absence of absolutely continuous spectrum of floquet operators the spectrum of the floquet operator associated with time periodic perturbations of discrete hamiltonians is considered if the gap between successive eigenvalues j of the unperturbed hamiltonian grows as j j 1 j and the multiplicity of j grows asj with 0 asj tends to infinity then the corresponding floquet operator possesses no absolutely continuous spectrum provided the perturbation is smooth enough

advances in the formulation of the rotation free basic shell triangle a family of rotation free three node triangular shell elements is presented the simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle an enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes this allows to compute an assumed linear membrane strain field which improves the in plane behaviour of the original element a simple and economic version of the element using a single integration point is presented the efficiency of the different rotation free shell triangles is demonstrated in many examples of application including linear and non linear analysis of shells under static and dynamic loads the inflation and de inflation of membranes and a sheet stamping problem

the bgk model with external confining potential existence long time behaviour and time periodic maxwellian equilibria we study global existence and long time behaviour for the inhomogeneous nonlinear bgk model for the boltzmann equation with an external confining potential for an initial datum f0 0 with bounded mass entropy and total energy we prove existence and strong convergence in l1 to a maxwellian equilibrium state by compactness arguments and multipliers techniques of particular interest is the case with an isotropic harmonic potential in which boltzmann himself found infinitely many time periodic maxwellian steady states this behaviour is shared with the boltzmann equation and other kinetic models for all these systems we study the multistability of the time periodic maxwellians and provide necessary conditions on f0 to identify the equilibrium state both in l1 and in lyapunov sense under further assumptions on f these conditions become also sufficient for the identification of the equilibrium in l1

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

stationary solutions for the ellipsoidal bgk model in a slab abstract we address the boundary value problem for the ellipsoidal bgk model of the boltzmann equation posed in a bounded interval the existence of a unique mild solution is established under the assumption that the inflow boundary data does not concentrate too much around the zero velocity and the gas is sufficiently rarefied

a vertex based finite volume method applied to non linear material problems in computational solid mechanics a vertex based finite volume fv method is presented for the computational solution of quasi static solid mechanics problems involving material non linearity and infinitesimal strains the problems are analysed numerically with fully unstructured meshes that consist of a variety of two and threedimensional element types a detailed comparison between the vertex based fv and the standard galerkin fe methods is provided with regard to discretization solution accuracy and computational efficiency for some problem classes a direct equivalence of the two methods is demonstrated both theoretically and numerically however for other problems some interesting advantages and disadvantages of the fv formulation over the galerkin fe method are highlighted

quasi periodic solutions for two level systems we consider the schrodinger equation for a class of two level atoms in a quasi periodic external field in the case in which the spacing 2 between the two unperturbed energy levels is small and we study the problem of finding quasi periodic solutions of a related generalized riccati equation we prove the existence of quasi periodic solutions of the latter equation for a cantor set of values of around the origin which is of positive lebesgue measure such solutions can be obtained from the formal power series by a suitable resummation procedure the set can be characterized by requesting infinitely many diophantine conditions of mel nikov type

a perturbative algorithm for quasi periodic linear systems close to constant coefficients a perturbative procedure is proposed to formally construct analytic solutions for a linear differential equation with quasi periodic but close to constant coefficients the scheme constructs the necessary linear transformations involved in the reduction process up to an arbitrary order in the perturbation parameter it is recursive can be implemented in any symbolic algebra package and leads to accurate analytic approximations sharing with the exact solution important qualitative properties this algorithm can be used in particular to carry out systematic stability analyses in the parameter space of a given system by considering variational equations

the cauchy problem for the bgk equation with an external force abstract in this paper we study the cauchy problem for the bgk equation with an external force firstly we establish an l existence result for this equation and obtain some weighted l estimates then by means of the regularizing effects to the initial datum we construct the approximate solutions and obtain some uniform estimates of the approximate solutions finally by using compactness method and passing to the limit we prove the existence theorems of the l 1 and l p solutions and establish the propagation properties of the l p moments

