extension of noncommutative soliton hierarchies a linear system which generates a moyal deformed two dimensional soliton equation as an integrability condition can be extended to a three dimensional linear system treating the deformation parameter as an additional coordinate the supplementary integrability conditions result in a first order differential equation with respect to the deformation parameter the flow of which commutes with the flow of the deformed soliton equation in this way a deformed soliton hierarchy can be extended to a bigger hierarchy by including the corresponding deformation equations we prove the extended hierarchy properties for the deformed akns hierarchy and specialize to the cases of deformed nls kdv and mkdv hierarchies corresponding results are also obtained for the deformed kp hierarchy a deformation equation determines a kind of seiberg witten map from classical solutions to solutions of the respective noncommutative deformed equation

3d tetrahedral unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric in this paper we present a 3d tetrahedral unstructured and anisotropic mesh generator that is not based on the delaunay frontal or octree method instead it proceeds by local optimizations and uses an anisotropic shape criterion to fit a metric field then we introduce a new 3d metric field that tightens the mesh around interfaces when the calculation domain is divided in several subdomains and a 3d metric field that places enough elements through each subdomain thickness without introducing too many nodes in the other directions finally we show some applications for material forming geometries

error estimation and anisotropic mesh refinement for 3d laminar aerodynamic flow simulations this article considers a posteriori error estimation and anisotropic mesh refinement for three dimensional laminar aerodynamic flow simulations the optimal order symmetric interior penalty discontinuous galerkin discretization which has previously been developed for the compressible navier stokes equations in two dimensions is extended to three dimensions symmetry boundary conditions are given which allow to discretize and compute symmetric flows on the half model resulting in exactly the same flow solutions as if computed on the full model using duality arguments an error estimation is derived for estimating the discretization error with respect to the aerodynamic force coefficients furthermore residual based indicators as well as adjoint based indicators for goal oriented refinement are derived these refinement indicators are combined with anisotropy indicators which are particularly suited to the discontinuous galerkin dg discretization two different approaches based on either a heuristic criterion or an anisotropic extension of the adjoint based error estimation are presented the performance of the proposed discretization error estimation and adaptive mesh refinement algorithms is demonstrated for 3d aerodynamic flows

lax pair and darboux transformation of a noncommutative u n principal chiral model we present a noncommutative generalization of the lax formalism of the u n principal chiral model in terms of a one parameter family of flat connections the lax formalism is further used to derive a set of parametric noncommutative backlund transformations and an infinite set of conserved quantities from the lax pair we derive a noncommutative version of the darboux transformation of the model

the two loop sunrise graph in two space time dimensions with arbitrary masses in terms of elliptic dilogarithms we present the two loop sunrise integral with arbitrary non zero masses in two space time dimensions in terms of elliptic dilogarithms we find that the structure of the result is as simple and elegant as in the equal mass case only the arguments of the elliptic dilogarithms are modified these arguments have a nice geometric interpretation

the two loop sunrise integral around four space time dimensions and generalisations of the clausen and glaisher functions towards the elliptic case we present the result for the finite part of the two loop sunrise integral with unequal masses in four space time dimensions in terms of the o e0 part and the o e1 part of the sunrise integral around two space time dimensions the latter two integrals are given in terms of elliptic generalisations of clausen and glaisher functions interesting aspects of the result for the o e1 part of the sunrise integral around two space time dimensions are the occurrence of depth two elliptic objects and the weights of the individual terms

the noncommutative akns system projection to matrix systems countable superposition and soliton like solutions starting from the recent work on noncommutative akns systems for functions with values in the bounded operators on a banach space it is shown how their formal 1 soliton solution depending on operator parameters can be mapped to solutions of matrix akns systems the main result is rather general solution formulas for matrix akns systems the most important applications are the countable superposition of matrix solitons and explicit expressions for the soliton like solutions of the classical akns system

non local continuity equations and binary darboux transformation of noncommutative anti self dual yang mills equations we present an infinite number of non local continuity equations of noncommutative anti self dual yang mills nc a sdym equations using the induction method of brezin et al 1979 phys lett b 82 442 and relate it to the lax pair and the parametric backlund transformation of the system from the lax pair we derive a binary darboux transformation to generate solutions of the nc a sdym equations

