a geometric setting for internal motions of the quantum three body system quantum mechanics for internal motions of the three body system is set up on the basis of the complex vector bundle theory the three body system is called a triatomic molecule in the born oppenheimer approximation the internal states of the molecule are described as cross sections in the complex vector bundle assigned by an eigenvalue of the square of the total angular momentum operator this bundle is equipped with a linear connection which is a natural consequence of a geometric interpretation of the so called eckart condition the coupling of the internal motion with the rotation is understood naturally in terms of this connection the internal hamiltonian operator is obtained which includes the internal motion rotation coupling and a centrifugal potential the complex vector bundle for the triatomic molecule proves to be a trivial bundle though the geometric setting for the internal motion is independent of whether the bundle is trivial or not

approximation theorems for the propagators of higher order abstract cauchy problems in this paper we present two quite general approximation theorems for the propagators of higher order in time abstract cauchy problems which extend largely the classical trotter kato type approximation theorems for strongly continuous operator semigroups and cosine operator functions then we apply the approximation theorems to deal with the second order dynamical boundary value problems

availability analysis for software system with intrusion tolerance this paper is devoted to analyzing the instantaneous availability of a typical software system with intrusion tolerance by formulating the system with a couple of ordinary differential and partial differential equations this paper describes the system as a time delay partial differential equation based on the time delay model both steady state availability and instantaneous availability are investigated the optimal policy for preventive patch management to maximize the steady state availability of the software system is obtained and its related availability criterions are also presented employing the finite difference scheme and trotter kato theorem we converted the time delay partial equation into a time delay ordinary equation as a result the instantaneous availability of the system is derived some numerical results are given to show the effectiveness of the method presented in the paper

cotangent bundle reduction and poincare birkhoff normal forms in this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free lie group action the canonical coordinates enable us to compute poincare birkhoff normal forms of relative equilibria using standard algorithms the case of simple mechanical systems with symmetries is studied in detail as examples we compute poincare birkhoff normal forms for a lagrangian equilateral triangle configuration of a three body system with a morse type potential and the stretched out configuration of a double spherical pendulum

determination of stress intensity factors by the finite element discretized symplectic method abstract when rewriting the governing equations in hamiltonian form analytical solutions in the form of symplectic series can be obtained by the method of separation of variable satisfying the crack face conditions in theory there exists sufficient number of coefficients of the symplectic series to satisfy any outer boundary conditions in practice the matrix relating the coefficients to the outer boundary conditions is ill conditioned unless the boundary is very simple e g circular in this paper a new two level finite element method using the symplectic series as global functions while using the conventional finite element shape functions as local functions is developed with the available classical finite elements and symplectic series the main unknowns are no longer the nodal displacements but are the coefficients of the symplectic series since the first few coefficients are the stress intensity factors post processing is not required a number of numerical examples as well as convergence studies are given

applications of numerical eigenfunctions in the fractal like finite element method the fractal like finite element method is an accurate and efficient method to determine the stress intensity factors sif around crack tips the use of a self similar mesh together with the william s eigenfunctions reduce the number of unknowns significantly the sifs are the primary unknowns and no post processing is required in all previous studies we used the analytic eigenfunction expression to perform the global transformation however the eigenfunction cannot be found analytically in general crack problems two dimensional axisymmetrical cracks are considered here the resulting static equilibrium equations in local co ordinates are non homogeneous ordinary differential equations for which the analytic eigenfunction cannot be found completely we use a finite difference method to determine all the eigenfunctions needed numerically our evaluated sif values show very close agreement with published results copyright 2004 john wiley sons ltd

classical adiabatic angles and quantal adiabatic phase a semiclassical connection is established between quantal and classical properties of a system whose hamiltonian is slowly cycled by varying its parameters round a circuit the quantal property is a geometrical phase shift gamma n associated with an eigenstate with quantum numbers n nl the classical property is a shift delta theta l i in the lth angle variable for motion round a phase space torus with actions i il the connection is delta theta l delta gamma delta nl two applications are worked out in detail the generalised harmonic oscillator with quadratic hamiltonian whose parameters are the coefficients of q2 qp and p2 and the rotated rotator consisting of a particle sliding freely round a non circular hoop slowly turned round once in its own plane

discretisation of an infinite delay equation in this paper a banach phase space containing bc 0 and contained in c 0 is defined with which existence of a solution and convergence of a discrete scheme are proved for an infinite delay differential equation