numerical comparison between the boltzmann and es bgk models for rarefied gases abstract rarefied gas flows obey the boltzmann equation but numerical simulations of this equation are not always possible so that simpler models have been introduced the es bgk equation is one of these models it gives the correct transport coefficients for the navier stokes approximation so that boltzmann or es bgk simulations are expected to give the same results for dense gases but in the case of a rarefied flow complete numerical comparisons are needed in this paper we present numerical comparisons between the two models in transitional regimes where the es bgk model is expected to be useful for reentry flows around a compression ramp and a plate we also emphasize that the es bgk model gives flow predictions closer to the boltzmann result than the simpler bgk model

optimal exponential decay for the linearized ellipsoidal bgk model in weighted sobolev spaces this paper deals with the asymptotic behavior of solution to the linearized ellipsoidal bgk model in torus we prove that the solution converges exponentially to the equilibrium in the weighted sobolev spaces with polynomial weight our exponential decay rate e lambda t is optimal in the sense that lambda 0 equals to the spectral gap of the linearized operator in the standard hilbert space our strategy is taking advantage of the quantitative spectral gap estimates in a smaller reference hilbert space the factorization method and the enlargement of the functional space for the associated semigroup

the tric shell element theoretical and numerical investigation the tric facet triangular shell element which is based on the natural mode method is seen under the light of the non consistent formulation proposed by bergan and co workers under this formulation the convergence requirements are fulfilled even with relaxed conditions on the conformity demands of the displacement shape functions the intrinsic connection between the non consistent formulation and the natural mode method is demonstrated establishing thus a rigorous theoretical foundation for the tric element under the perspective of the non consistent formulation tric s convergence characteristics are established by satisfying apriori the patch test due to its inherent properties and thus guaranteeing convergence to the exact solution furthermore the element s accuracy robustness and efficiency are tested in a number of judiciously selected numerical examples on benchmark plate and shell problems while a cpu time comparison with a pure displacement based isoparametric shell element demonstrates its computational merits

floquet spectrum for two level systems in quasiperiodic time dependent fields we study the time evolution ofn level quantum systems under quasiperiodic time dependent perturbations the problem is formulated in terms of the spectral properties of a quasienergy operator defined in an enlarged hilbert space or equivalently of a generalized floquet operator we discuss criteria for the appearance of pure point as well as continuous spectrum corresponding respectively to stable quasiperiodic dynamics and to unstable chaotic behavior we discuss two types of mechanisms that lead to instability the first one is due to near resonances while the second one is of topological nature and can be present for arbitrary ratios between the frequencies of the perturbation we treat explicitly an example of this type the stability of the pure point spectrum under small perturbations is proven using kam techniques

unitary transformations depending on a small parameter we formulate a unitary perturbation theory for quantum mechanics inspired by the liedeprit formulation of canonical transformations the original hamiltonian is converted into a solvable one by a transformation obtained through a magnus expansion this ensures unitarity at every order in a small parameter a comparison with the standard perturbation theory is provided we work out the scheme up to order ten with some simple examples

cauchy problem for the ellipsoidal bgk model of the boltzmann equation ellipsoidal bgk model es bgk is a generalized version of the original bgk model designed to reproduce the physically correct prandtl number in the navier stokes limit in this paper we study the cauchy problem for the es bgk model under the condition of finite initial mass energy and entropy equivalence type estimates for the temperature tensor are crucially used

quantum phase corrections from adiabatic iteration the phase change gamma acquired by a quantum state psi t rangle driven by a hamiltonian h 0 t which is taken slowly and smoothly round a cycle is given by a sequence of approximants gamma k obtained by a sequence of unitary transformations the phase sequence is not a perturbation series in the adiabatic parameter epsilon because each gamma k except gamma 0 contains epsilon to infinite order for spin frac 1 2 systems the iteration can be described in terms of the geometry of parallel transport round loops c k on the hamiltonian sphere non adiabatic effects transitions must cause the sequence of gamma k to diverge for spin systems with analytic h 0 t this happens in a universal way the loops c k are sinusoidal spirals which shrink as epsilon k until k sim epsilon 1 and then grow as k the smallest loop has a size exp 1 epsilon comparable with the non adiabaticity

