tunnelling in multidimensional systems tunnelling is examined from the point of view of analytic continuation of lagrangian manifolds this enables an intrinsically multidimensional approach to many problems we illustrate the ideas by computing energy splittings between congruent tori in a number of integrable systems the examples include an integrable but non separable system which cannot be solved by reduction to one dimensional systems the calculations can all be viewed in a geometrical way that does not make explicit use of separation of coordinates

local projection stabilization for the oseen problem and its interpretation as a variational multiscale method we propose to apply the recently introduced local projection stabilization to the numerical computation of the oseen equation at high reynolds number the discretization is done by nested finite element spaces using a priori error estimation techniques we prove the convergence of the method the a priori estimates are independent of the local peclet number and give a sufficient condition for the size of the stabilization parameters in order to ensure optimality of the approximation when the exact solution is smooth moreover we show how this method may be cast in the framework of variational multiscale methods we indicate what modeling assumptions must be made to use the method for large eddy simulations

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

polynomial poisson algebras for classical superintegrable systems with a third order integral of motion we present polynomial poisson algebras for the eight classical potentials in two dimensional euclidian space that separate in cartesian coordinates and allow a third order integral of motion some of the classical superintegrable potentials do not coincide with quantum ones but are their singular limits we show that all bounded trajectories in these potentials are periodic

golod shafarevich algebras free subalgebras and noetherian images abstract it is shown that golod shafarevich algebras of a reduced number of defining relations contain noncommutative free subalgebras in two generators and that these algebras can be homomorphically mapped onto prime noetherian algebras with linear growth it is also shown that golod shafarevich algebras of a reduced number of relations cannot be nil

finite element error analysis for a projection based variational multiscale method with nonlinear eddy viscosity the paper presents a finite element error analysis for a projection based variational multiscale vms method for the incompressible navier stokes equations in the vms method the influence of the unresolved scales onto the resolved small scales is modeled by a smagorinsky type turbulent viscosity

general nth order integrals of motion in the euclidean plane the general form of an integral of motion that is a polynomial of order n in the momenta is presented for a hamiltonian system in two dimensional euclidean space the classical and the quantum cases are treated separately emphasizing both the similarities and the differences between the two the main application will be to study nth order superintegrable systems that allow separation of variables in the hamilton jacobi and schrodinger equations respectively

non commutative graded algebras and their hilbert series abstract this paper begins to develop a theory of non commutative graded algebras and their hilbert series which parallels the extensive already existing theory on commutative graded algebras we begin with a discussion of order ideals of monomials which give us vector space bases for these algebras we develop next the concept of weak summand which is a special relationship a subalgebra may have to an algebra and the concept of strongly free sets strongly free sets have many properties which are analogous to the properties of regular sequences in commutative algebras we discuss combinatorially free sets which are useful for constructing examples of many of the preceding ideas we conclude with an application of our results to algebraic topology by showing that there is a finite cw complex with only nine positive dimensional cells whose loop space has an irrational poincare series

integrable heterogeneous nonlinear schrodinger equations with dielectric loss lie backlund symmetries the general cubic nonlinear schrodinger equation with heterogeneity and dielectric loss is i t 1 2 xx v x t iu x t r x t 2 all such equations are found that possess a third order lie backlund symmetry each of these equations are then transformed to the standard completely integrable cubic schrodinger equation u v 0 r 1 in addition some useful structure results are established for the time dependent symmetries of the linear 2 1 dimensional schrodinger equation with arbitrary real potential

the dissipative structure of variational multiscale methods for incompressible flows in this paper we present a precise definition of the numerical dissipation for the orthogonal projection version of the variational multiscale method for incompressible flows we show that only if the space of subscales is taken orthogonal to the finite element space this definition is physically reasonable as the coarse and fine scales are properly separated then we compare the diffusion introduced by the numerical discretization of the problem with the diffusion introduced by a large eddy simulation model results for the flow around a surface mounted obstacle problem show that numerical dissipation is of the same order as the subgrid dissipation introduced by the smagorinsky model finally when transient subscales are considered the model is able to predict backscatter something that is only possible when dynamic les closures are used numerical evidence supporting this point is also presented

