exploiting known structures to approximate normal cones the normal cone to a constraint set plays a key role in optimization theory algorithms and applications we consider the question of how to approximate the normal cone to a set under the assumption that the set is provided through an oracle function or collection of oracle functions but contains some exploitable structure we provide a new simplex gradient based approximation technique that works for sets defined through a finite number of oracle based functions we further present novel results showing that under a non degeneracy condition approximating normal cones to intersections of sets is possible by taking sums of approximations finally we provide numerical results that exemplify the accuracy of the simplex gradient approximation when it is applicable and the failure of this technique when a linear independence constraint qualification is not met

cluster expansion for abstract polymer models a new direct proof of convergence of cluster expansions for polymer contour models is given in an abstract setting it does not rely on kirkwood salsburg type equations or combinatorics of trees a distinctive feature is that at all steps the considered clusters contain every polymer at most once

diffusion in energy conserving coupled maps we consider a dynamical system consisting of subsystems indexed by a lattice each subsystem has one conserved degree of freedom energy the rest being uniformly hyperbolic the subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved we prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit

equilibrium phase transitions in coupled map lattices a pedestrian approach a class of piecewise linear coupled map lattices with simple symbolic dynamics is constructed it can be solved analytically in terms of the statistical mechanics of spin lattices the corresponding hamiltonian is written down explicitly in terms of the parameters of the map the approach follows the line of recent mathematical investigations but the presentation is kept elementary so that phase transitions in the dynamical model can be studied in detail although the method works only for map lattices with repelling invariant sets some of the conclusions i e the role of local curvature of the single site map and properties of the nearest neighbour coupling might play an important role for phase transitions in general dynamical systems

the clarke generalized gradient for functions whose epigraph has positive reach we consider the class of continuous functions that map an open set rn to r with an epigraph having locally positive reach with an additional property this class contains all finite convex and c1 1 functions but also ones that are not necessarily lipschitz continuous we provide a representation formula for the clarke generalized gradient of such functions using convex combinations and limits of gradients at differentiability points thus offering an alternative to the well known proximal normal formula by replacing a pointedness assumption by one of positive reach our proof consists of a detailed analysis of singularities using methods taken from both nonsmooth analysis and geometric measure theory and is based on an induction argument as an application we prove for a particular class of hamilton jacobi equations that an a e solution whose hypograph has positive reach and satisfies an additional property is indeed the unique viscosity solution

limit theorems for coupled analytic maps in 23 h h rugh gave a new and simpler method to prove existence of an invariant measure with nice mixing properties for some weakly coupled analytic maps he proved in fact a spectral gap property for a well defined transfer operator we modify this method to construct generalized transfer operators associated to potentials and preserve the spectral gap for small potentials this allows us to prove new limit results for these systems central limit theorem moderate deviations principle and a partial large deviations result these results are available under wide classes of observables and initial measures

coupled map lattices one step forward and two steps back abstract we discuss the general notion of lattice dynamical systems and some recent progress in their studies we suggest also the approach that allows to extract some information on the dynamics of finitely extended real systems from the results obtained on the infinitely extended artificial ones

minimum density hyperplanes associating distinct groups of objects clusters with contiguous regions of high probability density high density clusters is central to many statistical and machine learning approaches to the classification of unlabelled data we propose a novel hyperplane classifier for clustering and semi supervised classification which is motivated by this objective the proposed minimum density hyperplane minimises the integral of the empirical probability density function along it thereby avoiding intersection with high density clusters we show that the minimum density and the maximum margin hyperplanes are asymptotically equivalent thus linking this approach to maximum margin clustering and semi supervised support vector classifiers we propose a projection pursuit formulation of the associated optimisation problem which allows us to nd minimum density hyperplanes efficiently in practice and evaluate its performance on a range of benchmark data sets the proposed approach is found to be very competitive with state of the art methods for clustering and semi supervised classification

a note on an approximate lagrange multiplier rule in this note we show that for a large class of optimization problems the lagrange multiplier rule can be derived from the approximate multiplier rule