on the cauchy problem of the vlasov poisson bgk system global existence of weak solutions global in time existence of weak solutions to the cauchy problem of the three dimensional vlasov poisson bgk system is shown for initial data belonging to the space lp 3 3 with p 9 and having finite second order velocity moments this result solves partially the well posed problem for the vlasov poisson bgk system proposed by b perthame higher moments for kinetic equations the vlasov poisson and fokker planck cases math meth appl sci 13 441 452 1990

reduction of the semisimple 1 1 resonance the method of averaging is often used in hamiltonian systems of two degrees of freedom to find periodic orbits such periodic orbits can be reconstructed from the critical points of an associated reduced hamiltonian on a reduced space this paper details the construction of the reduced space and the reduced hamiltonian for the semisimple 1 1 resonance case the reduced space will be a 2 sphere in r3 and the reduced differential equations will be euler s equations restricted to this sphere the orbit projection from the energy surface in phase space to this sphere will be the hopf map the results of the paper are related to problems in physics on degeneracies due to symmetries of classical two dimensional harmonic oscillators and their quantum analogues for the hydrogen atom

a matrix free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction a new three dimensional 3d matrix free implicit unstructured multigrid finite volume fv solver for structural dynamics is presented in this paper the solver is first validated using classical 2d and 3d cantilever problems it is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates this method has been integrated into our existing fv compressible solver x lv y zhao et al an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects journal of computational physics 215 2 2006 661 690 based on the immersed membrane method imm x lv y zhao et al as mentioned above results for the interaction between the fluid and an immersed fixed free cantilever are also presented to demonstrate the potential of this integrated fluid structure interaction approach

the quantum normal form and its equivalents a quantum analog called the quantum normal form of the classical birkhoff gustavson normal form is presented the algebraic relationship between the quantum and birkhoff gustavson normal forms has been established by developing the latter using lie transforms it is shown that the birkhoff gustavson normal form can be obtained from the quantum normal form using an anharmonic oscillator and a henon heiles system as test cases the equivalence between the quantum normal form and the rayleigh schrodinger perturbation method is shown this equivalence provides an algebraic connection between the birkhoff gustavson normal form and the rayleigh schrodinger perturbation approach the question of weyl and torus quantizations of the birkhoff gustavson normal form is discussed in the light of the quantum normal form

a finite volume format for structural mechanics summary a general finite volume method fvm for the analysis of structural problems is presented it is shown that the fvm can be considered to be a particular case of finite elements with a non galerkin weighting for structural analysis this can readily be interpreted as equivalent to the unit displacement method which involves mainly surface integrals both displacement and mixed fv formulations are presented for static and dynamic problems the finite volume method fvm evolved in the early seventies via finite difference approxima tions on non orthogonal grids the popularity of the fvm has been extensive in the field of computational fluid dynamics cfd and heat transfer on the contrary in the field of computational solid mechanics csm the use of the fvm has never achieved such acceptance an early attempt to use fv concepts in csm is due to wilkins6 as an alternative approximation to derivatives in a cell in this he defines the average gradient of a function u in a volume l2 as using the well known divergence theorem such a definition of gradients can be written entirely in terms of function values at the boundary of a volume and has been used in the early hydrocodes of the lawrence livermore laboratory the reasons for the unpopularity of the fvm in structural mechanics is understandable as finite volumes are well known to be less accurate than galerkin based finite elements for self adjoint elliptic problems a comparison between fvm and fem has been recently pre sented by zienkiewicz and oiiate here the authors show that fvm and fem share concepts such as mesh discretization and interpolation giving precisely the same discretized systems of equations for some particular cases this coincidence also shows clearly for 2 d and 3 d structural problems as detailed in this work here however surface integrals are mostly involved and the number of computations can be shown to be proportional to the number of sides in the mesh this leads to an overall solution cost very similar to that of fe computations note that for a refined mesh of three node triangles the number of sides is 1 5 times that of elements this fact suggests that computational speed is not a prior one of the keys to the possible success of fvm in structural problems however the possibility of obtaining the element matrices and vectors in terms of computations along the element sides opens new possibilities for the solution of some structural problems which in the authors opinion may be worth exploring in detail indeed it

