on the efficiency of newton and broyden numerical methods in the investigation of the regular polygon problem of n 1 bodies numerical methods of finding the roots of a system of non linear algebraic equations are treated in this paper this paper attempts to give an answer to the selection of the most efficient method in a complex problem of celestial dynamics the so called ring problem of n 1 bodies we apply newton and broyden s method to these problems and we investigate by means of their use the planar equilibrium points the five equilibrium zones which are symbolized by a 1 a 2 b c 2 and c 1 by order of appearance from the center o to the periphery of the imaginary circle on which the primaries lie t j kalvouridis a planar case of the n 1 body problem the ring problem astrophys space sci 260 3 1999 309 325 and the attracting regions of the system the efficiency of these methods is studied through a comparative process the obtained results are demonstrated in figures and are discussed

how to decide continuity of rational functions on infinite words this paper presents a construction that computes the topological closure of a rational relation on infinite words besides showing that the closure is rational itself it also reduces the question of the continuity and the uniform continuity of a function to the question of the functionality of its closure which is known to be decidable in polynomial time

synchronized rational relations of finite and infinite words abstract the purpose of this paper is a comprehensive study of a family of rational relations both of finite and infinite words namely those that are computable by automata where the reading heads move simultaneously on the n input tapes and that we thus propose to call synchronized rational relations

fractal haar system abstract the haar system is an alternative to the classical fourier bases being particularly useful for the approximation of discontinuities the article tackles the construction of a set of fractal functions close to the haar set the new system holds the property of constitution of bases of the lebesgue spaces of p integrable functions on compact intervals likewise the associated fractal series of a continuous function is uniformly convergent the case p 2 owns some peculiarities and is studied separately

une extension aux mots infinis de la notion de transduction rationelle here is introduced an extension for infinite words of the classical notion of rational transduction we prove that this extension has the important property of mapping the adherence of a language of finite words into the adherence of an other language of finite words the set of such extensions is closed by composition and is exactly the family of the compositions of an inverse faithful sequential mapping and of a faithful sequential mapping

characteristic methods for multidimensional evolution equations abstract the stochastic method of characteristics is generalized to analyze multidimensional evolution equations of arbitrary order with constant coefficients a linear space of abstract characteristics is introduced along with algorithms for associating characteristics with evolutions a projection from a space of functions of characteristics to solutions of evolution equations plays the role of a generalized expectation the solution is then viewed as an average of initial data examples are presented which indicate possible extensions of the theory presented here to equations with variable coefficients

discontinuous galerkin spectral element approximations on moving meshes we derive and evaluate high order space arbitrary lagrangian eulerian ale methods to compute conservation laws on moving meshes to the same time order as on a static mesh we use a discontinuous galerkin spectral element method dgsem in space and one of a family of explicit time integrators such as adams bashforth or low storage explicit runge kutta the approximations preserve the discrete metric identities and the discrete geometric conservation law dgcl by construction we present time step refinement studies with moving meshes to validate the approximations the test problems include propagation of an electromagnetic gaussian plane wave a cylindrical pressure wave propagating in a subsonic flow and a vortex convecting in a uniform inviscid subsonic flow each problem is computed on a time deforming mesh with three methods used to calculate the mesh velocities from exact differentiation from the integration of an acceleration equation and from numerical differentiation of the mesh position

modified runge kutta methods for the numerical solution of odes with oscillating solutions three new modified runge kutta methods with minimal phase lag are developed for the numerical solution of ordinary differential equations with oscillating solutions these methods are based on the well known runge kutta method of dormand and prince rk5 4 t see 1 of order five numerical and theoretical results show that this new approach is more efficient compared with the well known classical fifth order runge kutta dormand and prince method

a homomorphic characterization of principal semiafls without using intersection with regular sets abstract if one is asked to consider which one of the semiafl or afl operations differs most from the remaining ones a set theoretically well grounded conclusion would be intersection with regular sets inspired by the result of culik fich and salomaa 1 we shall show that this operation can be replaced by catenation with an endmarker the basis of the paper is the result that any free principal semiafl m l can be presented in the form m l h 4 h 3 1 h 2 h 1 1 l each h 1 a homomorphism h 2 and h 3 free h 4 length preserving where is a symbol not in the alphabet of l if m l is not free we get the same representation but now h 4 erases the last letter from each word in h 3 1 h 2 1 h 1 1 l and is length preserving for other letters for the family of free regular languages we may choose l the last section contains a homomorphic characterization of regular star languages

