on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

universal vertex irf transformation for quantum affine algebras we construct a universal solution of the generalized coboundary equation in the case of quantum affine algebras which is an extension of our previous work to uq ar 1 this universal solution has a simple gauss decomposition which is constructed using sevostyanov s characters of twisted quantum borel algebras we show that in the evaluation representations it gives a vertex face transformation between a vertex type solution and a face type solution of the quantum dynamical yang baxter equation in particular in the evaluation representation of uq a1 1 it gives baxter s well known transformation between the 8 vertex model and the interaction round faces irf height model

poisson involutions spin calogero moser systems associated with symmetric lie subalgebras and the symmetric space spin ruijsenaars schneider models we develop a general scheme to construct integrable systems starting from realizations in symmetric coboundary dynamical lie algebroids and symmetric coboundary dynamical poisson groupoids the method is based on the successive use of dirac reduction and poisson reduction then we show that certain spin calogero moser systems associated with symmetric lie subalgebras can be studied in this fashion we also consider some spin generalized ruisjenaars schneider equations which correspond to the n soliton solutions of a 1 n affine toda field theory in this case we show how the equations are obtained from the dirac reduction of some hamiltonian system on a symmetric coboundary dynamical poisson groupoid

bounding zeros of h2 functions via concentrations it is well known that the zeros zj of a function in the classical hardy space h2 satisfy 1 zj however this sum can be arbitrarily large we bound this sum by a constant that depends on the concentration of the function a concept introduced by beauzamy and enflo

quasi bialgebras and dynamical r matrices abstract we study the relationship between general dynamical poisson groupoids and lie quasi bialgebras for a class of lie quasi bialgebras g naturally compatible with a reductive decomposition we extend the description of the moduli space of classical dynamical r matrices of etingof and schiffmann we construct in each gauge orbit an explicit analytic representative l can we translate the notion of duality for dynamical poisson groupoids into a duality for lie quasi bialgebras it is shown that duality maps the dynamical poisson groupoid for l can and g to the dynamical poisson groupoid for l can and the dual quasi bialgebra g

on error estimates of some higher order projection and penalty projection methods for navier stokes equations this paper is a continuation of our previous work 10 on projection methods we study first existing higher order projection schemes in the semidiscretized form for the navier stokes equations one error analysis suggests that the precision of these schemes is most likely plagued by the inconsistent neumann boundary condition satisfied by the pressure approximations we then propose a penalty projection scheme for which we obtain improved error estimates

triangular dynamical r matrices and quantization abstract we study some general aspects of triangular dynamical r matrices using poisson geometry we show that a triangular dynamical r matrix r h 2 g always gives rise to a regular poisson manifold using the fedosov method we prove that non degenerate triangular dynamical r matrices i e those such that the corresponding poisson manifolds are symplectic are quantizable and that the quantization is classified by the relative lie algebra cohomology h2 g h

new approach to the statistical properties of energy levels the joint distribution of energy eigenvalues of a hamiltonian is derived by means of the usual statistical laws of classical many body systems it makes a transition from the poisson type to the gaussian type depending on the value of a single parameter characteristic of the hamiltonian

surfaces and the sklyanin bracket we discuss the lie poisson group structures associated to splittings of the loop group lgl n due to sklyanin concentrating on the finite dimensional leaves of the associated poisson structure we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a invariant poisson structure in particular sklyanin s lie poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve the sklyanin structure is then equivalent to one considered by mukai tyurin and bottacin on a moduli space of sheaves on the poisson surface the abelianization procedure gives rise to natural darboux coordinates for these leaves as well as separation of variables for the integrable hamiltonian systems associated to invariant functions on the group

dynamical yang baxter equation and quantum vector bundles we develop a categorical approach to the dynamical yang baxter equation dybe for arbitrary hopf algebras in particular we introduce the notion of a dynamical extension of a monoidal category which provides a natural environment for quantum dynamical r matrices dynamical twists etc in this context we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits we build a dynamical twist for any pair of a reductive lie algebra and its levi subalgebra using this twist we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive lie groups

