an explicit description of the global attractor of the damped and driven sine gordon equation we prove that the size of the finite dimensional attractor of the damped and driven sine gordon equation stays small as the damping and driving amplitude become small a decomposition of finite dimensional attractors in banach space is found into a part that attracts all of phase space except sets whose finitedimensional projections have lebesgue measure zero and a partc that only attracts sets whose finite dimensional projections have lebesgue measure zero we describe the components of the attractor andc which is called the hyperbolic structure for the damped and driven sine gordon equation is low dimensional but the dimension ofc which is associated with transients is much larger we verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the attractor in this small parameter region

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

finite elements and terminal penalization for quadratic cost optimal control problems governed by ordinary differential equations we use the finite element method to compute optimal controls of systems governed by linear ordinary differential equations with a quadratic performance index as an application we use the penalty technique to solve terminal state optimal controllability problems numerical instabilities which are common in the use of penalty are minimized when the finite element method is applied to solve this problem convergence theorems are given and error and penalty parameter estimates are presented concrete examples for various situations are given to illustrate the theory

optimal structural design considering flexibility a topology optimization method based on the homogenization method the homogenization design method has been successfully applied to a variety of optimization problems such as the stiffness maximization problem and the eigen frequency maximization problem in this study a methodology to obtain the optimal structure design considering flexibility is developed as a new extension of the homogenization design method first flexibility is formulated using the mutual energy concept second a new multi objective function is proposed to obtain optimal solutions incorporating flexibility and stiffness next the topology optimization procedure is constructed using the homogenization method and sequential linear programming slp finally some examples are presented to confirm that the methodology presented here can provide flexible structures satisfying the problem specifications

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

explicit feature control in structural topology optimization via level set method abstract the present paper aims to address a long standing and challenging problem in structural topology optimization explicit feature control of the optimal topology the basic idea is to resort to the level set solution framework and impose constraints on the extreme values of the signed distance level set function used for describing the topology of the structure numerical examples are also presented and discussed to illustrate the effectiveness of the proposed approach

trace formulas for non self adjoint periodic schrodinger operators and some applications abstract recently a trace formula for non self adjoint periodic schrodinger operators in l 2 r associated with dirichlet eigenvalues was proved in differential integral equations 14 2001 671 700 here we prove a corresponding trace formula associated with neumann eigenvalues in addition we investigate dirichlet and neumann eigenvalues of such operators in particular using the dirichlet and neumann trace formulas we provide detailed information on location of the dirichlet and neumann eigenvalues for the model operator with the potential k e 2 i x where k c

the relaxation of two well energies with possibly unequal moduli the elastic energy of a multiphase solid is a function of its microstructure determining the infimum of the energy of such a solid and characterizing the associated optimal microstructures is an important problem that arises in the modeling of the shape memory effect microstructure evolution and optimal design mathematically the problem is to determine the relaxation under fixed phase fraction of a multiwell energy this paper addresses two such problems in the geometrically linear setting first in two dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy with arbitrary elastic moduli and transformation strains and provide a characterization of the optimal microstructures and the associated strain second in three dimensions we compute the relaxation under fixed phase fraction for a two well elastic energy when either 1 both elastic moduli are isotropic or 2 the elastic moduli are well ordered and the smaller elastic modulus is isotropic in both cases we impose no restrictions on the transformation strains we provide a characterization of the optimal microstructures and the associated strain we also compute a lower bound that is optimal except possibly in one regime when either 1 both elastic moduli are cubic or 2 the elastic moduli are well ordered and the smaller elastic modulus is cubic for moduli with arbitrary symmetry we obtain a lower bound that is sometimes optimal in all these cases we impose no restrictions on the transformation strains and whenever the bound is optimal we provide a characterization of the optimal microstructures and the associated strain in both two and three dimensions the quasiconvex envelope of the energy can be obtained by minimizing over the phase fraction we also characterize optimal microstructures under applied stress

isophasal isopolar and isospectral schrodinger operators and elementary complex analysis we explicitly construct super exponentially decaying potentials on d so that the associated schrodinger operators are isophasal and isopolar we use similar techniques to construct isospectral schrodinger operators with periodic potentials on and isospectral schrodinger operators on

