self dual codes and modules for finite groups in characteristic two using representation theoretical methods we investigate self dual group codes and their extensions in characteristic 2 we prove that the existence of a self dual extended group code heavily depends on a particular structure of the group algebra kg which can be checked by an easy to handle criteria in elementary number theory surprisingly in the binary case such a code is doubly even if the converse of gleason s theorem holds true i e the length of the code is divisible by 8 furthermore we give a short representation theoretical proof of an earlier result of sloane and thompson which states that a binary self dual group code is never doubly even if the sylow 2 subgroups of g are cyclic it turns out that exactly in the case of a cyclic or klein four group as sylow 2 subgroup doubly even group codes do not exist

a semi analytical approach to molecular dynamics despite numerous computational advances over the last few decades molecular dynamics still favors explicit and thus easily parallelizable time integrators for large scale numerical simulation as a consequence computational efficiency in solving its typically stiff oscillatory equations of motion is hampered by stringent stability requirements on the time step size in this paper we present a semi analytical integration scheme that offers a total speedup of a factor 30 compared to the verlet method on typical md simulation by allowing over three orders of magnitude larger step sizes by efficiently approximating the exact integration of the strong harmonic forces of covalent bonds through matrix functions far improved stability with respect to time step size is achieved without sacrificing the explicit symplectic time reversible or fine grained parallelizable nature of the integration scheme we demonstrate the efficiency and scalability of our integrator on simulations ranging from dna strand unbinding and protein folding to nanotube resonators

shadow hybrid monte carlo an efficient propagator in phase space of macromolecules shadow hybrid monte carlo shmc is a new method for sampling the phase space of large molecules particularly biological molecules it improves sampling of hybrid monte carlo hmc by allowing larger time steps and system sizes in the molecular dynamics md step the acceptance rate of hmc decreases exponentially with increasing system size n or time step t this is due to discretization errors introduced by the numerical integrator shmc achieves an asymptotic o n1 4 speedup over hmc by sampling from all of phase space using high order approximations to a shadow or modified hamiltonian exactly integrated by a symplectic md integrator shmc satisfies microscopic reversibility and is a rigorous sampling method shmc requires extra storage modest computational overhead and a reweighting step to obtain averages from the canonical ensemble this is validated by numerical experiments that compute observables for different molecules ranging from a small n alkane butane with four united atoms to a larger solvated protein with 14 281 atoms in these experiments shmc achieves an order magnitude speedup in sampling efficiency for medium sized proteins sampling efficiency is measured by monitoring the rate at which different conformations of the molecules dihedral angles are visited and by computing ergodic measures of some observables

lax pairs of time dependent gross pitaevskii equation we calculate the lax pairs of homogeneous and inhomogeneous one dimensional time dependent gross pitaevskii equations with time dependent scattering length the inhomogeneity corresponds to linear and quadratic potentials our approach introduces a systematic method of searching for the lax pair corresponding to a given differential equation we derive known lax pairs for the gross pitaevskii equation with homogeneous and quadratic potentials and time dependent scattering length we also derive new lax pairs corresponding to a gross pitaevskii equation with a linear potential using the resulting lax pairs the darboux transformation can be performed and exact solutions of the gross pitaevskii equation can be obtained for experimentally relevant cases such as solitonic solutions

a note on self dual group codes we classify group algebras over galois rings containing self dual ideals i e ideals c which satisfy c c sup spl perp with respect to the natural nondegenerate bilinear form given on group algebras

variational integrators for electric circuits in this contribution we develop a variational integrator for the simulation of stochastic and multiscale electric circuits when considering the dynamics of an electric circuit one is faced with three special situations 1 the system involves external control forcing through external controlled voltage sources and resistors 2 the system is constrained via the kirchhoff current kcl and voltage laws kvl 3 the lagrangian is degenerate based on a geometric setting an appropriate variational formulation is presented to model the circuit from which the equations of motion are derived a time discrete variational formulation provides an iteration scheme for the simulation of the electric circuit dependent on the discretization the intrinsic degeneracy of the system can be canceled for the discrete variational scheme in this way a variational integrator is constructed that gains several advantages compared to standard integration tools for circuits in particular a comparison to bdf methods which are usually the method of choice for the simulation of electric circuits shows that even for simple lcr circuits a better energy behavior and frequency spectrum preservation can be observed using the developed variational integrator