explorations of the extended nckp hierarchy a recently obtained extension xnckp of the moyal deformed kp hierarchy nckp hierarchy by a set of evolution equations in the moyal deformation parameters is further explored formulae are derived to compute these equations efficiently reductions of the xnckp hierarchy are treated in particular to the extended nckdv and ncboussinesq hierarchies furthermore a good part of the sato formalism for the kp hierarchy is carried over to the generalized framework in particular the well known bilinear identity theorem for the kp hierarchy expressed in terms of the formal baker akhiezer function extends to the xnckp hierarchy moreover it is demonstrated that n soliton solutions of the nckp equation are also solutions of the first few deformation equations this is shown to be related to the existence of certain families of algebraic identities

relations between elliptic multiple zeta values and a special derivation algebra we investigate relations between elliptic multiple zeta values and describe a method to derive the number of indecomposable elements of given weight and length our method is based on representing elliptic multiple zeta values as iterated integrals over eisenstein series and exploiting the connection with a special derivation algebra its commutator relations give rise to constraints on the iterated integrals over eisenstein series relevant for elliptic multiple zeta values and thereby allow to count the indecomposable representatives conversely the above connection suggests apparently new relations in the derivation algebra under https tools aei mpg de emzv we provide relations for elliptic multiple zeta values over a wide range of weights and lengths ar x iv 1 50 7 02 25 4v 1 he pth 8 j ul 2 01 5

with a cole hopf transformation to solutions of the noncommutative kp hierarchy in terms of wronski matrices in the case of the kp hierarchy where the dependent variable takes values in an arbitrary associative algebra it is known that there are solutions which can be expressed in terms of quasideterminants of a wronski matrix which solves the linear heat hierarchy we obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix kp hierarchies over and by use of a cole hopf transformation for this class of exact solutions we work out a correspondence with weakly nonassociative algebras

noncommutative burgers equation we present a noncommutative version of the burgers equation which possesses the lax representation and discuss the integrability in detail we find a noncommutative version of the cole hopf transformation and succeed in the linearization of it the linearized equation is the noncommutative diffusion equation and exactly solved we also discuss the properties of some exact solutions the result shows that the noncommutative burgers equation is completely integrable even though it contains infinite number of time derivatives furthermore we derive the noncommutative burgers equation from the noncommutative anti self dual yang mills equation by reduction which is an evidence for the noncommutative ward conjecture finally we present a noncommutative version of the burgers hierarchy by both the lax pair generating technique and sato s approach

about the self dual chern simons system and toda field theories on the noncommutative plane the relation of the noncommutative self dual chern simons ncsdcs system to the noncommutative generalizations of toda and of affine toda field theories is investigated more deeply this paper continues the programme initiated in cabrera carnero i 2005 j high energy phys jhep10 2005 071 where it was presented how it is possible to define toda field theories through second order differential equation systems starting from the ncsdcs system here we show that using the connection of the ncsdcs to the noncommutative chiral model exact solutions of the toda field theories can also be constructed by means of the noncommutative extension of the uniton method proposed in lee k m 2004 j high energy phys jhep08 2004 054 by lee particularly some specific solutions of the nc liouville model are explicitly constructed

anisotropic mesh adaptation for 3d flows on structured and unstructured grids abstract this paper presents a mesh optimization methodology in three dimensions mom3d an initial mesh is continually adapted during the solution process without the need for global remeshing the adaptation procedure uses an interpolation error estimate whose magnitude and direction are controlled by the hessian the matrix of second derivatives of the solution this metric error is projected over mesh edges and drives the nodal movement scheme as well as the edge refinement and coarsening strategies these operations yield highly anisotropic grids in which the mesh movement significantly contributes to the stretching and realignment of the edges along unidirectional features of flow problems the results presented have been chosen to illustrate some important points first the method is gauged on problems with exact solutions demonstrating good agreement between the error estimate and the true error as well as an equidistribution of the error the cost effectiveness of grid adaptation is then addressed by determining the size of an anisotropic grid that would be equivalent to that of a given non adapted finer grid for the same error level the capture of sharp discontinuities through highly anisotropic grids is illustrated on a transonic flow flow in a gas turbine combustor demonstrates how automatically generated meshes can sometimes cause convergence difficulties and how mesh adaptation can cure these ills finally the flow over a wing nacelle pylon configuration is studied to further validate the solver mesh adaptation capabilities by comparing the numerical results against experiments

