beta integers as natural counting systems for quasicrystals recently discrete sets of numbers the integers have been proposed as numbering tools in quasicrystalline studies indeed there exists a unique numeration system based on the irrational in which the integers are all real numbers with no fractional part these integers appear to be quite appropriate for describing some quasilattices relevant to quasicrystallography when precisely is equal to golden mean to or to i e when is one of the self similarity ratios observed in quasicrystalline structures as a matter of fact integers are natural candidates for coordinating quasicrystalline nodes and also the bragg peaks beyond a given intensity in corresponding diffraction patterns they could play the same role as ordinary integers do in crystallography in this paper we prove interesting algebraic properties of the sets when is a quadratic unit pv number a class of algebraic integers which includes the quasicrystallographic cases we completely characterize their respective meyer additive and multiplicative properties where f and g are finite sets and also their respective galois conjugate sets these properties allow one to develop a notion of a quasiring we hope that in this way we will initiate a sort of algebraic quasicrystallography in which we can understand quasilattices which be module on a quasiring in we give also some two dimensional examples with

an invariant imbedding method for computation of eigenlengths of singular two point boundary value problems in this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular tpbvp with the singularity at the origin the invariant imbedding yields a first order nonlinear equation called a riccati equation and also gives the initial conditions at the origin for this equation with the aid of theorem 8 in section 3 we numerically integrate the riccati equation to blowup which gives our computed eigenlength in closing we would like to comment on the numerical merits of the integration to blowup technique on the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results the feature of a variable step size is essential as the value of z approaches the actual eigenlength however it is desirable to have a priori estimate or bounds of the eigenlength similar to those of boland and nelson 2 for the nonsingular case the singular system however presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds hopefully an investigation of the matrix r z will yield these results

k block parallel addition versus 1 block parallel addition in non standard numeration systems parallel addition in integer base is used for speeding up multiplication and division algorithms k block parallel addition has been introduced by kornerup in 14 instead of manipulating single digits one works with blocks of xed length k the aim of this paper is to investigate how such notion inuences the relationship between the base and the cardinality of the alphabet allowing block parallel addition in this paper we mainly focus on a certain class of real bases the so called parry numbers we give lower bounds on the cardinality of alphabets of non negative integer digits allowing block parallel addition by considering quadratic pisot bases we are able to show that these bounds cannot be improved in general and we give explicit parallel algorithms for addition in these cases we also consider the d bonacci base which satises the equation x d x d 1 x d 2 x 1 if in a base being a d bonacci number 1 block parallel addition is possible on an alphabeta then a d 1 on the other hand there exists a k2 n such that k block parallel addition in this base is possible on the alphabetf0 1 2g which cannot be reduced in particular addition in the tribonacci base is 14 block parallel on alphabet f0 1 2g

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

analysis of warping deformation modes using higher order ancf beam element abstract most classical beam theories assume that the beam cross section remains a rigid surface under an arbitrary loading condition however in the absolute nodal coordinate formulation ancf continuum based beams this assumption can be relaxed allowing for capturing deformation modes that couple the cross section deformation and beam bending torsion and or elongation the deformation modes captured by ancf finite elements depend on the interpolating polynomials used the most widely used spatial ancf beam element employs linear approximation in the transverse direction thereby restricting the cross section deformation and leading to locking problems the objective of this investigation is to examine the behavior of a higher order ancf beam element that includes quadratic interpolation in the transverse directions this higher order element allows capturing warping and non uniform stretching distribution furthermore this higher order element allows for increasing the degree of continuity at the element interface it is shown in this paper that the higher order ancf beam element can be used effectively to capture warping and eliminate poisson locking that characterizes lower order ancf finite elements it is also shown that increasing the degree of continuity requires a special attention in order to have acceptable results because higher order elements can be more computationally expensive than the lower order elements the use of reduced integration for evaluating the stress forces and the use of explicit and implicit numerical integrations to solve the nonlinear dynamic equations of motion are investigated in this paper it is shown that the use of some of these integration methods can be very effective in reducing the cpu time without adversely affecting the solution accuracy