brownian time processes the pde connection ii and the corresponding feynman kac formula we delve deeper into our study of the connection of brownian time processes btps to fourth order parabolic pdes which we introduced in a recent joint article with w zheng probabilistically btps and their cousins btps with excursions form a unifying class of interesting stochastic processes that includes the celebrated ibm of burdzy and other new intriguing processes and is also connected to the markov snake of le gall btps also offer a new connection of probability to pdes that is fundamentally different from the markovian one they solve fourth order pdes in which the initial function plays an important role in the pde itself not only as initial data we connect two such types of interesting and new pdes to btps the first is obtained by running the btp and then integrating along its path and the second type of pdes is related to what we call the feynman kac formula for btps a special case of the second type is a step towards a probabilistic solution to linearized cahn hilliard and kuramoto sivashinsky type pdes which we tackle in an upcoming paper

approximation integration and differentiation of time functions using a set of orthogonal hybrid functions hf and their application to solution of first order differential equations abstract differential equations of different types and orders are of utmost importance for mathematical modeling of control system problems state variable method uses the concept of expressing n number of first order differential equations in vector matrix form to model and analyze synthesize control systems the present work proposes a new set of orthogonal hybrid functions hf which evolved from synthesis of sample and hold functions shf and triangular functions tf this hf set is used to approximate a time function in a piecewise linear manner with the mean integral square error mise much less than block pulse function based approximation which always provides staircase solutions the operational matrices for integration and differentiation in hf domain are also derived and employed for solving non homogeneous and homogeneous differential equations of the first order as well as state equations the results are compared with exact solutions the 4th order runge kutta method and its further improved versions proposed by simos 6 the presented hf domain theory is well supported by a few illustrations

variable stepsize chebyshev type methods for the integration of second order i v p s panovsky and richardson a family of implicit chebyshev methods for the numerical integration of second order differential equations j comput appl math 23 1988 35 51 presented a method based on chebyshev approximations for numerically solving the problem y f x y being the steplength constant coleman and booth analysis of a family of chebyshev methods for y f x y j comput appl math 44 1992 95 114 made an analysis of the above method and suggested the convenience to design a variable steplength implementation as far as we know this goal has not been achieved until now later on we extended the above method this journal 2003 and obtained a scheme for numerically solving the equation y 2gy g 2 w 2 f x y the question of how to extend these formulas to variable stepsize procedures is the primary topic of this paper

a note on ω regular languages abstract this paper is a continuation of the work by landweber 5 to clarify the relationship between classes of languages accepted by finite automata we study their properties in the topological framework introduced by boasson and nivat 1 we characterize the classes in terms of borel hierarchy and show some interrelation of them

fundamentals of higher order stochastic equations stochastic processes that correspond to equations of arbitrary differential order are obtained we show that stochastic paths in a complex extension of the original phase space allow an implementation of such higher order derivative terms the resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments however the correspondence has unusual properties only the analytic moments are convergent while non analytic moments such as the complex moduli are non convergent these results unify previous approaches that transform higher derivative equations into probabilistic stochastic equations larger ensembles are required as time steps are reduced giving these equations unusual convergence properties this type of process is relevant to the question of how to obtain stochastic quantum simulations using the wigner representation

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

a new theoretical approach to runge kutta methods runge kutta rk methods are treated here as linear multistage methods within the author s concept of a methods as a consequence the traditional difference in treatment of rk methods and linear multistep methods vanishes in particular it is not necessary to consider elementary differentials it is shown how the order of a rk method depends on the error constants of its stages and the conditions nonlinear equations for order p are generated from a very simple recursion

a study of rosenbrock type methods of high order this paper deals with the solution of nonlinear stiff ordinary differential equations the methods derived here are of rosenbrock type this has the advantage that they area stable or stiffly stable and nevertheless do not require the solution of nonlinear systems of equations we derive methods of orders 5 and 6 which require one evaluation of the jacobian and onelu decomposition per step we have written programs for these methods which use richardson extrapolation for the step size control and give numerical results

transfer function identification from impulse response via a new set of orthogonal hybrid functions hf the present work proposes a new set of hybrid functions hf which evolved from the synthesis of sample and hold functions shf and triangular functions tf traditional block pulse functions bpf still continue to be attractive to many researchers in the arena of control theory block pulse functions also gave birth to a few useful variants two such variants are shf and tf the former is efficient for analyzing sample and hold control systems while triangular functions established their superiority in obtaining piecewise linear solution of various control problems after developing the basic theory of hf a few square integrable functions are approximated via this set in a piecewise linear manner for such approximation it is shown the mean integral square error mise is much less than block pulse function based approximation the operational matrices for integration in hf domain are also derived finally this new set is employed for solving identification problem from impulse response data the results are compared with the solutions obtained via bpf shf tf etc and many illustrations are presented

