgescu using a new notion of an fm function between mv algebras we settle a half of their open problem about representation for some classes of tense mv algebras i e we show that any tense semisimple mv algebra is induced by a time frame analogously to classical works in this field of logic as a by product we obtain a new characterization of extremal states on mv algebras our method gives a general framework for representing functions with the so called jauch piron property including mv morphisms between mv algebras

natural discretizations for the divergence gradient and curl on logically rectangular grids abstract this is the first in series of papers creating a discrete analog of vector analysis on logically rectangular nonorthogonal nonsmooth grids we introduce notations for 2 d logically rectangular grids describe both cell valued and nodal discretizations for scalar functions and construct the natural discretizations of vector fields using the vector components normal and tangential to the cell boundaries we then define natural discrete analogs of the divergence gradient and curl operators based on coordinate invariant definitions and interpret these formulas in terms of curvilinear coordinates such as length of elements of coordinate lines areas of elements of coordinate surfaces and elementary volumes we introduce the discrete volume integral of scalar functions the discrete surface integral and a discrete analog of the line integral and prove discrete versions of the main theorems relating these objects these theorems include the following the discrete analog of relationship div a 0 if and only if a curl b curl a 0 if and only if a grad if a grad then the line integral does not depend on path and if the line integral of a vector function is equal to zero for any closed path then this vector is the gradient of a scalar function last we define the discrete operators div grad and curl in terms of primitive differencing operators based on forward and backward differences and primitive metric operators related to multiplications of discrete functions by length of edges areas of surfaces and volumes of 3 d cells these formulations elucidate the structure of the discrete operators and are useful when investigating the relationships between operators and their adjoints

a note on multisymplectic fourier pseudospectral discretization for the nonlinear schrodinger equation based on the multisymplectic fourier pseudospectral scheme for the nonlinear schrodinger nls equation we investigate some discrete properties corresponding to local conservation laws of the original equation the discrete normal conservation law is proved and the error estimation of local and global energy conservation laws are also obtained numerical experiments for cubic nls equation are provided to demonstrate the consistency between the theoretical analysis and the numerical results

operators on mv algebras and their representations a crucial problem concerning tense operators on mv algebras is their representations having an mv algebra with tense operators we can ask if there exists a frame yielding a representation of this mv algebra we solve this problem for semisimple mv algebras c 2013 elsevier b v all rights reserved

the symplectic evans matrix and the instability of solitary waves and fronts hamiltonian evolution equations which are equivariant with respect to the action of a lie group are models for physical phenomena such as oceanographic flows optical fibres and atmospheric flows and such systems often have a wide variety of solitary wave or front solutions in this paper we present a new symplectic framework for analyzing the spectral problem associated with the linearization about such solitary waves and fronts at the heart of the analysis is a multi symplectic formulation of hamiltonian partial differential equations where a distinct symplectic structure is assigned for the time and space directions with a third symplectic structure with two form denoted by omega associated with a coordinate frame moving at the speed of the wave this leads to a geometric decomposition and symplectification of the evans function formulation for the linearization about solitary waves and fronts we introduce the concept of the symplectic evans matrix a matrix consisting of restricted omega symplectic forms by applying hodge duality to the exterior algebra formulation of the evans function we find that the zeros of the evans function correspond to zeros of the determinant of the symplectic evans matrix based on this formulation we prove several new properties of the evans function restricting the spectral parameter lambda to the real axis we obtain rigorous results on the derivatives of the evans function near the origin based solely on the abstract geometry of the equations and results for the large lambda behaviour which use primarily the symplectic structure but also extend to the non symplectic case the lie group symmetry affects the evans function by generating zero eigenvalues of large multiplicity in the so called systems at infinity we present a new geometric theory which describes precisely how these zero eigenvalues behave under perturbation by combining all these results a new rigorous sufficient condition for instability of solitary waves and fronts is obtained the theory applies to a large class of solitary waves and fronts including waves which are biasymptotic to a nonconstant manifold of states as x tends to infinity to illustrate the theory it is applied to three examples a boussinesq model from oceanography a class of nonlinear schrodinger equations from optics and a nonlinear klein gordon equation from atmospheric dynamics

basic decomposition of elements and jauch piron effect algebras we show that every element of a complete atomic effect algebra e has a unique basic decomposition into a sum of a sharp element and unsharp multiples of isotropic atoms of e consequently for such effect algebras we obtain the smearing theorem for states establishing that every order continuous state existing on sharp elements of e can be extended to a state on e for a s complete separable atomic effect algebra e we prove that e is a unital and jauch piron effect algebra if and only if the set s e of all sharp elements of e is a unital jauch piron orthomodular lattice and for finite e s e is a boolean algebra

