integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity this article is devoted to the study of the asymptotic behavior of a class of energies defined on stochastic lattices under polynomial growth assumptions we prove that the energy functionals f varepsilon stored in the deformation of an varepsilon scaling of a stochastic lattice converge to a continuous energy functional when varepsilon goes to zero in particular the limiting energy functional is of integral type and deterministic if the lattice is ergodic we also generalize to systems and nonlinear settings well known results on stochastic homogenization of discrete elliptic equations as an application of the main result we prove the convergence of a discrete model for rubber towards the nonlinear theory of continuum mechanics we finally address some mechanical properties of the limiting models such as frame invariance isotropy and natural states

harnack inequalities and discrete continuous error estimates for a chain of atoms with two body interactions we consider deformations in r 3 of an infinite linear chain of atoms where each atom interacts with all others through a two body potential we compute the effect of an external force applied to the chain at equilibrium the positions of the particles satisfy an euler lagrange equation for large classes of potentials we prove that every solution is well approximated by the solution of a continuous model we establish an error estimate between the discrete and the continuous solution based on a harnack lemma of indepen dent interest finally we apply our results to some lennard jones potentials

effective cohesive behavior of layers of interatomic planes a simple model of cleavage in brittle crystals consists of a layer of material containing n atomic planes separating in accordance with an interplanar potential under the action of an opening displacement prescribed on the boundary of the layer the problem addressed in this work concerns the characterization of the constrained minima of the energy en of the layer as a function of as n becomes large these minima determine the effective or macroscopic cohesive law of the crystal the main results presented in this communication are i the computation of the limit e0 of en as n ii the characterization of the minimum values of e0 as a function of the macroscopic opening displacement iii a proof of uniform convergence of the minima of en for the case of nearest neighbor interactions and iv a proof of uniform convergence of the derivatives of en or tractions in the same case the scaling on which the present convergence analysis is based has the effect of separating the bulk and surface contributions to the energy it differs crucially from other scalings employed in the past in that it renders both contributions of the same order

finding einstein solvmanifolds by a variational method we use a variational approach to prove that any nilpotent lie algebra having a codimension one abelian ideal and anyone of dimension leq 5 admits a rank one solvable extension which can be endowed with an einstein left invariant riemannian metric a curve of 8 dimensional einstein solvmanifolds is also given

the energy of some microscopic stochastic lattices we introduce a notion of energy for some microscopic stochastic lattices such lattices are broad generalizations of simple periodic lattices for which the question of the definition of an energy was examined in a series of previous works 14 18 note that slightly more general deterministic geometries were also considered in 6 these lattices are involved in the modelling of materials whose microscopic structure is a perturbation in a sense made precise in the article of the periodic structure of a perfect crystal the modelling considered here is either a classical modelling where the sites of the lattice are occupied by ball like atomic systems that interact by pair potentials or a quantum modelling where the sites are occupied by nuclei equipped with an electronic structure spread all over the ambient space the corresponding energies for the infinite stochastic lattices are derived consistently with truncated systems of finite size by application of a thermodynamic limit process subsequent works 7 8 will be devoted to the macroscopic limits of the energies of such microscopic lattices thereby extending to a stochastic context the results of 4 5 such convergences in a stochastic setting in dimension 1 have been studied in 21 22 we will also study in 8 some variants and extensions of the stationary setting presented here

lagrangian aspects of the kirchhoff elastic rod the relation between euler s planar elastic curves and vortex filaments evolving by the localized induction equation lie of hydrodynamics was discovered by hasimoto in 1971 basic facts about an integrable case of kirchhoff elastic rods are described here which amplify the connection between the variational problem for rods and the soliton equation lie in particular it is shown that the centerline of the kirchhoff rod is an equilibrium for a linear combination of the first three conserved hamiltonians in the lie hierarchy

saint venant s problem and semi inverse solutions in nonlinear elasticity saint venant s problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross sectional resultants of force and moment using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite dimensional manifold in particular the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations moreover the set of semi inverse solutions can be analyzed locally

ground states of the 2d sticky disc model fine properties and n 3 4 law for the deviation from the asymptotic wulff shape we investigate ground state configurations for a general finite number n of parti cles of the heitmann radin sticky disc pair potential model in two dimensions exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic wulff shape which is a regular hexagon there are arbitrarily large n with ground state configura tions deviating from the nearest regular hexagon by a number of n 3 4 particles we also prove that for any n and any ground state configuration this deviation is bounded above by n 3 4 as a consequence we obtain an exact scaling law for the fluctuations about the asymptotic wulff shape in particular our results give a sharp rate of convergence to the limiting wulff shape

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

convergence of equilibria of three dimensional thin elastic beams a convergence result is proved for the equilibrium configurations of a three dimensional thin elastic beam as the diameter h of the cross section goes to zero more precisely we show that stationary points of the nonlinear elastic functional e whose energies per unit cross section are bounded by ch converge to stationary points of the limit of e h this corresponds to a nonlinear one dimensional model for inextensible rods describing bending and torsion effects the proof is based on the rigidity estimate for low energy deformations by friesecke james and muller 4 and on a compensated compactness argument in a singular geometry in addition possible concentration effects of the strain are controlled by a careful truncation argument

