a priori and a posteriori analysis of the quasinonlocal quasicontinuum method in 1d for a next nearest neighbour pair interaction model in a periodic domain a priori and a posteriori analyses of the quasinonlocal quasicontinuum method qnl qc are presented the results are valid for large deformations and essentially guarantee a one to one correspondence between atomistic solutions and qnl qc solutions the analysis is based on consistency error estimates in negative norms novel a priori and a posteriori stability estimates and a quantitative inverse function theorem

formulation and optimization of the energy based blended quasicontinuum method we formulate an energy based atomistic to continuum coupling method based on blending the quasicontinuum method for the simulation of crystal defects we utilize theoretical results from van koten and luskin 32 and ortner and van koten 24 to derive optimal choices of approximation parameters blending function and finite element grid for microcrack and di vacancy test problems and confirm our analytical predictions in numerical tests

a material frame approach for evaluating continuum variables in atomistic simulations we present a material frame formulation analogous to the spatial frame formulation developed by hardy whereby expressions for continuum mechanical variables such as stress and heat flux are derived from atomic scale quantities intrinsic to molecular simulation this formulation is ideally suited for developing an atomistic to continuum correspondence for solid mechanics problems we derive expressions for the first piola kirchhoff p k stress tensor and the material frame heat flux vector directly from the momentum and energy balances using localization functions in a reference configuration the resulting p k stress tensor unlike the cauchy expression has no explicit kinetic contribution the referential heat flux vector likewise lacks the kinetic contribution appearing in its spatial frame counterpart using a proof for a special case and molecular dynamics simulations we show that our p k stress expression nonetheless represents a full measure of stress that is consistent with both the system virial and the cauchy stress expression developed by hardy we also present an expanded formulation to define continuum variables from micromorphic continuum theory which is suitable for the analysis of materials represented by directional bonding at the atomic scale

admissibility conjecture and kazhdan s property t for quantum groups abstract we give a partial solution to a long standing open problem in the theory of quantum groups namely we prove that all finite dimensional representations of a wide class of locally compact quantum groups factor through matrix quantum groups admissibility conjecture for quantum group representations we use this to study kazhdan s property t for quantum groups with non trivial scaling group strengthening and generalising some of the earlier results obtained by fima kyed and soltan chen and ng daws skalski and viselter and brannan and kerr our main results are i all finite dimensional unitary representations of locally compact quantum groups which are either unimodular or arise through a special bicrossed product construction are admissible ii a generalisation of a theorem of wang which characterises property t in terms of isolation of finite dimensional irreducible representations in the spectrum iii a very short proof of the fact that quantum groups with property t are unimodular iv a generalisation of a quantum version of a theorem of bekka valette proven earlier for quantum groups with trivial scaling group which characterises property t in terms of non existence of almost invariant vectors for weakly mixing representations v a generalisation of a quantum version of kerr pichot theorem proven earlier for quantum groups with trivial scaling group which characterises property t in terms of denseness properties of weakly mixing representations

computational analysis of modeling error for the coupling of particle and continuum models by the arlequin method we propose in this paper a 1d model problem to study the convergence of surrogate approximations using an atomistic to continuum coupling method towards the solution of a full particle model the 1d problem consists of a collection of springs that exhibits a local defect materialized by a sudden change in the spring properties the surrogate model is obtained by the arlequin approach which introduces an overlap region in which the continuum and particle models are coupled together using lagrange multipliers the objective of the present work is to show via numerical experiments that the modeling error does indeed converge to zero as the distance of the overlap region from the defect and or its size are increased

interior regularity estimates for elliptic systems of difference equations we prove interior regularity estimates for a large class of difference approximations for elliptic systems of partial differential equations these estimates are analogous to those for the systems of differential equations from these regularity estimates we obtain estimates on the convergence of the solutions of the difference equations a theory of pseudo difference operators with order is used to prove the regularity results we also comment on factors to be considered in choosing a difference scheme for computations

model adaptivity for goal oriented inference using adjoints abstract an inverse problem seeks to infer unknown model parameters using observed data we consider a goal oriented inverse problem where the goal of inferring parameters is to use them in predicting a quantity of interest qoi recognizing that multiple models of varying fidelity and computational cost may be available to describe the physical system we formulate a goal oriented model adaptivity approach that leverages multiple models while controlling the error in the qoi prediction in particular we adaptively form a mixed fidelity model by using models of different levels of fidelity in different subregions of the domain taking the solution of the inverse problem with the highest fidelity model as our reference qoi prediction we derive an adjoint based third order estimate for the qoi error from using a lower fidelity model localization of this error then guides the formation of mixed fidelity models we demonstrate the method for example problems described by convection diffusion reaction models for these examples our mixed fidelity models use the high fidelity model over only a small portion of the domain but result in qoi estimates with small relative errors we also demonstrate that the mixed fidelity inverse problems can be cheaper to solve and less sensitive to the initial guess than the high fidelity inverse problems