on nesterov s nonsmooth chebyshev rosenbrock functions abstract we discuss two nonsmooth functions on r n introduced by nesterov we show that the first variant is partly smooth in the sense of lewis and that its only stationary point is the global minimizer in contrast we show that the second variant has 2 n 1 clarke stationary points none of them local minimizers except the global minimizer but also that its only mordukhovich stationary point is the global minimizer nonsmooth optimization algorithms from multiple starting points generate iterates that approximate all 2 n 1 clarke stationary points not only the global minimizer but it remains an open question as to whether the nonminimizing clarke stationary points are actually points of attraction for optimization algorithms

analyticity of the srb measure of a lattice of coupled anosov diffeomorphisms of the torus we consider the thermodynamic limit of a d dimensional lattice of hyperbolic dynamical systems on the 2 torus interacting via weak and nearest neighbor coupling we prove that the srb measure is analytic in the strength of the coupling the proof is based on symbolic dynamics techniques that allow us to map the srb measure into a gibbs measure for a spin system on a d 1 dimensional lattice this gibbs measure can be studied by an extension decimation of the usual cluster expansion techniques

large displacement analysis of thin walled beams with generic open section a finite element formulation for geometric non linear analysis of elastic straight thin walled beams is presented the updated lagrangian procedure is used for a stiffness matrix derivation to analyse large displacements or buckling behaviour of space frames the stiffness matrix is developed for a generic cross section and is equally efficient for symmetric monosymmetric or asymmetric open sections some non linear strain terms usually neglected in the literature have been considered resulting in a formulation which also includes the initial bimoment and torsion effect for all cross sections analysed the proposed method has been implemented and the results of sample problems are presented

spatio temporal large deviations principle for coupled circle maps we consider the d 1 dimensional an dynamical system constituted by weakly coupled expanding circle maps on zd together with the spatial shifts this viewpoint allows us to use thermodynamic formalism and to describe the asymptotic behavior of the system in this setup we obtain a volume lemma which describes the exponential behavior of the size under lebesgue measure of dynamical balls around any orbit and then a large deviations principle for the empirical measure associated to this dynamical system the proofs are direct we do not use the coding constructed by jiang for such systems

uniqueness of the srb measure for piecewise expanding weakly coupled map lattices in any dimension we prove the existence of a unique srb measure for a wide range of multidimensional weakly coupled map lattices these include piecewise expanding maps with diffusive coupling

uniqueness and exponential decay of correlations for some two dimensional spin lattice systems we consider a two dimensional lattice spin system which naturally arises in dynamical systems called coupled map lattice the configuration space of the spin system is a direct product of mixing subshifts of finite type the potential is defined on the set of all squares in z2 and decays exponentially with the linear size of the square via the polymer expansion technique we prove that for sufficiently high temperatures the limit gibbs distribution is unique and has an exponential decay of correlations

estimating tangent and normal cones without calculus we study simple estimates of the tangent and normal cones to a set with a nonsmooth boundary we seek estimates that are robust relative to the base point and that depend only on collections of nearby points known to lie inside and outside of the set

natural approach for geometric non linear analysis of thin walled frames a new stiffness matrix for geometrically non linear analysis of three dimensional beam columns with bisymmetrical thin walled i type cross sections is presented the displacement field is described in terms of natural and rigid body displacements and both cartesian and natural matrices are derived a novelty of the resulting method is the use of this dual description in all aspects of the analysis including the computer implementation a companion paper1 that introduces the plastic behaviour of the material also justifies the need for this dual approach as a means to separate nodal forces from stress resultants in the cross section of the 3d beam column element the current development uses the updated lagrangian formulation and corrections in the element matrix are made to properly consider the behaviour under finite rotation it is shown that accuracy and convergence of the solution process can be improved if the natural matrices derived here are used in a new force recovery process that is insensitive to rigid body displacements

rigid body considerations for non linear finite element analysis the purpose of the paper is to demonstrate how the concept of rigid body motions can be employed to derive the external stiffness matrix for an initially stressed finite element

a sequential quadratic programming algorithm for nonconvex nonsmooth constrained optimization we consider optimization problems with objective and constraint functions that may be nonconvex and nonsmooth problems of this type arise in important applications many having solutions at points of nondifferentiability of the problem functions we present a line search algorithm for situations when the objective and constraint functions are locally lipschitz and continuously differentiable on open dense subsets of mathbb r n our method is based on a sequential quadratic programming sqp algorithm that uses an ell 1 penalty to regularize the constraints a process of gradient sampling gs is employed to make the search direction computation effective in nonsmooth regions we prove that our sqp gs method is globally convergent to stationary points with probability one and illustrate its performance with a matlab implementation