linear conservation laws for odes abstract some physical systems described by odes have quantities that are conserved as the system evolves runge kutta formulas and linear multistep methods preserve linear conservation laws automatically a code for the integration of odes involves a number of algorithms in addition to the basic formula it is shown here that popular codes preserve linear conservation laws if they are used properly an application is made to differential algebraic equations of index 2 arising in the solution of the navier stokes equation by the method of lines

a multi purpose system for the numerical integration of ode s in this paper we describe the development of an experimental multi purpose integration system strongly modular the system allows the use of wide classes of multistep methods for solving ode s singulsr perturbation problems as well as implicit differential algebraic equations we focus our attention on a new family of collocation methods for nonstiff problems while these methods are strongly related to the stiffly stable bdf methods they are particularly well suited for the implementation of a type insensitive integration algorithm compound discretizations in which certain variables are integrated with the first method and others with the second can also be used a stability analysis is given and some implementation issues are discussed

on the implementation of arithmetic support functions for generalized signed digit number systems ordinary signed digit osd number representation systems have been defined for any radix r or 3 with digit values ranging over the set alpha 1 0 1 alpha where alpha is an arbitrary integer in the range r 2 0 and beta 0 explicitly zero detection sign detection and overflow handling are also treated in depth

a singular nonlinear boundary value problem membrane response of a spherical cap the nonlinear boundary value problem y n x 2 32y 2 lambda 2 8 quad 0 0 nu 0 are constants models the large deflection membrane response of a spherical cap and has been solved numerically by various techniques in the engineering literature we prove existence and uniqueness of a solution of this problem provide qualitative information about the solution and use this information to suggest an approach to numerical computation

stiffness characteristic and the applications of the method of lines on nonuniform grids the method of lines is investigated for the numerical solution of the stream function and vorticity form of the navier stokes equations on nonuniform grids stiffness characteristics of a linear one dimensional model equation are examined to establish the feasibility of applying the method to the vorticity equation in two dimensions the governing equations are transformed from the physical domain with a highly variable grid to a computational domain with a uniform grid the method of lines is used to solve only the vorticity equation and the successive over relaxation technique is used to solve the stream function equation it is observed that the transformed governing equations become stiffer with increased concentration of grid points and also as the number of grid points increases it is also observed that the differencing technique affects the stiffness characteristics the use of forward differencing is not feasible and backward differencing is preferable to central differencing for high reynolds numbers the results of specific applications for the solution of flow in curved wall diffusers and a driven cavity demonstrate that the method of lines under certain circumstances is feasible for the numerical solution of physical problems on domains covered with variable grids

on the exponential fitting of composite multiderivative linear multistep methods exponentially fitted composite multiderivative linear multistep methods of orders up to 5 are derived a numerical investigation of these formulae indicates that they may be a stable for all choices of the fitting parameters an important aspect of the class of integration formulae considered is that each formula contains a built in local error estimate a facility lacking in most other exponentially fitted formulae some preliminary numerical results are given and these show that for problems for which exponential fitting is appropriate the new formulae provide a large saving in computational effort compared with conventional ones

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

construction of algorithms for parallel addition in expanding bases via extending window method abstract an algebraic number c without conjugates of modulus 1 can serve as the base of a numeration system a allowing parallel addition i e the sum of two operands represented in with digits from a is calculated in constant time irrespective of the length of the operands to allow parallel addition sufficient redundancy must exist in the alphabet a the complexity of parallel addition algorithms depends heavily on the alphabet size smaller alphabet implies higher complexity of the algorithm we search for parallel addition algorithms on alphabets of the minimum possible size for a given base to handle high complexity of these algorithms we introduce a so called extending window method ewm an algorithm constructing parallel addition algorithms it can be applied on bases being expanding algebraic integers with all conjugates 1 in modulus the convergence of the ewm is not guaranteed nevertheless we developed tools for revealing non convergence and found some successful applications the ewm provides equal parallel addition algorithms as introduced earlier by a avizienis c y chow j e robertson or b parhami for integer bases and alphabets applying the ewm on complex bases with non integer alphabets a z delivers new results not found previously

variable order adams codes abstract variable step size variable order vsvo adams codes are very effective for solving initial value problems for first order systems of ordinary differential equations the theory of fixed order codes is classical but when the order is varied there is no theory explaining fundamental issues with realistic assumptions about order and step size selection we prove convergence approximate locally the behavior of the error and justify standard error estimators

