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

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

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

deformation quantizations with separation of variables on a kahler manifold we give a simple geometric description of all formal differentiable deformation quantizations on a kahler manifoldm such that for each open subsetu m multiplication from the left by a holomorphic function and from the right by an antiholomorphic function onu coincides with the pointwise multiplication by these functions we show that these quantizations are in 1 1 correspondence with the formal deformations of the original kahler metrics onm

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

a generalized minkowski problem with dirichlet boundary condition we prove the existence of hypersurfaces with prescribed boundary whose weingarten curvature equals a given function that depends on the normal of the hypersurface

quantum and classical pseudogroups part ii differential and symplectic pseudogroups the category of sympletic pseudospaces analogical to the category of pseudospaces in the sense of 2 is introduced and used to define symplectic pseudogroups structures analogical to pseudogroups 3 or quantum groups 4 it is shown that symplectic pseudogroups are in one to one correspondence with manin groups also introduced in this paper the set theoretical part of these structures has been described in i

a formal model of berezin toeplitz quantization we give a new construction of symbols of the differential operators on the sections of a quantum line bundle l over a kahler manifold m using the natural contravariant connection on l these symbols are the functions on the tangent bundle tm polynomial on fibres for high tensor powers of l the asymptotics of the composition of these symbols leads to the star product of a deformation quantization with separation of variables on tm corresponding to some pseudo kahler structure on tm surprisingly this star product is intimately related to the formal symplectic groupoid with separation of variables over m we extend the star product on tm to generalized functions supported on the zero section of tm the resulting algebra of generalized functions contains an idempotent element which can be thought of as a natural counterpart of the bergman projection operator using this idempotent we define an algebra of toeplitz elements and show that it is naturally isomorphic to the algebra of berezin toeplitz deformation quantization on m

formal symplectic groupoid the multiplicative structure of the trivial symplectic groupoid over d associated to the zero poisson structure can be expressed in terms of a generating function we address the problem of deforming such a generating function in the direction of a non trivial poisson structure so that the multiplication remains associative we prove that such a deformation is unique under some reasonable conditions and we give the explicit formula for it this formula turns out to be the semi classical approximation of kontsevich s deformation formula for the case of a linear poisson structure the deformed generating function reduces exactly to the cbh formula of the associated lie algebra the methods used to prove existence are interesting in their own right as they come from an at first sight unrelated domain of mathematics the runge kutta theory of the numeric integration of ode s

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

on gammelgaard s formula for a star product with separation of variables we show that gammelgaard s formula expressing a star product with separation of variables on a pseudo kahler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal fock space we prove this inversion formula directly and thus offer an alternative approach to gammelgaard s formula which gives more insight into the question why the directed graphs in his formula have no cycles

invariant imbedding applied to homogeneous two point boundary value problems with a singularity of the first kind it is shown that previous work of elder can be used to extend the version of invariant imbedding due to m r scott to homogeneous vector differential systems having a singularity of the first kind the boundary conditions considered consist of existence finite at the singularity and specified values for some subset of the dependent variables at a second point the important special case of a second order equation is discussed in some detail computational considerations are discussed and numerical examples are presented

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

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

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

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

weyl manifolds and deformation quantization this paper deals with non commutative objects based on the weyl algebra from the differential geometric point of view we propose to extend familiar notions from manifolds to non commutative objects in this algebraic approach the basic object is not a point of a space but certain sections of bundles

the dirichlet problem for monge ampere equations in non convex domains and spacelike hypersurfaces of constant gauss curvature in this paper we extend the well known results on the existence and regularity of solutions of the dirichlet problem for monge ampere equations in a strictly convex domain to an arbitrary smooth bounded domain in r as well as in a general riemannian manifold we prove for the nondegenerate case that a sufficient and necessary condition for the classical solvability is the existence of a subsolution for the totally degenerate case we show that the solution is in c11 q if the given boundary data extends to a locally strictly convex c2 function ori q as an application we prove some existence results for spacelike hypersurfaces of constant gauss kronecker curvature in minkowski space spanning a prescribed boundary

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

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

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

the dirichlet problem for weingarten hypersurfaces in lorentz manifolds we solve the dirichlet problem for strictly convex spacelike hypersurfaces of prescribed weingarten curvature under the main assumption that there exists an upper barrier we consider curvature functions that correspond in general to fully nonlinear elliptic partial differential equations which are not uniformly elliptic