analysis of a variational multiscale method for large eddy simulation and its application to homogeneous isotropic turbulence abstract in this paper a variational multiscale method based on local projection and grad div stabilization for large eddy simulation for the incompressible navier stokes problem is considered an a priori error estimate is given for a case with rather general nonlinear piecewise constant coefficients of the subgrid models for the unresolved scales of velocity and pressure then the design of the subgrid scale models is specified for the case of homogeneous isotropic turbulence and studied for the standard benchmark problem of decaying homogeneous isotropic turbulence

explicitly uncoupled vms stabilization of fluid flow abstract variational multiscale methods have proven to be an accurate and systematic approach to the simulation of turbulent flows many turbulent flows are solved by legacy codes or by ones written by a team of programmers and of great complexity so implementing a new approach to turbulence in such cases can be daunting we propose a new approach to inducing a vms treatment of turbulence in such cases the method adds a separate uncoupled and modular postprocessing step to each time step adding this step requires the ability to solve a stokes problem either on the same mesh or to solve and interpolate between the postprocessing step s mesh and the code s mesh we prove stability and convergence for the combination and quantify the vms dissipation induced numerical experiments confirming the theory are given in particular the performance of the two step modular vms method is comparable to a monolithic fully coupled vms method for the benchmark problem of decaying homogeneous turbulence

free subalgebras of graded algebras let k be a field and let a n 1an be a positively graded k algebra we recall that a is graded nilpotent if for every d 1 the subalgebra of a generated by elements of degree d is nilpotent we give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k subalgebra on two generators

optimal 5 step nilpotent quadratic algebras by the golod shafarevich theorem an associative algebra r given by n generators and d n23 homogeneous quadratic relations is not 5 step nilpotent we prove that this estimate is optimal namely we show that for every positive integer n there is an algebra r given by n generators and n23 homogeneous quadratic relations such that r is 5 step nilpotent

a connection between subgrid scale eddy viscosity and mixed methods we consider a new mixed method related to the evss method in computational viscoelasticity for the convection dominated convection diffusion equation in which stabilization is added then removed through the extra mixed variables this consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in u h by suitable choice of the mixed spaces a method of guermond is recovered exactly we show that for a different natural choice a new method results with global error estimates similar to both guermond s method and the streamline diffusion supg method

integrability of hamiltonians with third and fourth degree polynomial potentials the weak painleve property as a criterion of integrability is applied to the case of simple hamiltonians describing the motion of a particle in two dimensional polynomial potentials of degree three and four this allows a complete identification of all the integrable cases of cubic potentials in the case of quartic potentials although our results are not exhaustive some new integrable cases are discovered in both cases the integrability is explicited by a direct calculation of the second integral of motion of the system

an infinite dimensional affine nil algebra with finite gelfand kirillov dimension the famous 1960 s construction of golod and shafarevich yields infinite dimensional nil but not nilpotent algebras however these algebras have exponential growth here we construct an infinite dimensional nil but not locally nilpotent algebra which has polynomially bounded growth

golod shafarevich groups with property t and kac moody groups we construct golod shafarevich groups with property t and thus provide counterexamples to a conjecture stated in a recent paper of zelmanov ze2 explicit examples of such groups are given by lattices in certain topological kac moody groups over finite fields we provide several applications of this result including examples of residually finite torsion non amenable groups

necessary conditions for classical super integrability of a certain family of potentials in constant curvature spaces we formulate the necessary conditions for the maximal super integrability of a certain family of classical potentials defined in the constant curvature two dimensional spaces we give examples of homogeneous potentials of degree 2 on as well as their equivalents on and for which these necessary conditions are also sufficient we show explicit forms of the additional first integrals which can always be chosen as a polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

