oscillation of self adjoint second order vector difference equations to the parameter abstract this paper is concerned with oscillation of self adjoint second order vector difference equations with respect to a parameter properties of zeros and monotonicity of matrix valued solutions are studied the oscillation of two consecutive polynomials for vector valued solutions is discussed a separation theorem for matrix valued solutions is also obtained

eventual disconjugacy of self adjoint difference equations abstract given an eventually disconjugate second order self adjoint difference equation c n e n a n e n 1 0 where c n 0 and is not assumed to be positive this paper studies the eventual disconjugacy of the perturbed difference equation c n y n a n f n y n 1 0 analogous to a theorem of willett for corresponding differential equations a necessary and sufficient condition for eventual disconjugacy in terms of a convergent sequence of averaged sums of f n is obtained

riccati techniques and discrete oscillations abstract oscillation and nonoscillation criteria are obtained for the second order difference equation r n x n p n x n 1 0 where x n x n 1 x n is the forward difference operator and r n p n are sequences of real numbers the main techniques involved are riccati equations of the first and second levels

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

oscillation results for linear hamiltonian systems for linear hamiltonian systems this paper established some new oscillation criteria which improve previous results even for self adjoint second order differential systems the main tool used is a generalized riccati transformation

performance of the taylor series method for odes daes this paper revisits the use of the taylor series method for the numerical integration of odes and daes the numerical method is implemented using an efficient variable step variable order scheme several numerical tests comparing with well established numerical codes are presented

closed form expressions for the finite approximations of first and higher derivatives based on taylor series numerical differentiation formulas based on interpolating polynomials operators and lozenge diagrams can be simplified to one of the finite difference approximations based on taylor series in this paper we have presented closed form expressions of these approximations of arbitrary order for first and higher derivatives a comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses the comparison reveals that the central difference approximations can be used as digital differentiators because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator it is also observed that central difference approximations are in fact the same as maximally flat digital differentiators in the appendix a computer program written in mathematica is presented which can give the approximation of any order to the derivative of a function at a certain mesh point

new oscillation criteria for linear matrix hamiltonian systems abstract some new oscillation criteria are established for the matrix linear hamiltonian system x a t x b t y y c t x a t y under the hypothesis a t b t b t 0 and c t c t are n n real continuous matrix functions on the interval t 0 t 0 these results are sharper than some previous results even for self adjoint second order matrix differential systems

a general form of the generalized taylor s formula with some applications in this article a new general form of fractional power series is introduced some results of classical power series are circulated and proved to this new form a new general form of the generalized taylor s formula is also obtained applications including fractional power series solutions for fdes are analyzed results reveal that the new expansion is effective for formulating the solutions in this article a new general form of fractional power series is introduced in the sense of the caputo fractional derivative using this approach some results of the classical power series are circulated and proved to this fractional power series whilst a new general form of the generalized taylor s formula is also obtained some applications including fractional power series solutions for higher order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results the results reveal that the new fractional expansion is very effective straightforward and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components

lower bound for the spectrum and the presence of pure point spectrum of a singular discrete hamiltonian system in this paper criteria for limit point n case of a singular discrete hamiltonian system are established furthermore the lower bound of the essential spectrum is obtained and the present of pure point spectrum is discussed for such system by using the spectral theory of self adjoint operators in a hilbert space

a higher order local linearization method for solving ordinary differential equations the local linearization ll method for the integration of ordinary differential equations is an explicit one step method that has a number of suitable dynamical properties however a major drawback of the ll integrator is that its order of convergence is only two the present paper overcomes this limitation by introducing a new class of numerical integrators called the llt method that is based on the addition of a correction term to the ll approximation in this way an arbitrary order of convergence can be achieved while retaining the dynamic properties of the ll method in particular it is proved that the llt method reproduces correctly the phase portrait of a dynamical system near hyperbolic stationary points to the order of convergence the performance of the introduced method is further illustrated through computer simulations

numerical simulation of trajectory prescribed path control problems by the backward differentiation formulas the equations which describe trajectory prescribed path control problems naturally form nonlinear semiexplicit differential algebraic systems with index greater than one it is known that not all fully implicit systems may be solved stably by the k step backward differentiation formulas yet these methods do produce convergent numerical solutions to some semiexplicit systems in this note numerical results are presented for the simplest backward differentiation formula when applied to an index three semiexplicit system concerning the reentry of the space shuttle the application of this numerical method to a realistic problem illustrates some unresolved implementation difficulties

a three stage vsvo hermite birkhoff taylor ode solver the new variable step variable order ode solver hbt p of order p presented in this paper combines a three stage runge kutta method of order 3 with a taylor series method of order p 2 to solve initial value problems y f t y y t 0 y 0 where y r r d and f rxr d r d the order conditions satisfied by hbt p are formulated and they lead to vandermonde type linear algebraic systems whose solutions are the coefficients in the formulae for hbt p a detailed formulation of variable step hbt p in both fixed order and variable order modes is presented the new method and the taylor series method have similar regions of absolute stability to obtain high accuracy results at high order this method has been implemented in multiple precision