continuous adjoint approach for the spalart allmaras model in aerodynamic optimization a new variable solving the eikonal equation the accuracy of the sensitivity derivatives obtained with the complete turbulent approach is assessed by comparison with finite difference computations and the classical continuous adjoint with frozen viscosity showing substantial improvements in the convergence properties of the method and in the quality of the obtained gradients the validity of the overall methodology is illustrated with several design examples including the optimization of three dimensional geometries in combination with advanced freeform techniques for mesh deformation

time accurate anisotropic goal oriented mesh adaptation for unsteady flows we present a new algorithm for combining an anisotropic goal oriented error estimate with the mesh adaptation fixed point method for unsteady problems the minimization of the error on a functional provides both the density and the anisotropy stretching of the optimal mesh they are expressed in terms of state and adjoint this method is used for specifying the mesh for a time sub interval a global fixed point iterates the re evaluation of meshes and states over the whole time interval until convergence of the space time mesh applications to unsteady blast wave and acoustic wave euler flows are presented

a comparison of optimization based approaches for a model computational aerodynamics design problem abstract the objective of this paper is to compare three optimization based methods for solving aerodynamic design problems we use the euler equations for one dimensional duct flow as a model problem the optimization methods are i the black box method with finite difference gradients ii a modification where gradients are found by an algorithm based on the implicit function theorem and iii an all atonce method where the flow and design variables are simultaneously altered the three methods are applied to the model problem and compared for efficiency robustness and implementation difficulty we also show that the black box implicit gradient method is equivalent to applying the variational or optimal control approach to design optimization directly to the discretized analysis problem rather than to the continuous problem as is usually done the black box method with implicit gradients seems to provide a good compromise of features and can be retrofitted to most existing analysis codes to turn them into design codes although the all at once method was found to be less robust than the black box methods when it succeeded it was considerably more efficient

anisotropic norm oriented mesh adaptation for a poisson problem we present a novel formulation for the mesh adaptation of the approximation of a partial differential equation pde the discussion is restricted to a poisson problem the proposed norm oriented formulation extends the goal oriented formulation since it is equation based and uses an adjoint at the same time the norm oriented formulation somewhat supersedes the goal oriented one since it is basically a solution convergent method indeed goal oriented methods rely on the reduction of the error in evaluating a chosen scalar output with the consequence that as mesh size is increased more degrees of freedom only this output is proven to tend to its continuous analog while the solution field itself may not converge a remarkable quality of goal oriented metric based adaptation is the mathematical formulation of the mesh adaptation problem under the form of the optimization in the well identified set of metrics of a well defined functional in the new proposed formulation we amplify this advantage we search in the same well identified set of metrics the minimum of a norm of the approximation error the norm is prescribed by the user and the method allows addressing the case of multi objective adaptation like for example in aerodynamics adaptating the mesh for drag lift and moment in one shot in this work we consider the basic linear finite element approximation and restrict our study to l 2 norm in order to enjoy second order convergence numerical examples for the poisson problem are computed

adjoint sensitivity analysis for time dependent partial differential equations with adaptive mesh refinement a new adjoint sensitivity analysis approach is presented for time dependent partial differential equations with adaptive mesh refinement the new approach called adda combines the best features of both the adjoint of the discretization ad and discretization of the adjoint da approaches it removes the obstacles of applying ad to adaptive methods and in contrast to da requires for its use only a minimal amount of knowledge about the formulation of adjoint pdes and their boundary conditions the effectiveness and efficiency of adda are demonstrated for several numerical examples