an adaptive strategy for the control of modeling error in two dimensional atomic to continuum coupling simulations an adaptive approach to control modeling error in multiscale simulations that involve molecular and continuum scales is presented the modeling error is defined as the difference between the solution of a reference particle model which is considered intractable in practice and the solution of a manageable multiscale surrogate problem based on the arlequin framework the method relies on computable error estimates of the modeling error in specific outputs of interest that require the solution of an adjoint problem these are the so called goal oriented error estimates which are used to adapt the surrogate model i e to find the optimal configuration of the overlap region between the molecular and continuum models in order to deliver approximations of quantities of interest within some preset accuracy performance of the adaptive strategy is demonstrated on two dimensional numerical examples

the steep nekhoroshev s theorem revising nekhoroshev s geometry of resonances we provide a fully constructive and quantitative proof of nekhoroshev s theorem for steep hamiltonian systems proving in particular that the exponential stability exponent can be taken to be 1 2n alpha 1 cdots alpha n 2 alpha i s being nekhoroshev s steepness indices and n ge 3 the number of degrees of freedom on the base of a heuristic argument we conjecture that the new stability exponent is optimal

property t discrete quantum groups and subfactors with triangle presentations abstract kazhdan s property t has been studied for several discrete group like structures including standard invariants of jones subfactors and discrete quantum groups we prove a uk type spectral gap criterion for property t in this setting this allows us to construct a family of property t discrete quantum groups as well as subfactor standard invariants given by generators and a triangle presentation these are the first examples of property t discrete quantum groups that are not directly related to a discrete group with property t

some estimates for h p k refinement in isogeometric analysis in this paper we propose a theoretical study of the approximation properties of nurbs spaces which are used in isogeometric analysis we obtain error estimates that are explicit in terms of the mesh size h the degree p and the global regularity measured by the parameter k our approach covers the approximation with global regularity from c 0 up to c k 1 with 2k 1 p notice that the interesting case of higher regularity up to k p is still open however our results give an indication of the role of the smoothness k in the approximation properties and offer a first mathematical justification of the potential of isogeometric analysis based on globally smooth nurbs

drinfeld center and representation theory for monoidal categories motivated by the relation between the drinfeld double and central property t for quantum groups given a rigid c tensor category mathcal c and a unitary half braiding on an ind object we construct a representation of the fusion algebra of mathcal c this allows us to present an alternative approach to recent results of popa and vaes who defined c algebras of monoidal categories and introduced property t for them as an example we analyze categories mathcal c of hilbert bimodules over a ii1 factor we show that in this case the drinfeld center is monoidally equivalent to a category of hilbert bimodules over another ii1 factor obtained by the longo rehren construction as an application we obtain an alternative proof of the result of popa and vaes stating that property t for the category defined by an extremal finite index subfactor n subset m is equivalent to popa s property t for the corresponding se inclusion of ii1 factors in the last part of the paper we study muger s notion of weakly monoidally morita equivalent categories and analyze the behavior of our constructions under the equivalence of the corresponding drinfeld centers established by schauenburg in particular we prove that property t is invariant under weak monoidal morita equivalence

analysis of an energy based atomistic continuum approximation of a vacancy in the 2d triangular lattice we present an a priori error analysis of a practical energy based atomistic continuum coupling method a v shapeev multiscale model simul 9 3 905 932 2011 in two dimensions for finite range pair potential interactions in the presence of vacancy defects we establish first order consistency and stability of the method from which we obtain a priori error estimates in the h1 norm and the energy in terms of the mesh size and the smoothness of the atomistic solution in the continuum region from these error estimates we obtain heuristics for an optimal choice of the atomistic region and the finite element mesh as well as convergence rates in terms of the number of degrees of freedom our analytical predictions are supported by extensive numerical tests

finite element analysis of cauchy born approximations to atomistic models this paper is devoted to a new finite element consistency analysis of cauchy born approximations to atomistic models of crystalline materials in two and three space dimensions through this approach new atomistic cauchy born models are introduced and analyzed these intermediate models can be seen as first level atomistic quasicontinuum approximations in the sense that they involve only short range interactions the analysis and the models developed herein are expected to be useful in the design of coupled atomistic continuum methods in more than one dimension taking full advantage of the symmetries of the atomistic lattice we show that the consistency error of the models considered both in energies and in dual w1 p type norms is mathcal o varepsilon 2 where varepsilon denotes the interatomic distance in the lattice