a new projection based stabilized method for steady convection dominated convection diffusion equations abstract we propose a new projection based stabilized method for the steady convection dominated convection diffusion equations this method directly increases the l 2 stability instead of the h 1 stability which the variational multiscale vms methods do numerical analysis and numerical experiments illustrate and confirm that our new method has effective numerical performances on convection dominated convection diffusion problems

quasiseparation of variables in the schrodinger equation with a magnetic field we consider a two dimensional integrable hamiltonian system with a vector and scalar potential in quantum mechanics contrary to the case of a pure scalar potential the existence of a second order integral of motion does not guarantee the separation of variables in the schrodinger equation we introduce the concept of quasiseparation of variables and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra

the proper quantum analogue of the birkhoff gustavson method of normal forms a deformation approach resolves problems concerning the ordering of operators and the commutation of quantization with canonical transformation this completes the programme of robnik 1984 to develop a purely algebraic method for the quantization of non integrable hamiltonian systems about a point of stable equilibrium

on a classical constrained system and its quantum mechanical counterpart the authors discuss a nonintegrable system of three coupled anharmonic oscillators with certain constraints so that it is transformed into an integrable one after having quantized the system with constraints and the classically equivalent unconstrained integrable system using the bfv algorithm and the ebk quantization method respectively the equivalence of the two quantization procedures is shown by explicitly performing the reduction to the physical subspace on the quantum level

a variational multiscale method for turbulent flow simulation with adaptive large scale space in turbulent flows it is only feasible to simulate large flow structures variational multiscale vms methods define these flow structures by projections onto appropriate function spaces this paper presents a finite element vms method which chooses the large scale projection space adaptively the adaption controls the influence of an eddy viscosity model and it is based on the size of the so called resolved small scales the adaptive procedure is described in detail numerical studies at a turbulent channel flow and a turbulent flow around a cylinder are presented it is shown that the method selects the large scale space in a reasonable way and that appropriately chosen parameters improve the results compared to the basic method which uses the same local large scale space in the whole domain and for all times

computational error analysis of a discontinuous galerkin discretization applied to large eddy simulation of homogeneous turbulence a computational error assessment of large eddy simulation les in combination with a discontinuous galerkin finite element method is presented for homogeneous isotropic decaying turbulence the error landscape database approach is used to quantify the total simulation error that arises from the use of the smagorinsky eddy viscosity model in combination with the galerkin discretization we adopt a modified hllc flux allowing an explicit control over the dissipative component of the numerical flux the optimal dependence of the smagorinsky parameter on the spatial resolution is determined for second and third order accurate galerkin methods in particular the role of the numerical dissipation relative to the contribution from the smagorinsky dissipation is investigated we observed an exchange of dissipation principle in the sense that an increased numerical dissipation implied a reduction in the optimal smagorinsky parameter the predictions based on galerkin discretization with fully stabilized hllc flux were found to be less accurate than when a central discretization with mainly smagorinsky dissipation was used this was observed for both the second and third order galerkin discretization suggesting to emphasize central discretization of the convective nonlinearity and stabilization that mimics eddy viscosity as sub filter dissipation

error analysis and iterative solvers for navier stokes projection methods with standard and sparse grad div stabilization abstract this paper shows that use of a recently introduced sparse grad div stabilization can increase the accuracy of projection methods for solving the navier stokes equations sparse grad div stabilization has recently been introduced as an alternative to standard grad div stabilization which has a sparser matrix representation for both sparse and standard grad div stabilized projection methods we prove error estimates and provide numerical experiments which reveal that both stabilizations can cause a significant decrease in the error we then compare iterative solvers for the linear systems of equations arising from the use of both of the stabilizations a theoretical analysis of a simplified model problem as well as numerical tests show that iterative solvers perform better for systems arising from sparse grad div compared to standard grad div stabilized systems

free subalgebras of division algebras over uncountable fields we study the existence of free subalgebras in division algebras and prove the following general result if a is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0 then either the quotient division algebra of a contains a free algebra on two generators or it is left algebraic over every maximal subfield as an application we prove that if k is an uncountable algebraically closed field and a is a finitely generated k algebra that is a domain of gk dimension strictly less than 3 then either a satisfies a polynomial identity or the quotient division algebra of a contains a free k algebra on two generators

