minimizers for a double well problem with affine boundary conditions this paper is concerned with the existence of minimizers for functionals having a double well integrand with affine boundary conditions such functionals are related to the so called kohn strang functional which arises in optimal shape design problems in electrostatics or elasticity they are known to be not quasi convex and therefore existence of minimizers is in general guaranteed only for their quasi convex envelopes we generalize previous results in 1 and give necessary and sufficient conditions on the affine boundary conditions for existence of minimizers our method relies on the computation of the quasiconvexification of these functionals by using homogenization theory we also prove by a general argument that their rank one convexifications coincide with their quasi convexifications

anisotropic singular perturbations the vectorial case we obtain the l 1 limit of the sequence where e e is the family of anisotropic perturbations of the nonconvex functional of vector valued functions the proof relies on the blow up argument introduced by fonseca and muller

fine properties of functions with bounded deformation the paper is concerned with the fine properties of functions in the space of functions with bounded deformation we analyse the set of lebesgue points and the set where these functions have one sided approximate limits moreover following the analogy with we decompose the symmetric distributional derivative into an absolutely continuous part a jump part and a cantor part the main result of the paper is a structure theorem for functions showing that these parts of the derivative can be recovered from the corresponding ones of the one dimensional sections moreover we prove that functions are approximately differentiable in almost every point of their domain

a quantum cup length estimate for symplectic fixed points a new lower bound for the number of fixed points of hamiltonian automorphisms of closed symplectic manifolds m is established the new estimate extends the previously known estimates to the class of weakly monotone symplectic manifolds we prove for arbitrary closed symplectic manifolds with rational symplectic class that the cup length estimate holds true if the hofer energy of the hamiltonian automorphism is sufficiently small for arbitrary energy and on weakly monotone symplectic manifolds we define an analogon to the cup length based on the quantum cohomology ring of m providing a quantum cup length estimate

shells structure in uncorrelated scale free networks abstract we study structure of shells in uncorrelated scale free networks using probabilistic arguments we obtain explicit expression for the distance distribution i e average number of nodes at the l th shell for different ranges of the degree exponent to overcome the analytical difficulties when 2 3 we show that the heterogeneous network can be approximated by a disassortative ordered network and the average degree of neighbors of a node must depends on shells we also deduce the mean distance between nodes l as the distance at which distance distribution is maximum taking number of nodes large we retrieve the known scaling forms for the different ranges of mainly the small world and the ultra small world behaviors very good accordance with simulations is also found the expressions of l involve all the network s parameters and can be used as good approximations of mean distance in real world problems

statistical analysis of articulation points in configuration model networks an articulation point ap in a network is a node whose deletion would split the network component on which it resides into two or more components aps are vulnerable spots that play an important role in network collapse processes which may result from node failures attacks or epidemics therefore the abundance and properties of aps affect the resilience of the network to these collapse scenarios we present analytical results for the statistical properties of aps in configuration model networks in order to quantify their abundance we calculate the probability p i in rm ap that a random node i in a configuration model network with p k k is an ap we also obtain the conditional probability p i in rm ap k that a random node of degree k is an ap and find that high degree nodes are more likely to be aps than low degree nodes using bayes theorem we obtain the conditional degree distribution p k k rm ap over the set of aps and compare it to p k k we propose a new centrality measure based on aps each node can be characterized by its articulation rank r which is the number of components that would be added to the network upon deletion of that node for nodes which are not aps the articulation rank is r 0 while for aps r ge 1 we obtain a closed form expression for the distribution of articulation ranks p r r configuration model networks often exhibit a coexistence between a giant component and finite components to examine the distinct properties of aps on the giant and on the finite components we calculate the probabilities presented above separately for the giant and the finite components we apply these results to ensembles of configuration model networks with a poisson exponential and power law degree distributions the implications of these results are discussed in the context of common attack scenarios and network dismantling processes