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

the schwarz alternating method in solid mechanics abstract we advance the schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics we prove that the schwarz alternating method converges to the solution of the problem on the entire domain and that the convergence rate is geometric provided that each of the subdomain problems is well posed i e their corresponding energy density functions are quasi convex it is shown that the use of a newton type method for the solution of the resultant nonlinear system leads to two kinds of block linearized systems depending on the treatment of the dirichlet boundary conditions the first kind is a symmetric block diagonal linear system in which each diagonal block is the tangent stiffness of each subdomain i e the off diagonal blocks are all zero and the coupling terms appear only on the right hand side the second kind is a nonsymmetric block system with off diagonal coupling terms several variants of the schwarz alternating method are proposed for the first kind of linear system including one in which the schwarz alternating iterations and the newton iterations are combined into a single scheme this version of the method is particularly attractive as it lends itself to a minimally intrusive implementation into existing finite element codes finally we demonstrate the performance of the proposed variants of the schwarz alternating method on several one dimensional and three dimensional examples

the planetary n body problem symplectic foliation reductions and invariant tori the 6n dimensional phase space of the planetary 1 n body problem after the classical reduction of the total linear momentum is shown to be foliated by symplectic leaves of dimension 6n 2 invariant for the planetary hamiltonian mathcal h such foliation is described by means of a new global set of darboux coordinates related to a symplectic partial reduction of rotations on each symplectic leaf mathcal h has the same form and it is shown to preserve classical symmetries further sets of darboux coordinates may be introduced on the symplectic leaves so as to achieve a complete total reduction of rotations next by explicit computations it is shown that in the reduced settings certain degeneracies are removed in particular full torsion is checked both in the partially and totally reduced settings as a consequence a new direct proof of arnold s theorem arnold in russ math surv 18 6 85 191 1963 on the stability of planetary system both in the partially and in the totally reduced setting is easily deduced producing diophantine lagrangian invariant tori of dimension 3n 1 and 3n 2 finally elliptic lower dimensional tori bifurcating from the secular equilibrium are easily obtained

ccap for universal discrete quantum groups we show that the discrete duals of the free orthogonal quantum groups have the haagerup property and the completely contractive approximation property analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite dimensional c algebras the proof relies on the monoidal equivalence between free orthogonal quantum groups and su q 2 quantum groups on the construction of a sufficient supply of bounded central functionals for su q 2 quantum groups and on the free product techniques of ricard and xu our results generalize previous work in the kac setting due to brannan on the haagerup property and due to the second author on the ccap

stability instability and error of the force based quasicontinuum approximation due to their algorithmic simplicity and high accuracy force based model coupling techniques are popular tools in computational physics for example the force based quasicontinuum qcf approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model in this paper we present a detailed stability and error analysis of this method our optimal order error estimates provide a theoretical justification for the high accuracy of the qcf approximation they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region the main challenge we need to overcome is the fact that the linearized qcf operator is typically not positive definite moreover we prove that no uniform inf sup stability condition holds for discrete versions of the w1 p w1 q duality pairing with 1 p 1 q 1 if 1 p however we were able to establish an inf sup stability condition for a discrete version of the w1 w1 1 duality pairing which leads to optimal order error estimates in a discrete w1 norm

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

stability of a force based hybrid method with planar sharp interface we study a force based hybrid method that couples an atomistic model with a cauchy born elasticity model with sharp transition interface we identify stability conditions that guarantee the convergence of the hybrid scheme to the solution of the atomistic model with second order accuracy as the ratio between lattice parameter and the characteristic length scale of the deformation tends to zero convergence is established for hybrid schemes with planar sharp interface for systems without defects with general finite range atomistic potential and simple lattice structure the key ingredients of the proof are regularity and stability analysis of elliptic systems of difference equations we apply the results to atomistic to continuum scheme for a two dimensional triangular lattice with planar interface

coupled atomistic continuum simulations using arbitrary overlapping domains we present a formulation for coupling atomistic and continuum simulation methods for application to both quasistatic and dynamic analyses in our formulation a coarse scale continuum discretization is assumed to cover all parts of the computational domain with atomistic crystals introduced only in regions of interest the geometry of the discretization and crystal are allowed to overlap arbitrarily our approach uses interpolation and projection operators to link the kinematics of each region which are then used to formulate a system potential energy from which we derive coupled expressions for the forces acting in each region a hyperelastic constitutive formulation is used to compute the stress response of the defect free continuum with constitutive properties derived from the cauchy born rule a correction to the cauchy born rule is introduced in the overlap region to minimize fictitious boundary effects features of our approach will be demonstrated with simulations in one two and three dimensions

