the duality in octonions and the triality principle we show that the existence of an a 2 type subalgebra of e 6 which is dual to another one of g 2 type leads to a definition of the product of octonions on an 8 dimensional subspace of e 6 thisproduct is then only expressed in terms of the lie bracket of e 6 the well knowntriality principle becomes an easy consequence of this definition and g 2 acting by the adjoint action isshown to be the algebra of derivations of the octonions the real octonions are obtained from two specific real forms of e 6

a new two step p stable hybrid obrechkoff method for the numerical integration of second order ivps in this paper we introduce a class of new p stable hybrid obrechkoff methods for the numerical solution of second order initial value problems we generate a two step symmetric p stable hybrid obrechkoff method of order 8 with two off step symmetric points and phase lag of order at least 2 and for special cases this order can be increased up to 6 the numerical results obtained by the new method for some problems show its superiority in efficiency accuracy and stability

a qualitative analysis of two and three step methods for stable second order systems a variety of two and three step direct integration methods have been proposed for solving stable systems of linear second order ordinary differential equations we will assess their suitability using as our main criterion their ability to reproduce stable numerical solutions for stable positive definite and symmetric differential systems in addition we determine which of these methods have the ability to reproduce the weak and strong damping properties associated with these differential systems next we investigate their suitability for solving unsymmetric linear gyroscopic systems and finally we show that for a wider class of unsymmetric but stable differential systems only a very restricted class of methods may be stable through this investigation we isolate all the potentially useful two step methods as a subset of the newmark methods similarly we identify a class of useful second order three step methods and this latter investigation provides new insight into the popular houbolt and wilson

the method of lines and exponential fitting when the method of lines is used for solving time dependent partial differential equations finite differences are commonly employed to obtain the semidiscrete equations usually if the solution is expected to be smooth symmetric difference formulae are chosen for approximating the spatial derivatives these difference formulae are almost invariably based on lagrange type differentiation formulae however if it is known in advance that periodic components of given frequency are dominating in the solution more accurate difference formulae based on exponentials with imaginary exponent are available this paper derives such formulae and presents numerical results which clearly indicate that the accuracy can be improved considerably by exploiting additional knowledge on the frequencies of the solution

a 6 4 optimized embedded runge kutta nystrom pair for the numerical solution of periodic problems in this paper an optimization of the non fsal embedded rkn 6 4 pair with six stages of moawwad el mikkawy el desouky rahmo is presented the new method is derived after applying phase fitting and amplification fitting and has variable coefficients the preservation of the algebraic order is verified and the principal term of the local truncation error is evaluated furthermore periodicity analysis is performed which reveals that the new method is almost p stable the efficiency of the new method is measured via the integration of several initial value problems

a four step trigonometric fitted p stable obrechkoff method for periodic initial value problems in this paper we present a new p stable obrechkoff four step method which greatly improves the performance of our previous obrechkoff four step method and extends its application range by trigonometric fitting we extend the interval of periodicity of the previous four step method from about h2 16 to infinity and at the same time we keep all its advantage in the accuracy and efficiency we have tested the new method by four well known problems 1 the test equation 2 stiefel and bettis problem 3 duffing equation without damping and 4 bessel equation the numerical results show that the new method is more accurate than any previous method it also has great advantage in stability and efficiency

a rational construction of lie algebras of type e7 abstract we give an explicit construction of lie algebras of type e 7 out of a lie algebra of type d 6 with some restrictions up to odd degree extensions every lie algebra of type e 7 arises this way for lie algebras that admit a 56 dimensional representation we provide a more symmetric construction based on an observation of manivel the input is seven quaternion algebras subject to some relations

high orderp stable formulae for the numerical integration of periodic initial value problems recently there has been considerable interest in the approximate numerical integration of the special initial value problemy f x y for cases where it is known in advance that the required solution is periodic the well known class of stormer cowell methods with stepnumber greater than 2 exhibit orbital instability and so are often unsuitable for the integration of such problems an appropriate stability requirement for the numerical integration of periodic problems is that ofp stability however lambert and watson have shown that ap stable linear multistep method cannot have an order of accuracy greater than 2 in the present paper a class of 2 step methods of runge kutta type is discussed for the numerical solution of periodic initial value problems p stable formulae with orders up to 6 are derived and these are shown to compare favourably with existing methods