an inverse scattering formalism for higher order differential operators abstract an inverse scattering formalism is presented for certain linear ordinary differential operators of arbitrary even order n which generalizes the gel fand levitan theory for sturm liouville operators under suitable hypotheses it is proved that for given n 2 the coefficients in these operators are uniquely determined by a single spectral matrix our proof makes use of a transformation due to m k fage which generalizes the povzner levitan transformation for sturm liouville operators

beyond conventional runge kutta methods in numerical integration of odes and daes by use of structures and local models there are two parts in this paper in the first part we consider an overdetermined system of differential algebraic equations daes we are particularly concerned with hamiltonian and lagrangian systems with holonomic constraints the main motivation is in finding methods based on gauss coefficients preserving not only the constraints symmetry symplecticness and variational nature of trajectories of holonomically constrained hamiltonian and lagrangian systems but also having optimal order of convergence the new class of s s gauss lobatto specialized partitioned additive runge kutta spark methods uses greatly the structure of the daes and possesses all desired properties in the second part we propose a unified approach for the solution of ordinary differential equations odes mixing analytical solutions and numerical approximations the basic idea is to consider local models which can be solved efficiently for example analytically and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction in order to preserve also symmetry we define the new class of symmetrized runge kutta methods with local model srklm

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

characterizations of rational ω languages by means of right congruences abstract by means of right congruences we characterize in a unified way different classical classes of rational languages a new congruence associated with an automaton is introduced named cycle congruence the family of rational languages which have by morphism a unique minimal recognizing automaton is characterized it appears that for such a minimal automaton the cycle congruence coincides with the syntactic congruence of the recognized language we prove that the other rational languages have an infinite number of minimal automaton

two characterizations of rational adherences resume nous presentons deux caracterisations des adherences en termes de langages finis d homomorphismes strictement alphabetiques et d homomorphismes uniformes inverses nous en deduisons une caracterisation similaire pour les langages rationnels

determinization of transducers over finite and infinite words we study the determinization of transducers over finite and infinite words the first part of the paper is devoted to finite words we recall the characterization of subsequential functions due to choffrut we describe here a known algorithm to determinize a transducer in the case of infinite words we consider transducers with all their states final we give an effective characterization of sequential functions over infinite words we describe an algorithm to determinize transducers over infinite words this part contains the main novel results of the paper

generalization of the diffusion equation by using the maximum entropy principle by using the so called maximum entropy principle in information theory one derives a generalization of the fokker planck kolmogorov equation which applies when the n first transition moments of the process are proportional to t while the other ones can be neglected

two step fourth order p stable methods with phase lag of order six for y t y abstract we present a new family of two step fourth order methods which when applied to the test equation y 2y 0 are at once p stable and have a phase lag of order h6 h h h is the step size

the three subfamilies of rational ω languages closed under ω transduction abstract the transductions defined by gire as bimorphisms coincide with the compositions of non erasing morphisms and inverse non erasing morphisms we prove that there are exactly three subfamilies of rational languages which are closed under transduction the family of rational adherences the family of rational languages whose complementary is deterministic and the whole family of rational languages

numerical methods for ordinary differential equations in the 20th century abstract numerical methods for the solution of initial value problems in ordinary differential equations made enormous progress during the 20th century for several reasons the first reasons lie in the impetus that was given to the subject in the concluding years of the previous century by the seminal papers of bashforth and adams for linear multistep methods and runge for runge kutta methods other reasons which of course apply to numerical analysis in general are in the invention of electronic computers half way through the century and the needs in mathematical modelling of efficient numerical algorithms as an alternative to classical methods of applied mathematics this survey paper follows many of the main strands in the developments of these methods both for general problems stiff systems and for many of the special problem types that have been gaining in significance as the century draws to an end

explicit runge kutta nystro m methods with reduced phase errors for computing oscillating solutions we construct explicit runge kutta nystrom methods for the integration of first and second order differential equations having an oscillatory solution special attention is paid to the phase errors or dispersion of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model rk n methods are constructed which are dispersive of orders up to 10 whereas the algebraic order of accuracy is only 2 or 3 application of these methods to equations describing free and weakly forced oscillations and to semidiscretized hyperbolic equations reveals that the phase errors can significantly be reduced