a simple fem formulation for large deflection 2d frame analysis based on position description this study presents a simple formulation to treat large deflections by the finite element method the proposed formulation does not use the displacement concept it considers position as the main unknown variable of the problem the strain determination is performed directly from the proposed position concept a non dimensional space is created and relative curvature and fibers length are calculated for both reference and deformed configurations and used to directly compute the strain energy at general points this procedure is very simple and presents good results as shown in the example section the simplicity of the resulting formulation may be considered as its main attribute two dimensional frame problems are analyzed by the proposed formulation

numerical results from large deflection beam and frame problems analysed by means of elliptic integrals numerical evaluations of elliptic integral solutions of some large deflection beam and frame problems are presented the values are given in tabular form with up to six significant figures the numerical technique used for evaluating the elliptic integrals is described

two numerical methods for optimizing matrix stability consider the affine matrix family a x a0 m 1 xkak mapping a design vector x r m into the space of n n real matrices we are interested in the question of how to choose x to optimize the stability of the matrix a x a typical motivation is that one wishes to control the stability of the dynamical system a x z a classic example is stabilization by output feedback we take two approaches let x denote the spectral abscissa the largest real part of the eigenvalues of a matrix x as is well known this quantity bounds the asymptotic decay rate of the trajectories of the associated dynamical system z xz our first approach to optimizing stability is to directly minimize the function a x the spectral abscissa x is a continuous but non smooth in fact non lipschitz function of the matrix argument x and finding a global minimizer of a x is hard we introduce a novel random gradient bundle method for approximating local minimizers motivated by recent work on nonsmooth analysis of the function x our second approach is to minimize a related function a x where is a robustness parameter in 0 1 one motivation for the definition of the robust spectral abscissa x is that it bounds transient peaks as well as asymptotic decay of trajectories of xz the function x is lipschitz but non convex for 0 1

clarke subgradients for directionally lipschitzian stratifiable functions using a geometric argument we show that under a reasonable continuity condition the clarke subdifferential of a semi algebraic or more generally stratifiable directionally lipschitzian function admits a simple form the normal cone to the domain and limits of gradients generate the entire clarke subdifferential the characterization formula we obtain unifies various apparently disparate results that have appeared in the literature our techniques also yield a simplified proof that closed semialgebraic functions on rn have a limiting subdifferential graph of uniform local dimension n

ising type and other transitions in one dimensional coupled map lattices with sign symmetry we consider a one dimensional lattice of expanding antisymmetric maps 1 1 1 1 with nearest neighbor diffusive coupling for such systems it is known that if the coupling parameter e is small there is unique stationary in time state which is chaotic in space time a disputed question is whether such systems can exhibit ising type phase transitions as e grows beyond some critical value ec we present results from computer experiments which give definite indication that such a transition takes place the mean square magnetization appears to diverge as e approaches some critical value with a critical exponent around 0 9 we also study other properties of the coupled map system

equilibrium states for random lattice models of hyperbolic type abstract we study the structural stability of random coupled map lattice models of hyperbolic type under certain metrics furthermore by describing the thermodynamic formalism of the underlying random spin lattice system we prove the existence of equilibrium states for equi holder continuous random functions on lattice models under the conditions of random weak interaction and translation invariance and present some partial results on the uniqueness of equilibrium states

convected systems for curved structural elements this paper presents a new approach to the use of the updated lagrangian formulation with convected systems the development of this new method is similar to the updated lagrangian formulation presented by bathe et al 4 the main difference is that here the tensors are written in their components in the curvilinear convected system while bathe ramm and wilson employ cartesian components a special form of the natural approach5 is also introduced in this formulation to simplify the element computations the matrices of a two dimensional curved beam column element are derived as an example of this new formulation finally a numerical evaluation is performed to evaluate critically the new curved element and the method proposed here

dynamical systems on lattices with decaying interaction ii hyperbolic sets and their invariant manifolds abstract this is the second part of the work devoted to the study of maps with decay in lattices here we apply the general theory developed in fontich et al 2011 3 to the study of hyperbolic sets in particular we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map we also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds

scaling behavior of nonhyperbolic coupled map lattices coupled map lattices of nonhyperbolic local maps arise naturally in many physical situations described by discretized reaction diffusion equations or discretized scalar field theories as a prototype for these types of lattice dynamical systems we study diffusively coupled tchebyscheff maps of nth order which exhibit strongest possible chaotic behavior for small coupling constants a we prove that the expectations of arbitrary observables scale with sqrt of a in the low coupling limit contrasting the hyperbolic case which is known to scale with a moreover we prove that there are log periodic oscillations of period ln n2 modulating the sqrt of a dependence of a given expectation value we develop a general 1st order perturbation theory to analytically calculate the invariant one point density show that the density exhibits log periodic oscillations in phase space and obtain excellent agreement with numerical results

floquet spectrum of weakly coupled map lattices we consider weakly coupled analytic expanding circle maps on the lattice d for d 1 with small coupling strength e and summable decay of the two sites coupling we study the spectrum of the associated perron frobenius transfer operators e on a suitable banach space perturbation theory applied to the difference of a high iterate n 0 of the transfer operators 0 n 0 and e n 0 yields localisation of the full spectrum of e n 0 the time transfer operator e n 0 commutes with the spatial translations and we provide a description of part of their joint eigenvalues more precisely of the nonresonant spectrum of e n 0 restricted to the eigenspaces of the spatial translations in particular we exhibit smooth curves of eigenvalues and eigenspaces of e n 0 as functions of the eigenvalues e i crystal momenta of the spatial translations

on the definition of srb measures for coupled map lattices we consider srb measures of coupled map lattices the emphasis is given to a definition according to which a srb measure is an invariant probability measure whose projections onto finite dimensional subsystems are absolutely continuous with respect to the lebesgue measure we show that coupled map lattices which are close to an uncoupled expanding map have typically an infinite number of srb measures in particular we give a counterexample to the bricmont kupiainen conjecture

approximating subdifferentials by random sampling of gradients many interesting real functions on euclidean space are differentiable almost everywhere all lipschitz functions have this property but so for example does the spectral abscissa of a matrix a non lipschitz function in practice the gradient is often easy to compute we investigate to what extent we can approximate the clarke subdifferential of such a function at some point by calculating the convex hull of some gradients sampled at random nearby points

variational analysis of non lipschitz spectral functions we consider spectral functions f where f is any permutation invariant mapping from cn to r and is the eigenvalue map from the set of n n complex matrices to cn ordering the eigenvalues lexicographically for example if f is the function maximum real part then f is the spectral abscissa while if f is maximum modulus then f is the spectral radius both these spectral functions are continuous but they are neither convex nor lipschitz for our analysis we use the notion of subgradient extensively analyzed in variational analysis r t rockafellar and r j b wets springer 1998 we show that a necessary condition for y to be a subgradient of an eigenvalue function f at x is that y commutes with x we also give a number of other necessary conditions for y based on the schur form and the jordan form of x in the case of the spectral abscissa we refine these conditions and we precisely identify the case where subdifferential regularity holds we conclude by introducing the notion of a semistable program maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non positive semistable programming is a nonconvex generalization of semidefinite programming using our analysis we derive a necessary condition for a local maximizer of a semistable program and we give a generalization of the complementarity condition familiar from semidefinite programming

a robust gradient sampling algorithm for nonsmooth nonconvex optimization let f be a continuous function on rl n and suppose f is continuously differentiable on an open dense subset such functions arise in many applications and very often minimizers are points at which f is not differentiable of particular interest is the case where f is not convex and perhaps not even locally lipschitz but is a function whose gradient is easily computed where it is defined we present a practical robust algorithm to locally minimize such functions based on gradient sampling no subgradient information is required by the algorithm when f is locally lipschitz and has bounded level sets and the sampling radius eps is fixed we show that with probability 1 the algorithm generates a sequence with a cluster point that is clarke eps stationary furthermore we show that if f has a unique clarke stationary point bar x then the set of all cluster points generated by the algorithm converges to bar x as eps is reduced to zero numerical results are presented demonstrating the robustness of the algorithm and its applicability in a wide variety of contexts including cases where f is not locally lipschitz at minimizers we report approximate local minimizers for functions in the applications literature which have not to our knowledge been obtained previously when the termination criteria of the algorithm are satisfied a precise statement about nearness to clarke eps stationarity is available a matlab implementation of the algorithm is posted at http www cs nyu edu overton papers gradsamp alg