numerical algorithms for highly oscillatory dynamic system based on commutator free method in the present paper an efficiently improved modified magnus integrator algorithm based on commutator free method is proposed for the second order dynamic systems with time dependent high frequencies firstly the second order dynamic systems are transferred to the frame of reference by introducing new variable so that highly oscillatory behaviour inherited from the entries then the modified magnus integrator method based on local linearization is appropriately designed for solving the above new form and some optimized strategies for reducing the number of function evaluations and matrix operations are also suggested finally several numerical examples for highly oscillatory dynamic systems such as airy equation bessel equation mathieu equation are presented to demonstrate the validity and effectiveness of the proposed method

symmetric semisimple modules of group algebras over finite fields and self dual permutation codes abstract a module of a finite group over a finite field with a symmetric non degenerate bilinear form which is invariant by the group action is called a symmetric module in this paper a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained and some applications to the self dual permutation codes are shown

hyperbolic modules and cyclic subgroups abstract let g be a finite group of odd order f a finite field of odd characteristic p and b a finite dimensional symplectic f g module we show that b is f g hyperbolic i e it contains a self perpendicular f g submodule iff it is f n hyperbolic for every cyclic subgroup n of g

an averaging technique for highly oscillatory hamiltonian problems in this paper we are concerned with the numerical solution of highly oscillatory hamiltonian systems with a stiff linear part we construct an averaged system whose solution remains close to the exact one over bounded time intervals possesses the same adiabatic and hamiltonian invariants as the original system and is nonstiff we then investigate its numerical approximation through a method which combines a symplectic integration scheme and an acceleration technique for the evaluation of time averages developed in e cances et al numer math 100 2005 pp 211 232 eventually we demonstrate the efficiency of our approach on two test problems with one or several frequencies

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

a stochastic immersed boundary method for fluid structure dynamics at microscopic length scales in modeling many biological systems it is important to take into account flexible structures which interact with a fluid at the length scale of cells and cell organelles thermal fluctuations of the aqueous environment become significant in this work it is shown how the immersed boundary method of c s peskin the immersed boundary method acta num 11 2002 1 39 for modeling flexible structures immersed in a fluid can be extended to include thermal fluctuations a stochastic numerical method is proposed which deals with stiffness in the system of equations by handling systematically the statistical contributions of the fastest dynamics of the fluid and immersed structures over long time steps an important feature of the numerical method is that time steps can be taken in which the degrees of freedom of the fluid are completely underresolved partially resolved or fully resolved while retaining a good level of accuracy error estimates in each of these regimes are given for the method a number of theoretical and numerical checks are furthermore performed to assess its physical fidelity for a conservative force the method is found to simulate particles with the correct boltzmann equilibrium statistics it is shown in three dimensions that the diffusion of immersed particles simulated with the method has the correct scaling in the physical parameters the method is also shown to reproduce a well known hydrodynamic effect of a brownian particle in which the velocity autocorrelation function exhibits an algebraic 3 2 decay for long times b j alder t e wainwright decay of the velocity autocorrelation function phys rev a 1 1 1970 18 21 a few preliminary results are presented for more complex systems which demonstrate some potential application areas of the method specifically we present simulations of osmotic effects of molecular dimers worm like chain polymer knots and a basic model of a molecular motor immersed in fluid subject to a hydrodynamic load force the theoretical analysis and numerical results show that the immersed boundary method with thermal fluctuations captures many important features of small length scale hydrodynamic systems and holds promise as an effective method for simulating biological phenomena on the cellular and subcellular length scales