high order multi symplectic schemes for the nonlinear klein gordon equation in this paper we construct multi symplectic schemes with any order of accuracy for the nonlinear klein gordon equation by concatenating the symplectic schemes for odes some existing schemes such as the preissman scheme and the leap frog scheme and new multi symplectic schemes are constructed we also show that the composition method which plays a crucial role in finding the high order symplectic integrators for the odes can also be applied to construct high order multi symplectic schemes for pdes extension of the concatenating method to more than one space dimension is also discussed numerical experiments are presented to show the order and the efficiency of the constructed multi symplectic schemes

geometric integrators for classical spin systems practical structure preserving methods for integrating classical heisenberg spin systems are discussed two new integrators are derived and compared including 1 a symmetric energy and spin length preserving integrator based on a red black splitting of the spin sites combined with a staggered timestepping scheme and 2 a lie poisson symplectic integrator based on hamiltonian splitting the methods are applied to both 1d and 2d lattice models and are compared with the commonly used explicit runge kutta projected runge kutta and implicit midpoint schemes on the bases of accuracy conservation of invariants and computational expense it is shown that while any of the symmetry preserving schemes improves the integration of the dynamics of solitons or vortex pairs compared to runge kutta or projected runge kutta methods the staggered red black scheme is far more efficient than the other alternatives

splitting multisymplectic integrators for maxwell s equations in the paper we describe a novel kind of multisymplectic method for three dimensional 3 d maxwell s equations splitting the 3 d maxwell s equations into three local one dimensional lod equations then applying a pair of symplectic runge kutta methods to discretize each resulting lod equation it leads to splitting multisymplectic integrators we say this kind of schemes to be lod multisymplectic scheme lod ms the discrete conservation laws convergence dispersive relation dissipation and stability are investigated for the schemes theoretical analysis shows that the schemes are unconditionally stable non dissipative and of first order accuracy in time and second order accuracy in space as a reduction we also consider the application of lod ms to 2 d maxwell s equations numerical experiments match the theoretical results well they illustrate that lod ms is not only efficient and simple in coding but also has almost all the nature of multisymplectic integrators

multi symplectic spectral discretizations for the zakharov kuznetsov and shallow water equations abstract the time evolution of multi symplectic pdes with periodic boundary conditions in one and two space dimensions is considered we introduce the idea of a multi symplectic fourier transform for systems on periodic domains leading to a semi discretization on fourier space and to the concept of multi symplecticity on fourier space the spatial discretization leads to a discrete lattice model with certain discrete conservation laws imposed on it a discrete wave model a by product of the semi discretization is that it leads automatically to a system of hamiltonian odes in time when truncated we show that the one dimensional shallow water equations and the two dimensional zakharov kuznetsov equation are multi symplectic and derive spectral discretizations for these systems and present numerical experiments

on the multi symplectic structure of boussinesq type systems i derivation and mathematical properties the boussinesq equations are known since the end of the xixst century however the proliferation of various boussinesq type systems started only in the second half of the xxst century today they come under various flavours depending on the goals of the modeller at the beginning of the xxist century an effort to classify such systems at least for even bottoms was undertaken and developed according to both different physical regimes and mathematical properties with special emphasis in this last sense on the existence of symmetry groups and their connection to conserved quantities of particular interest are those systems admitting a symplectic structure with the subsequent preservation of the total energy represented by the hamiltonian in the present paper a family of boussinesq type systems with multi symplectic structure is introduced some properties of the new systems are analyzed their relation with already known boussinesq models the identification of those systems with additional hamiltonian structure as well as other mathematical features like well posedness and existence of different types of solitary wave solutions the consistency of multi symplectic systems with the full euler equations is also discussed

characterization of homogeneity in orthocomplete atomic effect algebras we prove that homogeneity can be characterized by the decompositions of sharp elements in orthocomplete atomic effect algebras especially an orthocomplete atomic effect algebra is homogeneous if and only if the non zero coefficient of an atom in any atomic decomposition of all sharp elements is the same if and only if the non zero coefficient of an atom in any atomic decomposition of the unit element is the same which confirm the name homogeneous as an application we prove the state smearing theorem for orthocomplete atomic homogeneous effect algebras

an efficient agglomerative clustering algorithm using a heap abstract an efficient algorithm for agglomerative clustering is presented the algorithm uses a heap in which distances of all pairs of clusters are stored then the nearest pair of clusters is given by the element of the root node of the binary tree corresponding to the heap updating the heap at each stage of the hierarchy is easily implemented by shifting up or down the elements of the heap along the path of the heap tree the computation time of the algorithm is at most o n2 log n when n objects are going to be classified