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

on the lusternik schnirelmann category of symplectic manifolds and the arnold conjecture in arn appendix 9 arnold proposed a beautiful conjecture concerning the relation between the number of fixed points of certain i e exact or hamiltonian selfdiffeomorphisms of a closed symplectic manifold m and the minimum number of critical points of any smooth c function on m the first author succeeded in proving this form of the arnold conjecture r2 under the hypothesis that and c1 vanish on all spherical homology classes and that there is equality between the lusternik schnirelmann category of m and the dimension of m in this paper we use a fundamental property of category weight to show that for any closed symplectic manifold whose symplectic form vanishes on the image of the hurewicz map the required equality holds thus we show that the original form of the arnold conjecture holds for all symplectic manifolds having 2 m 0 c1 2 m

revealing the microstructure of the giant component in random graph ensembles the microstructure of the giant component of the erd s renyi network and other configuration model networks is analyzed using generating function methods while configuration model networks are uncorrelated the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree degree correlations of all orders we present exact analytical results for the degree distributions as well as higher order degree degree correlations on the giant components of configuration model networks we show that the degree degree correlations are essential for the integrity of the giant component in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component to demonstrate the importance and broad applicability of these results we apply them to the study of the distribution of shortest path lengths on the giant component percolation on the giant component and spectra of sparse matrices defined on the giant component we show that by using the degree distribution on the giant component one obtains high quality results for these properties which can be further improved by taking the degree degree correlations into account this suggests that many existing methods currently used for the analysis of the whole network can be adapted in a straightforward fashion to yield results conditioned on the giant component

a variational view of partial brittle damage evolution under time dependent loading an elastic material undergoes the simplest form of damage in which it passes from its original state to a weaker elastic state elaborating on prior work 14 we establish the existence of a relaxed variational evolution where at each time the two states of the material combine to form a fine mixture optimal from the standpoint of the applied load at that time yet preserving the irreversibility of the damaging process

threshold based quasi static brittle damage evolution we introduce models for static and quasi static damage in elastic materials based on a strain threshold and then investigate the relationship between these threshold models and the energy based models introduced in francfort and marigo eur j mech a solids 12 149 189 1993 and francfort and garroni ration mech anal 182 1 125 152 2006 a somewhat surprising result is that while classical solutions for the energy models are also threshold solutions this is shown not to be the case for nonclassical solutions that is solutions with microstructure a new and arguably more physical definition of solutions with microstructure for the energy based model is then given in which the energy minimality property is satisfied by sequences of sets that generate the effective elastic tensors rather than by the tensors themselves we prove existence for this energy based problem and show that these solutions are also threshold solutions a by product of this analysis is that all local minimizers in both the classical setting and for the new microstructure definition are also global minimizers

concentration versus oscillation effects in brittle damage this work is concerned with an asymptotic analysis in the sense of gamma convergence of a sequence of variational models of brittle damage in the context of linearized elasticity the study is performed as the damaged zone concentrates into a set of zero volume and at the same time and to the same order varepsilon the stiffness of the damaged material becomes small three main features make the analysis highly nontrivial at varepsilon fixed minimizing sequences of each brittle damage model oscillate and develop microstructures as varepsilon to 0 concentration and saturation of damage are favoured and the competition of these phenomena translates into a degeneration of the growth of the elastic energy which passes from being quadratic at varepsilon fixed to being of linear growth type in the limit consequently homogenization effects interact with singularity formation in a nontrivial way which requires new methods of analysis in particular the interaction of homogenization with singularity formation in the framework of linearized elasticity appears to not have been considered in the literature so far we explicitly identify the gamma limit in two and three dimensions for isotropic hooke tensors the expression of the limit effective energy turns out to be of hencky plasticity type we further consider the regime where the divergence remains square integrable in the limit which leads to a tresca type model

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

the gradient theory of phase transitions for systems with two potential wells on generalise la theorie du gradient des transitions de phase au cas a valeur vectoriel on considere la famille des perturbations e e u w u dx e 2 u 2 dx de la fonctionnelle non convexe e 0 u w u dx ou w r n r supporte deux phases et n 1

lagrangian intersections critical points and qcategory fundacao para a ciencia e a tecnologia fct programa operacional ciencia tecnologia inovacao pocti