type insensitive ode codes based on implicit a stable formulas a special concept of stiffness is appropriate for implict a stable formulas it is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ode using this development a variety of effective ode solvers could be made insensitive to the type of problem i e the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

efficient and robust implementation of the tlismni method abstract the dynamics of large scale and complex multibody systems mbs that include flexible bodies and contact impact pairs is governed by stiff equations because explicit integration methods can be inefficient and often fail in the case of stiff problems the use of implicit numerical integration methods is recommended in this case this paper presents a new and efficient implementation of the two loop implicit sparse matrix numerical integration tlismni method proposed for the solution of constrained rigid and flexible mbs differential and algebraic equations the tlismni method has desirable features that include avoiding numerical differentiation of the forces allowing for an efficient sparse matrix implementation and ensuring that the kinematic constraint equations are satisfied at the position velocity and acceleration levels in this method a sparse lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is used to solve for all the accelerations and lagrange multipliers the generalized coordinate partitioning or recursive methods can be used to satisfy the constraint equations at the position and velocity levels in order to improve the efficiency and robustness of the tlismni method the simple iteration and the jacobian free newton krylov approaches are used in this investigation the new implementation is tested using several low order formulas that include hilber hughes taylor hht l stable park a stable trapezoidal and a stable bdf methods the hht method allows for including numerical damping discussion on which method is more appropriate to use for a certain application is provided the paper also discusses tlismni implementation issues including the step size selection the convergence criteria the error control and the effect of the numerical damping the use of the computer algorithm described in this paper is demonstrated by solving complex rigid and flexible tracked vehicle models railroad vehicle models and very stiff structure problems the results obtained using these low order formulas are compared with the results obtained using the explicit adams bashforth predictor corrector method using the tlismni method which does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation for solving complex and stiff structure problems leads to significant computational cost saving as demonstrated in this paper in some problems it was found that the new tlismni implementation is 35 times faster than the explicit adams bashforth method

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

sensitivity analysis of flexible multibody systems using composite materials components in this paper a general formulation for the computation of the first order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented the direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables which are the ply orientations of the laminated in order to appraise the benefits of the approach suggested and to highlight the risks of the procedure the analytical sensitivities are compared with the numerical results obtained by using the finite difference method for the beam composite material elements the section properties and their sensitivities are found using an asymptotic procedure that involves a two dimensional 2 d finite element analysis of their cross section the equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems the equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations velocities and the sensitivities of accelerations and velocities are integrated in time using a multi step multi order integration algorithm through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure with respect to finite difference approaches or to the direct implementation of the analytic sensitivities are discussed copyright 2008 john wiley sons ltd

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

a note on the relative efficiency of adams methods abstract in several pieces of the most effective mathematical software for non stiff and mildly stiff initial value problems in ordinary differential equations the classical adams methods are implemented in a particular mode one reason for the choice of mode is sometimes explained in terms of the associated scaled absolute stability regions for the methods in this note it is shown that the same choice of mode remains appropriate from this point of view when certain families of generalized adams type methods are considered

an algorithm for starting multistep methods the present paper deals with the problem of starting multistep methods we take into consideration an adams bashforth moulton pece pair with the predictor of order q and the corrector of order q 1 to start this method q 1 starting values are necessary in addition to y0 a well known result from theory says that the order of convergence of the whole integration is q 1 if all those starting values are accurate of that same order present production codes start with a predictor of order 1 and a corrector of order 2 at the first step and then proceed step by step each time raising the order by 1 until all the necessary starting values have been obtained but in this manner all the starting errors keep of order 3 and so the whole integration converges no faster than that order this drawback is normally compensated for by taking very small step sizes in the starting phase the general algorithm we propose furnishes at a reasonably low cost the necessary number q 1 of starting values each of the appropriate order q 1 whatever q might be it is independent of the particular multistep formula considered and is mainly designed to be used for high values of q q 10 where the alternative strategies are too expensive or do not exist at all the numerical results reported show the validity of our approach 2003 elsevier science ltd all rights reserved

approximation of weak adjoints by reverse automatic differentiation of bdf methods we shed light on the relation between the discrete adjoints of multistep backward differentiation formula bdf methods and the solution of the adjoint differential equation to this end we develop a functional analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations we devise a petrov galerkin finite element fe interpretation of the bdf method and its discrete adjoint scheme obtained by reverse internal numerical differentiation we show how the fe approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation we also show convergence of the discrete adjoints to the classical adjoints on the inner time interval finally we give numerical results for non adaptive and fully adaptive bdf schemes the presented framework opens the way to carry over techniques on global error estimation from fe methods to bdf methods