phase lag analysis of implicit runge kutta methods we analyse the phase errors introduced by implicit runge kutta methods when a linear inhomogeneous test equation is integrated it is shown that the homogeneous phase errors dominate if long interval integrations are performed homogeneous dispersion relations for the special class of dirk methods are derived and a few high order dispersive dirk methods are constructed these methods are applied to systems of linear differential equations with oscillating solutions and compared with the conventional dirk methods of norsett and crouzeix

runge kutta interpolants with minimal phase lag abstract we develop the basic theory for the construction of explicit scaled runge kutta methods srk for the integration of first order differential equations having an oscillatory solution based on this theory and on the theory of runge kutta methods with minimal phase lag we construct an embedded runge kutta 10 12 method and an explicit scaled runge kutta method numerical results indicate the efficiency of the new algorithms

construction of a complex valued fractional brownian motion of order n in this paper a brownian motion of order n n 2 is defined by a probabilistic approach different from hochberg s and mandelbrot s this process is constructed from sums of independent r 1 n valued random variables rv where r 1 n z c zn r many properties of the real standard brownian motion are generalized at order n but in the case n 2 it is interesting to describe the brownian motion of order n on the algebra b r 1 n r where b r 1 n is the algebra generated by sets of type a 0 h z c zn 0 hn her this algebra is totally different from b r r thus this study shows the fractal nature of the brownian motion of order n and given invariance scale self similarity properties then a stochastic integral and an ito taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average all these results can be obtained via nonstandard analysis methods infinitesimal time discretization finally one rem

differential equations satisfied by the components with respect to the cyclic group of order n of some special functions let f be a complex function of the variable z admitting a laurent expansion in an annulus c with center in the origin for an arbitrary positive integer n ricci s theorem asserts that the function f can be written as the sum of n functions f n k k 0 1 n 1 defined byf n k z 1n l 0n 1exp 2i klnfzexp2i ln z c in this paper we present a technique which starting from a suitable differential equation satisfies by the function f provides a differential equation satisfied by the functions f n k some known special functions are treated for illustration

a variable mesh c splage method of accuracy o k2hl 1 khl hl3 for 1d nonlinear parabolic equations a new high accuracy two level implicit cubic spline formulation for the solution of nonlinear parabolic equation u x email protected x t u u x u t is presented the strategy of the cubic spline alternating group explicit c splage method is developed the proposed c splage method requires only 3 spatial grid points and applicable to singular problems the convergence analysis of the proposed c splage method is given the obtained c splage method is compared to the corresponding successive over relaxation sor method both in terms of stability and performance

diagonally implicit runge kutta nystro m methods for oscillatory problems implicit runge kutta nystrom rkn methods are constructed for the integration of second order differential equations possessing an oscillatory solution based on a linear homogeneous test model we analyse the phase errors or dispersion introduced by these methods and derive so called dispersion relations diagonally implicit rkn methods of relatively low algebraic order are constructed which have a high order of dispersion up to 10 application of these methods to a number of test examples linear as well as nonlinear yields a greatly reduced phase error when compared with conventional dirkn methods

solving directly special fourth order ordinary differential equations using runge kutta type method in this paper an explicit runge kutta method for solving directly fourth order ordinary differential equations odes is constructed and denoted as rkfd we present a relevant colored tree theory and the associated b series theory for the order conditions based on the order conditions a three stage fourth order rkfd method and a three stage fifth order rkfd method are constructed numerical illustrations are presented to show the efficiency of the new rkfd methods by comparing them with other existing runge kutta nystrom rkn and runge kutta rk methods in the scientific literature after converting the problem into a system of second order odes and a system of first order odes respectively

the inverse problem for ordinary differential operators on the line on etudie le probleme inverse sur la ligne pour l operateur p d n q n 2 d n 2 q 0 d d idx pour tout ordre n 2 et sans hypothese de propriete d auto adjoint

squaring transducers an efficient procedure for deciding functionality and sequentiality we describe here a construction on transducers that give a new conceptual proof for two classical decidability results on transducers it is decidable whether a finite transducer realizes a functional relation and whether a finite transducer realizes a sequential relation a better complexity follows then for the two decision procedures