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

on hill s equation with a singular complex valued potential 1 introduction the differential equation y00 qy ey is called hill s equation if q is a realvalued periodic locally integrable function of a real variable it is very well known that the spectrum of the operator associated with this equation consists of a countably in nite number of compact intervals of the real line these intervals are called spectral bands and are generally separated by open intervals called gaps in some instances however all but a nite number of gaps are empty resulting in a spectrum consisting of a nite number of compact bands and one closed interval extending to negative in nity in this case one calls q a nite band or nite gap potential the relationship of nite band potentials with the hierarchy of korteweg de vries equations has been the reason for a heightened interest in them during the past twenty years for recent accounts see for instance the monographs by dickey 8 and belokolos et al 3 the rst non trivial example of a nite band potential was given by ince in 1940 18 consider the lame potentials q x yg g 1 x c jq q0 where g 2n and jq q0 denotes weierstrass s elliptic function with fundamental half periods q and q0 if q is real q0 is pure imaginary and c q0 then q is realvalued periodic and continuous ince s result when translated to the present language shows that q then is a nite band potential with g non empty gaps between the bands this however raises immediately the question of what happens when c 6 q0 or when the half periods are more general in these cases one encounters potentials which are complex valued but continuous as well as potentials which have singularities of the form yg g 1 x and are realor complex valued before giving an overview of the content of the paper i want to give a brief account of the history of the subject let q be a real locally integrable periodic function with period p l the differential expression d dx q and h the self adjoint operator in l r associated with l then consider the following four statements a there are precisely 2g 1 real numbers e2g e0 such that the spectrum of h is given by

complex hill s equation and the complex periodic korteweg de vries equations on etudie la classe spectrale des operateurs differentiels l d 2 dx 2 q avec un potentiel q de periode un on traite des potentiels complexes et des operateurs auto adjoints

multiplier methods for nonlinear optimal control error estimates are derived for an augmented lagrangian approximation to an optimal control problem convex control constraints are treated explicitly while a lagrange multiplier is introduced for the nonlinear system dynamics the nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations since the natural coercivity assumption is formulated in a hilbert space where the cost is not differentiable this discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions the paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite element discretization

topology design with optimized self adaptive materials significant performance improvements can be obtained if the topology of an elastic structure is allowed to vary in shape optimization problems we study the optimal shape design of a two dimensional elastic continuum for minimum compliance subject to a constraint on the total volume of material the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology relaxation of the optimization problem via quasiconvexification or homogenization methods is required the effect of relaxation is to introduce a perforated microstructure that must be optimized simultaneously with the macroscopic distribution of material

central limit theorem in negative curvature we prove a central limit theorem for the distance of the brownian point on the universal cover of a compact negatively curved riemannian manifold the technical point is a contraction property for the leafwise brownian motion along the stable foliation

topology design of structures using a dual algorithm and a constraint on the perimeter dual optimization algorithms are well suited for the topology design of continuum structures in discrete variables since in these problems the number of constraints is small in comparison to the number of design variables the raw dual algorithm which was originally proposed for the minimum compliance design problem worked well when a perimeter constraint was added in addition to the volume constraint however if the perimeter constraint was gradually relaxed by increasing the upper bound on the allowable perimeter the algorithm tended to behave erratically recently a simple strategy has been suggested which modifies the raw dual algorithm to make it more robust in the absence of the perimeter constraint in particular the problem of checkerboarding which is frequently observed with the use of lower order finite elements is eliminated in this work we show how the perimeter constraint can be incorporated in this improved algorithm so that it not only provides a designer with a control over the topology but also generates good topologies irrespective of the value of the upper bound on the perimeter

foliations the ergodic theorem and brownian motion regarding the leaves of a compact foliated riemannian manifold as the orbits of a dynamical system we will prove a birkhoff ergodic theorem asserting that the spatial average of a function equals its time average along random paths in the leaves there are several unexpected developments in the foundations first there always are measures on any compact foliated riemannian manifold m f with respect to which an adequate brownian motion along the leaves ergodic theorem holds second relative to these measures which are called harmonic there is a liouville type theorem for any bounded bore1 function on the manifold which is harmonic on each leaf namely for almost all leaves relative to any one of these harmonic measures this leaf harmonic function is constant on each leaf third the harmonic measures have a very reasonable local characterization in any foliation coordinate system they are transversal sums of h l dx l where h l is a positive harmonic function on the leaf l and dx l is the riemannian measure on l these statements comprise theorem 1 which is stated more precisely later in the introduction the first fact is surprising because up to now the only measures attached to foliations were the holonomy invariant transverse measures of plante 141 and ruelle sullivan 151 and some foliations do not admit these the second fact is unexpected because there are foliations where all the leaves are hyperbolic planes and each hyperbolic plane being conformal to the disk admits 285 0022 1236183 3 00