new family of modal methods for calculating eigenvector derivatives a new family of modal methods for the calculation of eigenvector derivatives in non sell adjoint systems with a singular coefficient matrix is developed the family contains the modal and modified modal methods as a subset in the family the component of the mth eigenvector in the expansion of the derivative of the jth eigenvector is multiplied by various powers of the eigenvalue ratio j m thereby accelerating convergence the family of methods is applied to a sell adjoint example problem namely a cantilever beam whose root depth is used as the design variable to which the sensitivity of the first four mode shapes is sought

geometry over composition algebras projective geometry abstract the purpose of this article is to introduce projective geometry over composition algebras the analogue of projective spaces and grassmannians over them are defined it will follow from this definition that the projective spaces are in correspondence with jordan algebras and that the points of a projective space correspond to rank one matrices in the jordan algebra a second part thus studies properties of rank one matrices i also give an explicit description of the simply connected chevalley group of type e 6 over the integers

an optimized runge kutta method for the solution of orbital problems we present a new explicit runge kutta method with algebraic order four minimum error of the fifth algebraic order the limit of the error is zero when the step size tends to zero infinite order of dispersion and eighth order of dissipation the efficiency of the newly constructed method is shown through the numerical results of a wide range of methods when these are applied to well known periodic orbital problems

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

highly accurate modal method for calculating eigenvector derivatives in viscous damping systems a highly accurate modal superposition method for computing complex eigenvector derivatives in viscous damp ing systems has been developed when higher modes are truncated the conventional modal superposition method cannot give an accurate solution and the errors may become significant in this paper calculating the derivatives is regarded as calculating the structural response to harmonic exciting using multiple modal accelerations and shifted poles highly accurate results would be obtained when only few modes are used all of the available modal superposition methods would be directly obtained from the proposed method numerical examples show that it achieves better calculation efficiency than all of the available modal methods and nelson s method when more than one eigenvector derivative is of interest moreover the presented method can be used to improve response calculations and substructure syntheses igenvector derivatives with respect to system parameters are widely used in structural design structural stability anal ysis finite element model updating and structural control since the earlier work of fox and kapoor 1 many methods have been developed 2 3 all of the available methods could be categorized into three groups modal method 1 3 4 direct method 1 4 and iter ation method 5 6 most of the iteration methods suffer from slow convergence rates and are less efficient recently ting6 improved the effectiveness of the method because the direct methods give out the exact solutions they are widely used however when the derivatives of a large number of eigenvectors are demanded the cal culation is very time consuming here the modal methods are more suitable the modal methods employ a modal superposition idea there fore the accuracy is dependent on the number of modes used in calculation when the complete set of eigenvectors is used the ex act solution could be obtained to guarantee the accuracy the clas sical modal method1 needs higher eigenvectors recently wang4 presented a modified modal method in which a modal acceleration type approach was used to approximate the contribution of trun cated higher modes therefore the modified modal method con verges faster than the classical method ma and hagiwara7 im proved the modified method with an appropriate shift value better convergence than that of the modified method is achieved and sys tems having zero eigenvalues can be directly analyzed to further improve the convergence the improved modified modal method8 10 and accurate modal superposition method9 have been proposed by several researchers the basic original idea of the aforementioned improved modal methods was found in the hu s book 11 the idea was used to improve the convergency of dynamic flexibility in series form for no damping systems all of the aforementioned methods were used for real eigen vectors for the cases of complex eigenvectors in viscous damp ing systems the three groups of methods were presented 12 13 the classical modal method for the complex eigenvectors which was similar to the method presented in ref 1 for real eigenvectors suffers from heavy modal truncation errors recently akgun14 presented a new family of modal methods for the calculation of the derivatives in non self adjo int systems the modal accelera tion approach was used to improve the convergence in this paper calculating the derivatives in viscous damping systems is regarded as calculating a structural response to a harmonic exciting using multiple modal accelerations and shifted poles a highly accurate modal method for calculating the derivatives has been developed there are clear mathematical and physical meanings in conduct ing the proposed method from which all of the available meth ods would be directly obtained furthermore fewer eigenvectors even only the specific eigenvector that is being considered are re quired for the predetermined accuracy as a result the method is more efficient in computation than akgun s method and nelson s method when more than one eigenvector derivative is of inter est numerical examples show that the proposed method is cor rect and drastically improves solution accuracy the method can directly be applied in systems with zero eigenvalues although the method is developed to calculate eigenvectors derivatives it can be used to improve response calculations and substructure syntheses