oscillation criteria for second order functional difference equations with summation small coefficient we consider the second order functional difference equation a n n p n x g n 0 where n p n are sequences of integers with a n 0 0 g n n limn g n p n 0 and p n 0 we obtain some necessary conditions for equation 1 exists nonoscillatory solutions and some sufficient conditions for equation 1 is oscillatory

validated solutions of initial value problems for ordinary differential equations compared to standard numerical methods for initial value problems ivps for ordinary differential equations odes validated methods for ivps for odes have two important advantages if they return a solution to a problem then 1 the problem is guaranteed to have a unique solution and 2 an enclosure of the true solution is produced the authors survey taylor series methods for validated solutions of ivps for odes describe several such methods in a common framework and identify areas for future research

status of the differential transformation method abstract further to a recent controversy on whether the differential transformation method dtm for solving a differential equation is purely and solely the traditional taylor series method it is emphasized that the dtm is currently used often only as a technique for analytically calculating the power series of the solution in terms of the initial value parameters sometimes a piecewise analytic continuation process is implemented either in a numerical routine e g within a shooting method or in a semi analytical procedure e g to solve a boundary value problem emphasized also is the fact that at the time of its invention the currently used basic ingredients of the dtm that transform a differential equation into a difference equation of same order that is iteratively solvable were already known for a long time by the traditional taylor method users notably in the elaboration of software packages numerical routines for automatically solving ordinary differential equations at now the defenders of the dtm still ignore the though much better developed studies of the traditional taylor method users who in turn seem to ignore similarly the existence of the dtm the dtm has been given an apparent strong formalization set on the same footing as the fourier laplace or mellin transformations though often used trivially it is easily attainable and easily adaptable to different kinds of differentiation procedures that has made it very attractive hence applications to various problems of the taylor method and more generally of the power series method including noninteger powers has been sketched it seems that its potential has not been exploited as it could be after a discussion on the reasons of the misunderstandings which have caused the controversy the preceding topics are concretely illustrated it is concluded that for the sake of clarity when the dtm is applied to odes it should be mentioned that the dtm exactly coincides with the traditional taylor method contrary to what is currently written

disconjugacy criteria for matrix difference equations abstract discrete inequalities are given and applied to the matrix jacobi equation 2yt 1 ptyt 0 to develop criteria for disconjugacy that take into account the cancellation of positive and negative parts of pt

weighted means and oscillation conditions for second order matrix differential equations on considere le systeme d equations differentielles d ordre 2 p t y t q t y t 0 pour t 0 ou p et q sont des matrices symetriques n n reelles sur 0 et y un vecteur colonne on suppose que p t est definie positive pour tout t tel que p t 1 on donne des conditions d oscillations

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

linear oscillation of second order nonlinear difference equations with damped term abstract in this paper we shall establish relations between the oscillation of second order nonlinear difference equations with damped term and the oscillation of their linear limiting equations

oscillation of self adjoint linear matrix hamiltonian systems abstract by means of monotone functionals defined on suitable matrix spaces and new methods oscillation criteria for self adjoint linear hamiltonian matrix system of the form x x a x x x b x y x y x c x x x a x y x are obtained there results are sharper than some previous results even for self adjoint second order matrix differential systems in particular our theorems are generalizations and improvements of many existing results including those obtained by sun j math anal appl 279 2003 651 658

a sixth order a stable explicit one step method for stiff systems this paper presents a sixth order explicit one step method which is proved to be a stable the numerical experiments demonstrate that the new method is suitable for stiff systems

oscillation results for second order scalar and matrix difference equations we consider the self adjoint second order scalar difference equation rn xn pnxn 1 0 1 and the matrix system rn xn pnxn 1 0 2 where rn 0 pn 0 rn 0 pn 0 are sequences of real numbers d d hermitian matrices with rn 0 rn 0 we obtain some new oscillation criteria through change of variable techniques combined with consideration of 1 and 2 as perturbations of the cases pn 0 pn 0

oscillation theorems for second order advanced functional difference equations abstract we consider the second order advanced functional difference equation a n n p n g n 0 where a n 0 s n 0 1 a s p n 0 p n 0 g n n 1 g n is a monotone increasing integer sequence we obtain some new oscillation criteria through an appropriate riccati equation