classical solutions for the ellipsoidal bgk model with fixed collision frequency abstract we establish the existence of global in time smooth solutions for the ellipsoidal bgk model which is a variant of the bgk model for the boltzmann equation designed to yield the correct prandtl number in the hydrodynamic approximation at the navier stokes level for this we carefully design a function space which captures the growth of the solution in a weighted sobolev norm and show that the ellipsoidal relaxation operator is lipschitz continuous in the induced metric this approach is restricted to the case when the collision frequency does not depend on the macroscopic field but no smallness on the initial data is required

nonlinearly coupled oscillators in quantum mechanics a normal form approach quantum mechanical hamiltonians of the form of n harmonic oscillators coupled via an interaction of the form e times a polynomial in the position and momenta variables are studied in a rigorous hilbert space setting in particular normal form theory is used to define the mth approximation to the associated schrodinger initial value problem and it is shown that it deviates in norm from the exact solution by a term of order em 1 t t time provided only that the initial vector is confined to an appropriate dense subspace the main concentration is on the case in which there exist no resonances of order m between the frequencies of the n oscillators but the case of two oscillators in 1 1 resonance is also taken up

normal form and representation theory representation theory of lie algebras is called upon to develop a procedure for normalizing a dynamical system with two degrees of freedom in the neighborhood of an equilibrium when the hamiltonian h x y x y in the coordinates x y and their conjugate momenta x y is of the type h x2 y2 2 v x y x y the potential energy v being a sum of homogeneous polynomials in the phase variables of degree strictly greater than 2 the fact that the resulting potential v is a polynomial in the new coordinates x y and the angular momentum g x y y x implies that the normalization is a rotation in the configuration space from a fixed frame to an ideal frame the technique is intended for normalizing an hamiltonian in equilibrium at the origin when the lie derivative associated with the quadratic part is not semisimple e g the planar restricted problem of three bodies at the equilateral equilibrium l4 when the basic frequencies are equal routh s singular case

weighted l bounds and uniqueness for the boltzmann bgk model we prove rather general l bounds for hydrodynamical fields in terms of weighted l norms of the kinetic density we use these estimates to prove l bounds and uniqueness for the solution of the bgk equation which is a simple relaxation model introduced by bhatnagar gross krook to mimic boltzmann flows

singular continuous spectrum under rank one perturbations and localization for random hamiltonians we consider a selfadjoint operator a and a selfadjoint rank one projection p onto a vector which is cyclic for a in terms of the spectral measure d a we give necessary and sufficient conditions for a p to have empty singular continuous spectrum or to have only point spectrum for a e we apply these results to questions of localization in the one and multi dimensional anderson models

proof of dynamical localization for perturbations of discrete 1d schrodinger operators with uniform electric fields we present a proof of discrete spectrum and dynamical localization for small perturbations of discrete one dimensional schrodinger operators with uniform electric fields the proof of dynamical localization is based on the kam technique

an edge based finite volume scheme for saturated unsaturated groundwater flow abstract the application of a computationally efficient finite volume method for modelling groundwater flow in saturated unsaturated soils is presented the flow equation is solved by applying a vertex centred edge based finite volume scheme and the discretisation of the flow domain is achieved using structured and unstructured grids an acceleration algorithm is employed which utilises an explicit non linear loop for the primary variable update the scheme is employed to solve a two dimensional groundwater flow problem and the results obtained are compared against published solutions it is shown that the edge based finite volume scheme captures the saturation front more efficiently than the published finite element solution comput geosci 27 2 2001 179 moreover an improvement in the solution is shown by the application of a compact stencil scheme to calculate second derivative terms as compared to the standard edge based finite volume scheme