practical complete modal space and its applications the orthogonality relationship between the column vectors of the equivalent matrix of high order modes and the mass and stiffness matrices does not exist this limitation reduces the computational efficiency in addition this nonorthogonality behavior may sometimes reduce the precision for certain applications of the pcms method to improve the orthogonality relationship between the equivalent high order mode matrix and the mass and stiffness matrices an optimal gram schmidt orthogonalization method can be used in the analysis

symmetric multistep methods with zero phase lag for periodic initial value problems of second order differential equations we present in this paper two step and four step symmetric multistep methods involving a parameter p to solve a special class of initial value problems associated with second order ordinary differential equations in which the first derivative does not appear explicitly it is shown that the methods have zero phase lag when p is chosen as 2 times the frequency of the given initial value problem the periodicity intervals are given in terms of expressions involving the parameter p as p increases the periodicity intervals increase and for large p the methods are almost p stable

an explicit numerov type method for second order differential equations with oscillating solutions in this paper a new explicit numerov type method is introduced the construction is based on a modification of a sixth order explicit numerov type method recently developed by tsitouras ch tsitouras explicit numerov type methods with reduced number of stages comput math appl 45 2003 37 42 two free parameters are added in order to nullify the phase lag and the amplification the method is useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize numerical results obtained for well known test problems show the efficiency of the new method

some groups of transformations defined by jordan algebras i it is well known that the lie algebra of derivations of the exceptional jordan algebra m 3 8 over an algebraically closed field of characteristic 0 or over the field of reals is the exceptional lie algebra f 4 of killing cartan and that the lie algebra of linear transformations in m 3 8 leaving a certain cubic norm form defined on m 3 8 invariant is the exceptional lie algebra e 6 chevalley schafer 7 freudenthal 11 the program envisaged in the present series of papers is the extension of these results and their group analogues to arbitrary central simple jordan algebras the present paper gives the definition of the groups and lie algebras in the general case and gives their determination for special jordan algebras with some restrictions on the base field in a second paper we plan to consider the groups of automorphisms of arbitrary reduced exceptional simple jordan algebras and in a third paper we hope to study the generalized groups of types e 6 which arise from these jordan algebras

derivatives of complex eigenvectors using nelson s method 6shyy w an adaptive grid method for navier stokes flow computation applied mathematics and computation vol 21 no 3 1987 pp 201 219 7jeng y n andliou s c modi edmultipleone dimensional adaptive grid method numerical heat transfer part b vol 15 no 2 1989 pp 241 247 8jeng y n and lee z s revisit to the modi ed muliple onedimensional adaptive grid method numerical heat transfer part b vol 29 no 3 1996 pp 305 323 9bockelie m j and smith r e an adaptive grid method for computing the high speed 3d viscous flow about a re entry vehicle aiaa paper 92 2685 june 1992 10yee h c and harten a implicit tvd schemes for hyperbolic conservation laws in curvilinear coordinates aiaa journal vol 25 no 2 1987 pp 266 274 11jeng y n and liou y c adaptive grid generation by elliptic equations with grid control at all of the boundaries numerical heat transfer part b vol 23 no 2 1993 pp 135 151

a quartic c 3 spline collocation method for solving second order initial value problems abstract this paper introduces and analyzes a new formula for integrating special second order periodic and nonperiodic initial value problems in ordinary differential equations the presented formula is based on quartic c3 splines as an approximation to the exact solution of the initial value problem y x f x y y 0 y0 y 0 y 0 it turns out that the proposed method is a continuous extension of the well known fourth order numerov s method and hence possesses nonvanishing intervals of periodicity and absolute stability

high order p stable multistep methods abstract recently numerical integration of the special ivp y f t y whose solutions are of periodic type have been considered of great interest p stability is the appropriate requirement for numerical methods integrating periodic problems as established by lambert and watson 1976 a p stable linear multistep method has order no greater than two in this paper two step multi stage linear methods are proposed for numerical integration of periodic problems and p stable methods of this type obtained in a simple form up to order eight are shown finally those methods are successfully tested