from molecular models to continuum mechanics we present here a limiting process allowing us to write some continuum mechanics models as a natural asymptotic of molecular models the approach is based on the hypothesis that the macroscopic displacement is equal to the microscopic one we carry out the corresponding calculations in the case of two body energies including higher order terms and also in the case of thomas fermi type models

construction and sharp consistency estimates for atomistic continuum coupling methods with general interfaces a two dimensional model problem we present a new variant of the geometry reconstruction approach for the formulation of atomistic continuum coupling methods a c methods for many body nearest neighbor interactions on the two dimensional triangular lattice we show that patch test consistent a c methods can be constructed for arbitrary interface geometries moreover we prove that all methods within this class are first order consistent at the atomistic continuum interface and second order consistent in the interior of the continuum region

justification of the cauchy born approximation of elastodynamics we present sharp convergence results for the cauchy born approximation of general classical atomistic interactions for static problems with small data and for dynamic problems on a macroscopic time interval

sharp nekhoroshev estimates for the three body problem around periodic orbits abstract we construct a local nekhoroshev like result of stability with sharp constants for the planar three body problem both in the planetary and in the restricted circular case by using the periodic averaging technique our constructions can be generalized to any near integrable hamiltonian system whose unperturbed hamiltonian is quasi convex the dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account this allows for a deep analytical understanding of the limits of such techniques in insuring nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases finally two examples with concrete values are considered one for the planetary case and one for the restricted case

bridging techniques in a multi scale modeling of pattern formation abstract bridging techniques between microscopic and macroscopic models are discussed in the case of wrinkling analysis the considered macroscopic models are related to envelope equations of ginzburg landau type but generally they are not valid up to the boundary to this end a multi scale approach is considered the reduced model is implemented in the bulk while the full model is applied near the boundary and these two models are coupled with the arlequin method ben dhia 1998 this paper focuses on the definition of the coupling model and the transition between two scales especially a new nonlocal bridging technique is presented and compared with another recent one hu et al 2011 the present method can also be seen as a guide for coupling techniques involving other reduced order models

electronic structure of smoothly deformed crystals cauchy born rule for the nonlinear tight binding model the electronic structure of a smoothly deformed crystal is analyzed using a minimalist model in quantum many body theory the nonlinear tight binding model an extension of the classical cauchy born rule for crystal lattices is established for the electronic structure under sharp stability conditions a nonlinear elasticity model is rigorously derived the onset of instability is briefly examined 2010 wiley periodicals inc

annular representation theory for rigid c tensor categories abstract we define annular algebras for rigid c tensor categories providing a unified framework for both ocneanu s tube algebra and jones affine annular category of a planar algebra we study the representation theory of annular algebras and show that all sufficiently large full annular algebras for a category are isomorphic after tensoring with the algebra of matrix units with countable index set hence have equivalent representation theories annular algebras admit a universal c algebra closure analogous to the universal c algebra for groups these algebras have interesting corner algebras indexed by some set of isomorphism classes of objects which we call centralizer algebras the centralizer algebra corresponding to the identity object is canonically isomorphic to the fusion algebra of the category and we show that the admissible representations of the fusion algebra of popa and vaes are precisely the restrictions of arbitrary non degenerate representations of full annular algebras this allows approximation and rigidity properties defined for categories by popa and vaes to be interpreted in the context of annular representation theory this perspective also allows us to define higher weight approximation properties based on other centralizer algebras of an annular algebra using the analysis of annular representations due to jones and reznikoff we identify all centralizer algebras for the tlj categories for 2

a continuum to atomistic bridging domain method for composite lattices the bridging domain method is an overlapping domain decomposition approach for coupling finite element continuum models and molecular mechanics models in this method the total energy is decomposed into atomistic and continuum parts by complementary weight functions applied to each part of the energy in the coupling domain to enforce compatibility the motions of the coupled atoms are constrained by the continuum displacement field using lagrange multipliers for composite lattices this approach is suboptimal because the internal modes of the lattice are suppressed by the homogeneous continuum displacement field in the coupling region to overcome this difficulty we present a relaxed bridging domain method in this method the atom set is divided into primary and secondary atoms the relative motions between them are often called the internal modes only the primary atoms are constrained in the coupling region which succeed in allowing these internal modes to fully relax several one and two dimensional examples are presented which demonstrate improved accuracy over the standard bridging domain method copyright 2009 john wiley sons ltd

weakly nonlinear schrodinger equation with random initial data it is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data while for the general case a proof of the kinetic limit remains open we report on first progress as wave equation we consider the nonlinear schrodinger equation discretized on a hypercubic lattice since this is a hamiltonian system a natural choice of random initial data is distributing them according to the corresponding gibbs measure with a chemical potential chosen so that the gibbs field has exponential mixing the solution t x of the nonlinear schrodinger equation yields then a stochastic process stationary in x d and t if denotes the strength of the nonlinearity we prove that the space time covariance of t x has a limit as 0 for t 2 with fixed and sufficiently small the limit agrees with the prediction from kinetic theory