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

a comparative analysis of painleve lax pair and similarity transformation methods in obtaining the integrability conditions of nonlinear schrodinger equations we derive the integrability conditions of nonautonomous nonlinear schrodinger equations using the lax pair and similarity transformation methods we present a comparative analysis of these integrability conditions with those of the painleve method we show that while the painleve integrability conditions restrict the dispersion nonlinearity and dissipation gain coefficients to be space independent and the external potential to be only a quadratic function of position the lax pair and the similarity transformation methods allow for space dependent coefficients and an external potential that is not restricted to the quadratic form the integrability conditions of the painleve method are retrieved as a special case of our general integrability conditions we also derive the integrability conditions of nonautonomous nonlinear schrodinger equations for two and three spacial dimensions

phase engineering modulational instability and solitons of gross pitaevskii type equations in 1 1 dimensions motivated by recent proposals of collisionally inhomogeneous bose einstein condensates becs which have a spatially modulated scattering length we introduce a phase imprint into the macroscopic order parameter governing the dynamics of becs with spatiotemporal varying scattering length described by a cubic gross pitaevskii gp equation and then suitably engineer the imprinted phase to generate the modified gp equation also called the cubic derivative nonlinear schrodinger nls equation this equation describes the dynamics of condensates with two body attractive and repulsive interactions in a time varying quadratic external potential we then carry out a theoretical analysis which invokes a lens type transformation that converts the cubic derivative nls equation into a modified nls equation with only explicit temporal dependence our analysis suggests a particular interest in a specific time varying potential with the strength of the magnetic trap similar to 1 t t 2 for a time varying quadratic external potential of this kind an explicit expression for the growth rate of a purely growing modulational instability is presented and analyzed we point out the effect of the imprint parameter and the parameter t on the instability growth rate as well as on the solitary waves of the becs doi 10 1103 physreve 87 022914

a uniformly accurate ua multiscale time integrator pseudospectral method for the dirac equation in the nonrelativistic limit regime we propose and rigourously analyze a multiscale time integrator fourier pseudospectral mti fp method for the dirac equation with a dimensionless parameter varepsilon in 0 1 which is inversely proportional to the speed of light in the nonrelativistic limit regime i e 0 varepsilon ll 1 the solution exhibits highly oscillatory propagating waves with wavelength o varepsilon 2 and o 1 in time and space respectively due to the rapid temporal oscillation it is quite challenging in designing and analyzing numerical methods with uniform error bounds in varepsilon in 0 1 we present the mti fp method based on properly adopting a multiscale decomposition of the solution of the dirac equation and applying the exponential wave integrator with appropriate numerical quadratures by a careful study of the error propagation and using the energy method we establish two independent error estimates via two different mathematical approaches as h m 0 frac tau 2 varepsilon 2 and h m 0 tau 2 varepsilon 2 where h is the mesh size tau is the time step and m 0 depends on the regularity of the solution these two error bounds immediately imply that the mti fp method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth and uniformly in time with linear convergence rate at o tau for all varepsilon in 0 1 and optimally with quadratic convergence rate at o tau 2 in the regimes when either varepsilon o 1 or 0 varepsilon lesssim tau numerical results are reported to demonstrate that our error estimates are optimal and sharp finally the mti fp method is applied to study numerically the convergence rates of the solution of the dirac equation to those of its limiting models when varepsilon to0

an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations in this chapter we derive and analyse an energy preserving and symmetric scheme for nonlinear hamiltonian wave equations which can exactly preserve the energy of the underlying hamiltonian wave equations to this end we first define and discuss the bounded operator argument functions on the underlying domain we then introduce an operator variation of constants formula based on which we present an energy preserving scheme for nonlinear hamiltonian wave equations the scheme preserves the energy of the original continuous hamiltonian system exactly in comparison with the existing work on this topic such as the well known average vector field avf formula for hamiltonian ordinary differential equations the energy preserving scheme avoids the semi discretisation of spatial derivatives and exactly preserves the hamiltonian of the original continuous hamiltonian wave equation this point is very significant in comparison with the avf formula since the avf formula can preserve only the energy of hamiltonian ordinary differential equations hence the main theme of this chapter is to establish a scheme which can exactly preserve the energy of the nonlinear hamiltonian wave equation the chapter is also accompanied by some examples

on error bounds for the gautschi type exponential integrator applied to oscillatory second order differential equations this paper studies a numerical method for second order oscillatory differential equations in which high frequency oscillations are generated by a linear time and or solution dependent part for constant linear part it is known that the method allows second order error bounds independent of the product of the step size with the frequencies and is therefore a long time step method most real world problems are not of that kind and it is important to study more general equations the analysis in this paper shows that one obtains second order error bounds even in the case of a time and or solution dependent linear part if the matrix is evaluated at averaged positions