singularp version finite elements for stress intensity factor computations the finite element analysis of linear elastic fracture mechanics problems is complicated by the presence of the singular and finite non singular stress distributions in the crack tip region the availability of a constant stress term in addition to the singular term in the standard h version singular finite elements is insufficient to model the finite nonsingular stress zone a p version singular finite element capable of modelling the higher order non singular stress terms in addition to the singular term and the constant term is presented the formulation for the displacement substitution technique for computing the stress intensity factors using singular p version triangular finite elements is developed unlike the standard h version formulation the stress intensity factors computed using the p version displacement substitution technique do not depend on the specific arrangement and length of the quarter point elements and require simple mesh designs as well as fewer number of degrees of freedom numerical studies comparing the convergence of the stress intensity factors computed by the p version method against other available alternatives such as the h version method and the contour integral method are presented to demonstrate the effectiveness of the present developments 1997 by john wiley sons ltd

explicitly solvable non linear optimal control problems the method of collective hamiltonians is applied to the study of optimal control problems on manifolds this approach enables us to single out exactly solvable situations various examples are considered necessary results from hamiltonian formalism are given including an industrial technique for the construction of completely integrable systems

a new numerical method for backward parabolic problems in the maximum norm setting a new method for solving numerically backward parabolic problems is proposed as usual for this kind of ill posed problems it is assumed that an a priori bound for the solution is available the algorithm consists of two basic steps first a standard forward integration is performed in order to approximate the solution at suitable future time levels second a holomorphic recovery procedure is carried out providing the required approximations for the preceding times the analysis is valid in the maximum norm setting and rigorous estimates are derived among other advantages the method can also be applied to nonlinear problems and it produces a continuous output some numerical illustrations are presented

isoholonomic problems and some applications we study the problem of finding the shortest loops with a given holonomy we show that the solutions are the trajectories of particles in yang mills potentials theorem 4 or equivalently the projections of kaluza klein geodesics theorem 2 applications to quantum mechanics berry s phase sect 3 and the optimal control of deformable bodies sect 6 are touched upon

exact stability chart of an elastic beam subjected to delayed feedback abstract the stability of an elastic beam is studied when the beam is subjected to a longitudinal force governed by a feedback loop the setup of the simplified mechanical model is motivated by a basic problem of electroacoustics the corresponding governing equation is the 1d wave equation with delayed boundary conditions by means of the d alembert solution the system can be transformed into a delay differential equation of neutral type that includes two time delays the intricate stability chart is constructed analytically in the parameter plane of the gain parameter and the ratio of the time delays the complete chart extends the numerical results of the literature while it also draws attention to the numerical difficulties of finite degree of freedom approximations and to the extreme sensitivity of the dynamics with respect to the time delay parameters of the system

numerical methods for a class of singular integro differential equations based on semigroup approximation in this paper computational methods for singular integro differential equations of neutral type are discussed the study is based on a well posed state space formulation and uses semigroup theoretical framework to analyze the convergence of semi and fully discrete numerical schemes feasibility of the fully discrete scheme is demonstrated by applying it to abel volterra equations of the first kind and to a singular neutral functional differential equation

avoiding the order reduction of runge kutta methods for linear initial boundary value problems a new strategy to avoid the order reduction of runge kutta methods when integrating linear autonomous nonhomogeneous initial boundary value problems is presented the solution is decomposed into two parts one of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction a numerical illustration is given this idea applies to practical problems where spatial discretization is also required leading to the full order both in space and time

a finite element for variable order singularities based on the displacement formulation an element for analysis of variable order one point singularities in plane problems has been proposed the element meets all the convergence requirements viz the rigid body mode the constant strain condition and the interelement compatibility it can be easily incorporated in any package based on the displacement formulation four examples from fracture mechanics are presented to demonstrate its performance the examples involve mechanical or thermal loadings the accuracy of results in all the examples is very good

super singular element method for two dimensional crack analysis super singular element method ssem is an accurate method devoted to stress singularity analysis accomplished by stress analysts who have little or no experience of fracture mechanics the basic i

differentiability and convergence rates of approximating semigroups for retarded functional differential equations in this paper the averaging approximations i e semidiscrete finite difference scheme for linear retarded functional differential equations are considered in the context of the semigroups on the hilbert space r n times l 2 it is shown that the approximating semigroups are all differentiable uniformly with respect to the index n determining the mesh size this fact makes it possible to prove the existence of a uniform exponential decay or growth rate for the approximating semigroups which can be made arbitrarily close to the decay respectively growth rate of the original semigroup more importantly the uniform differentiability makes it possible to establish convergence rates that improve with time and to prove the convergence of the approximating semigroups in the uniform operator topology the dependence of the convergence rate on the initial conditions and system parameters is also established