on the roots of determinants of a class of holomorphic matrix valued functions abstract we consider matrix valued functions that are holomorphic in the unit disk that are cauchy transforms of finite matrix valued measures for non zero roots z j of the determinants of such functions we provide estimates for z j 1 1 and z j 1 1 2 1 2

a family of hyperbolic spin calogero moser systems and the spin toda lattices in this paper we continue to develop a general scheme to study a broad class of integrable systems naturally associated with the coboundary dynamical lie algebroids in particular we present a factorization method for solving the hamiltonian flows we also present two important classes of new examples a family of hyperbolic spin calogero moser systems and the spin toda lattices to illustrate our factorization theory we show how to solve these hamiltonian systems explicitly 2004 wiley periodicals inc

classical r matrices and the feigin odesskii algebra via hamiltonian and poisson reductions we present a formula for a classical r matrix of an integrable system obtained by hamiltonian reduction of some free field theories using pure gauge symmetries the framework of the reduction is restricted only by the assumption that the respective gauge transformations are lie group ones our formula is in terms of dirac brackets and some new observations on these brackets are made we apply our method to derive a classical r matrix for the elliptic calogero moser system with spin starting from the higgs bundle over an elliptic curve with marked points in the paper we also derive a classical feigin odesskii algebra by a poisson reduction of some modification of the higgs bundle over an elliptic curve this allows us to include integrable lattice models in a hitchin type construction

the euler implicit explicit scheme for the 2d time dependent navier stokes equations with smooth or non smooth initial data this paper considers the stability and convergence results for the euler implicit explicit scheme applied to the spatially discretized two dimensional 2d time dependent navier stokes equations a galerkin finite element spatial discretization is assumed and the temporal treatment is implicit explict scheme which is implicit for the linear terms and explicit for the nonlinear term here the stability condition depends on the smoothness of the initial data u 0 h i e the time step condition is c 0 in the case of a 2 log h c 0 in the case of a 1 and th 2 c 0 in the case of a 0 for mesh size h and some positive constant c 0 we provide the h 2 stability of the scheme under the stability condition with a 0 1 2 and obtain the optimal h 1 l 2 error estimate of the numerical velocity and the optimal l 2 error estimate of the numerical pressure under the stability condition with a 1 2

solutions of the quantum dynamical yang baxter equation and dynamical quantum groups the quantum dynamical yang baxter qdyb equation is a useful generalization of the quantum yang baxter qyb equation this generalization was introduced by gervais neveu and felder unlike the qyb equation the qdyb equation is not an algebraic but a difference equation with respect to a matrix function rather than a matrix the qdyb equation and its quasiclassical analogue the classical dynamical yang baxter equation arise in several areas of mathematics and mathematical physics conformal field theory integrable systems representation theory the most interesting solution of the qdyb equation is the elliptic solution discovered by felder in this paper we prove the first classification results for solutions of the qdyb equation these results are parallel to the classification of solutions of the classical dynamical yang baxter equation obtained in our previous paper all solutions we found can be obtained from felder s elliptic solution by a limiting process and gauge transformations fifteen years ago the quantum yang baxter equation gave rise to the theory of quantum groups namely it turned out that the language of quantum groups hopf algebras is the adequate algebraic language to talk about solutions of the quantum yang baxter equation in this paper we propose a similar language originating from felder s ideas which we found to be adequate for the dynamical yang baxter equation this is the language of dynamical quantum groups or hopf algebroids which is the quantum counterpart of the language of dynamical poisson groupoids introduced in our previous paper

a class of integrable spin calogero moser systems we introduce a class of spin calogero moser systems associated with root systems of simple lie algebras and give the associated lax representations with spectral parameter and fundamental poisson bracket relations the associated integrable models called integrable spin calogero moser systems in the paper and their lax pairs are then obtained via poisson reduction and gauge transformations for lie algebras of a n type this new class of integrable systems includes the usual calogero moser systems as subsystems our method is guided by a general framework which we develop here using dynamical lie algebroids

intrinsic construction of invariant functions on simple lie algebras abstract an algorithm for constructing primitive adjoint invariant functions on a complex simple lie algebra is presented the construction is intrinsic in the sense that it does not resort to any representation a primitive invariant function on the whole lie algebra is obtained by lifting a coordinate function on a kostant slice of the lie algebra such an intrinsic construction of invariant functions is most useful for the bigger exceptional lie algebras such as the e s the maple implementation of this algorithm is outlined at the end and will be applied to these exceptional lie algebras in a future work