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

nonautonomous integrable nonlinear schrodinger equations with generalized external potentials inhomogeneous nonlinear schrodinger equations are constructed systematically using the covariance with respect to the darboux transformation the darboux invariants and the spectral parameter are functions of time and space which result in equations describing nonautonomous solitons moving in generalized external potentials all the known inhomogeneous nonlinear schrodinger equations as well as new ones of a more generalized form can be derived from the constructed equation one and two soliton solutions are explicitly constructed using the darboux transformation

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

metabolic modules of finite group algebras over finite fields of characteristic two abstract let g be a finite group and let f be a finite field of characteristic 2 we introduce f special subgroups and f special elements of g in the case where f contains a p th primitive root of unity for each odd prime p dividing the order of g e g it is the case once f is a splitting field for all subgroups of g the f special elements of g coincide with real elements of odd order we prove that a symmetric fg module v is metabolic if and only if the restriction v d of v to every f special subgroup d of g is metabolic and also if and only if the characteristic polynomial on v defined by every f special element of g is a square of a polynomial over f some immediate applications to characters self dual codes and witt groups are given

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

on tensor induction for representations of finite groups abstract it is well known that if d is an irreducible complex representation of a finite group g then every direct summand of the restriction of d to a subgroup h must have degree at least as large as the degree of d divided by the index g h moreover d is induced from h if and only if the restriction does have a direct summand whose dimension is equal to this quotient this paper explores the possibility of an analogous result for tensor induction under the additional assumption that d is faithful quasi primitive and not a tensor product of projective representations of degree greater than 1 and that the fitting subgroup f g is not in the centre z g the main question is this if the restriction has a projective tensor factor whose degree is the g h th root of the degree of d does it follow that d is tensor induced from h among other results examples are given to show that the answer can be negative when the index is 2 an affirmative answer is proved for normal subgroups of odd index and also for arbitrary subgroups of odd prime index as might be expected the key lies in the study of f g z g as a symplectic module over a finite prime field in particular in exploring the connection between ordinary induction and form induction of such modules

an iterative solution method for solving f a x b using krylov subspace information obtained for the symmetric positive definite matrix a abstract the conjugate gradients method generates successive approximations x i for the solution of the linear system ax b where a is symmetric positive definite and usually sparse it will be shown how intermediate information obtained by the conjugate gradients cg algorithm or by the closely related lanczos algorithm can be used to solve f a x b iteratively in an efficient way for suitable functions f the special case f a a 2 is discussed in particular we also consider the problem of solving ax b for different right hand sides b a variant on a well known algorithm for that problem is proposed which does not seem to suffer from the usual loss of orthogonality in the standard cg and lanczos algorithms

long time step methods for oscillatory differential equations considered are numerical integration schemes for nondissipative dynamical systems in which multiple time scales are present it is assumed that one can do an explicit separation of the rhs forces into fast forces and slow forces such that i the fast forces contain the high frequency part of the solution ii the fast forces are conservative and iii the reduced problem consisting only of the fast forces can be integrated much more cheaply than the full problem the fast forces are allowed to have low frequency components particular applications for which the schemes are intended include n body problems for which most of the forces are slow and nonlinear wave phenomena for which the fast forces can be propagated by spectral methods the assumption of cheap integration of fast forces implies that the overall cost of integration is primarily determined by the step size used to sample the slow forces a long time step method is one in which this step size exceeds half the period of the fastest normal mode present in the full system an existing method that comes close to qualifying is the impulse method also known as verlet i and r respa it is shown that it might fail though for a couple of reasons first it suffers a serious loss of accuracy if the step size is near a multiple of the period of a normal mode and second it is unstable if the step size is near a multiple of half the period of a normal mode proposed in this paper is a mollified impulse method having an error bound that is independent of the frequency of the fast forces it is also shown to possess superior stability properties theoretical results are supplemented by numerical experiments the method is efficient and reasonably easy to implement