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

noncompact homogeneous einstein manifolds attached to graded lie algebras in this paper we study the nilradicals of parabolic subalgebras of semisimple lie algebras and the natural one dimensional solvable extensions of them we investigate the structures curvatures and einstein conditions of the associated nilmanifolds and solvmanifolds we show that our solvmanifold is einstein if the nilradical is two step new examples of einstein solvmanifolds with three step and four step nilradicals are also given

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

atomic to continuum passage for nanotubes a discrete saint venant principle and error estimates we consider general infinite nanotubes of atoms in mathbb r 3 where each atom interacts with all the others through a two body potential at the equilibrium the positions of the atoms satisfy a euler lagrange equation when there are no exterior forces and for a suitable geometry a particular family of nanotubes is the set of perfect nanotubes at the equilibrium when exterior forces are applied on the nanotube we compare the nanotube to nanotubes of the previous family in part i of the paper this quantitative comparison is formulated in our first main result as a discrete saint venant principle as a corollary we also give a liouville classification result our saint venant principle can be derived for a large class of potentials including the lennard jones potential when the perfect nanotubes at the equilibrium are stable the approach is designed to be applicable to nanotubes that can have general shapes like for instance carbon nanotubes or dna under the oversimplified assumption that all the atoms are identical in part ii of the paper we derive from our saint venant principle a macroscopic mechanical model for general nanotubes we prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting but no bending we establish error estimates between the discrete and the continuous solution more precisely we give two error estimates one at the microscopic level and one at the macroscopic level

variational formulation of softening phenomena in fracture mechanics the one dimensional case we show that discrete models of atoms subject to nearest neighbour non linear interactions approximate continua allowing for softening and fracture a detailed study of local minima and stationary points is carried out scale effects are discussed

normal hyperbolicity of center manifolds and saint venant s principle the concept of normal hyperbolicity of center manifolds is generalized to infinite dimensional differential equations in particular to elliptic problems in cylindrical domains it is shown that all solutions u staying close to the center manifold for t l l satisfy an estimate of the form left u t tilde u t right leqslant ce alpha l t where c and are independent of l and is a solution on the center manifold these results are applied to saint venant s principle for the static deformation of nonlinearly elastic prismatic bodies the use of the center manifold permits the effective treatment of the general case of non zero resultant forces and moments acting on each cross section

parabolic subgroups of semisimple lie groups and einstein solvmanifolds in this paper we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple lie algebras these solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type we show that the ricci curvatures of our solvmanifolds coincide with the restrictions of the ricci curvatures of the ambient symmetric spaces consequently all of our solvmanifolds are einstein which provide a large number of new examples of noncompact homogeneous einstein manifolds we also show that our solvmanifolds are minimal but not totally geodesic submanifolds of symmetric spaces

testing the markov property with high frequency data this paper develops a framework to nonparametrically test whether discrete valued irregularly spaced financial transactions data follow a markov process for that purpose we consider a specific optional sampling in which a continuous time markov process is observed only when it crosses some discrete level this framework is convenient for it accommodates the irregular spacing that characterizes transactions data under such an observation rule the current price duration is independent of a previous price duration given the previous price realization a simple nonparametric test then follows by examining whether this conditional independence property holds monte carlo simulations suggest that the asymptotic test has huge size distortions though a bootstrap based variant entails reasonable size and power properties in finite samples as for an empirical illustration we investigate whether bid ask spreads follow markov processes using transactions data from the new york stock exchange the motivation lies on the fact that asymmetric information models of market microstructures predict that the markov property does not hold for the bid ask spread we robustly reject the markov assumption for two out of the five stocks under scrutiny finally it is reassuring that our results are consistent with two alternative measures of asymmetric information

a discrete mechanics approach to the cosserat rod theory part 1 static equilibria a theory of discrete cosserat rods is formulated in the language of discrete lagrangian mechanics by exploiting kirchhoff s kinetic analogy the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the lagrangian function of a dynamical system the equilibrium equations are derived from a variational principle using a formulation that involves null space matrices in this formulation no lagrange multipliers are necessary to enforce orthonormality of the directors noether s theorem relates first integrals of the equilibrium equations to lie group actions on the configuration bundle so called symmetries the symmetries relevant for rod mechanics are frame indifference isotropy and uniformity we show that a completely analogous and self contained theory of discrete rods can be formulated in which the arc length is a discrete variable ab initio in this formulation the potential energy density is defined directly on pairs of points along the arc length of the rod in analogy to veselov s discrete reformulation of lagrangian mechanics a discrete version of noether s theorem then identifies exact first integrals of the discrete equilibrium equations these exact conservation properties confer the discrete solutions accuracy and robustness as demonstrated by selected examples of application

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

on the crystallization of 2d hexagonal lattices it is a fundamental problem to understand why solids form crystals at zero temperature and how atomic interaction determines the particular crystal structure that a material selects in this paper we focus on the zero temperature case and consider a class of atomic potentials v v 2 v 3 where v 2 is a pair potential of lennard jones type and v 3 is a three body potential of stillinger weber type for this class of potentials we prove that the ground state energy per particle converges to a finite value as the number of particles tends to infinity this value is given by the corresponding value for a optimal hexagonal lattice optimized with respect to the lattice spacing furthermore under suitable periodic or dirichlet boundary condition we show that the minimizers do form a hexagonal lattice