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

iterative penalty methods for the steady navier stokes equations abstract this paper presents one level and two level iterative penalty finite element methods to approximate the solutions of steady navier stokes equations first one level iterative penalty finite element method is applied to solve the steady navier stokes equations numerically and its h 1 and l 2 error estimates are derived then two level iterative penalty scheme is given and its error estimates are obtained for velocity and pressure finally the numerical results are displayed to verify the theoretical analysis

penalty finite element method for stokes problem with nonlinear slip boundary conditions abstract the penalty finite element method for stokes problem with nonlinear slip boundary conditions based on the finite element subspace v h m h which satisfies the discrete inf sup condition is investigated in this paper since this class of nonlinear slip boundary conditions include the subdifferential property the weak variational formulation associated with stokes problem is variational inequality under some regularity assumptions we obtain the optimal h 1 and l 2 error estimates between u and u h and between u and u h e where the error orders are e h for h 1 error and e h 2 for l 2 error

characteristic classes and hitchin systems general construction we consider topologically non trivial higgs g bundles over riemann surfaces g with marked points and the corresponding hitchin systems we show that if g is not simply connected then there exists a finite number of different sectors of the higgs bundles endowed with the hitchin hamiltonians they correspond to different characteristic classes of the underlying bundles defined as elements of h 2 sigma g mathcal z g mathcal z g is a center of g we define the conformal version cg of g an analog of gl n for sl n and relate the characteristic classes with degrees of cg bundles we describe explicitly bundles in the genus one g 1 case if 1 has one marked point and the bundles are trivial then the hitchin systems coincide with calogero moser cm systems for the nontrivial bundles we call the corresponding systems the modified calogero moser mcm systems their phase space has the same dimension as the phase space of the cm systems with spin variables but less number of particles and greater number of spin variables starting with these bundles we construct lax operators quadratic hamiltonians and define the phase spaces and the poisson structure using dynamical r matrices the latter are completion of the classification list of etingof varchenko corresponding to the trivial bundles to describe the systems we use a special basis in the lie algebras that generalizes the basis of t hooft matrices for sl n we find that the mcm systems contain the standard cm subsystems related to some unbroken subalgebras the configuration space of the cm particles is the moduli space of the stable holomorphic bundles with non trivial characteristic classes

quantum groupoids and dynamical categories abstract in this paper we realize the dynamical categories introduced in our previous paper as categories of modules over bialgebroids we study the bialgebroids arising in this way we define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to the dynamical categories we show that dynamical twists over an arbitrary base give rise to bialgebroid twists we prove that the classical dynamical r matrices over an arbitrary base manifold are in one to one correspondence with a special class of coboundary lie bialgebroids

the boundary penalty method for the diffusion equation subject to the specification of mass in this paper we concern with a new boundary penalty finite element method for the one dimensional time dependent diffusion equation subject to the specification of mass instead of solving the diffusion equation directly we will use the finite element method to solve a boundary penalty problem which is derived from the diffusion equation and analyse the error estimate of the semi discrete and fully discrete approximations when the penalty parameter e is sufficiently small the numerical experiments show that our method is very effective and supports our analysis

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

on error estimates of projection methods for navier stokes equations first order schemes in this paper projection methods or fractional step methods are studied in the semi discretized form for the navier stokes equations in a two or three dimensional bounded domain error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method a modified projection scheme which leads to improved error estimates is also proposed

concentration at a radius for hardy class functions abstract in this paper we establish the fundamental properties of concentration at a radius for functions in the classical hardy space on the unit disk for f z which is not identically zero and given r 0 r 1 the concentration is defined via the ratio of the norm of f r z to the norm of f z using the mahler measure of f z we obtain information on the distribution of the zeros of f z in terms of the concentration ratio in the last section of the paper we examine the sharpness of concentration estimates for the blaschke factor