numerical solution of third order linear differential equations using generalized one shot operational matrices in orthogonal hybrid function domain abstract the present work proposes a new set of orthogonal hybrid functions hf which evolved from synthesis of orthogonal sample and hold functions shf and triangular functions tf the hf set is used to approximate a time function in a piecewise linear manner for such approximation the mean integral square error mise is much less than block pulse function based approximation which always provides staircase solutions and it is found that hf based approximation is a strong contender of approximations based upon orthogonal polynomials like legendre polynomials the respective mise s and elapsed times for mise computations for these three kinds of approximations are also studied in detail the one shot operational matrices for integration of different orders in hf domain are also derived these matrices are employed for more accurate multiple integration an example is treated to determine efficiency of these matrices finally these matrices are employed for solving third order non homogeneous differential equations followed by a numerical example the results obtained via matlab are compared with the exact solution as well as the results obtained via 4th order runge kutta method

a low order embedded runge kutta method for periodic initial value problems sideridis a b and t e simos a low order embedded runge kutta method for periodic initial value problems journal of computational and applied mathematics 44 1992 23 5 244 an embedded runge kutta fehlberg method is developed it should be noted that this embedded method is procuced using the runge kutta fehlberg method with algebraic order four to estimate a truncation phase lag error of algebraic order three the numerical results indicate that this new method is efficient for the numerical solution of differential equations with periodic solution using variable stepsize

time domain goal oriented adaptivity using pseudo dual error representations abstract goal oriented adaptive algorithms produce optimal grids to solve challenging engineering problems recently a novel error representation using unconventional pseudo dual problems for goal oriented adaptivity in the context of frequency domain wave propagation problems has been developed in this paper we extend this error representation to the case of time domain problems we express the entire problem in weak form in order to derive the adjoint formulation and apply goal oriented adaptivity one dimensional 1d numerical results show that upper bounds for the new error representation are sharper than the classical ones therefore this new error representation can be used to design more efficient goal oriented adaptive methodologies

the concept of b convergence a useful concept of convergence for nonlinear stiff ode s will be developed which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem this concept will be discussed for three simple methods for stiff systems the implicit euler scheme the implicit midpoint rule and the implicit trapezoidal rule

a fourth order runge kutta method based on bdf type chebyshev approximations in this paper we consider a new fourth order method of bdf type for solving stiff initial value problems based on the interval approximation of the true solution by truncated chebyshev series it is shown that the method may be formulated in an equivalent way as a runge kutta method having stage order four the method thus obtained have good properties relatives to stability including an unbounded stability domain and large a value concerning a a stability a strategy for changing the step size based on a pair of methods in a similar way to the embedding pair in the runge kutta schemes is presented the numerical examples reveals that this method is very promising when it is used for solving stiff initial value problems

a fokker planck equation of fractional order with respect to time by combining the maximum entropy principle with some considerations related to derivatives of fractional order one is led to suggest a fokker planck of fractional order with respect to time which could be related to dynamical systems subject to fractional brownian motion the relation with the process associated with the equation p t 1 n 1 2np x2n is exhibited

a noumerov type method with minimal phase lag for the integration of second order periodic initial value abstract in a recent paper 2 we gave a noumerov type method with minimal phase lag for the integration of second order initial value problems y f t y y t0 y0 y t0 y 0 however the method given there is implicit we show here the interesting result that if the noumerov type methods of 2 are made explicit with the help of the classical second order method then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size 1 40320 h 6 and also a slightly larger interval of periodicity of size 2 75 than the phase lag of size 1 12096 h 6 and interval of periodicity of size 2 71 for the implicit method 2 more interestingly it turns out that noumerov made explicit of chawla 3 also has less frequency distortion than the implicit noumerov method we shall assume a familiarity with the notation and discussion given in 2

product formulas for solutions of initial value partial differential equations i abstract families of operators which approximate semi groups or evolution systems generated by partial differential operators are constructed product formulas are used to recover these semi groups or evolution systems through product integrals conditions on generators are provided under which its semi group or evolution system can be approximated in this way by families of specific types of operators

an explicit sixth order method with phase lag of order eight for y f t y abstract recently thomas 6 gave a two step sixth order method with phase lag 1 101478 67 h8 for the numerical integration of periodic initial value problems y t y y t 0 y 0 y t 0 y 0 however thomas method is implicit it possesses an interval of periodicity of size 2 77 and for nonlinear problems her method requires 6i 1 function evaluations for i modified newton iterations for the solution of the resulting implicit equations at each step in the present paper we present a new two step sixth order method which also has phase lag of order eight but with a smaller constant given by 1 3 628 800 h8 in contrast with thomas method our method is explicit possesses a larger interval of periodicity of size 4 63 and it is decidedly more economical since it involves only six function evaluations per step numerical experiments confirm the superiority of our present method over thomas method