goal oriented adaptive mesh refinement for the quasicontinuum approximation of a frenkel kontorova model abstract the quasicontinuum approximation e b tadmor m ortiz r phillips quasicontinuum analysis of defects in solids philos mag a 73 6 1996 1529 1563 is a method to reduce the atomistic degrees of freedom of a crystalline solid by piecewise linear interpolation from representative atoms that are nodes for a finite element triangulation in regions of the crystal with a highly nonuniform deformation such as around defects every atom must be a representative atom to obtain sufficient accuracy but the mesh can be coarsened away from such regions to remove atomistic degrees of freedom while retaining sufficient accuracy we present an error estimator and a related adaptive mesh refinement algorithm for the quasicontinuum approximation of a generalized frenkel kontorova model that enables a quantity of interest to be efficiently computed to a predetermined accuracy

a coupled molecular dynamics and extended finite element method for dynamic crack propagation a multiscale method is presented which couples a molecular dynamics approach for describing fracture at the crack tip with an extended finite element method for discretizing the remainder of the domain after recalling the basic equations of molecular dynamics and continuum mechanics the discretization is discussed for the continuum subdomain where the partition of unity property of finite element shape functions is used since in this fashion the crack in the wake of its tip is naturally modelled as a traction free discontinuity next the zonal coupling method between the atomistic and continuum models is recapitulated finally it is discussed how the stress has been computed in the atomic subdomain and a two dimensional computation is presented of dynamic fracture using the coupled model the result shows multiple branching which is reminiscent of recent results from simulations on dynamic fracture using cohesive zone models

theory based benchmarking of the blended force based quasicontinuum method we formulate an atomistic to continuum coupling method based on blending atomistic and continuum forces our precise choice of blending mechanism is informed by theoretical predictions we present a range of numerical experiments studying the accuracy of the scheme focusing in particular on its stability these experiments confirm and extend the theoretical predictions and demonstrate a superior accuracy of b qcf over energy based blending schemes

a priori error analysis of two force based atomistic continuum models of a periodic chain the force based quasicontinuum qcf approximation is a non conservative atomistic continuum hybrid model for the simulation of defects in crystals we present an a priori error analysis of the qcf method applied to a one dimensional periodic chain that is valid for an arbitrary interaction range large deformations and takes coarse graining into account our main tool in this analysis is a new concept of atomistic stress moreover we formulate a new atomistic continuum coupling mechanism based on coupling stresses instead of forces and extend the a priori analysis to this new method we show that the new method has several theoretical advantages over the original qcf method

an optimal order error analysis of the one dimensional quasicontinuum approximation we derive a model problem for quasicontinuum approximations that allows a simple yet insightful analysis of the optimal order convergence rate in the continuum limit for both the energy based quasicontinuum approximation and the quasi nonlocal quasicontinuum approximation for simplicity the analysis is restricted to the case of second neighbor interactions and is linearized about a uniformly stretched reference lattice the optimal order error estimates for the quasi nonlocal quasicontinuum approximation are given for all strains up to the continuum limit strain for fracture the analysis is based on an explicit treatment of the coupling error at the atomistic to continuum interface combined with an analysis of the error due to the atomistic and continuum schemes using the stability of the quasicontinuum approximation

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

quantum groups property t and weak mixing for second countable discrete quantum groups and more generally second countable locally compact quantum groups with trivial scaling group we show that property t is equivalent to every weakly mixing unitary representation not having almost invariant vectors this is a generalization of a theorem of bekka and valette from the group setting and was previously established in the case of low dual by daws skalski and viselter our approach uses spectral techniques and is completely different from those of bekka valette and daws skalski viselter by a separate argument we furthermore extend the result to second countable nonunimodular locally compact quantum groups which are shown in particular not to have property t generalizing a theorem of fima from the discrete setting we also obtain quantum group versions of characterizations of property t of kerr and pichot in terms of the baire category theory of weak mixing representations and of connes and weiss in terms of the prevalence of strongly ergodic actions

atomistic continuum blending with ghost force correction we combine the ideas of atomistic continuum energy blending and ghost force correction to obtain an energy based atomistic continuum coupling scheme which has for a range of benchmark problems the same convergence rates as optimal force based coupling schemes we present the construction of this new scheme numerical results exploring its accuracy in comparison with established schemes and a rigorous error analysis for an instructive special case