rattle a velocity version of the shake algorithm for molecular dynamics calculations an algorithm called rattle for integrating the equations of motion in molecular dynamics calculations for molecular models with internal constraints is presented the algorithm is similar to shake which is one of the standard methods for performing such calculations rattle calculates the positions and velocities at the next time from the positions and velocities at the present time step without requiring information about the earlier history like shake it is based on the verlet algorithm and retains the simplicity of using cartesian coordinates for each of the atoms to describe the configuration of a molecule with internal constraints rattle guarantees that the coordinates and velocities of the atoms in a molecule satisfy the internal constraints at each time step rattle has two advantages over shake on computers of fixed precision it is of higher precision than shake since it deals directly with the velocities it is easier to modify rattle for use with the recently developed constant temperature and constant pressure molecular dynamics methods and with the nonequilibrium molecular dynamics methods that make use of rescaling of the atomic velocities

the extended estabrook wahlquist method abstract variable coefficient korteweg de vries vckdv modified korteweg de vries vcmkdv and nonlinear schrodinger nls equations have a long history dating from their derivation in various applications a technique based on extended lax pairs has been devised recently to derive time and space dependent coefficient generalizations of various such lax integrable nlpde hierarchies which are thus more general than almost all cases considered earlier via methods such as the painleve test bell polynomials and similarity methods however this technique although operationally effective has the significant disadvantage that for any integrable system with spatiotemporally varying coefficients one must guess a generalization of the structure of the known lax pair for the corresponding system with constant coefficients motivated by the somewhat arbitrary nature of the above procedure we embark in this paper on an attempt to systematize the derivation of lax integrable systems with variable coefficients we consider the estabrook wahlquist ew prolongation technique a relatively self consistent procedure requiring little prior information however this immediately requires that the technique be significantly generalized in several ways including solving matrix partial differential equations instead of algebraic ones as the structure of the lax pair is systematically computed and also in solving the constraint equations to deduce the explicit forms for various coefficient matrices the new and extended ew technique which results is illustrated by algorithmically deriving generalized lax integrable versions of the nls generalized fifth order kdv mkdv and derivative nonlinear schrodinger dnls equations we also show how this method correctly excludes the existence of a nontrivial lax pair for a nonintegrable nlpde such as the variable coefficient cubic quintic nls

heterogeneous multiscale methods for stiff ordinary differential equations the heterogeneous multiscale methods hmm is a general framework for the numerical approximation of multiscale problems it is here developed for ordinary differential equations containing differe

explicit symplectic integrators using hessian vector products in 1991 rowlands proposed an effectively fourth order effectively two stage explicit symplectic integrator based on using a hessian vector product to modify the force evaluation in the leapfrog method and evidence indicates that for modest accuracy this method is highly competitive here we explore the possible existence of even more efficient fourth order explicit symplectic integrators also based on the use of hessian vector products and the concept of effective order first it is shown that the cost of a force evaluation plus a hessian vector product is less than twice the cost of the force alone for a sum of two body interactions then a new method is found that is generally better than both the method of rowlands and that of calvo according to both a theoretical measure of the error and limited numerical experiments the basic motivation behind the new method is quite simple do a hessian vector computation only every other step significantly cutting costs while only marginally increasing the error the idea of effective order means that we allow for both the possibility of preprocessing the initial values before application of the basic method and the possibility of postprocessing the values obtained by the basic method but at output points only for some applications processing is unnecessary but in any case processing has been shown to be possible at low additional cost the derivation of the new method illustrates how to simplify by the use of h series the determination of parameters for methods of increased effective accuracy