comparison of several methods for calculating vibration mode shape derivatives four methods for the calculation of derivatives of vibration mode shapes eigenvectors with respect to design parameters are described these are finite difference method modal method a modified modal method and nelson s method the methods are implemented in a general purpose commercial finite element program and applied to the following test problems a cantilever beam and a stiffened cylinder with a cutout design variables are a beam tip mass a beam root height and specific dimensions of the cylinder model the methods are compared on the basis of central processor cp seconds required to obtain the derivatives and two of the methods are also evaluated for the rapidity of convergence data is presented showing the amount of cp time used to compute the first four eigenvector derivatives for each example problem a scalar measure of the error in the mode shape derivative is defined and numerical results illustrating the rapidity of convergence of the approximate derivative to the exact derivative are presented results indicate an advantage in using nelson s method because this method is exact and requires less cp time especially when derivatives with respect to several design variables are computed

natural frequency and mode shape sensitivities of damped systems part ii multiple natural frequencies abstract an efficient algorithm with proven numerical stability is derived for computation of eigenvalue and eigenvector derivatives of damped vibratory systems with multiple eigenvalues in the proposed method adjacent eigenvectors and orthonormal condition are used to compose an algebraic equation whose order is n m n m where n is the number of co ordinates and m the number of multiplicity of a multiple natural frequency the mode shape derivatives of the damped systems can be obtained by solving the algebraic equation the method can be consistently applied to both structural systems with structural design parameters and mechanical systems with lumped design parameters as an example of a structural system to demonstrate the theory of the proposed methods and its possibilities in the case of multiple eigenvalues the finite element model of the cantilever beam is considered and also a 5 dof mechanical system in the case of a non proportionally damped system the design parameter of the cantilever beam is its height and that of the 5 dof mechanical system is a spring

spatial discretization of hyperbolic equations with periodic solutions we investigate the cauchy problem for hyperbolic equations for which the frequencies of the main fourier components in the solution are located in a given frequency interval difference formulae for the spatial derivatives are constructed that are tuned to the given intervals of frequencies numerical examples illustrating these special discretizations are given both for linear and non linear problems

freudenthal triple systems by root system methods abstract for certain lie algebras g we can use a grading g g 2 g 1 g 0 g 1 g 2 and define a quartic form and a skew symmetric bilinear form on g 1 thereby constructing a freudenthal triple system the structure of the freudenthal triple system is examined using root system methods available in the lie algebra context in the cases g e 8 where g 1 is the minuscule representation of e 7 and g d 4 we determine the groups stabilizing the quartic form and both the quartic and bilinear forms

an adapted explicit hybrid method of numerov type for the numerical integration of perturbed oscillators in 1971 scheifele has designed a method for the integration of perturbed oscillators by refining the classical method of power series in this paper we examine how we can extend the algorithm of scheifele for the construction of a fourth order hybrid method without explicit first derivatives an application to some popular test problems such as the satellite problem shows the properties of the developed algorithm

a new class of two step p stable tfpl methods for the numerical solution of second order ivps with oscillating solutions abstract a new class of two step linear symmetric methods is introduced for the numerical solution of second order initial value problems that have highly oscillatory solutions in this class for the first time in the literature we calculate the coefficients of the method by a combination of a trigonometrically fitted tf method and a vanishing of the phase lag and some of its derivatives vsdpl method and we construct a new class of methods that has the properties of tf methods and vsdpl methods which we call the tfpl method this method is of algebraic order 8 and has an important p stability property the main structure of the method is multiderivative and the combined phases were applied to expand the stability interval and to achieve p stability the advantage of the method in comparison with similar methods in terms of efficiency accuracy and stability is shown by its implementation in some important problems the undamped duffing equation etc

a trigonometrically fitted explicit hybrid method for the numerical integration of orbital problems abstract in this paper we present a kind of trigonometrically fitted explicit two step hybrid method which achieves algebraic order six the new method is zero dissipative phase fitted and almost p stable numerical experiments from its application to well known periodic orbital problems show that the new method is more competitive when comparing with the codes proposed by franco j m franco a class of explicit two step hybrid methods for second order ivps j comput appl math 187 2006 41 57 anastassi and simos z a anastassi t e simos a trigonometrically fitted runge kutta method for the numerical solution of orbital problems new astron 10 2005 301 309

stage reduction on p stable numerov type methods of eighth order we present an implicit hybrid two step method for the solution of second order initial value problem it costs only six function evaluations per step and attains eighth algebraic order the method satisfy the p stability property requiring one stage less we conclude dealing with implementation issues for the methods of this type and give some first pleasant results from numerical tests