finite volumes and finite elements two good friends in this paper a comparison between the finite element and the finite volume methods is presented in the context of elliptic convective diffusion and fluid flow problems the paper shows that both procedures share a number of features like mesh discretization and approximation moreover it is shown that in many cases both techniques are completely equivalent

a reduction theorem for time dependent schrodinger operator with finite differentiable unbounded perturbation in this paper we consider the time dependent schrodinger operator with a temporal quasi periodic perturbation which is some function of class cl for any l 100 3n 2 1 by using the kam technique we establish a reduction theorem for the above operator and prove that the invariant tori of the nearly integrable hamiltonian system with unbounded small perturbation are linearly stable both for d d 1 and the limiting case d d 1

non linear diffusions as limit of kinetic equations with relaxation collision kernels kinetic transport equations with a given confining potential and non linear relaxation type collision operators are considered general monotone energy dependent equilibrium distributions are allowed with a chemical potential ensuring mass conservation existence and uniqueness of solutions is proved for initial data bounded by equilibrium distributions the diffusive macroscopic limit is carried out using compensated compactness theory the results are drift diffusion equations with non linear diffusion the most notable examples are of the form partial t rho nabla cdot nabla rho m rho nabla v ranging from porous medium equations to fast diffusion with the exponent satisfying 0 m 5 3 in mathbb r 3

fractional diffusion limit for collisional kinetic equations this paper is devoted to diffusion limits of linear boltzmann equations when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion equation in this paper we consider situations in which the equilibrium distribution function is a heavy tailed distribution with infinite variance we then show that for an appropriate time scale the small mean free path limit gives rise to a fractional diffusion equation

birkhoff gustavson normal form in classical and quantum mechanics a quantum analogue of the classical transformations to the birkhoff gustavson normal form is derived with unitary transformations taking over the role of canonical transformations the author finds a striking similarity to lie transformations in the classical case the author suggests that the quantum normal form is identical to the rayleigh schrodinger perturbation series and thus usually only asymptotic to the eigenvalues the convergence question of both classical and quantum normal forms is discussed in some detail

floquet hamiltonians with pure point spectrum we consider floquet hamiltonians of the type k f i partial t h 0 beta v omega t whereh0 a selfadjoint operator acting in a hilbert space has simple discrete spectrume1 0 for a given 0 t v t is 2 periodic andr times strongly continuously differentiable as a bounded operator on and are real parameters and the periodic boundary condition is imposed in time we show roughly that providedr is large enough small enough and non resonant then the spectrum ofkf is pure point the method we use relies on a successive application of the adiabatic treatment due to howland and the kam type iteration settled by bellissard and extended by combescure both tools are revisited adjusted and at some points slightly simplified

quantum averaging for driven systems with resonances abstract we discuss the effects of resonances in driven quantum systems within the context of quantum averaging techniques in the floquet representation we consider in particular iterative methods of kam type and the extensions needed to take into account resonances the approach consists in separating the coupling terms into resonant and nonresonant components at a given scale of time and intensity the nonresonant part can be treated with perturbative techniques which we formulate in terms of kam type unitary transformations that are close to the identity these can be interpreted as averaging procedures with respect to the dynamics defined by effective uncoupled hamiltonians the resonant parts are treated with a different kind of unitary transfomations that are not close to the identity and are adapted to the structure of the resonances they can be interpreted as a renormalization of the uncoupled hamiltonian that yields an effective dressed hamiltonian around which a perturbation expansion can be developed the combination of these two ingredients provides a strongly improved approximation technique