continuous mesh framework part i well posed continuous interpolation error in the context of mesh adaptation riemannian metric spaces have been used to prescribe orientation density and stretching of anisotropic meshes but such structures are only considered to compute lengths in adaptive mesh generators in this article a riemannian metric space is shown to be more than a way to compute a length it is proven to be a reliable continuous mesh model in particular we demonstrate that the linear interpolation error can be evaluated continuously on a riemannian metric space from one hand this new continuous framework proves that prescribing a riemannian metric field is equivalent to the local control in the mathbf l 1 norm of the interpolation error this proves the consistency of classical metric based mesh adaptation procedures on the other hand powerful mathematical tools are available and are well defined on riemannian metric spaces calculus of variations differentiation optimization dots whereas these tools are not defined on discrete meshes

space time adaptive solution of inverse problems with the discrete adjoint method this paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent adaptive grid adaptive step models discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation however this approach brings several important challenges the space time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations a reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter grid transfer operators moreover the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations this work shows that several of these potential issues can be avoided through a multi level optimization strategy using discontinuous galerkin dg hp adaptive discretizations paired with runge kutta rk time integration we extend the concept of dual adjoint consistency to space time rk dg discretizations which are then shown to be well suited for the adaptive solution of time dependent inverse problems furthermore we prove that dg mesh transfer operators on general meshes are also dual consistent this allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation ad potentially speeding up development of large scale simulation codes the theoretical analysis is supported by numerical results reported for a two dimensional non stationary inverse problem

continuous mesh framework part ii validations and applications this paper gives a numerical validation of the continuous mesh framework introduced in part i a loseille and f alauzet siam j numer anal 49 2011 pp 38 60 we numerically show that the interpolation error can be evaluated analytically once analytical expressions of a mesh and a function are given in particular the strong duality between discrete and continuous views for the interpolation error is emphasized on two dimensional and three dimensional examples in addition we show the ability of this framework to predict the order of convergence given a specific adaptive strategy defined by a sequence of continuous meshes the continuous mesh concept is then used to devise an adaptive strategy to control the mathbf l p norm of the continuous interpolation error given the mathbf l p norm of the continuous interpolation error we derive the optimal continuous mesh minimizing this error this exemplifies the potential of this framework as we use a calculus of variations that is not defined on the space of discrete meshes anisotropic adaptations on analytical functions correlate the optimal predicted theoretical order of convergence the extension to a solution of nonlinear pdes is also given comparisons with experiments show the efficiency and the accuracy of this approach

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

fully anisotropic goal oriented mesh adaptation for 3d steady euler equations this paper studies the coupling between anisotropic mesh adaptation and goal oriented error estimate the former is very well suited to the control of the interpolation error it is generally interpreted as a local geometric error estimate on the contrary the latter is preferred when studying approximation errors for pdes it generally involves non local error contributions consequently a full and strong coupling between both is hard to achieve due to this apparent incompatibility this paper shows how to achieve this coupling in three steps first a new a priori error estimate is proved in a formal framework adapted to goal oriented mesh adaptation for output functionals this estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error second the error estimate is applied to the set of steady compressible euler equations which are solved by a stabilized galerkin finite element discretization a goal oriented error estimation is derived it involves the interpolation error of the euler fluxes weighted by the gradient of the adjoint state associated with the observed functional third rewritten in the continuous mesh framework the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations the optimal continuous mesh is then derived analytically thus it can be used as a metric tensor field to drive the mesh adaptation from a numerical point of view this method is completely automatic intrinsically anisotropic and does not depend on any a priori choice of variables to perform the adaptation 3d examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency

continuous adjoint sensitivities for optimization with general cost functionals on unstructured meshes a continuous adjoint approach is developed to obtain the sensitivity derivatives for the euler equations the completederivationofthecostateequations andtheirtransversality boundary conditionsarepresented both the state and the costate equations are second order e nite volume discretized for unstructured meshes and they are coupled with a constrained optimization algorithm also integrated into the overall methodology are a geometry parameterization method for the shape optimization and a dynamic unstructured mesh method for the shape evolution and the consequent volume mesh adaptations for the proof of concept three transonic airfoil optimization problems are presented this method accepts general cost functionals which are not necessarily functions of pressure only it is also shown that a switch to the natural coordinate system in conjunction with the reduction of the governing state equation to the control surface results in sensitivity integrals that are only a function of the tangential derivativesof the state variables thisapproach eliminates the need fornormal derivativecomputations that can be erroneous