examples of weakly amenable discrete quantum groups abstract we prove that the free orthogonal and free unitary quantum groups f o n and f u n are weakly amenable and that their cowling haagerup constant is equal to 1 this is achieved by estimating the completely bounded norm of the projections on the coefficients of irreducible representations of their compact duals an argument of monoidal equivalence then allows us to extend this result to quantum automorphism groups of finite spaces and even yields some examples of weakly amenable non unimodular discrete quantum groups with the haagerup property

an analysis of node based cluster summation rules in the quasicontinuum method we investigate two examples of node based cluster summation rules that have been proposed for the quasicontinuum qc method a force based approach and an energy based approach which is a generalization of the nonlocal qc method we show that even for the case of nearest neighbor interaction in a one dimensional periodic chain both of these approaches create large errors that cannot be removed by increasing the cluster size when used with graded and more generally nonsmooth meshes we offer some suggestions for how the accuracy of cluster summation rules may be improved

kinetic limit for wave propagation in a random medium we study crystal dynamics in the harmonic approximation the atomic masses are weakly disordered in the sense that their deviation from uniformity is of the order e the dispersion relation is assumed to be a morse function and to suppress crossed recollisions we then prove that in the limit e 0 the disorder averaged wigner function on the kinetic scale time and space of order e 1 is governed by a linear boltzmann equation

analysis of energy based blended quasi continuum approximations the development of patch test consistent quasi continuum energies for multidimensional crystalline solids modeled by many body potentials remains a challenge the original quasi continuum energy qce r miller and e tadmor model simul mater sci eng 17 2009 053001 has been implemented for many body potentials in two and three space dimensions but it is not patch test consistent we propose that by blending the atomistic and corresponding cauchy born continuum models of qce in an interfacial region with thickness of a small number k of blended atoms a general blended quasi continuum energy bqce can be developed with the potential to significantly improve the accuracy of qce near lattice instabilities such as dislocation formation and motion in this paper we give an error analysis of the blended quasi continuum energy bqce for a periodic one dimensional chain of atoms with next nearest neighbor interactions our analysis includes the optimization of the blending function for an improved convergence rate we show that the ell 2 strain error for the nonblended qce energy which has low order o varepsilon 1 2 where varepsilon is the atomistic length scale m dobson and m luskin siam j numer anal 47 2009 pp 2455 2475 p ming and j z yang multiscale model simul 7 2009 pp 1838 1875 can be reduced by a factor of k 3 2 for an optimized blending function where k is the number of atoms in the blending region the qce energy has been further shown to suffer from a o 1 error in the critical strain at which the lattice loses stability m dobson m luskin and c ortner j mech phys solids 58 2010 pp 1741 1757 we prove that the error in the critical strain of bqce can be reduced by a factor of k 2 for an optimized blending function thus demonstrating that the bqce energy for an optimized blending function has the potential to give an accurate approximation of the deformation near lattice instabilities such as crack growth

on the convergence of adaptive nonconforming finite element methods for a class of convex variational problems we formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems the class of minimization problems we admit includes highly singular problems for which no euler lagrange equation or inequality is available as a consequence our arguments only use the structure of the energy functional we are nevertheless able to prove convergence of an adaptive algorithm using even refinement indicators that are not reliable error indicators

discrete crystal elasticity and discrete dislocations in crystals this article is concerned with the development of a discrete theory of crystal elasticity and dislocations in crystals the theory is founded upon suitable adaptations to crystal lattices of elements of algebraic topology and differential calculus such as chain complexes and homology groups differential forms and operators and a theory of integration of forms in particular we define the lattice complex of a number of commonly encountered lattices including body centered cubic and face centered cubic lattices we show that material frame indifference naturally leads to discrete notions of stress and strain in lattices lattice defects such as dislocations are introduced by means of locally lattice invariant but globally incompatible eigendeformations the geometrical framework affords discrete analogs of fundamental objects and relations of the theory of linear elastic dislocations such as the dislocation density tensor the equation of conservation of burgers vector kroner s relation and mura s formula for the stored energy we additionally supply conditions for the existence of equilibrium displacement fields we show that linear elasticity is recovered as the limit of harmonic lattice statics as the lattice parameter becomes vanishingly small we compute the limit of dilute dislocation distributions of dislocations and we show that the theory of continuously distributed linear elastic dislocations is recovered as the limit of the stored energy as the lattice parameter and burgers vectors become vanishingly small

a posteriori error estimation and adaptive algorithm for atomistic continuum coupling in two dimensions atomistic continuum coupling methods aim to achieve optimal balance between accuracy and efficiency adaptivity is the key for the efficient implementation of such methods in this paper we carry