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

long term stable time integration scheme for dynamic analysis of planar geometrically exact timoshenko beams abstract in this paper an energy momentum method for geometrically exact timoshenko type beam is proposed the classical time integration schemes in dynamics are known to exhibit instability in the non linear regime the so called timoshenko type beam with the use of rotational degree of freedom leads to simpler strain relations and simpler expressions of the inertial terms as compared to the well known bernoulli type model the treatment of the bernoulli model has been recently addressed by the authors in this present work we extend our approach of using the strain rates to define the strain fields to in plane geometrically exact timoshenko type beams the large rotational degrees of freedom are exactly computed the well known enhanced strain method is used to avoid locking phenomena conservation of energy momentum and angular momentum is proved formally and numerically the excellent performance of the formulation will be demonstrated through a range of examples

hyperbolic knots spanning accidental seifert surfaces of arbitrarily high genus a method for constructing hyperbolic knots each of which bounds accidental incompressible seifert surfaces of arbitrarily high genus is given

minimal genus in 4 manifolds with a free circle action abstract let n be a closed irreducible 3 manifold and assume n is not a graph manifold we improve for all but finitely many s 1 bundles m over n the adjunction inequality for the minimal complexity of embedded surfaces this allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many s 1 bundles over a large class of 3 manifolds

varying the componentwise order of the multistep methods in solving odes and its absolute stability most variable order variable stepsize codes for solving odes vary the order along the interval of integration but keep the order constant across the system however some codes that do partitioning of the odes into stiff and nonstiff subsystems vary the order for each equation in the system in this paper the reasons for varying the order componentwise are given and some of its salient features are illustrated using a numerical example also given is the direct proof of the existence of a region of absolute stability when different order adams explicit methods are used to solve a system of odes

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

explicit parallel two step runge kutta nystrom methods abstract the aim of this paper is to design a class of two step runge kutta nystrom methods of arbitrarily high order for the special second order equation y t f y t for use on parallel computers starting with an s stage implicit two step runge kutta nystrom method of order p with k p 2 implicit stages we apply the highly parallel predictor corrector iteration process in p ec me mode in this way we obtain an explicit two step runge kutta nystrom method that has order p for all m and that requires k m 1 right hand side evaluations per step of which each k evaluation can be computed in parallel by a number of numerical experiments we show the superiority of the parallel predictor corrector methods proposed in this paper over both sequential and parallel methods available in the literature

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

using nonconvergence of iteration to partition odes given is a variation of partitioning odes when using the multistep method essentially the nonconvergence property is made use of for partitioning the criteria used are explained by using a test system of two linear equations and generalisation is made to a general system of first order odes finally some numerical results are given that include systems greater than two

rkn type parallel block pc methods with lagrange type predictors abstract this paper describes the construction of block predictor corrector methods based on runge kutta nystrom correctors our approach is to apply the predictor corrector method not only at step points but also at off step points block points so that in each step a whole block of approximations to the exact solution at off step points is computed in the next step these approximations are used to obtain a high order predictor formula using lagrange interpolation by suitable choice of the abscissas of the off step points a much more accurately predicted value is obtained than by predictor formulas based on last step values since the block of approximations at the off step points can be computed in parallel the sequential costs of these block predictor corrector methods are comparable with those of a conventional predictor corrector method furthermore by using runge kutta nystrom corrector methods the computation of the approximation at each off step point is also highly parallel application of the resulting block predictor corrector methods to a few widely used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature

construction of variable stepsize multistep formulas a systematic way of extending a general fixed stepsize multistep formula to a minimum storage variable stepsize formula has been discovered that encompasses fixed coefficient interpolatory variable coefficient variable step and fixed leading coefficient as special cases in particular it is shown that the interpolatory stepsize changing technique of nordsieck leads to a truly variable stepsize multistep formula which has implications for local error estimation and formula changing and it is shown that the variable step stepsize changing technique applicable to the adams and backward differentiation formulas has a reasonable generalization to the general multistep formula in fact it is shown how to construct a variable order family of variable coefficient formulas finally it is observed that the first dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration

implicit schemes for differential equations abstract implicit high order accurate temporal integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only the ordinary differential equations are solved using various accurate integration methods the schemes are free from extraneous boundary and or initial conditions and are stable for large courant number