an explicit four step phase fitted method for the numerical integration of second order initial value problems abstract an explicit four step method with phase lag of infinite order is developed for the numerical integration of second order initial value problems extensive numerical testing indicates that this new method can be generally more efficient than other four step methods

second derivative methods applied to implicit first and second order systems a wide variety of methods have been proposed for solving implicit second order systems of differential equations such as arise in structural and mechanical vibration analyses here we discuss a class of methods based on second derivative formulae some of which have previously been used by enright 1 we show them to have desirable analytic and implementation properties we relate these methods to others recently proposed by brusa and nigro and by serbin showing that we obtain algorithms which have all the desirable properties of their methods and which can also be extended to classes of practical problems which were not considered by these other authors since implicit first order systems are also important in practice for example in finite element analyses of heat conduction problems we demonstrate the efficient implementation of the second derivative formulae in this case too

novel modal method for efficient calculation of complex eigenvector derivatives a novel modal superposition method is presented for calculating eigenvector derivatives in self adjoint or non self adjoint systems the method is especially appropriate for determination of derivatives of many eigenvectors and requires only one eigenvalue and its corresponding right and left eigenvectors two parameters for the eigenvalue under consideration are introduced to obtain a generic derivative formula which contains most of the available modal superposition methods as a branch matrices of the modified system are decomposed with available lower modes to avoid repeatedly solving equations for different eigenvalues the contribution of truncated higher modes is expanded in a continued product of a few matrices polynomials and one generalized power series whereby rapid convergence can be achieved easily even in the vicinity of the convergent boundary without additional computation cost all other modal superposition coefficients can be neglected intrinsically with the determination of introduced parameters but a simple condition for coefficients associated with lower order eigenvalues should be satisfied in a special case numerical examples show that the method is efficient and can give results comparable to the exact solutions the method is applicable to various damped systems and closed mode cases

fine gradings on simple exceptional jordan pairs and triple systems abstract we give a classification up to equivalence of the fine group gradings by abelian groups on the jordan pairs and triple systems of types bi cayley and albert under the assumption that the base field is algebraically closed of characteristic different from 2 the associated weyl groups are computed we also determine for each fine grading on the bi cayley and albert pairs the induced grading on the exceptional simple lie algebra given by the tits kantor koecher construction

sdirk methods for stiff odes with oscillating solutions new sdirk methods specially adapted to the numerical solution of stiff systems of odes which are assumed to possess oscillating solutions are obtained our interest is centered on the dispersion phase errors and the dissipation numerical damping of the dominant components in the numerical oscillations when these methods are applied to a homogeneous linear test model two a stable methods with algebraic order 3 and higher order of dissipation are obtained among the members of a family of methods proposed by van der houwen and sommeijer 1989 some numerical results are presented to show the efficiency of the new methods when they are compared with other methods presented in van der houwen and sommeijer 1989

a class of two step p stable methods for the accurate integration of second order periodic initial value problems abstract in this paper we consider a two parameter family of two step methods for the accurate numerical integration of second order periodic initial value problems by applying the methods to the test equation y 2y 0 we determine the parameters so that the phase lag frequency distortion of the method is minimal the resulting method is a p stable method with a minimal phase lag 6h6 42000 the superiority of the method over the other p stable methods is illustrated by a comparative study of the phase lag errors and by illustrating with a numerical example

damping and phase analysis for some methods for solving second order ordinary differential equations we consider numerical methods for initial value problems for second order systems of ordinary differential equations analysing them by applying them to the test equation we discuss conditions which ensure an oscillatory numerical solution and the desirability of such a property we also use a slightly more general test equation to derive conditions which ensure that the numerical forced oscillation is in phase with the true forced oscillation to illustrate the theory we consider the damping and phase properties of some frequently used methods

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

an optimal algorithm for certain boundary value problem abstract the o h 4 finite difference scheme for the second derivative u x leads to a coherent pentadiagonal matrix which is factorized into two tridiagonal matrices this factorization is used to derive an optimal algorithm for solving a linear system of equations with the pentadiagonal matrix as an application a nonlinear system of ordinary differential equations is approximated by an o h 4 convergent finite difference scheme this scheme is solved by the implicit iterative method applying the algorithm at each iteration a mathematica module designed for the purpose of testing and using the method is attached

an l stable rosenbrock method for step by step time integration in structural dynamics abstract a single step two stage l stable rosenbrock method is given in a form suitable for step by step integration of linear and non linear structural dynamics problems the method has second order accuracy and is a true single step method in that it does not require initial values of acceleration for each time step the method requires a single formation of the tangential stiffness matrix a single lu decomposition and two linear system substitutions but does not require iteration analysis and numerical examples are used to show the good stability and accuracy properties of the method and the importance of l stability when integrating numerically stiff problems in structural dynamics