three dimensional anisotropic mesh adaptation for phase change problems a three dimensional adaptive strategy for the finite element simulation of phase change problems is presented discussed and validated a semi phase field formulation is used for the solution of the stefan problem the adaptive method is based on the definition of edge length using a solution dependent metric and produces strongly anisotropic meshes numerical results illustrating the performance and accuracy of the proposed method are presented

an optimization based framework for anisotropic simplex mesh adaptation we present a general framework for anisotropic h adaptation of simplex meshes given a discretization and any element wise localizable error estimate our adaptive method iterates toward a mesh that minimizes error for a given degrees of freedom utilizing mesh metric duality we consider a continuous optimization problem of the riemannian metric tensor field that provides an anisotropic description of element sizes first our method performs a series of local solves to survey the behavior of the local error function this information is then synthesized using an affine invariant tensor manipulation framework to reconstruct an approximate gradient of the error function with respect to the metric tensor field finally we perform gradient descent in the metric space to drive the mesh toward optimality the method is first demonstrated to produce optimal anisotropic meshes minimizing the l 2 projection error for a pair of canonical problems containing a singularity and a singular perturbation the effectiveness of the framework is then demonstrated in the context of output based adaptation for the advection diffusion equation using a high order discontinuous galerkin discretization and the dual weighted residual dwr error estimate the method presented provides a unified framework for optimizing both the element size and anisotropy distribution using an a posteriori error estimate and enables efficient adaptation of anisotropic simplex meshes for high order discretizations

a new approach to deformation equations of noncommutative kp hierarchies partly inspired by sato s theory of the kadomtsev petviash vili kp hierarchy we start with a quite general hierarchy of linear ordinary differential equatio ns in a space of matrices and derive from it a matrix riccati hierarchy the latter is then shown to exhibit an underlying weakly nonassociative wna algebra structure from which we can conclude refering to previous work that any solution of the riccati system also solves the potential kp hierarchy in the corresponding matrix algebra we then turn to the case where the components of the matrices are multiplied using a generalized star product associated with the deformation parameters there are additional symmetries flow equations which enlarge the respective kp hierarchy they have a compact formulation in terms of the wna structure we also present a formulation of the kp hierarchy equations themselves as deformation flow equations

tetrahedral mesh optimisation and adaptivity for steady state and transient finite element calculations abstract a method for optimising a pre existing mesh of tetrahedral finite elements is described it is based on a series of mesh connectivity and node position searches of the landscape defining mesh quality a riemannian metric reflecting the a posteriori error measure is used to calculate element size and shape a functional is defined which embodies both shape and size quality of an element with respect to the metric and is used to gauge mesh quality a heuristic based local search strategy is adopted local in the sense that it has no hill climbing abilities the paper presents applications of the method to complex steady state and time dependent problems which highlight its anisotropic feature capturing abilities numerical evidence is provided which suggests that the computational complexity time of the proposed algorithm varies linearly with the number of elements or nodes of the finite element mesh

contribution to the optimal shape design of two dimensional internal flows with embedded shocks we explore the praticability of optimal shape design for flows modeled by the euler equations we define a functional whose minimum represents the optimality condition the gradient of the functional with respect to the geometry is calculated with the lagrange multipliers which are determined by solving acostateequation the optimization problem is then examined by comparing the performance of several gradient based optimization algorithms in this formulation the flow field can be computed to an arbitrary order of accuracy finally some results for internal flows with embedded shocks are presented including a case for which the solution to the inverse problem does not belong to the design space

continual lie algebras and noncommutative counterparts of exactly solvable models noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of saveliev vershik continual lie algebras examples of noncommutative liouville and sin h gordon equations are given the simplest soliton solution to the noncommutative sine gordon equation is found