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

quantum averaging iii time dependent systems we present new perturbation algorithms for time dependent quantum systems these new methods make use of a time dependent averaging method and are based on analogies with the poincare von zeipel pvz and kolmogorov arnold moser kam expansions application to a simple example shows that the pvz as well as the kam expansion are superior to the standard quantum mechanical time dependent perturbation method dyson expansion

convergence of a weighted particle method for solving the boltzmann b g k equation we consider in this paper the numerical solution of the b g k model for the boltzmann equation the numerical method used here was introduced by mas gallic transport theory statist phys 16 1987 pp 885 887 using the results of perthame and pulvirenti arch rational mech anal 125 1993 pp 289 295 we first derive a uniqueness result for the b g k model and also regularity properties of the solution which are necessary for our analysis we then prove following mas gallic and poupaud transport theory statist phys 17 1988 pp 311 345 the convergence of the numerical approximation for the b g k model finally some numerical experiments are presented

reducibility for wave equations of finitely smooth potential with periodic boundary conditions abstract in the present paper the reducibility is derived for the wave equations with finitely smooth and time quasi periodic potential subject to periodic boundary conditions more exactly the linear wave equation u t t u x x m u e v 0 t u x x v t x u 0 x r 2 z can be reduced to a linear hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set where v is finitely smooth in t x quasi periodic in time t with diophantine frequency r n and v 0 is finitely smooth and quasi periodic in time t with diophantine frequency r n moreover it is proved that the corresponding wave operator possesses the property of pure point spectra and zero lyapunov exponent

a finite point method in computational mechanics applications to convective transport and fluid flow the paper presents a fully meshless procedure fo solving partial differential equations the approach termed generically the finite point method is based on a weighted least square interpolation of point data and point collocation for evaluating the approximation integrals some examples showing the accuracy of the method for solution of adjoint and non self adjoint equations typical of convective diffusive transport and also to the analysis of compressible fluid mechanics problem are presented

method of averaging and the quantum anharmonic oscillator the krylov bogoliubov method of averaging is applied to the time dependent quantum anharmonic oscillator a regular perturbation expansion contains secular terms the averaging approximation does not and as a result has a validity over larger time intervals a new variant of the usual averaging transformation is used and rigorous error bounds are derived rigorous averaging methods have been applied extensively to ordinary differential equations but our work appears to be the first generalization to partial differential equations

quantum averaging ii kolmogorov s algorithm a complete analogue of kolmogorov s perturbation algorithm in classical mechanics is presented for perturbations of self adjoint operators the resulting perturbation theory is different from the usual rayleigh schr dinger or kato rellich perturbation theory

on the sign of the second order energy shift in the rayleigh schrodinger perturbation theory for a highly excited state the second order energy shift in the rayleigh schrodinger perturbation theory is most likely to be positive for large quantum numbers i e for excited states within the wilson sommerfeld approximation good for large quantum numbers we show on the contrary that the sufficient condition for the second order shift to be negative is that the total potential be symmetric and both the unperturbed and perturbed potentials be monotonically increasing

discretisation procedures for multi physics phenomena procedures are described for solving the equations governing a multi physics process finite volume techniques are used to discretise using the same unstructured mesh the equations of fluid flow heat transfer with solidification and solid deformation these discretised equations are then solved in an integrated manner the computational mechanics environment physica which facilitates the building of multi physics models is described comparisons between model predictions and experimental data are presented for the casting of metal components

geometrically nonlinear formulation for thin shells without rotation degrees of freedom abstract a finite element formulation is presented for modelling geometrically nonlinear thin shells which exploits standard lagrange finite element basis functions without introducing rotation degrees of freedom the classical regularity requirements associated with thin bending problems are circumvented by introducing special integrals over inter element edges the use of lagrange finite element basis functions and the absence of rotation degrees of freedom make the formulation relatively simple and discontinuities in material properties and non smooth shell geometry can be incorporated trivially the variational problem can be exactly linearised leading to an efficient newton raphson solution process the performance of the approach is demonstrated via a range of numerical benchmarks for both geometrically linear and nonlinear problems it is shown that cubic elements perform particularly well

global existence to the bgk model of boltzmann equation abstract we present an existence and a stability proof for solutions to the bgk model of boltzmann equation f v x f f m f t 0 x r n v r n m f p 2 t n 2 exp v u 2 2t p pu p u 2 pt 1 x r n 1 v v 2 f t x v dv it relies on the strong compactness of u t and on a new estimate on the third moment of v dv we also prove the entropy relation for 1