convergence of a penalty finite element approximation for an obstacle problem this study establishes an error estimate for a penalty finite element approximation of the variational inequality obtained by a class of obstacle problems by special identification of the penalty term we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality the rate of convergence of the penalization is where is the penalty parameter to obtain the error of finite element approximation we apply the results obtained by brezzi hager and raviart for the mixed finite element method to the variational inequality

a schauder and riesz basis criterion for non self adjoint schrodinger operators with periodic and antiperiodic boundary conditions under the assumption that v l2 0 dx we derive necessary and sufficient conditions in terms of spectral data for non self adjoint schrodinger operators d2 dx2 v in l2 0 dx with periodic and antiperiodic boundary conditions to possess a riesz basis of root vectors i e eigenvectors and generalized eigenvectors spanning the range of the riesz projection associated with the corresponding periodic and antiperiodic eigenvalues we also discuss the case of a schauder basis for periodic and antiperiodic schrodinger operators d2 dx2 v in lp 0 dx p 1

dual approximations in optimal control we analyze a dual approximation for the solution to an optimal control problem the differential equation is handled with a lagrange multiplier while other constraints are treated explicitly an algorithm for solving the dual problem is presented

the center manifold and bifurcations of damped and driven sine gordon breathers abstract the generic bifurcations of breathers in the damped and driven sine gordon equation are investigated both numerically and analytically the linear stability analysis and information from periodic spectral theory suggest that three modes are relevant for the system they correspond to frequency and temporal phase changes and to the flat pendulum using these modes nonautonomous amplitude equations are derived and compared with numerical simulations of the perturbed sine gordon equation

continuous approximation of material distribution for topology optimization in this paper we propose a checkerboard free topology optimization method without introducing any additional constraint parameter this aim is accomplished by the introduction of finite element approximation for continuous material distribution in a fixed design domain that is the continuous distribution of microstructures or equivalently design variables is realized in the whole design domain in the context of the homogenization design method hdm by the discretization with finite element interpolations by virtue of this continuous fe approximation of design variables discontinuous distribution like checkerboard patterns disappear without any filtering schemes we call this proposed method the method of continuous approximation of material distribution camd to emphasize the continuity imposed on the material field two representative numerical examples are presented to demonstrate the capability and the efficiency of the proposed approach against some classes of numerical instabilities copyright 2004 john wiley sons ltd

scale effects in the optimal design of a microstructured medium against buckling abstract certain classes of problems in optimal structural design lead naturally to the introduction of periodic microstructured media as the basis material for the construction of a mechanical element the unit cell size of these microstructures cannot be arbitrarily small as suggested by the pertaining optimization analyses up to date and has to be related to the structure s overall dimensions one physically important mechanism that provides such microstructure size limitations is elastic buckling an analytically tractable model of an infinite periodic rectangular planar frame with axially compressed beams is used to study the optimal buckling loads for any given design one can find a critical stress above which buckling instability occurs in addition one can also find the region in the design space for which the optimal critical mode is a global one i e its characteristic length is much larger than the unit cell size in this region of the design space one can safely use the homogenized material properties to describe the medium for they provide all the information needed to predict a global buckling instability in addition to the detailed parametric study for the model problem investigated here implications for the optimal design against buckling of more general structures are also briefly discussed

geometry of kdv 4 abel sums jacobi variety and theta function in the scattering case the study of the general projective curve x defined by the vanishing of an irreducible polynomial p x y originated in the discovery of addition formulas for integrals v r x y dx of rational functions on such a curve

chaotic perturbations of kdv i rational solutions abstract rational pole solutions of a perturbed kdv equation describing nonlinear ion acoustic plasma waves become chaotic in time when a small perturbation is periodically driven a mcgehee transformation blows up a degenerate stationary point at infinity and the smale birkhoff homoclinic theorem adapted to submanifolds in phase permits the use of the melnikov method with pole solutions being homoclinic orbits