the baum connes conjecture for free orthogonal quantum groups we prove an analogue of the baum connes conjecture for free orthogonal quantum groups more precisely we show that these quantum groups have a element and that 1 it follows that free orthogonal quantum groups are k amenable we compute explicitly their k theory and deduce in the unimodular case that the corresponding reduced c algebras do not contain nontrivial idempotents our approach is based on the reformulation of the baum connes conjecture by meyer and nest using the language of triangulated categories an important ingredient is the theory of monoidal equivalence of compact quantum groups developed by bichon de rijdt and vaes this allows us to study the problem in terms of the quantum group suq 2 the crucial part of the argument is a detailed analysis of the equivariant kasparov theory of the standard podle

analysis of boundary conditions for crystal defect atomistic simulations numerical simulations of crystal defects are necessarily restricted to finite computational domains supplying artificial boundary conditions that emulate the effect of embedding the defect in an effectively infinite crystalline environment this work develops a rigorous framework within which the accuracy of different types of boundary conditions can be precisely assessed we formulate the equilibration of crystal defects as variational problems in a discrete energy space and establish qualitatively sharp regularity estimates for minimisers using this foundation we then present rigorous error estimates for i a truncation method dirichlet boundary conditions ii periodic boundary conditions iii boundary conditions from linear elasticity and iv boundary conditions from nonlinear elasticity numerical results confirm the sharpness of the analysis

on the l2 and the h1 couplings for an overlapping domain decomposition method using lagrange multipliers in this paper a comparison of the l 2 and the h couplings is made for an overlapping domain decomposition method using lagrange multipliers the analysis of the local equations arising from the formulation of the coupling of two mechanical models shows that continuous weight functions are required for the l 2 coupling term whereas both discontinuous and continuous weight functions can be used for the h 1 coupling the choice of the lagrange multiplier space is discussed and numerically studied the paper ends with some numerical examples of an end loaded cantilever beam and a cracked plate under tension and shear it is shown that the continuity enforced with the h 1 coupling leads to a link with a flexibility that can be beneficial for coupling a very coarse mesh with a very fine one to limit the effect of the volume coupling on the global response a narrow coupling zone is recommended in this case volume coupling tends to a surface coupling especially with a l 2 coupling

a priori and a posteriori w 1 infty error analysis of a qc method for complex lattices in this paper we derive regularity results for equilibria of multilattices under an external force and prove a priori and a posteriori error estimates for a multiscale numerical method for computing such equilibria the estimates are derived in a w 1 infty norm in one space dimension one of the features of our analysis is that we establish an equivalent way of formulating the coarse grained problem which greatly simplifies derivation of the error bounds both a priori and a posteriori we illustrate our error estimates with numerical experiments

finite difference approximations of generalized solutions we consider finite difference schemes approximating the dirichlet problem for the poisson equation we provide scales of error estimates in discrete sobolev like norms assuming that the generalized solution belongs to a nonnegative order sobolev space

on an h type mesh refinement strategy based on minimization of interpolation errors abstract an adaptive finite element method is proposed which involves an automatic mesh refinement in areas of the mesh where local errors are determined to exceed a pre assigned limit the estimation of local errors is based on interpolation error bounds and extraction formulas for highly accurate estimates of second derivatives applications to several two dimensional model problems are discussed the results indicate that the method can be very effective for both regular problems and problems with strong singularities

convergence of a force based hybrid method in three dimensions we study a force based hybrid method that couples an atomistic model with the cauchy born elasticity model we show that the proposed scheme converges to the solution of the atomistic model with second order accuracy since the ratio between lattice parameter and the characteristic length scale of the deformation tends to 0 convergence is established for the three dimensional system without defects with general finite range atomistic potential and simple lattice structure the proof is based on consistency and stability analysis general tools for stability analysis are developed in the framework opseudodifference operators in arbitrary dimensions 2012 wiley periodicals inc

irreducible unitary representations of quantum lorentz group a complete classification of irreducible unitary representations of a one parameter deformations q l 2 c 0 q 1 ofsl 2 c is given it shows that in spite of a popular belief the representation theory fors q l 2 c is not a smooth deformation of the one forsl 2 c

cauchy born rule and the stability of crystalline solids static problems we study the connection between atomistic and continuum models for the elastic deformation of crystalline solids at zero temperature we prove under certain sharp stability conditions that the correct nonlinear elasticity model is given by the classical cauchy born rule in the sense that elastically deformed states of the atomistic model are closely approximated by solutions of the continuum model with stored energy functionals obtained from the cauchy born rule the analysis is carried out for both simple and complex lattices and for this purpose we develop the necessary tools for performing asymptotic analysis on such lattices our results are sharp and they also suggest criteria for the onset of instabilities of crystalline solids