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

singular continuous floquet operator for systems with increasing gaps abstract consider the floquet operator of a time independent quantum system periodically perturbed by a rank one kick acting on a separable hilbert space e ih0te i t where t and are the period and the coupling constant respectively assume the spectrum of the self adjoint operator h0 is pure point simple bounded from below and the gaps between the eigenvalues n grow like n 1 n cnd with d 2 under some hypotheses on the arithmetical nature of the eigenvalues and the vector cyclic for h0 we prove the floquet operator of the perturbed system has purely singular continuous spectrum

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

cauchy problem for the ellipsoidal bgk model for polyatomic particles abstract we establish the existence and uniqueness of mild solutions for the polyatomic ellipsoidal bgk model which is a relaxation type kinetic model describing the evolution of polyatomic gaseous system at the mesoscopic level

time dependent normal form theory and schrodinger initial value problems time dependent normal form theory is introduced and studied in the context of schrodinger initial value problems associated with hamiltonians having time dependent perturbations to arrive at approximations that deviate in norm from the exact solution by a term of order em 1 1 t provided the initial vector is restricted to an appropriate dense subspace in addition to the abstract theory within a rigorous hilbert space formalism it is applied to the quantum mechanical hamiltonians of the form n harmonic oscillators coupled via an interaction of the form e times sum of products of bounded continuous functions with polynomials in the position and momenta variables satisfying a nonresonance condition in particular first order approximation is explicitly constructed for time evolution of a system describing anharmonic oscillator which changes trigonometrically with time

a weak weak formulation for large displacements beam statics a finite volumes approximation this paper addresses the problem of the numerical solution of beam statics undergoing large displacements a kinematic analysis outlines the beam geometrical model through the definition of its lagrangian co ordinate and strain parameters a definition of the stress parameters a constitutive law and an expression for the strain energy of the beam are then provided under the hypothesis of small strain the equations governing the beam equilibrium are introduced and their weak form is derived these equations are then proved to be equivalent to the primal and mixed form of principle of virtual work the numerical approximation is introduced by applying the bidiscontinuous finite elements method on the linearized weak form the weak weak formulation is attained by using the lowest interpolation order both for test and trial functions on two staggered decompositions of the space domain some numerical examples prove the capability of present formulation in handling actual problems

long time approximation to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics a simple normal form approach is used to obtain uniform long time approximations to the evolution of a resonant or nonresonant anharmonic oscillator system governed by a hamiltonian he which is a self adjoint operator acting in the hilbert space h l2 r 2 and is given formally by h0 ev here h0 denotes the hamiltonian of one dimensional harmonic oscillators whose coupling is represented by ev where v is an operator of multiplication by a smooth function of at most polynomial growth at infinity and e 0 is a small parameter we consider the general situation in which 1 of the frequencies of these oscillators are rationally independent imposing a standard diophantine condition on the independent frequencies if 2 under these assumptions which are stated in a mathematically precise way in the paper an nth order approximant n t e to the exact solution t e of the schrodinger equation id t e dt he t e satisfying the initial condition 0 e 0 is constructed inductively 0 being an arbi

lie series approach to the evolution of resonant and nonresonant anharmonic oscillators in quantum mechanics a version of a normal form method of kummer and gompa analogous to the lie series methods of classical mechanics is used to rigorously construct uniform long time approximations to the quantum evolution of a system of resonant or nonresonant anharmonic oscillators the hamiltonian he governing the system is a self adjoint operator in the hilbert space h l2 r 2 given formally by h0 ev here h0 denotes the hamiltonian of one dimensional simple harmonic oscillators and ev their anharmonic coupling e 0 being a small parameter and v an operator of multiplication by a smooth function of polynomial growth at infinity we treat the case in which an arbitrary number 1 of the frequencies of these oscillators are rationally independent imposing a standard diophantine condition on the independent frequencies if 2 under these assumptions stated precisely in the paper an nth order approximant n t e to the exact solution t e of the schrodinger equation id t e dt he t e satisfying the initi