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

quintic c 2 spline integration initial value equation problems abstract in this paper we propose a global collocation method for the integration of the special second order ordinary initial value problem ivp y f x y the presented method is based on quintic c2 splines s x as an approximation to the exact solution y x of the ivp analysis of stability shows that the method possesses 0 36 54 110 2 as interval of periodicity and absolute stability moreover the method has phase lag of order four with actual phase lag h4 18 6 error bounds in the uniform norm for s i y i o h 4 i 0 1 2 if y c6 0 b together with illustrative test examples will also be considered

a generator of high order embedded p stable methods for the numerical solution of the schro dinger equation a generator of new embedded p stable methods of order 2n 2 where n is the number of layers used by the embedded methods for the approximate numerical integration of the one dimensional schrodinger equation is developed in this paper these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for one dimensional differential equations of the schrodinger type show the validity of the developed theory

p stable symmetric super implicitmethods for periodic initial value problems the idea of super implicit methods requiring not just past and present but also future values was suggested by fukushima recently here we construct p stable super implicit methods for the solution of second order initial value problems the benefit of such methods is realized when using vector or parallel computers

sensitivity analysis methods for coupled acoustic structural systems part i modal sensitivities an analytical method is proposed for calculating the sensitivities of the eigenvalues and eigenvectors of coupled acoustic structural systems for the purpose of reducing vehicle interior noise the concept of the left and right eigenvectors with regard to the eigenvalue problems of coupled systems is presented first and four propositions concerning this concept are then proved based on these propositions the formulations for the eigenvalue and eigenvector sensitivity analysis of coupled systems are obtained it is shown that the new method provides better convergence than the modal method and modified modal method extensions of these methods to coupled systems and it achieves better calculation efficiency than the extended nelson s method when more than one eigenvector is being considered the theoretical results and computation equations are verified through application of the method to a coupled acoustic box structure and an actual vehicle interior noise problem

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

a new family of explicit linear two step singularly p stable obrechkoff methods for the numerical solution of second order ivps abstract according to lambert and watson 6 theorem 4 says that the linear multistep p stable methods can not be explicit they must be implicit and being implicit is the essential condition to obtain important feature of p stability few explicit p stable methods have been created in which they are nonlinear or at most p stable for the first time in literature in this paper we create a new family of explicit linear two step singularly p stable methods with phase lag of order infinity for the numerical solution of initial value problems of second order ordinary differential equations finally the numerical results obtained by the new family for some well known problems show its superiority in efficiency accuracy convergency and stability

p stable high order super implicit and obrechkoff methods for periodic initial value problems this paper discusses the numerical solution of periodic initial value problems two classes of methods are discussed super implicit and obrechkoff the advantage of obrechkoff methods is that they are high order one step methods and thus will not require additional starting values on the other hand they will require higher derivatives of the right hand side in cases when the right hand side is very complex we may prefer super implicit methods we develop a super implicit p stable method of order 12 and obrechkoff method of order 18

freudenthal triple systems in arbitrary characteristic abstract let q v k be the form of degree 4 and h the alternating bilinear form that together give rise to a freudenthal triple system when the characteristic of k is different from 2 and 3 we give a form of degree 4 on a central cover of v over an arbitrary field in arbitrary characteristic and determine the group of similitudes of this group coincides with the intersection of the group of similitudes of q and the isometry group of h when the characteristic is different from 2 and 3 and has the expected structure of a suitable levi subgroup in general we extend known results about the stabilizers of elements of v and orbits of the group of similitudes and show in particular the existence of an exotic set of orbits when char k 2 and k is not perfect

an exponentially fitted and trigonometrically fitted method for the numerical solution of periodic initial value problems abstract an explicit symmetric multistep method is presented in this paper the new method is exponentially fitted and trigonometrically fitted and is of algebraic order eight the effectiveness of the exponential fitting is proved by the application of the new method and the classical one with constant coefficients to well known periodic problems

