quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

numerical methods for the topology optimization of structures that exhibit snap through the inclusion of non linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response e g the geometric non linear response of compliant mechanisms in previous work the non linear response is computed by standard non linear elastic finite element analysis here we incorporate a load displacement constraint method to traverse non linear equilibrium paths with limit points to design structures that exhibit snap through behaviour to accomplish this we modify the basic arc length algorithm and embed this analysis into the topology optimization problem copyright 2002 john wiley sons ltd

robust topology optimization of structures with imperfect geometry based on geometric nonlinear analysis abstract topology optimization often leads to structures consisting of slender elements which are particularly sensitive to geometric imperfections such imperfections might affect the structural stability and induce large displacement effects in these slender structures this paper therefore presents a robust approach to topology optimization which accounts for geometric imperfections and their potentially detrimental influence on the structural stability geometric nonlinear effects are incorporated in the optimization by means of a total lagrangian finite element formulation in the minimization of end compliance geometric imperfections are modeled as a vector valued random field in the design domain the resulting uncertain performance of the design is taken into account by minimizing a weighted sum of the mean and standard deviation of the compliance in the robust optimization problem these stochastic moments are typically estimated by means of sampling methods such as monte carlo simulation however these methods require multiple independent nonlinear finite element analyses in each design iteration of the optimization algorithm an efficient solution algorithm which uses adjoint differentiation in a second order perturbation method is therefore developed to estimate the stochastic moments during the optimization two applications with structures that exhibit different types of structural instabilities are examined in both cases it is demonstrated by means of an extensive monte carlo simulation that the deterministic design is very sensitive to imperfections while the design obtained by means of the proposed method is much more robust

pbw bases of coideal subalgebras and a freeness theorem let h be a character hopf algebra every right coideal subalgebra u that contains the coradical has a pbw basis which can be extended up to a pbw basis of h if additionally u is a bosonization of an invariant with respect to the left adjoint action subalgebra then h is a free left and right u module with a free pbw basis over u these results remain valid if h is a braided hopf algebra generated by a categorically ordered subset of primitive elements if the ground field is algebraically closed the results are still true provided that h is a pointed hopf algebra with commutative coradical and is generated over the coradical by a direct sum of finite dimensional yetter drinfeld submodules of skew primitive elements

algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

primitive ideals of cq sl n abstract the primitive ideals of the quantum group c q sl n are classified in the case where q is a non zero complex number which is not a root of unity it is shown that the orbits in prim c q sl n under the action of the character group h c n 1 are parameterized naturally by w w where w is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals of c q sl n and symplectic leaves of the associated poisson algebraic group sl n c

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

element connectivity parameterization for topology optimization of geometrically nonlinear structures in the current element density based topology optimization element stiffness is penalized to yield either solid or void or very weak materials but low density elements appear during and even after optimization iterations especially when nonlinear problems involving large deformation are considered low density finite elements cause serious numerical problems as their tangent stiffness matrices lose positive definiteness to completely eliminate the problem caused by low density elements we propose a new method all finite elements are kept solid throughout the optimization process zero length elastic links are introduced to parameterize inter element connectivity the link stiffness is penalized in this approach the design variables are defined on the links and vary from 0 and 1 corresponding to the unconnected and rigidly connected states since the finite elements used to discretize the analysis domain always remain solid typical numerical problems encountered by the standard element density based formulation disappear to implement the present method several issues such as the handling of the mass constraint and raster imaging are also investigated

a method to compute statistics of large noise induced perturbations of nonlinear schrodinger solitons we describe in detail the application of importance sampling to numerical simulations of large noise induced perturbations in soliton based optical transmission systems governed by the nonlinear schrodinger equation the method allows one to concentrate the samples in monte carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors and it yields computational speedups of several orders of magnitude over standard monte carlo simulations we demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude frequency timing and phase fluctuations in a prototypical soliton based communication system

robust truss topology design via semidefinite programming we present and motivate a new model of the truss topology design problem where the rigidity of the resulting truss with respect both to given loading scenarios and small occasional loads is optimized it is shown that the resulting optimization problem is a semidefinite program we derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

complete eigenfunctions of linearized integrable equations expanded around a soliton solution complete eigenfunctions for an integrable equation linearized around a soliton solution are the key to the development of a direct soliton perturbation theory in this article we explicitly construct such eigenfunctions for a large class of integrable equations including the kdv nls and mkdv hierarchies we establish the striking result that the linearization operators of all equations in the same integrable hierarchy share the same complete set of eigenfunctions furthermore these eigenfunctions are precisely the squared eigenfunctions of the associated eigenvalue problem the key step in our derivation is to show that the linearization operator of an integrable equation can be factored into a function of the integro differential operator which generates the integrable equation and the linearization operator of the lowest order integrable equation in the same hierarchy we also obtain similar results for the adjoint linearization operator of an integrable equation even though our analysis is conducted only for the kdv nls and mkdv hierarchies similar results are expected for other integrable hierarchies as well we further explicitly present the complete eigenfunctions for the kdv nls and mkdv hierarchy equations and give their inner products thus they can be readily used to develop a direct soliton perturbation theory for any of those hierarchy equations

on the chern character of θ summable fredholm modules we show that the entire cyclic cohomology class given by the jaffe lesniewski osterwalder formula is the same as the class we had constructed earlier as the chern character of summable fredholm modules