the analytic structure of dynamical systems and self similar natural boundaries abstract in this paper we investigate the analytic complex time structure of the movable singularities for several dynamical systems in general it is found that there exists a direct connection between the occurencce of a certain type of multiple valuedness of the singularities and the existence of a class of remarkable self similar natural boundaries for these systems an asymptotic description of the distribution of singularities in the natural boundary is developed this provides a description of the fine scale structure of these natural boundaries that agrees closely with the numerical calculations

pryce pre analysis adapted to some dae solvers abstract the ode solver hbt 12 5 of order 12 t nguyen ba h hao h yagoub r vaillancourt one step 5 stage hermite birkho taylor ode solver of order 12 appl math comput 211 2009 313 328 doi 10 1016 j amc 2009 01 043 which combines a taylor series method of order 9 with a runge kutta method of order 4 is expanded into the dae solver hbt 12 5dae of order 12 dormand prince s dp 8 7 13m is also expanded into the dae solver dp 8 7 dae pryce structural pre analysis extended odes and ode first order forms are adapted to these dae solvers with a stepsize control based on local error estimators and a modified pryce algorithm to advance integration hbt 12 5dae uses only the first nine derivatives of the unknown variables as opposed to the first 12 derivatives used by the taylor series method t12dae of order 12 numerical results show the advantage of hbt 12 5dae over t12dae dp 8 7 dae and other known dae solvers

a restriction theorem for metivier groups abstract in the spirit of an earlier result of d muller on the heisenberg group we prove a restriction theorem on a certain class of two step nilpotent lie groups our result extends that of muller also in the framework of the heisenberg group

projector based integration of daes with the taylor series method using automatic differentiation automatic or algorithmic differentiation ad opens new possibilities to analyze and solve daes by projector based methods in this paper we present a new approach to compute consistent initial values and integrate daes up to index two considering the nonlinear dae in each time step as a nonlinear system of equations for taylor expansions these systems will be solved by the newton kantorowitsch method whereas the resulting linear systems are decoupled using the splitting techniques related to the tractability index concept this approach provides a description of the inherent ode that allows an application of the classical taylor series method to the integration of initial value problems linear and nonlinear daes with index up to two are examined and solved numerically

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

analytic structure of the henon heiles hamiltonian in integrable and nonintegrable regimes the solutions of the henon heiles hamiltonian are investigated in the complex time plane the use of the painleve property i e the property that the only movable singularities exhibited by the solution are poles enables successful prediction of the values of the nonlinear coupling parameter for which the system is integrable special attention is paid to the structure of the natural boundaries that are found in some of the nonintegrable regimes these boundaries have a remarkable self similar structure whose form changes as a function of the nonlinear coupling

sensitivity analysis of odes daes using the taylor series method this paper studies the applicability of the taylor method for the sensibility analysis of odes and daes extended automatic differentiation rules are introduced for the calculus of partial derivatives of taylor series the numerical method is implemented using an efficient variable step variable order scheme finally some numerical tests are presented showing the benefits of the formulation

implicit extension oftaylor series method with numerical derivatives for initial value problems the taylor series method is one of the earliest analytic numeric algorithms for approximate solution of initial value problems for ordinary differential equations the main idea of the rehabilitation of this algorithms is based on the approximate calculation of higher derivatives using well known finite difference technique for the partial differential equations the approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval integration this property offers different facility for adaptive error control this paper describes several explicit taylor series algorithms with numerical derivatives and their implicit extension and examines its consistency and stability properties the implicit extension based on a collocation term added to the explicit truncated taylor series and the approximate solution obtained as a continuously differentiable piecewise polynomials function some numerical test results is presented to prove the efficiency of these new old algorithm

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

weyl titchmarsh theory for a class of discrete linear hamiltonian systems abstract this paper is concerned with establishing the weyl titchmarsh theory for a class of discrete linear hamiltonian systems over a half line fundamental properties of solutions regular spectral problems and the corresponding maximal and minimal operators are first studied matrix disks are constructed and proved to be nested and converge to a limiting set some precise relationships among the rank of the matrix radius of the limiting set the number of linearly independent square summable solutions and the defect indices of the minimal operator are established based on the above results a classification of singular discrete linear hamiltonian systems is given in terms of the defect indices of the minimal operator and several equivalent conditions on the cases of limit point and limit circle are obtained respectively especially several problems in the limit point case are more carefully investigated including fundamental properties of square summable solutions properties of the weyl function which is the unique element in the limiting set in this case and inhomogeneous boundary problems self adjointness of the corresponding hamiltonian operator relationship between the spectrum of the hamiltonian operator and the analyticity of the weyl function as well as the dependence of the spectrum on the boundary data in which some interesting separation results for the spectrum are obtained finally another set of four equivalent conditions on the limit point case are established