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

equilibrium considerations of the updated lagrangian formulation of beam columns with natural concepts bathe and co workers 1 2 updated lagrangian formulation is reviewed for a two dimensional beam column element from this review a physical interpretation and properties are derived for the matrices of the method a variant of argyris and co workers 4 5 natural concept is introduced in the updated lagrangian formulation and a new method to derive element matrices arises a detailed derivation of the matrices of the twodimensional beam column element is presented to clarify further the ul formulations explicit expressions for these matrices computed with the use of symbolic manipulation 17 are also presented to illustrate the concepts discussed finally numerical results are shown to compare and evaluate different methods and matrices

the dimension of semialgebraic subdifferential graphs a b s t r a c t a corollary of a celebrated theorem of minty is that the subdifferential graph of a closed convex function on r n has uniform local dimension n in contrast there exist nonconvex closed functions whose subdifferentials have large graphs we consider how far minty s corollary extends to functions that are nonconvex but semi algebraic

non linear analysis of thin walled structures using plate elements this paper presents a simple and efficient rectangular thin plate element for elastic geometric non linear analysis of thin walled structures based on the updated lagrangian formulation and small strain hypothesis this element can accurately predict the non linear pre and post buckling load deflection paths of a plate structure using symbolic manipulation the linear and geometric stiffness matrices for the rectangular thin plate element are derived explicitly thereby eliminating the need for numerical integration the element contains 30 degrees of freedom d o f 14 d o f for the in plane membrane action and 16 d o f for the out of plane bending action several examples involving the large deflection behaviour of thin walled structures are presented to demonstrate the accuracy efficiency and versatility of the method

dynamical systems on lattices with decaying interaction i a functional analysis framework abstract we consider weakly coupled map lattices with a decaying interaction that is we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites we develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds this framework is very well suited to study systematically invariant objects one obtains that the invariant objects are essentially local we use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties i e the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets in the companion paper by fontich et al 2011 10

infinite dimensional srb measures we review the basic steps leading to the construction of a sinai ruelle bowen srb measure for an infinite lattice of weakly coupled expanding circle maps and we show that this measure has exponential decay of space time correlations first using the perron frobenius operator one connects the dynamical system of coupled maps on a d dimensional lattice to an equilibrium statistical mechanical model on a lattice of dimension d 1 this lattice model is for weakly coupled maps in a high temperature phase and we use a general but very elementary method to prove exponential decay of correlations at high temperatures

on the spectra of randomly perturbed expanding maps we consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing we do this by proving the robustness of the spectra of their perron frobenius operators

equilibrium measures for coupled map lattices existence uniqueness and finite dimensional approximations we consider coupled map lattices of hyperbolic type i e chains of weakly interacting hyperbolic sets attractors over multi dimensional lattices we describe the thermodynamic formalism of the underlying spin lattice system and then prove existence uniqueness mixing properties and exponential decay of correlations of equilibrium measures for a class of holder continuous potential functions with a sufficiently small holder constant we also study finite dimensional approximations of equilibrium measures in terms of lattice systems approximations and lattice spin systems d approximations we apply our results to establish existence uniqueness and mixing property of srb measures as well as obtain the entropy formula

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

semi algebraic functions have small subdifferentials we prove that the subdifferential of any semi algebraic extended real valued function on mathbf r n has n dimensional graph we discuss consequences for generic semi algebraic optimization problems

sinai ruelle bowen measures for lattice dynamical systems for weakly coupled expanding maps on the unit circle bricmont and kupiainen showed that the sinai ruelle bowen srb measure exists as a gibbs state via thermodynamic formalism we prove that this srb measure is indeed the unique equilibrium state for a holder continuous potential function on the infinite dimensional phase space for a more general class of lattice systems that are small perturbations of the uncoupled map lattice we present the variational principle the entropy formula and the formula for the potential function for the srb measures for coupled map lattices with nearest neighbor interactions we give an explicit formula of the potential function for the srb measure and consequently obtain the entropy in terms of coupling parameters

the entropy formula for srb measures of lattice dynamical systems we give a detailed proof of the entropy formula for srb measures of coupled hyperbolic attractors over integer lattices we show that the topological pressure for the potential function of the srb measure is zero