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

numerical algorithms for the direct spectral transform with applications to nonlinear schrodinger type systems we implement two different algorithms for computing numerically the direct zakharov shabat eigenvalue problem on the infinite line the first algorithm replaces the potential in the eigenvalue problem by a piecewise constant approximation which allows one to solve analytically the corresponding ordinary differential equation the resulting algorithm is of second order in the step size the second algorithm uses the fourth order runge kutta method we test and compare the performance of these two algorithms on three exactly solvable potentials we find that even though the runge kutta method is of higher order this extra accuracy can be lost because of the additional dependence of its numerical error on the eigenvalue this limits the usefulness of the runge kutta algorithm to a region inside the unit circle around the origin in the complex plane of the eigenvalues for the computation of the continuous spectrum density this limitation is particularly severe as revealed by the spectral decomposition of thel2 norm of a solution to the nonlinear schrodinger equation we show that no such limitations exist for the piecewise constant algorithm in particular this scheme converges uniformly for both continuous and discrete spectrum components

interpolation scheme for fictitious domain techniques and topology optimization of finite strain elastic problems abstract the focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches with structures undergoing large displacements numerical instability in the finite element simulations can often be observed due to excessive distortion in low stiffness regions a new energy interpolation scheme is proposed in order to stabilize the numerical simulations the elastic energy density in the solid and void regions is interpolated using the elastic energy densities for large and small deformation theory respectively the performance of the proposed method is demonstrated for a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms the effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated as well numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for the simulated cases results in stable topology optimization of structures undergoing large displacements

robust topology optimization minimization of expected and variance of compliance robust topology optimization has long been considered computationally intractable as it combines two highly expensive computational strategies this paper considers simultaneous minimization of expectancy and variance of compliance in the presence of uncertainties in loading magnitude using exact formulations and analytically derived sensitivities it shows that only a few additional load cases are required which scales in polynomial time with the number of uncertain load cases the approach is implemented using the level set topology optimization method shape sensitivities are derived using the adjoint method several examples are used to investigate the effect of including variance in robust compliance optimization

an extended stochastic finite element method for solving stochastic partial differential equations on random domains abstract recently a new strategy was proposed to solve stochastic partial differential equations on random domains it is based on the extension to the stochastic framework of the extended finite element method x fem this method leads by a direct calculus to an explicit solution in terms of the variables describing the randomness on the geometry it relies on two major points the implicit representation of complex geometries using random level set functions and the use of a galerkin approximation at both stochastic and deterministic levels in this article we detail the basis of this technique from theoretical and technical points of view several numerical examples illustrate the efficiency of this method and compare it to other approaches

amplitude energy fluctuations of optical pulses and statistics of the exit distance abstract we consider the propagation of an isolated optical pulse in the presence of additive noise we apply an approach based on the fokker planck equation to determine the probability distribution functions of the propagation distance at which the pulse energy and the pulse amplitude leave a certain region

safe and effective importance sampling abstract we present two improvements on the technique of importance sampling first we show that importance sampling from a mixture of densities using those densities as control variates results in a useful upper bound on the asymptotic variance that bound is a small multiple of the asymptotic variance of importance sampling from the best single component density this allows one to benefit from the great variance reductions obtainable by importance sampling while protecting against the equally great variance increases that might take the practitioner by surprise the second improvement is to show how importance sampling from two or more densities can be used to approach a zero sampling variance even for integrands that take both positive and negative values

robust structural topology optimization considering boundary uncertainties in classical deterministic topology optimization the effect of the possible boundary variations on the performance of the structure is not taken into account which may lead to designs that are very sensitive to manufacturing errors as a consequence the performance of the real structure may be far from optimal and even not meet the design requirements in the present paper structural topology optimization considering the uncertainty of boundary variations is considered via level set approach in order to make the optimal designs less sensitive to the possible boundary variations we choose the compliance and fundamental frequency of structure enduring the worst case perturbation as the objective function for ensuring the robustness of the optimal solution with use of the schwarz inequality the original bi level optimization problem is transformed to a single level optimization problem which can be solved efficiently numerical examples demonstrate the effectiveness of the proposed approach

topology optimization of non linear elastic structures and compliant mechanisms abstract a linear elastic response is assumed in most structural topology optimization problems while this assumption is valid for a wide variety of problems it is not valid for structures undergoing large displacements the elastic structural analysis used here accommodates geometric and material non linearities the material density field is filtered to enforce a length scale on the field variation and is penalized to remove less effective intermediate densities the filtering scheme is embedded in the structural analysis to resolve the non existent solution to the solid void topology problem in this way we know precisely what optimization problem is being solved the structural topology optimization formulation is also used to design compliant mechanisms

an iterative stochastic method for simulating large deviations and rare events we describe an iterative method capable of determining large deviations responsible for rare events of interest in lightwave systems with additive noise the method makes use of the singular value decomposition svd to efficiently compute the most important directions in state space and a stochastic optimization scheme known as the cross entropy ce method to determine the most probable manner in which these large deviations arise information from the svd and ce steps of the method provides a basis for performing importance sampling with monte carlo simulation allowing one to determine the probabilities of the rare events associated with such large deviations we apply the combined method to investigate some of the mechanisms affecting large amplitude fluctuations in optical systems

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

statistics of soliton bearing systems with additive noise we present a consistent method to calculate the probability distribution of soliton parameters in systems with additive noise even though the noise is weak we are interested in probabilities of large fluctuations generally non gaussian which are beyond perturbation theory our method is a development of the instanton formalism method of optimal fluctuation based on a saddle point approximation in the path integral we first