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

discrete invariant imbedding for singular boundary value problems with a regular singularity we describe a discrete invariant imbedding method for solving a two point boundary value problem in the interval 0 b for a linear second order ordinary differential equation with a singularity of the first kind at x 0 by employing the series expansion on 0 d where d is near the singularity we first replace it by a regular problem on some interval d b the discrete invariant imbedding method is then described to solve the problem over the reduced interval the stability analysis of the method is discussed some model problems are solved and the numerical results compared with those of other methods

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

formal symplectic groupoid of a deformation quantization we give a self contained algebraic description of a formal symplectic groupoid over a poisson manifold m to each natural star product on m we then associate a canonical formal symplectic groupoid over m finally we construct a unique formal symplectic groupoid with separation of variables over an arbitrary kahler poisson manifold

fedosov products and quantum momentum maps the purpose of this paper is to study various aspects of star products on a symplectic manifold related to the fedosov method by introducing the notion of quantum exponential maps we give a characterization of fedosov connections as an application a geometric realization is obtained for the equivalence between an arbitrary product and a fedosov one every fedosov product is shown to be a vey product consequently we find that every product is equivalent to a vey product a classical result of lichnerowicz quantization of a hamiltonian g space and in particular quantum momentum maps are studied lagrangian submanifolds are also studied under a deformation quantization

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

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

numerical solution of singular boundary value problems by invariant imbedding abstract an invariant imbedding method is proposed for numerically solving singular boundary value problems with a regular singular point at one end of the interval the singular problem is first reduced to a regular boundary value problem by using series solution in the vicinity of the singular point to subtract out the singularity the regular boundary value problem is then solved by employing invariant imbedding method some numerical examples have been included to demonstrate the efficiency of the method

closed weingarten hypersurfaces in semi riemannian manifolds the existence of closed hypersurfaces of prescribed curvature in semi riemannian manifolds is proved provided there are barriers

fedosov quantization of lagrange finsler and hamilton cartan spaces and einstein gravity lifts on co tangent bundles we provide a method of converting lagrange and finsler spaces and their legendre transforms to hamilton and cartan spaces into almost kahler structures on tangent and cotangent bundles in particular cases the hamilton spaces contain nonholonomic lifts of pseudo riemannian einstein metrics on effective phase spaces this allows us to define the corresponding fedosov operators and develop deformation quantization schemes for nonlinear mechanical and gravity models on lagrange and hamilton fedosov manifolds

cubic spline for solving singular two point boundary value problems in this paper we present another version of cubic spline method for numerically solving singular two point boundary value problems for certain ordinary differential equation having singular coefficients to remove the singularity we first use series solution in the vicinity of the singular point and the boundary condition at a point x d in the vicinity of the singularity is derived the resulting regular bvp is then efficiently treated by employing cubic spline for finding the numerical solution some examples have been included and comparison of the numerical results made with other methods

algebraic index theorem we prove the atiyah singer index theorem where the algebra of pseudodifferential operators is replaced by an arbitrary deformation quantization of the algebra of functions on a symplectic manifold

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

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

quantum and classical pseudogroups part i union pseudogroups and their quantization union pseudogroups structures analogical to pseudogroups in the sense of 1 are defined using the category dual to the category of groupoids instead of the category of pseudospaces in the sense of 2 it is shown that these structures are equivalent to double groups in the sense of 3 moreover it is shown that a quantization procedure associates with each finite union pseudogroup a quantum pseudogroup therefore for each finite double group there is a finite pseudogroup

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

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

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

symplectic groupoids and poisson manifolds on definit la notion de groupoide symplectique on donne la variete de poisson de ce groupoide symplectique on presente une construction

an eigenlength problem arising from a model of patch formation abstract the patch formation model of del grosso gerardi and marchetti 1 is reformulated as an eigenlength problem the latter is then solved numerically by using recent results of elder 2 3 in which the method of invariant imbedding is extended to linear problems having a singularity of the first kind the results agree very well with those previously obtained 4 for the eigenvalue problem by means of the frobenius series sturmian theory and a certain monotone iteration