jordan algebras exceptional groups and bhargava composition abstract we consider an integral version of the freudenthal construction relating jordan algebras and exceptional algebraic groups we show how this construction is related to higher composition laws of m bhargava in number theory we propose an algorithmic approach to studying orbit spaces of groups underlying bhargava s higher composition laws using this method we discover two new examples of spaces sharing similar properties and indicate several more examples of spaces where such composition laws may be introduced

accelerated subspace iteration for eigenvector derivatives an accelerated subspace iteration method for calculating eigenvector derivatives has been developed factors affecting the effectiveness and the reliability of the subspace iteration are identified and effective strategies concerning these factors are presented the method has been implemented and the results of a demonstration problem are presented

a new four step p stable obrechkoff method with vanished phase lag and some of its derivatives for the numerical solution of radial schrodinger equation abstract in this paper we present a new four step implicit p stable obrechkoff method of tenth algebraic order for solving one dimensional second order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable schrodinger equation numerical stability and phase properties of the new methods are analyzed numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature

a new hybrid imbedded variable step procedure for the numerical integration of the schrodinger equation a new imbedded variable step procedure is developed for the numerical integration of the radial schrodinger equation the new imbedded method is based on p stable methods of exponential order eight ten 12 and 14 numerical results indicate that the new procedure is more efficient than similar variable step procedures

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

isotropic trialitarian algebraic groups abstract by modifying a construction from knus et al we construct all isotropic algebraic groups of type 3 d 4 and 6 d 4 over an arbitrary field of characteristic 2 we also provide a nice isomorphism criterion for such groups the results of this paper extend the main results of allison using entirely different methods to fields of nonzero characteristic and algebraic groups

families of two step fourth order p stable methods for second oder differential equations this paper deals with a class of symmetric hybrid two step fourth order p stable methods for the numerical solution of special second order initial value problems such methods were proposed independently by cash 1 and chawla 3 and normally require three function evaluations per step the purpose of this paper is to point out that there are some values of the free parameters available in the methods proposed which can reduce this work we study two classes of such methods the first is the class of economical methods see definition 3 1 which reduce this work to two function evaluations per step and the second is the class of efficient methods see definition 3 2 which reduce this work with respect to implementation for nonlinear problems we report numerical experiments to illustrate the order acuracy and implementational aspects of these two classes of methods

embedded methods for the numerical solution of the schrodinger equation abstract new methods for the approximate numerical integration of the one dimensional schrodinger equation are developed in this paper complete phase lag analysis of the new methods is included these new methods are called embedded methods because of a simple natural error control mechanism numerical results obtained for a one dimensional schrodinger equation show the validity of the developed theory

an explicit hybrid method of numerov type for second order periodic initial value problems abstract we consider a two parameter family of explicit hybrid methods of numerov type for the numerical integration of second order intial value problems when these methods are applied to the linear equation y 2 y 0 0 we determine the parameters so that the phase lag frequency distortion of the method is minimal the resulting method has algebraic order 4 and a small frequency distortion of size 1 3628800 v 8 v h h being the step size and in addition it possesses an interval of periodicity of size 4 63 which is larger than the interval of periodicity corresponding to the explicit method of chawla and rao 1986 the application of this method to equations describing free and weakly forced oscillations reveals its superiority over other methods

multistep numerical methods based on the scheifele g functions with application to satellite dynamics in 1971 scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series this method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas in spite of the good behavior of the scheifele method it is only practical for use in a few particular cases in which the force function is very simple given the complexity of the preliminary calculations needed to obtain the recurrence formulas this problem is resolved in this article by converting the g functions method into a multistep scheme the key property is that those algorithms are able to integrate without truncation error harmonic oscillators involving various frequencies and that for perturbed problems the local error contains the small perturbation parameter as a factor therefore the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi periodic ones an application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of earth satellites

efficient computation of many eigenvector derivatives using dynamic flexibility method a new approach for computing eigenvector derivatives with repeated eigenvalues of a complex structural system has been successfully developed this method utilizes a dynamic flexibility technique unlike other published methods this method requires one only to solve the system linear algebraic equations once regardless of the number of eigenvector derivatives needed to be computed tremendous computer time can be saved using this method theoretical derivation of this method is presented numerical examples are also presented to verify the analysis this method can be a powerful tool for the design engineer to compute many eigenvector derivatives of a large complex structural system