vsvo formulation of the taylor method for the numerical solution of odes this paper presents an analysis of the taylor method for the numerical solution of odes when a very high precision of the solution is required some theoretical properties of the taylor method are considered from the practical point of view a variable stepsize variable order vsvo scheme is presented and its utility is discussed with several examples to reach the goal of high precision the use of multiprecision libraries is considered finally some numerical tests based on the test problems given in 1 and on a set of important problems in dynamical systems and astrodynamics are presented showing the benefits of the vsvo formulation especially for high precision demands compared with a well established runge kutta code

oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term abstract some riccati type difference inequalities are given for the second order nonlinear difference equations with nonlinear neutral term and using these inequalities we obtain some oscillation criteria for the above equation

averaging techniques for self adjoint matrix equations on a measure chain abstract we apply averaging techniques to establish oscillation and nonoscillation criteria for the self adjoint matrix dynamic equation lx t p t x t q t x t 0 on a measure chain t these results extend and unify earlier results for the differential and difference equation case

introduction to automatic differentiation and matlab object oriented programming an introduction to both automatic differentiation and object oriented programming can enrich a numerical analysis course that typically incorporates numerical differentiation and basic matlab computation automatic differentiation consists of exact algorithms on floating point arguments this implementation overloads standard elementary operators and functions in matlab with a derivative rule in addition to the function value for example sin u will also compute cos u ast u prime where u and u prime are numerical values these methods are mostly one line programs that operate on a class of value and derivative objects providing a simple example of object oriented programming in matlab using the new as of release 2008a class definition structure the resulting powerful tool computes derivative values and multivariable gradients and is applied to newton s method for root finding in both single and multivariable settings to compute higher order derivatives of a single variable function another class of series objects keeps taylor polynomial coefficients up to some order overloading multiplication on series objects is a combination discrete convolution of coefficients this idea leads to algorithms for other operations and functions on series objects a survey of more advanced topics in automatic differentiation includes an introduction to the reverse mode our implementation is forward mode and considerations in arbitrary order multivariable series computation

a class of runge kutta formulae of order three and four with reduced evaluations of function traditionally an explicit runge kutta method of order three requires three evaluations of function and an explicit runge kutta method of order four requires four evaluations of function in this paper a class of runge kutta processes of order three and order four with reduced evaluations of function are proposed the numerical experiments shows that these new runge kutta processes are more attractive than the traditional ones

atomft solving odes and daes using taylor series abstract taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long taylor series the series terms are generated recursively using the techniques of automatic differentiation the atomft system includes a translator to transform statements of the system of odes into a fortran 77 object program that is compiled linked with the atomft runtime library and run to solve the problem we review the use of the atomft system for nonstiff and stiff odes the propagation of global errors and applications to differential algebraic equations arising from certain control problems to boundary value problems to numerical quadrature and to delay problems

disconjugacy disfocality and oscillation of second order difference equations abstract we obtain necessary and sufficient conditions for the disconjugacy disfocality or oscillation of the second order linear difference equation the conditions are given explicitly in terms of the coefficients in the equation

on riesz means of eigenfunction expansions for the kohn laplacian in a recent article c sogge improved some classical results by l hormander about the l p convergence of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in this paper we shall deal with the analogous problem for the model operator of a non elliptic but subelliptic differential operator the kohn laplacian on the heisenberg group

breaking the limits the taylor series method abstract this paper discusses several examples of ordinary differential equation ode applications that are difficult to solve numerically using conventional techniques but which can be solved successfully using the taylor series method these results are hard to obtain using other methods such as runge kutta or similar schemes indeed in some cases these other schemes are not able to solve such systems at all in particular we explore the use of the high precision arithmetic in the taylor series method for numerically integrating odes we show how to compute the partial derivatives how to propagate sets of initial conditions and finally how to achieve the brouwer s law limit in the propagation of errors in long time simulations the tides software that we use for this work is freely available from a website

a restriction theorem for the heisenberg group since the discovery by e m stein see fefferman 5 that the fourier transform of an lp function on r has a well defined restriction to the unit sphere s which is square integrable on sn 1 if p is close enough to 1 various new restriction theorems have been proved see e g 10 22 27 29 the importance of such theorems in harmonic analysis see e g 5 6 24 28 as well as in the theory of partial differential equations 18 19 25 has become evident it has turned out 29 that the crucial step in proving the stein tomas restriction theorem is the following let a denote the surface measure of s n 1 and let k denote the fourier transform of a then the operator af f k is bounded from lp to lp if p is sufficiently close to 1 9 can be interpreted in terms of the laplacian a on rn let a fox dex be the spectral decomposition of a then formally 9 const f106 1 x dex where 6 denotes dirac s point measure therefore it is natural to study analogues for 9 where a is replaced by a more general positive differential operator l in the case of second order elliptic operators on compact manifolds this has been done by sogge 25

oscillation of linear hamiltonian systems abstract in this paper sufficient conditions have been obtained for the oscillation of a class of linear hamiltonian systems