a new family of runge kutta type methods for the numerical integration of perturbed oscillators our task in this paper is to present a new family of methods of the runge kutta type for the numerical integration of perturbed oscillators the key property is that those algorithms are able to integrate exactly without truncation error harmonic oscillators and that for perturbed problems the local error contains the perturbation parameter as a factor some numerical examples show the excellent behaviour when they compete with runge kutta nystrom type methods

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

blocks of group algebras are derived simple a derived version of maschke s theorem for finite groups is proved the derived categories bounded or unbounded of all blocks of the group algebra of a finite group are simple in the sense that they admit no nontrivial recollements this result is independent of the characteristic of the base field

dynamique des applications holomorphes propres de domaines reguliers et probleme de l injectivite this paper deals with proper holomorphic self maps of smoothly bounded pseudoconvex domains in open image in new window we study the dynamical properties of their extension to the boundary and show that their non wandering sets are always contained in the weakly pseudoconvex part of the boundary in the case of complete circular domains we combine this fact with an entropy degree argument to show that the maps are automorphisms some of our results remain true in open image in new window

on proper holomorphic mappings between rigid polynomial domains in cn 1 abstract we study the behavior of the branch locus of proper holomorphic mappings between nondegenerate rigid polynomial domains in c n 1 nonnecessary pseudoconvex in particular we show that it depends only on the first domain this paper generalizes publ mat 45 2001 69 77 in the nonpseudoconvex case

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

on multisymplecticity of partitioned runge kutta methods previously it has been shown that discretizing a multi hamiltonian pde in space and time with partitioned runge kutta methods gives rise to a system of equations that formally satisfy a discrete multisymplectic conservation law however these previous studies use the same partitioning of the variables into two parts in both space and time this gives rise to a large number of cases to be considered each with its own set of conditions to be satisfied we present here a much simpler set of conditions covering all of these cases where the variables are partitioned independently in space and time into an arbitrary number of parts in general it is not known when such a discretization of a multi hamiltonian pde will give rise to a well defined numerical integrator however a numerical integrator that is explicit will typically be well defined in this paper we give sufficient conditions on a multi hamiltonian pde for a lobatto iiia iiib discretization in space to give rise to explicit odes and an algorithm for constructing these odes

long time energy conservation of numerical methods for oscillatory differential equations we consider second order differential systems where high frequency oscillations are generated by a linear part we present a frequency expansion of the solution and we discuss two invariants of the system that determine the coefficients of the frequency expansion these invariants are related to the total energy and the oscillatory harmonic energy of the original system for the numerical solution we study a class of symmetric methods that discretize the linear part without error we are interested in the case where the product of the step size with the highest frequency can be large in the sense of backward error analysis we represent the numerical solution by a frequency expansion where the coefficients are the solution of a modified system this allows us to prove the near conservation of the total and the oscillatory energy over very long time intervals

high order closed newton cotes trigonometrically fitted formulae for the numerical solution of the schrodinger equation abstract in this paper we investigate the connection between closed newton cotes formulae trigonometrically fitted methods symplectic integrators and efficient integration of the schrodinger equation the study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry see the relevant literature and the references here in this paper we study the closed newton cotes formulae and we write them as symplectic multilayer structures based on the closed newton cotes formulae we also develop trigonometrically fitted symplectic methods an error analysis for the one dimensional schrodinger equation of the new developed methods and a comparison with previous developed methods is also given we apply the new symplectic schemes to the well known radial schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems

explicit multi symplectic extended leap frog methods for hamiltonian wave equations in this paper we study the integration of hamiltonian wave equations whose solutions have oscillatory behaviors in time and or space we are mainly concerned with the research for multi symplectic extended runge kutta nystrom erkn discretizations and the corresponding discrete conservation laws we first show that the discretizations to the hamiltonian wave equations using two symplectic erkn methods in space and time respectively lead to an explicit multi symplectic integrator eleap frogi then we derive another multi symplectic discretization using a symplectic erkn method in time and a symplectic partitioned runge kutta method which is equivalent to the well known stormer verlet method in space eleap frogii these two new multi symplectic schemes are extensions of the leap frog method the numerical stability and dispersive properties of the new schemes are analyzed numerical experiments with comparisons are presented where the two new explicit multi symplectic methods and the leap frog method are applied to the linear wave equation and the sine gordon equation the numerical results confirm the superior performance and some significant advantages of our new integrators in the sense of structure preservation

new modified runge kutta nystrom methods for the numerical integration of the schrodinger equation in this work we construct new runge kutta nystrom methods especially designed to integrate exactly the test equation y w 2y we modify two existing methods the runge kutta nystrom methods of fifth and sixth order we apply the new methods to the computation of the eigenvalues of the schrodinger equation with different potentials such as the harmonic oscillator the doubly anharmonic oscillator and the exponential potential

on multi symplectic partitioned runge kutta methods for hamiltonian wave equations many conservative pdes such as various wave equations schrodinger equations kdv equations and so on allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes in this note we show the discretization to hamiltonian wave equations in space and time using two symplectic partitioned runge kutta methods respectively leads to multi symplectic integrators which preserve a symplectic conservation law under some conditions we discuss the energy and momentum conservative property of partitioned runge kutta methods for the wave equations with a quadratic potential

proper self maps of weakly pseudoconvex domains this theorem was proved by burns and shnider i 5 in the case that d is strongly pseudoconvex in the more general case pin6uk 1 9 corollary 2 has reduced the problem of showing that a proper mapping f d d is biholomorphic to proving that f is unbranched i e that det f 0 our interest in the theorem above stems from the fact that if f d t d 2 is a proper holomorphic mapping of a weakly pseudoconvex domain d 1 onto a second domain dz it is possible for f to branch if d1 d 2 this is contrary to what may occur if d 1 is assumed to be strongly pseudoconvex see alexander 1 pin6uk 9 10 fornaess 1 8 bell 1 3 diederich and fornaess 1 7 it has recently been proved in 1 4 and in i 6 that the mapping f in the theorem extends smoothly to this fact is basic to our proof the theorem may be shown to remain valid if 112 is replaced by a stein manifold see 2 but si ace the arguments of 4 and 6 do not carry over automatically to this case the methods used in 1 2 are more technical

the bergman kernel function and proper holomorphic mappings it is proved that a proper holomorphic mapping f between bounded complete reinhardt domains extends holomorphically past the boundary and that if in addition f 1 0 01 then f is a polynomial mapping the proof is accomplished via a transformation rule for the bergman kernel function under proper holomorphic mappings

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