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

efficient calculation of chemically reacting flow a semi implicit finite volume formulation is used to jrvstudy flow with chemical reactions in this formulation the source terms resulting from the chemical reaction are treated implicitly and the resulting system of partial differential equations is sovled using two time stepping schemes the first is based on the runge kutta method and the second on an adams predictor corrector method results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated furthermore analysis and computation indicate that the rungekutta method is more efficient than the adams methods for these systems of differential equations finally an adaptive time stepping scheme is developed to study problems involving shock ignition calculations for a hydrogen air system agree well with other methods

on noncharacteristic boundary conditions for discrete hyperbolic initial boundary value problems abstract the stability properties of some discrete hyperbolic initial boundary value problems involving two dependent variables and noncharacteristic boundary conditions are examined approximations based on the lax wendroff leap frog and crank nicolson methods are considered the most accurate approximations are found to be the least stable the results for the model problems can be used in the stability analysis of larger problems where characteristics occur singly or in pairs in particular discretizations of the shallow water and gas dynamics equations which involve boundary conditions written in natural variables are considered some computations demonstrating the stability results are discussed an appendix discussing numerical stability verification techniques is included

an efficient approach for the numerical simulation of multibody systems the field of kinematics and dynamics of mechanical systems has progressed from a manual graphics art to a highly developed discipline in analytical geometry and dynamics various general purpose formulations for the dynamic analysis of constrained mechanical systems cms lead to mixed differential algebraic equations daes called the euler lagrange equations during the past fifteen years many contributions have been made to the theory of computational kinematics and dynamics of cms also called mutibody dynamics the recent advances in computer hardware and software have tremendously revolutionized the analysis of cms there are various numerical approaches for solving general vector fields in the previous paper appl math comput 92 1998 153 193 a complete and detailed analysis of various approaches for the numerical solution of vector fields is given in this paper we extend the algorithms presented in the above mentioned reference to solve the euler lagrange equations of motion for cms the numerical experiments suggest that perturbation approach performs better than the other approaches

a two loop sparse matrix numerical integration procedure for the solution of differential algebraic equations application to multibody systems abstract in this paper a two loop implicit sparse matrix numerical integration tlismni procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed the proposed method ensures that the kinematic constraint equations are satisfied at the position velocity and acceleration levels in this method a sparse lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and lagrange multipliers the independent coordinates and velocities are then identified and integrated using htt or newmark formulas expressed in this paper in terms of the independent accelerations only the constraint equations at the position level are then used within an iterative newton raphson procedure to determine the dependent coordinates the dependent velocities are determined by solving a linear system of algebraic equations in order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements a two loop iterative method is proposed equally important the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms numerical examples are presented in order to demonstrate the use of the new integration procedure

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

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

splittings of knot groups let k be a knot of genus g if k is fibered then it is well known that the knot group pi k splits only over a free group of rank 2g we show that if k is not fibered then pi k splits over non free groups of arbitrarily large rank furthermore if k is not fibered then pi k splits over every free group of rank at least 2g however pi k cannot split over a group of rank less than 2g the last statement is proved using recent results of agol przytycki wise and wise

parallel addition in non standard numeration systems we consider numeration systems where digits are integers and the base is an algebraic number b such that b 1 and b satisfies a polynomial where one coefficient is dominant in a certain sense for this class of bases b we can find an alphabet of signed digits on which addition is realizable by a parallel algorithm in constant time this algorithm is a kind of generalization of the one of avizienis we also discuss the question of cardinality of the used alphabet and we are able to modify our algorithm in order to work with a smaller alphabet we then prove that b satisfies this dominance condition if and only if it has no conjugate of modulus 1 when the base b is the golden mean we further refine the construction to obtain a parallel algorithm on the alphabet 1 0 1 this alphabet cannot be reduced any more