periodic solutions of the kdv equation in this paper we construct a large family of special solutions of the kdv equation which are periodic in x and almost periodic in t these solutions lie on n dimensional tori very likely they are dense among all solutions the special solutions are characterized variationally they minimize fn u subject to the constraints fj u aj j 1 n 1 here fj denote the remarkable sequence of conserved functionals discovered by kruskal and zabusky the above minimum problem was originally suggested by them in exploring the manifold of solutions of this minimum problem we make essential use of gardner s discovery that these functionals are in involution with respect to a suitable poisson bracket gardner greene kruskal and miura have shown that the eigenvalues of the schrodinger operator are conserved functionals if the potential is a function of t and satisfies the kdv equation in section 6 a new set of conserved quantities is constructed which serve as a link between the eigenvalues of the schrodinger operator and the fj another result in section 6 is a slight sharpening of an earlier result of the author and j moser for the special solutions constructed above all but 2n 1 eigenvalues of the schrodinger operator are double the simplest class of special solutions n 1 are cnoidal waves in an appendix m hyman describes the results of computing numerically the next simplest case n 2 these calculations show that the shape of these solutions recurs exactly after a finite time in a shifted position the theory verifies this fact

on a theorem of hochstadt in fact hochstadt gives an explicit formula for the discriminant which will be reproduced below since hochstadt s work there has been a considerable interest in finite band potentials i e potentials q such that the spectrum of the operator associated with d2 dx 2 q consists of finitely many regular analytic arcs due to their close relationship with the korteweg de vries hierarchy and hence with integrable systems however the interest is not restricted to real valued potentials for instance

vortex blob methods applied to interfacial motion we develop a boundary integral method for computing the motion of an interface separating two incompressible inviscid fluids the velocity integral is regularized so that the vortex sheet on the interface is replaced by a sum of blobs of vorticity the regularization allows control of physical instabilities we design a class of high order blob methods and analyze the errors numerical tests suggest that the blob size should be scaled with the local spacing of the interfacial markers for a vortex sheet in one fluid with a first order kernel we obtain a spiral roll up similar to krasny j comput phys 65 1986 292 but the higher order kernels lead to more detailed structure we verify the accuracy of the new method by computing a liquid gas interface with rayleigh taylor instability we then apply the method to the more difficult case of rayleigh taylor flow separating two fluids of positive density a case for which the regularization appears to be essential as found by kerr and tryggvason both j comput phys 76 1988 48 75 1988 253 we use a blob regularization in certain local terms in the evolution equations as well as in the velocity integral we find strong evidence that improved spatial resolution with fixed blob size leads to a converged regularized solution without numerical instabilities however it is not clear that there is a weak limit as the regularization is decreased

stability of finite element models for distributed parameter optimization and topology design abstract we address a problem of numerical instability that is often encountered in finite element solutions of distributed parameter optimization and variable topology shape design problems we show that the cause of this problem is numerical rather than physical in nature we consider a two field distributed parameter optimization problem involving a design field and a response field and show that the optimization problem corresponds to a mixed variational problem an improper selection of the discrete function spaces for these two fields leads to grid scale anomalies in the numerical solutions to optimization problems similar to those that are sometimes encountered in mixed formulations of the stokes problem we present a theoretical framework to explain the cause of these anomalies and present stability conditions for discrete models the general theoretical framework is specialized to analyze the stability of specific optimization problems and stability results for various mixed finite element models are presented we propose patch tests that are useful in identifying unstable elements

topological design for vibrating structures the topological optimization technique using micro scale voids with the homogenization method for stress analysis of perforated structures has been applied to solve stiffness maximization problem with success here the homogenization design technique is extended to design problems concerning vibrating structures an extended optimization algorithm is also derived to maximize a set of eigenvalues as well as to identify the topology design for specified eigenvalues to characterize forced vibration of a structure

approximations to the multiplier method we analyze approximations to the multiplier method for solving an equality constrained optimization problem the multiplier method replaces the constrained problem by the unconstrained optimization of a penalized lagrangian over a hilbert space we consider the effect of replacing this hilbert space by a subspace the analysis is closely connected to an analysis of the galerkin approximation to the original optimization problem our estimates are essentially corollaries of a theorem which bounds the error in the galerkin approximation to a nonlinear equation some applications to problems in optimal control are discussed