an algebraic scheme associated with the noncommutative kp hierarchy and some of its extensions a well known ansatz trace method for soliton solutions turns the equations of the non commutative kp hierarchy and those of certain extensions into families of algebraic sum identities we develop an algebraic formalism in particular involving a mixable shuffle product to explore their structure more precisely we show that the equations of the non commutative kp hierarchy and its extension xnckp in the case of a moyal deformed product as derived in previous work correspond to identities in this algebra furthermore the moyal product is replaced by a more general associative product this leads to a new even more general extension of the non commutative kp hierarchy relations with rota baxter algebras are established

conserved quantities in the noncommutative principal chiral model with wess zumino term we construct a noncommutative extension of the u n principal chiral model with wess zumino term and obtain an infinite set of local and non local conserved quantities for the model using the iterative procedure of brezin et al 1979 phys lett b 82 442 we also present the equivalent description as a lax formalism of the model we expand the fields perturbatively and derive zeroth and first order equations of motion zero curvature condition iteration method lax formalism local and non local conserved quantities

admitting the inadmissible adjoint formulation for incomplete cost functionals in aerodynamic optimization we derive the adjoint equations for problems in aerodynamic optimization which are improperly considered as inadmissible for example a cost functional which depends on the density rather than on the pressure is considered inadmissible for an optimization problem governed by the euler equations we show that for such problems additional terms should be included in the lagrangian functional when deriving the adjoint equations these terms are obtained from the restriction of the interior pde to the control surface demonstrations of the explicit derivation of the adjoint equations for inadmissible cost functionals are given for the potential euler and navier stokes equations

systematic continuous adjoint approach to viscous aerodynamic design on unstructured grids doi 10 2514 1 24859 a continuous adjoint approach to aerodynamic design for viscous compressible flow on unstructured grids is developed sensitivity gradients arecomputed using tools ofshapedeformation of boundaryintegrals theresulting expressions involve second order derivatives of the flow variables that require numerical solvers with greater than second order accuracy a systematic way of reducing the order of these terms is presented the accuracy of the sensitivity derivatives is assessed by comparison with finite difference computations and the validity of the overall methodology is illustrated with several design examples

adjoint based three dimensional error prediction and grid adaptation engineering computational uid dynamics cfd analysis and design applications focus on output functions e g lift drag errors in these output functions are generally unknown and conservatively accurate solutions may be computed computable error estimates can oer the possibility to minimize computational work for a prescribed error tolerance such an estimate can be computed by solving the ow equations and the linear adjoint problem for the functional of interest the computational mesh can be modied to minimize the uncertainty of a computed error estimate this robust mesh adaptation procedure automatically terminates when the simulation is within a user specied error tolerance this procedure for estimating and adapting to error in a functional is demonstrated for three dimensional euler problems an adaptive mesh procedure that links to a computer aided design cad surface representation is demonstrated for wing wing body and extruded high lift airfoil congurations the error estimation and adaptation procedure yielded corrected functions that are as accurate as functions calculated on uniformly rened grids with ten times as many grid points

commuting flows and conservation laws for noncommutative lax hierarchies we discuss commuting flows and conservation laws for lax hierarchies on noncommutative spaces in the framework of the sato theory on commutative spaces the sato theory has revealed essential aspects of the integrability for wide class of soliton equations which are derived from the lax hierarchies in terms of pseudodifferential operators noncommutative extension of the sato theory has been already studied by the author and toda and the existence of various noncommutative lax hierarchies are guaranteed in this paper we present conservation laws for the noncommutative lax hierarchies with both space space and space time noncommutativities and prove the existence of infinite number of conserved densities we also give the explicit representations of them in terms of lax operators our results include noncommutative versions of kp kdv boussinesq coupled kdv sawada kotera modified kdv equation and so on

a posteriori error estimates based on hierarchical bases the authors present an analysis of an a posteriors error estimator based on the use of hierarchical basis functions the authors analyze nonlinear nonselfadjoint and indefinite problems as well as the selfadjoint positive definite case because both the analysis and the estimator itself are quite simple it is easy to see how various approximations affect the quality of the estimator as examples the authors apply the theory to some scalar elliptic equations and the stokes system of equations