phase fitted and amplification fitted two step hybrid methods for y f x y this paper provides a theoretical framework for a new type of phase fitted and amplification fitted two step hybrid ftsh methods which is introduced by the author in h van de vyver a phase fitted and amplification fitted explicit two step hybrid method for second order periodic initial value problems internat j modern phys c 17 2006 663 675 the methods constitute a modification of dissipative two step hybrid methods in the sense that two free parameters are added to eliminate the phase lag and the amplification error the methods are useful only when a good estimate of the frequency of the problem is known in advance the parameters depend on the product of the estimated frequency and the stepsize the algebraic order zero stability stability and phase properties are examined the theory is illustrated with sixth order explicit ftsh methods numerical results carried out on an assortment of test problems show the relevance of the theory

structurable algebras and groups of type e6 and e7 abstract it is well known that every group of type f 4 is the automorphism group of an exceptional jordan algebra and that up to isogeny all groups of type 1 e 6 with trivial tits algebras arise as the isometry groups of norm forms of such jordan algebras we describe a similar relationship between groups of type e 6 and groups of type e 7 and use it to give explicit descriptions of the homogeneous projective varieties associated to groups of type e 7 with trivial tits algebras

obrechkoff versus super implicit methods for the solution of first and second order initial value problems abstract this paper discusses the numerical solution of first order initial value problems and a special class of second order ones those not containing first derivative two classes of methods are discussed super implicit and obrechkoff we will show equivalence of super implicit and obrechkoff schemes the advantage of obrechkoff methods is that they are high order one step methods and thus will not require additional starting values on the other hand they will require higher derivatives of the right hand side in case the right hand side is complex we may prefer super implicit methods the disadvantage of super implicit methods is that they in general have a larger error constant to get the same error constant we require one or more extra future values we can use these extra values to increase the order of the method instead of decreasing the error constant one numerical example shows that the super implicit methods are more accurate than the obrechkoff schemes of the same order

adaptive two step methods for semidiscretized wave equations adaptive two step direct integration methods are constructed for the integration of second order semidiscrete evolution equations possessing oscillatory solutions the methods are based on a class of adaptive multistep methods for a semilinear test model whose frequency is known they are constructed following the notion of diagonally implicit rk methods by using efficient rational approximations to cos v v 0 our interest is centered on the dispersion or phase errors of the dominant components in the numerical oscillations when these methods are applied to a linear homogeneous test model two step methods which have high order of dispersion up to 12 whereas the algebraic order is relatively low 2 or 4 are derived applications of these methods to linear as well as non linear test models and to semidiscretized hyperbolic equations reveal a good behaviour with regard to error propagation when they are compared with other conventional methods

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

explicit two step methods with minimal phase lag for the numerical integration of special second order initial value problems and their application to the one dimensional schro dinger equation abstract two step sixth order methods with phase lag of order eight ten and twelve are developed for the numerical integration of the special second order initial value sproblem an application to the one dimensional schrodinger equation on the resonance problem indicates that these new methods are generally more accurate than methods developed by chawla rao and neta this journal 1986 1987

some new implicit two step multiderivative methods for solving special second order ivp s abstract the aim of this paper is to select from the huge family of implicit two step multiderivative methods those which have better characteristics as regards the error order and error constant stability and easy of use with most of the coefficients equal to zero a few of these multiderivative methods that are a 0 stable and a stable have been obtained this feature makes them useful for solving stiff problems the stability intervals or stability regions for the methods chosen are presented some numerical tests are included the numerical test concerning the wave equation shows the good performance of the methods of lower order but a 0 stable in comparison with the poor performance of higher order methods

an eight step semi embedded predictor corrector method for orbital problems and related ivps with oscillatory solutions for which the frequency is unknown our new linear symmetric semi embedded predictor corrector method sepcm presented here is based on the multistep symmetric method of quinlan and tremaine 1990 with eight steps and eighth algebraic order and constructed to solve numerically the two dimensional kepler problem it can also be used to integrate related ivps with oscillatory solutions for which the frequency is unknown firstly we present a sepcm see panopoulos and simos 2013 36 37 in pair form this form has the advantage that reduces the computational expense from this form we construct a new symmetric eight step method the new scheme has constant coefficients and algebraic order ten we tested the efficiency of our newly developed scheme against some well known methods from the literature we measure the efficiency of the methods and conclude that the new scheme is the most efficient of all the compared methods and for all the problems solved

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