liouville integrability of a class of integrable spin calogero moser systems and exponents of simple lie algebras in previous work we introduced a class of integrable spin calogero moser systems associated with the classical dynamical r matrices with spectral parameter as classified by etingof and varchenko for simple lie algebras here the main purpose is to establish the liouville integrability of these systems by a uniform method based on evaluating the primitive invariants of chevalley on the lax operators with spectral parameter as part of our analysis we will develop several results concerning the algebra of invariant polynomials on simple lie algebras and their expansions

analysis of locally stabilized mixed finite element methods for the stokes problem in this paper a locally stabilized finite element formulation of the stokes problem is analyzed a macroelement condition which is sufficient for the stability of locally stabilized mixed methods based on a piecewise constant pressure approximation is introduced by satisfying this condition the stability of the qi po quadrilateral and the pi po triangular element can be established

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

unconditional stability and error estimates of modified characteristics fems for the navier stokes equations the paper is concerned with the unconditional stability and convergence of characteristics type methods for the time dependent navier stokes equations we present optimal error estimates in l 2 l2 and h 1 h1 norms for a typical modified characteristics finite element method unconditionally while all previous works require certain time step restrictions the analysis is based on an iterated characteristic time discrete system with which the error function is split into a temporal error and a spatial error with a rigorous analysis to the characteristic time discrete system we prove that the difference between the numerical solution and the solution of the time discrete system is tau independent where tau denotes the time stepsize thus numerical solution in w 1 infty w1 is bounded and optimal error estimates can be obtained in a traditional way numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time dependent navier stokes equations the approach used in this paper can be easily extended to many other characteristics based methods

optimal error estimate of the penalty finite element method for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations the time discretization of the penalty navier stokes equations is based on the backward euler scheme the spatial discretization of the time discretized penalty navier stokes equations is based on a finite element space pair x h m h which satisfies some approximate assumption an optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e at and h are sufficiently small

iterative methods in penalty finite element discretization for the steady mhd equations this paper characterizes one penalty finite element method for the incompressible mhd equations the method is an interesting combination of the classic iterative schemes stokes newton and oseen iterations with two different finite element pairs p1b p1 p1b and p1 p0 p1 moreover the rigorous analysis of stability and error estimate for the proposed methods are given finally the applicability and effectiveness of the presented schemes are illustrated in several numerical experiments

error estimate of the p 1 nonconforming finite element method for the penalized unsteady navier stokes equations we consider a finite element method for the penalty formulation of the time dependent navier stokes equations usually the improper choice of the finite element space will lead that the error estimate inversely depends on the penalty parameter epsilon we use the classical p 1 nonconforming finite element space for the spatial discretization optimal epsilon uniform error estimations for both velocity and pressure are obtained

geometry and classificatin of solutions of the classical dynamical yang baxter equation the classical yang baxter equation cybe is an algebraic equation central in the theory of integrable systems its nondegenerate solutions were classified by belavin and drinfeld quantization of cybe led to the theory of quantum groups a geometric interpretation of cybe was given by drinfeld and gave rise to the theory of poisson lie groups

a penalty finite element method based on the euler implicit explicit scheme for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations where the time discretization is based on the euler implicit explicit scheme with some implicit linear terms and an explicit nonlinear term and the finite element spatial discretization is based on the p 1b p 1 element pair which satisfies the discrete inf sup condition this method allows us to separate the computation of the velocity from the computation of the pressure with a larger time step size dt so that the numerical velocity u e h n and the pressure p e h n are easily computed an optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter e the time step size dt and the mesh size h satisfy the following stability conditions ec 1 1 dt k 1 1 and h 2 b 1 dt respectively for some positive constants c 1 k 1 and b 1 finally some numerical tests to confirm the theoretical results of the penalty finite element method are provided

numerical analysis of a penalty approach for the solution of a transient eddy current problem abstract the aim of this paper is to propose and analyze a numerical method to solve transient eddy current problems formulated in terms of the magnetic field intensity space discretization is based on nedelec edge elements while a backward euler scheme is used for time discretization the curl free constraint in the dielectric domain is imposed by means of a penalty strategy convergence of the penalized problem as the penalty parameter goes to zero is proved for the continuous and the discrete problems for the latter uniformly in the discretization parameters optimal order error estimates for the convergence of the discrete penalized problem with respect to the penalty and the discretization parameters are also proved finally some numerical tests are reported to assess the performance of this approach