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

a conservative spectral method for several two dimensional nonlinear wave equations a conservative spectral method is proposed to solve several two dimensional nonlinear wave equations the conventional fast fourier transform is used to approximate the spatial derivatives and a three level difference scheme with a free parameter is to advance the solution in time our time discretization is semi implicit in the sense that the linear terms are treated implicitly while the nonlinear terms are evaluated only by previous time levels and thus treated explicitly however the cost of the algorithm is no greater than that of a fully explicit method because the linear boundary value problem that must be solved at each time step is almost trivial in a spectral spatial discretization a linear stability analysis shows that the method leads to a less restrictive stability condition than the corresponding explicit one the method is conservative and the ratio of the numerical dispersion to the physical dispersion is of the order o t2 applications of our method to the kadomtsev petviashvili and the zakharov kuznetsov equations exhibit excellent results

the multi symplecticity of partitioned runge kutta methods for hamiltonian pdes in this article we consider partitioned runge kutta prk methods for hamiltonian partial differential equations pdes and present some sufficient conditions for multi symplecticity of prk methods of hamiltonian pdes

local discontinuous galerkin methods for nonlinear dispersive equations we develop local discontinuous galerkin dg methods for solving nonlinear dispersive partial differential equations that have compactly supported traveling waves solutions the so called compactons the schemes we present extend the previous works of yan and shu on approximating solutions for linear dispersive equations and for certain kdv type equations we present two classes of dg methods for approximating solutions of such pdes first we generate nonlinearly stable numerical schemes with a stability condition that is induced from a conservation law of the pde an alternative approach is based on constructing linearly stable schemes i e schemes that are linearly stable to small perturbations the numerical simulations we present verify the desired properties of the methods including their expected order of accuracy in particular we demonstrate the potential advantages of using dg methods over pseudospectral methods in situations where discontinuous fronts and rapid oscillations co exist in a solution

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

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

on the hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms we reconsider the problem of the hamiltonian interpolation of symplectic mappings following moser s scheme we prove that for any mapping e analytic and e close to the identity there exists an analytic autonomous hamiltonian system he such that its time one mapping he differs from e by a quantity exponentially small in 1 e this result is applied in particular to the problem of numerical integration of hamiltonian systems by symplectic algorithms it turns out that when using an analytic symplectic algorithm of orders to integrate a hamiltonian systemk one actually follows exactly namely within the computer roundoff error the trajectories of the interpolating hamiltonian he or equivalently of the rescaled hamiltonian ke e 1he which differs fromk but turns out to be e5 close to it special attention is devoted to numerical integration for scattering problems

nonlinear stability analysis of area preserving integrators linear stability analysis is inadequate for integrators designed for nondissipative systems such as hamiltonian systems in which nonlinear effects are often decisive mathematical theory exists kam theory for rigorous analysis of small perturbations from equilibria but it needs to be expressed in a form that is more easily applicable to the study of area preserving maps we have pursued this obtaining a completely rigorous nonlinear stability analysis for elliptic equilibria based on the moser twist theorem and a result of cabral and meyer nonlinearity 12 1999 pp 1351 1362 together with the theory of normal forms for hamiltonian systems the result is a determination of necessary and sufficient conditions for stability these conditions are sharpened for the case of reversible maps and applied to the symplectic members of the newmark family of integrators which includes the leapfrog the implicit midpoint and the stormer cowell methods nonlinear stability limits are more severe than those of linear theory as an example the leapfrog scheme actually has a step size limitation of 71 of that predicted by linear analysis

meager projections in orthocomplete homogeneous effect algebras abstract we define meager projections on homogeneous effect algebras and discuss their properties as an application we prove that if e is an orthocomplete homogeneous effect algebra such that s e is lattice ordered then e is a lattice effect algebra which gives an affirmative answer to an open problem stated by mesiar and stup anova

on the stability of symplectic and energy momentum algorithms for non linear hamiltonian systems with symmetry abstract this paper presents a detailed comparison of two implicit time integration schemes for a simple non linear hamiltonian system with symmetry the motion of a particle in a central force field the goal is to establish analytical and numerical results pertaining to the stability properties of the implicit mid point rule the proto typical implicit symplectic method and a particular energy momentum conserving scheme and to compare the two schemes with respect to accuracy while all results presented herein are within the context of a simple model problem the problem was constructed so as to exhibit key features typical of more complex systems with symmetry such as those arising in non linear solid mechanics namely the presence of large and relatively slow overall motions together with high frequency internal motions