stratified dynamical systems and their boundary behaviour for three bodies in space with insight into small vibrations the centre of mass system for many particles is stratified into strata by the rotation group action the principal stratum consists of nonlinear configurations the collinear configurations form a lower dimensional stratum classical mechanics for many particles with nonlinear configurations and for those with collinear configurations are set up on the tangent or cotangent bundles over respective strata and can be reduced by the use of rotational symmetry a question arises as to how a many body system behaves in a neighbourhood of a collinear configuration the system may make a vibration to bend its collinear configuration which is a motion taking place across the boundary of the principal stratum this paper deals with the behaviour of those boundaries for three bodies in space the equations of motion for small vibrations as boundary behaviour at a collinear configuration will be given as a limit of those equations of motion for nonlinear configurations by means of two key facts that the isotropy subgroup may act non trivially on the tangent space at the collinear configuration and that vibrations take place in a constant plane in space further the perturbation of small vibrations is studied by applying moser s averaging method to show that small vibrations may give rise to finite rotations after a sufficiently large number of periods

dynamic analysis of frames by a two level finite element method abstract a two level finite element technique of constructing a frame super element is introduced to reduce the computational efforts for solving large scale frame problems the ordinary finite element method is used first to yield matrices for the beam members then the nodal displacements of all the nodes are related to those of a small number of selected joints master nodes in the frame by means of global finite element interpolating functions thus the order of the overall matrices is greatly reduced and the frame may be considered as a super element to be connected to other elements by means of the master nodes the accuracy of solution may be improved either with finer subdivision or by taking more master nodes inside each super element

method of lines approximations of delay differential equations we consider a special type of numerical methods for delay differential equations ddes by introducing a new independent variable an initial value problem for ddes is converted into an initial boundary value problem for the convection equation thus it is also possible to get an approximate solution to the dde problem by solving the initial boundary value problem with a suitable numerical method instead of solving the original problem in this paper we study a family of method of lines mol approximations to the problem which is obtained by applying runge kutta rk methods for space discretization and prove their convergence under the assumption that the rk methods satisfy a condition known as an algebraic characterization of a stability the result is also confirmed by numerical experiments moreover we show that the condition derives several stability properties of the mol approximations

holonomy reduced dynamics of triatomic molecules whereas it is easy to reduce the translational symmetry of a molecular system using e g jacobi coordinates the situation is much more involved for rotational symmetry in this paper we address the latter problem using holonomy reduction we suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection using the fact that for the special case of the three body problem the holonomy group is so 2 as opposed to so 3 like in systems with more than three bodies we obtain a holonomy reduced configuration space of topology the dynamics then takes place on the cotangent bundle over the holonomy reduced configuration space on this phase space there is an s1 symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method using a theorem by arnold it follows that the resulting symmetry reduced phase space is again a natural mechanical phase space i e a cotangent bundle this is different from what is obtained from the usual approach where symplectic reduction is used from the outset this difference is discussed in some detail and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established

modelling and analysis of repairable systems with preventive maintenance the time dependent solution of a kind of repairable system with preventive maintenance is investigated in the paper with total probability formula we show that the behavior of the system can be described as a group of ordinary differential equations coupled with partial differential equations which can be formulated as a time delay equation in an appropriate banach space based on the time delay equation this paper presents a difference scheme as an approximating method to solve the time dependent solution which is necessary for analyzing the instantaneous availability of the repairable system some numerical examples are shown to illustrate the effectiveness of this approach

the trotter kato theorem and approximation of pdes we present formulations of the trotter kato theorem for approximation of linear c 0 semigroups which provide very useful framework when convergence of numerical approximations to solutions of pdes are studied applicability of our results is demonstrated using a first order hyperbolic equation a wave equation and stokes equation as illustrative examples

rotational vibrational energy spectra of triatomic molecules near relative equilibria complete hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry j math phys 44 4411 2003 in this article the full hamiltonian is specialized for a rigid and for a semirigid molecule for the rigid molecule all internal coordinates are fixed at constants so that the hamiltonian operator comes to take an ordinary matrix form and accordingly the schrodinger equation becomes an algebraic eigenvalue equation the eigenvalues then provide rotational energy spectra of the rigid molecule for the semirigid molecule the full hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter indeed the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondege

boundary controllability of a linear semi discrete 1 d wave equation derived from a mixed finite element method in this article one discusses the controllability of a semi discrete system obtained by discretizing in space the linear 1 d wave equation with a boundary control at one extremity it is known that the semi discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero see 8 here we introduce a new semi discrete model based on a mixed finite element method with two different basis functions for the position and velocity we show that the controls obtained with these semi discrete systems can be chosen uniformly bounded in l 2 0 t and in such a way that they converge to the hum control of the continuous wave equation i e the minimal l 2 norm control we illustrate the mathematical results with several numerical experiments