a robust dual algorithm for topology design of structures in discrete variables dual optimization algorithms for the topology optimization of continuum structures in discrete variables are gaining popularity in recent times since in topology design problems the number of constraints is small in comparison to the number of design variables good topologies can be obtained for the minimum compliance design problem when the perimeter constraint is imposed in addition to the volume constraint however when the perimeter constraint is relaxed the dual algorithm tends to give bad results even with the use of higher order finite element models as we demonstrate in this work since a priori one does not know what a good value of the perimeter to be specified is it is essential to have an algorithm which generates good topologies even in the absence of the perimeter constraint we show how the dual algorithm can be made more robust so that it yields good designs consistently in the absence of the perimeter constraint in particular we show that the problem of checkerboarding which is frequently observed with the use of lower order finite elements is eliminated

barycenters of measures transported by stochastic flows we investigate the evolution of barycenters of masses transported by stochastic flows the state spaces under consideration are smooth affine manifolds with certain convexity structure under suitable conditions on the flow and on the initial measure the barycenter zt is shown to be a semimartingale and is described by a stochastic differential equation for the hyperbolic space the barycenter of two independent brownian particles is a martingale and its conditional law converges to that of a brownian motion on the limiting geodesic on the other hand for a large family of discrete measures on suitable cartan hadamard manifolds the barycenter of the measure carried by an unstable brownian flow converges to the busemann barycenter of the limiting measure

numerical solution of some initial optimal control problems using the reproducing kernel hilbert space technique in this paper we present a general technique for solving a class of linear nonlinear optimal control problems in fact an analytical solution of the state variable is represented in the form of a

coupled deformations of elastic curves and surfaces an equilibrium theory for the coupled finite deformations of elastic curves and surfaces is described possible wrinkling of the curve or surface is taken into account by using a relaxed version of the theory obtained from minimum energy considerations the relaxed theory admits a dual formulation leading to extremum principles and uniqueness of the equilibrium stress distribution a number of examples are treated using spatial finite differences together with a dynamic relaxation method in which equilibrium configurations are obtained in the long time limit of a damped dynamical problem

singularities of the complex korteweg de vries flows on etudie le probleme isospectral de l equation de hill complexe etant donne 2g 1 valeurs propres simples dans le plan complexe trouver la classe spectrale correspondante q on considere la caracterisation de la classe isospectrale et ses sections singulieres on construit q a l aide des flots non lineaires de korteweg de vries dans c 1 les fonctions regulieres de periode 1

theoremes limite pour les systemes lineaires a coefficients markoviens nous montrons un theoreme central limite et une estimee des grandes deviations pour les solutions d equations differentielles stochastiques lineaires a coefficients markoviens ces resultats sont obtenus dans le cadre general des processus markoviens multiplicatifs ils s appliquent alors par exemple au flot derive d un flot stochastique sur une variete compacte et aux produits de matrices aleatoires en dependance markovienne

distributed parameter optimization and topology design for non linear thermoelasticity abstract topology optimization has been the focus of considerable attention in the shape optimization community in recent years since significant performance improvements can be obtained if the topology is allowed to vary in shape optimization problems attention has mainly focussed on the topology design of an elastic continuum for minimum compliance subject to a volume constraint the macroscopic version of this problem is not well posed if no restrictions are placed on the structure topology the most widely used method for making the problem has been the so called homogenization method which introduces microstructure to the design space though there are a number of advantages to this method it also suffers from some drawbacks optimal designs generated using an optimal microstructure are difficult to manufacture while using sub optimal microstructures reverts the problem back to the original ill posed problem in addition extension of the homogenization method to problems involving non linear material behavior is quite difficult to address some of these issues a new method for making the compliance optimization problem well posed has recently been proposed this method introduces an additional constraint on the perimeter of the solid regions in the design to make the problem well posed since microstructure is not introduced the designs are easily manufacturable preliminary results for topology design using the perimeter method for problems where the material behavior is linear elastic have already been reported in the literature we show that the perimeter method can be used even when the material behavior involves material and geometric non linearities we formulate the distributed parameter optimization and topology design problems using the perimeter method for non linear thermoelasticity finite element optimization procedures based on this formulation are developed and numerical solutions for the distributed parameter optimization problem are presented though we use the compliance as the objective function one could optimize any objective function by making minor modifications in the method outlined here