bicomplex formulation and moyal deformation of 2 1 dimensional fordy kulish systems using bicomplex formalism we construct generalizations of fordy kulish systems of matrix nonlinear schrodinger equations on two dimensional space time in two respects firstly we obtain corresponding equations in three space time dimensions secondly a moyal deformation is applied to the space time coordinates and the ordinary product of functions replaced by the moyal product in a suitable way both generalizations preserve the existence of an infinite set of conservation laws

functional representations of integrable hierarchies we consider a general framework for integrable hierarchies in lax form and derive certain universal equations from which functional representations of particular hierarchies such as kp discrete kp mkp akns i e formulations in terms of functional equations are systematically and quite easily obtained the formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative typically matrix algebra the obtained functional representations can be understood as noncommutative analogues of fay identities for the kp hierarchy

from akns to derivative nls hierarchies via deformations of associative products using deformations of associative products derivative nonlinear schrodinger dnls hierarchies are recovered as akns type hierarchies since the latter can also be formulated as gelfand dickey type lax hierarchies a recently developed method to obtain functional representations can be applied we actually consider hierarchies with dependent variables in any possibly noncommutative associative algebra e g an algebra of matrices of functions this also covers the case of hierarchies of coupled derivative nls equations

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

an a priori anisotropic goal oriented error estimate for viscous compressible flow and application to mesh adaptation abstract we present a goal oriented error analysis for the calculation of low reynolds steady compressible flows with anisotropic mesh adaptation the error analysis is of a priori type its central principle is to express the right hand side of the error equation often referred as the local error as a function of the interpolation error of a collection of fields present in the nonlinear partial differential equations this goal oriented error analysis is the extension of 39 done for inviscid flows to laminar viscous flows by adding viscous terms the main benefits of this approach in comparison to other error estimates in the literature is that the optimal anisotropy of the mesh directly appears in the error analysis and is not obtained from an ad hoc variable nor a local analysis as a consequence an optimum is obtained and the convergence of the mesh adaptive process is very fast i e generally the convergence is obtained after 5 to 10 mesh adaptation cycle then using the continuous mesh framework an optimal metric is analytically obtained from the error estimation applications to mesh adaptive calculations of flows past airfoils are presented

optimal control of flow with discontinuities optimal control of the 1 d riemann problem of euler equations is studied with the initial values for pressure and density as control parameters the least squares type cost functional employs either distributed observations in time or observations calculated at the end of the assimilation window existence of solutions for the optimal control problem is proven smooth and nonsmooth optimization methods employ the numerical gradient respectively a subgradient of the cost functional obtained from the adjoint of the discrete forward model the numerical flow obtained with the optimal initial conditions obtained from the nonsmooth minimization matches very well with the observations the algorithm for smooth minimization converges for the shorter time horizon but fails to perform satisfactorily for the longer time horizon except when the observations corresponding to shocks are detected and removed

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

samples of noncommutative products in certain differential equations a set of associative noncommutative products is considered in different differential equations of the ordinary and partial types a method of separation of variables is considered for a large set of those systems the products involved include for example some products and some products based on nijenhuis tensors which are embedded in the differential equations of the laplace poisson lax and schr dinger styles a comment on the products of reshetikhin jambor sykora type is also given in relation to products of vey type

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

motivic multiple zeta values and superstring amplitudes the structure of tree level open and closed superstring amplitudes is analyzed for the open superstring amplitude we find a striking and elegant form which allows one to disentangle its expansion into several contributions accounting for different classes of multiple zeta values this form is bolstered by the decomposition of motivic multiple zeta values i e the latter encapsulate the expansion of the superstring amplitude moreover a morphism induced by the coproduct maps the expansion onto a non commutative hopf algebra this map represents a generalization of the symbol of a transcendental function in terms of elements of this hopf algebra the expansion assumes a very simple and symmetric form which carries all the relevant information equipped with these results we can also cast the closed superstring amplitude into a very elegant form

mixed tate motives over z we prove that the category of mixed tate motives over z is spanned by the motivic fundamental group of p 1 minus three points we prove a conjecture by m hoffman which states that every multiple zeta value is a q linear combination of n 1 n r where n i 2 3