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

random matrix theory in semiclassical quantum mechanics of chaotic systems the statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles the author explains this correspondence including the fact that long ranged spectral statistics show deviations from the results of random matrix theory the method depends on the ambiguity of quantisation of a given classical system although the energy levels depend on the particular quantisation used the spectral statistics are assumed to depend only on the classical motion the ambiguity of quantisation can be represented by a small perturbation acting on the hamiltonian this perturbation executes a random walk which causes its matrix elements to undergo a random walk if the matrix elements are completely uncorrelated the spectral statistics are those of random matrix theory in the case of classically chaotic systems the matrix elements are constrained by sum rules related to classical periodic orbits this leads to deviations of the long ranged spectral statistics from the predictions of random matrix theory

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

classical versus quantum integrability classical integrability and quantum integrability are compared for two degrees of freedom hamiltonian systems we use c number representatives for quantum operators and the moyal bracket for the commutator three different cases are found i the c number representative of the quantum mechanical second invariant is identical to the classical second invariant ii o 2 corrections are needed in the classical second invariant to obtain the quantum invariant and iii also the potential must be deformed by an o 2 term several examples from the henon heiles and holt families of integrable potentials are included

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

integrable and superintegrable quantum systems in a magnetic field integrable quantum mechanical systems with magnetic fields are constructed in two dimensional euclidean space the integral of motion is assumed to be a first or second order hermitian operator contrary to the case of purely scalar potentials quadratic integrability does not imply the separation of variables in the schrodinger equation moreover quantum and classical integrable systems do not necessarily coincide the hamiltonian can depend on the planck constant in a nontrivial manner

construction of new integrable hamiltonians in two degrees of freedom a new procedure for deriving integrable hamiltonians and their constants of the motion is introduced we term this procedure the truncation program integrable hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter e the relationship between this program and the whittaker program in two degrees of freedom is discussed integrable hamiltonians occurring in the whittaker program a generalization of whittaker s work possess constants of the motion which are polynomials in the momentum coordinates many previously known integrable hamiltonians are derived a new family of integrable double resonance hamiltonians and a new family of integrable hamiltonians of the form p21 p22 2 v q1 q2 are derived

a finite element filtered eddy viscosity method for the navier stokes equations with large reynolds number fluid turbulence is commonly modeled by the navier stokes equations with a large reynolds number the simulation of turbulence model is known to be very difficult we study artificial spectral viscosity models that render the simulation of turbulence tractable the models introduce several parameters we show that the models have solutions that converge in certain parameter limits to solutions of the navier stokes equations we also show using the mathematical analysis how effective choice for the parameter can be made the direct computational simulation of turbulence flow is a formidable task due to the disparate scales that have to be resolved turbulence modeling attempts to mitigate this situation by accounting for the effects of small scale behavior on that at large scales without explicitly esolving the small scales one such approach is to add viscosity to the problem the smagorin sky and ladyzhenskaya models and other eddy viscosity models are examples of this approach unfortunately this approach usually results in over dampening at the large scales i e large scale structures are unphysically smeared out to mitigate this fault of eddy viscosity modeling filtered eddy viscosity methods that add the artificial viscosity only to the high frequency modes were developed in the context of spectral methods we apply the filtered eddy viscosity idea to finite element methods with hierarchical basis functions we prove the existence and uniqueness of the finite element approximation and its convergence to a weak solution of the navier stokes system

painleve conjecture revisited the discovery of new integrable two dimensional hamiltonian systems is reported the analytic structure of the solutions makes necessary the generalization of the painlev e conjecture a widely used integrability criterion such a generalization is presented which the authors believe should replace the usual conjecture for two dimensional hamiltonian systems it is indeed compatible with all the systems already found and in addition leads to still new integrable cases