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

distribution of shortest path lengths in subcritical erdős renyi networks networks that are fragmented into small disconnected components are prevalent in a large variety of systems these include the secure communication networks of commercial enterprises government agencies and illicit organizations as well as networks that suffered multiple failures attacks or epidemics the structural and statistical properties of such networks resemble those of subcritical random networks which consist of finite components whose sizes are nonextensive surprisingly such networks do not exhibit the small world property that is typical in supercritical random networks where the mean distance between pairs of nodes scales logarithmically with the network size unlike supercritical networks whose structure has been studied extensively subcritical networks have attracted relatively little attention a special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies the overall statistics of the network can be obtained by a summation over all the components with suitable weights we use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths dspl on components of given sizes and topologies in subcritical erd ifmmode mbox h o else h o fi s r enyi er networks from this expansion we obtain an exact analytical expression for the dspl of the entire subcritical network in the asymptotic limit the dspl which accounts for all the pairs of nodes that reside on the same finite component fc is found to follow a geometric distribution of the form p mathrm fc l ensuremath ell ll ensuremath infty 1 ensuremath c c ensuremath ell ensuremath 1 where cl1 is the mean degree using computer simulations we calculate the dspl in subcritical er networks of increasing sizes and confirm the convergence to this asymptotic result we also obtain exact asymptotic results for the mean distance ensuremath langle l ensuremath rangle mathrm fc and for the standard deviation of the dspl ensuremath sigma l mathrm fc and show that the simulation results converge to these asymptotic results using the duality relations between subcritical and supercritical er networks we obtain the dspl on the nongiant components of er networks above the percolation transition

damage and fracture evolution in brittle materials by shape optimization methods this paper is devoted to a numerical implementation of the francfort marigo model of damage evolution in brittle materials this quasi static model is based at each time step on the minimization of a total energy which is the sum of an elastic energy and a griffith type dissipated energy such a minimization is carried over all geometric mixtures of the two healthy and damaged elastic phases respecting an irreversibility constraint numerically we consider a situation where two well separated phases coexist and model their interface by a level set function that is transported according to the shape derivative of the minimized total energy in the context of interface variations hadamard method and using a steepest descent algorithm we compute local minimizers of this quasi static damage model initially the damaged zone is nucleated by using the so called topological derivative we show that when the damaged phase is very weak our numerical method is able to predict crack propagation including kinking and branching several numerical examples in 2d and 3d are discussed

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

the degenerate special lagrangian equation abstract this article introduces the degenerate special lagrangian equation dsl and develops the basic analytic tools to construct and study its solutions the dsl governs geodesics in the space of positive graph lagrangians in c n existence of geodesics in the space of positive lagrangians is an important step in a program for proving existence and uniqueness of special lagrangians moreover it would imply certain cases of the strong arnold conjecture from hamiltonian dynamics we show the dsl is degenerate elliptic we introduce a space time lagrangian angle for one parameter families of graph lagrangians and construct its regularized lift the superlevel sets of the regularized lift define subequations for the dsl in the sense of harvey lawson we extend the existence theory of harvey lawson for subequations to the setting of domains with corners and thus obtain solutions to the dirichlet problem for the dsl in all branches moreover we introduce the calibration measure which plays a role similar to that of the monge ampere measure in convex and complex geometry the existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the dsl have a well defined length in the space of positive lagrangians

cuplength estimates on lagrangian intersections on demontre l estimation suivante sur des intersections lagrangiennes si l est une sous variete lagrangienne de p avec 2 p l 0 et l est obtenue a partir de l par un diffeomorphisme exact de p alors le nombre d elements de l l est superieure ou egale a la longueur de calotte de p

an iterated graph construction and periodic orbits of hamiltonian delay equations abstract according to the arnold conjectures and floer s proofs there are non trivial lower bounds for the number of periodic solutions of hamiltonian differential equations on a closed symplectic manifold whose symplectic form vanishes on spheres we use an iterated graph construction and lagrangian floer homology to show that these lower bounds also hold for certain hamiltonian delay equations

on analytical applications of stable homotopy the arnold conjecture critical points we prove the arnold conjecture for closed symplectic manifolds with 2 m 0 and catm dimm furthermore we prove an analog of the lusternik schnirelmann theorem for functions with generalized hyperbolicity property

distance distribution in configuration model networks we present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks the results which are based on recursion equations are shown to be in good agreement with numerical simulations for networks with degenerate binomial and power law degree distributions the mean mode and variance of the distribution of shortest path lengths are also evaluated these results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution illuminating the connection between the two distributions