a note concerning quantum integrability heuristic arguments are presented supporting the conjecture that almost all quantum hamiltonians are integrable in the sense that there exist n n number of freedoms mutually commuting observables which in analogy with the classical action variables can be chosen to be the number operators this follows from perturbational considerations the series may converge for almost all perturbations that preserve the discreteness of the spectra because a quantum small denominator almost always uniformly satisfies the condition of sufficient irrationality the radius of convergence vanishes if h cross 0 the classical limit as h cross to 0 of the quantum integrals of motion generically does not exist

spectrum for quantum duffing oscillator and small divisor equation with large variable coefficient by establishing a new estimate of the solutions for the small divisor equation with large variable coefficients and using the kam technique we prove a reduction theorem that entails the pure point nature of the floquet spectrum of the quantum duffing oscillator with a small perturbation temporal quasi periodic with nonresonant frequencies 2010 wiley periodicals inc

pure point spectrum of the floquet hamiltonian for the quantum harmonic oscillator under time quasi periodic perturbations we prove that the 1 d quantum harmonic oscillator is stable under spatially localized time quasi periodic perturbations on a set of diophantine frequencies of positive measure this proves a conjecture raised by enss veselic in their 1983 paper ev in the general quasi periodic setting the motivation of the present paper also comes from construction of quasi periodic solutions for the corresponding nonlinear equation

a stabilized finite point method for analysis of fluid mechanics problems in this paper a meshless procedure termed the finite point method for solving convection diffusion and fluid flow type problems is presented the method is based on the use of a weighted least square interpolation procedure together with point collocation for evaluating the approximation integrals special emphasis is given to the stabilization of the convective terms and the neumann boundary condition which has been found to be essential to obtain accurate results some examples of application to diffusive and convective transport and compressible flow problems using quadratic fp interpolations are presented

a novel pressure velocity formulation and solution method for fluid structure interaction problems in the standard approach for simulating fluid structure interaction problems the solution of the set of equations for solids provides the three displacement components while the solution of equations for fluids provides the three velocity components and pressure in the present paper a novel reformulation of the elastodynamic equations for hookean solids is proposed so that they contain the same unknowns as the navier stokes equations namely velocities and pressure a separate equation for pressure correction is derived from the constitutive equation of the solid material the system of equations for both media is discretised using the same method finite volume on collocated grids and the same iterative technique simple algorithm is employed for the pressure velocity coupling with this approach the continuity of the velocity field at the interface is automatically satisfied a special pressure correction procedure that enforces the compatibility of stresses at the interface is also developed the new method is employed for the prediction of pressure wave propagation in an elastic tube computations were carried out with different meshes and time steps and compared with available analytic solutions as well as with numerical results obtained using the flugge equations that describe the deformation of thin shells for all cases examined the method showed very good performance

lp solutions to the cauchy problem of the bgk equation the bgk model of the boltzmann equation plays an important role in the kinetic theory of rarefied gases some existence and uniqueness results of solutions to its cauchy problem were established for large initial data under various circumstances see for example perthame b global existence to the bgk model of boltzmann equation j differ equations 82 191 205 1989 in this paper by establishing some weighted lp estimates of the hydrodynamical quantities of a gas we prove the existence theorem of the lp solutions to the cauchy problem and establish the propagation properties of the lp moments for this kind of solutions

absolute continuity of the floquet spectrum for a nonlinearly forced harmonic oscillator we prove that the floquet spectrum of the time periodic schrodinger equation corresponding to a mildly nonlinear resonant forcing is purely absolutely continuous for suitably small

time quasi periodic unbounded perturbations of schrodinger operators and kam methods we eliminate by kam methods the time dependence in a class of linear differential equations in l2 subject to an unbounded quasi periodic forcing this entails the pure point nature of the floquet spectrum of the operator h0 ep t for e small here h0 is the one dimensional schrodinger operator p2 v v x x 2 for x the time quasi periodic perturbation p may grow as x 2 2 and the frequency vector is non resonant the proof extends to infinite dimensional spaces the result valid for quasiperiodically forced linear differential equations and is based on kuksin s estimate of solutions of homological equations with non constant coefficients