two level penalty finite element methods for the stationary incompressible magnetohydrodynamics problem in the paper we consider the penalty finite element method for the stationary incompressible magnetohydrodynamics mhd problem with a factor of penalty parameter stability and convergence of numerical solutions are established furthermore two level penalty methods are also developed for the mhd problem our methods consist of solving a nonlinear mhd problem by the usual penalty method on a coarse mesh with mesh size h and then a linearized mhd problem based on the stokes newton and oseen iterations on a fine mesh is solved by the penalty method with mesh size h h h respectively stability and error estimates of numerical solutions in two level penalty methods are presented finally some numerical tests are provided to demonstrate the effectiveness of the developed algorithms

a pressure poisson stabilized finite element method for the non stationary stokes equations to circumvent the inf sup condition in this article stabilized finite element methods are considered for the non stationary stokes equations based on some lowest equal order finite elements space pair x h m h which do not satisfy the discrete inf sup condition the stability of two kinds of methods is derived under some regularity assumptions then the convergence of the penalty method and the pressure poisson stabilized method is compared the result shows that the former error limits the order of approximation to o e h e and the latter yields the optimal error estimate o h

a finite element penalty method for the linearized viscoelastic oldroyd fluid motion equations abstract in this paper a fully discrete finite element penalty method is considered for the two dimensional linearized viscoelastic fluid motion equations arising from the oldroyd model for the non newton fluid flows with the finite element method for the spatial discretization and the backward euler scheme for the temporal discretization the velocity and pressure are decoupled in this method which leads to a large reduction of the computational scale under some realistic assumptions the unconditional stability of the fully discrete scheme is proved moreover the optimal error estimates are obtained which are better than the existing results finally some numerical results are given to verify the theoretical analysis the difference between the motion of the newton and non newton fluid is also observed

a generalisation of the calogero moser system abstract a generalised calogero moser many body system in which the particles possess extra internal degrees of freedom is introduced and solved by the method of olshanetsky and perelomov further the hierarchy of flows commuting with this system is obtained from a hierarchy of linear systems possessing sl n symmetry by the usual method of reduction thus they are shown to be integrable

r matrix structure of hitchin system in tyurin parameterization we present a classical r matrix for the hitchin system without marked points on an arbitrary non degenerate algebraic curve of genus g 2 using tyurin parameterization of holomorphic vector bundles

conformal blocks on elliptic curves and the knizhnik zamolodchikov bernard equations we give an explicit description of the vector bundle of wzw conformal blocks on elliptic curves with marked points as a subbundle of a vector bundle of weyl group invariant vector valued theta functions on a cartan subalgebra we give a partly conjectural characterization of this subbundle in terms of certain vanishing conditions on affine hyperplanes in some cases explicit calculations are possible and confirm the conjecture the friedan shenker flat connection is calculated and it is shown that horizontal sections are solutions of bernard s generalization of the knizhnik zamolodchikov equation

the non dynamical r matrix structure for the elliptic calogero moser model in this paper we construct a new lax operator for the elliptic calogero moser model with general from the classical dynamical twisting in which the corresponding r matrix is purely numerical non dynamic the non dynamical r matrix structure of this lax operator is obtained which is the elliptic symmetric r matrix

quantization of classical dynamical r matrices with nonabelian base we construct some classes of dynamical r matrices over a nonabelian base and quantize some of them by constructing dynamical pseudo twists in the sense of xu this way we obtain quantizations of r matrices obtained in earlier work of the second author with schiffmann and varchenko a part of our construction may be viewed as a generalization of the donin mudrov nonabelian fusion construction we apply these results to the construction of equivariant star products on poisson homogeneous spaces which include some homogeneous spaces introduced by de concini

matrix factorizations and integrable systems on montre que les algorithmes qr lij et de cholesky pour calculer les valeurs propres de matrices reelles sont les evaluations a temps entier de flots hamiltoniens completement integrables