existence and stability of a screw dislocation under anti plane deformation we formulate a variational model for a geometrically necessary screw dislocation in an anti plane lattice model at zero temperature invariance of the energy functional under lattice symmetries renders the problem non coercive nevertheless by establishing coercivity with respect to the elastic strain and a concentration compactness principle we prove the existence of a global energy minimizer and thus demonstrate that dislocations are globally stable equilibria within our model

analysis of blended atomistic continuum hybrid methods we present a comprehensive error analysis of two prototypical atomistic to continuum coupling methods of blending type the energy based and the force based quasicontinuum methods our results are valid in two and three dimensions for finite range many body interactions e g eam type and in the presence of lattice defects we consider point defects and dislocations the two key ingredients in the analysis are 1 new force and energy consistency error estimates and 2 a new technique for proving energy norm stability of a c couplings that requires only the assumption that the exact atomistic solution is a stable equilibrium

comparison of several staggered atomistic to continuum concurrent coupling strategies abstract in this contribution several staggered schemes used to couple continuum mechanics cm and molecular mechanics mm are proposed the described approaches are based on the atomistic to continuum correspondence obtained by spatial averaging in the spirit of irving and kirkwood and noll similarities between this and other concurrent coupling schemes are indicated thus providing a broad overview of different approaches in the field the schemes considered here are decomposed into the surface type displacement or traction boundary conditions and the volume type the latter restricts the continuum displacement field and possibly its gradient in some sense to the atomistic discrete displacements using lagrange multipliers a large strain cm formulation incorporating lagrange multipliers and a strategy to solve the resulting coupled linear system using an iterative solver is presented finally the described coupling methods are numerically examined using two examples uniaxial deformation and a plate with a hole relaxed under surface tension accuracy and convergence rates of each method are reported it was found that the displacement surface coupling scheme and the lagrangian volume scheme based on either discrete displacements or the h 1 norm derived from continuous displacement fields provide the best performance

comparison of unitary duals of drinfeld doubles and complex semisimple lie groups we determine a substantial part of the unitary representation theory of the drinfeld double of a q deformation of a compact lie group in terms of the complexification of the compact lie group using this we show that the dual of every q deformation of a higher rank compact lie group has central property t we also determine the unitary dual of sl q n mathbb c

a bridging technique to analyze the influence of boundary conditions on instability patterns in this paper we present a new numerical technique that permits to analyse the effect of boundary conditions on the appearance of instability patterns envelope equations of landau ginzburg type are classically used to predict pattern formation but it is not easy to associate boundary conditions for these macroscopic models indeed envelope equations ignore boundary layers that can be important for instance in cases where the instability starts first near the boundary in this work the full model is considered close to the boundary an envelope equation in the core and they are bridged by the arlequin method 1 simulation results are presented for the problem of buckling of long beams lying on a non linear elastic foundation

finite element approximations of nonlinear elastic waves in this paper we study finite element methods for a class of problems of nonlinear elastodynamics we discretize the equations in space using galerkin methods for the temporal discretization the construction of our schemes is based on rational approximations of cos x and ex we analyze semidiscrete as well as secondand fourth order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal order l2 error estimates for some schemes a taylor type technique is used so that only linear systems of equations need be solved at each time step in the proofs we need various estimates for a nonlinear elliptic projection the proofs of which are also established in the paper

the arlequin method as a flexible engineering design tool by superposing and gluing models the arlequin method offers an extended modelling framework for the design of engineering structures this paper aims at developing the numerical aspects of the approach and at showing how it can be used with great flexibility and in a consistent manner to change locally a global mechanical model the capabilities of the arlequin method and the effectiveness of the implemented numerical tools are exemplified by 1 d 2 d and 3 d numerical applications

metastability and dynamics of discrete topological singularities in two dimensions a γ convergence approach this paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions based on convergence we consider discrete systems described by scalar functions defined on a square lattice and governed by periodic interaction potentials our main motivation comes from xy spin systems described by the phase parameter and screw dislocations described by the displacement function for these systems we introduce a discrete notion of vorticity as the lattice spacing tends to zero we derive the first order limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices as a byproduct of this analysis we show that such systems exhibit increasingly many metastable configurations of singularities therefore we propose a variational approach to the depinning and dynamics of discrete vortices based on minimizing movements we show that letting first the lattice spacing and then the time step of the minimizing movements tend to zero the vortices move according with the gradient flow of the renormalized energy as in the continuous ginzburg landau framework

force based atomistic continuum blending for multilattices we formulate the blended force based quasicontinuum method for multilattices and develop rigorous error estimates in terms of the approximation parameters choice of atomistic region blending region and continuum finite element mesh balancing the approximation parameters yields a convergent atomistic continuum multiscale method for multilattices with point defects including a rigorous convergence rate in terms of the computational cost the analysis is illustrated with numerical results for a stone wales defect in graphene