a locking free fem for cavitation computation in nearly incompressible nonlinear elasticity abstract a reduced selective integration finite element method coupled with a gradient flow method is introduced for the cavitation computation in nearly incompressible nonlinear elasticity numerical experiments show that for the iso parametric biquadratic finite element approximation of the displacement field the method is locking free and efficient for the nearly incompressible cavitation computation and has the same optimal convergence rates as in the compressible case for the elastic energy as well as the deformation and stress field if and only if the reduced integration is applied to the bulk energy term using no more than 2 gaussian quadrature nodes in each dimension

the radial volume derivative and the critical boundary displacement for cavitation we study the displacement boundary value problem of minimizing the total energy e mathbf u stored in a nonlinearly elastic body occupying a spherical domain b in its reference configuration over possibly discontinuous radial deformations mathbf u of the body subject to affine boundary data mathbf u mathbf x lambda mathbf x for mathbf x in partial b for a given value of lambda we define what we call the radial volume derivative at lambda denoted by g lambda which measures the stability or instability of the underlying homogeneous deformation mathbf u h lambda mathbf x equiv lambda mathbf x to the formation of holes we give conditions under which the critical boundary displacement lambda mathrm crit for radial cavitation is the unique solution of g lambda 0 moreover we prove that the radial volume derivative g lambda can be approximated by the corresponding volume derivative for a punctured ball b epsilon containing a pre existing cav

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

what tasks can be performed with an uncalibrated stereo vision system this article studies the following question when is it possible to decide on the basis of images of point features observed by an imprecisely modeled two camera stereo vision system whether or not a prescribed robot positioning task has been precisely accomplished results are shown for three camera model classes injective cameras weakly calibrated projective cameras and uncalibrated projective cameras in particular given a weakly calibrated stereo pair it is shown that a positioning task can be precisely accomplished if and only if the task specification is invariant to projective transformations it is shown that injective and uncalibrated projective cameras can accomplish fewer tasks but are still able to accomplish tasks involving point coincidences the same formal framework is applied to the problem of determining the set of tasks which can be precisely accomplished with the well known position based control architecture it is shown that for any class of camera models the set of tasks which can be precisely accomplished using a position based control architecture is a subset of the complete set of tasks which can be decided on the set but includes all positioning tasks based on point coincidences two ways of extending the idea of position based control to accomplish more tasks are also presented

on critical points of the σ2 energy over a space of measure preserving maps abstract let x r n be a bounded lipschitz domain and consider the 2 energy functional f 2 u x x u u 2 d x over the space of admissible maps a x u w 1 4 x r n u x x det u 1 for l n a e in x a good measure of how much a map u stretches areas of 2 dimensional sub manifolds of the domain x is the norm of u u 2 t x 2 t x analogously to u 2 the dirichlet energy density that is a measure of length s stretching these kinds of functionals also were arisen as a physical model describing the strong interactions of quantum fields which was introduced by t skyrme in 1961 in this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the euler lagrange equations associated with f 2 x over a x in particular we present a novel characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions which follows very closely to the ideas that have been introduced by the second author in his recent paper shahrokhi dehkordi and taheri 2009 17 indeed we show that in even dimensions the latter system of equations admits infinitely many smooth solutions modulo isometries amongst such maps in odd dimensions this number reduces to one the result relies on a careful analysis of the full versus the restricted euler lagrange equations we investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

radially symmetric motions of nonlinearly viscoelastic bodies under live loads this paper treats radially symmetric motions of nonlinearly viscoelastic circular cylindrical and spherical shells subjected to the live loads of centrifugal force and time dependent hydrostatic pressures the governing equations are exact versions of those for 3 dimensional continuum mechanics so shell does not connote an approximate via some shell theory these motions are governed by quasilinear third order parabolic hyperbolic equations having but one independent spatial variable the principal part of such a partial differential equation is determined by a general family of nonlinear constitutive equations the presence of strains in two orthogonal directions requires a careful treatment of constitutive restrictions that are physically natural and support the analysis the interaction of geometrically exact formulations the compatible use of general constitutive equations for material response and the presence of live loads show how these factors play crucial roles in the behavior of solutions in particular for different kinds of live loads there are thresholds separating materials that produce qualitatively different dynamical behavior the analysis using classical methods covers infinite time blowup for cylindrical shells subject to centrifugal forces infinite time blowup for cylindrical shells subject to steady and time dependent hydrostatic pressures finite time blowup for spherical shells subject to steady and time dependent hydrostatic pressures and the preclusion of total compression this paper concludes with a sketch using some modern methods of the existence of regular solutions until the time of blowup

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

spherically symmetric two phase deformations and phase transition zones this paper is concerned with characterization and stability assessment of two phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material we study general properties of equilibrium two phase spherically symmetric deformations then we specialize to phase transformations of a solid sphere that is subjected to an all round tension pressure two material models are used to demonstrate a variety of transformation behaviours and some common features for both materials we construct phase transition zones ptzs formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface then we demonstrate how the ptz can be used for the prediction of the number of two phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the ptz we show that even in the simplest case of one interface the solution is not unique two equilibrium two phase solutions as well as one uniform one phase solution are found under the same boundary conditions for the three solutions we construct their load deformation diagrams and compare the associated total energies the stability of the two phase states with respect to radial and small wavelength perturbations is also examined we observe how unstable solutions are related with the ptz

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

matching two perspective views a computational approach to image matching is described it uses multiple attributes associated with each image point to yield a generally overdetermined system of constraints taking into account possible structural discontinuities and occlusions in the algorithm implemented intensity edgeness and cornerness attributes are used in conjunction with the constraints arising from intraregional smoothness field continuity and discontinuity and occlusions to compute dense displacement fields and occlusion maps along the pixel grids the intensity edgeness and cornerness are invariant under rigid motion in the image plane in order to cope with large disparities a multiresolution multigrid structure is employed coarser level edgeness and cornerness measures are obtained by blurring the finer level measures the algorithm has been tested on real world scenes with depth discontinuities and occlusions a special case of two view matching is stereo matching where the motion between two images is known the algorithm can be easily specialized to perform stereo matching using the epipolar constraint

a numerical method for detecting singular minimizers a numerical method for computing minimizers in one dimensional problems of the calculus of variations is described such minimizers may have unbounded derivatives even when the integrand is smooth and regular in such cases because of the lavrentiev phenomenon standard finite element methods may fail to converge to a minimizer the scheme proposed is shown to converge to an absolute minimizer and is tested on an example the effect of quadrature is analyzed the implications for higher dimensional problems and in particular for fracture in nonlinear elasticity are discussed

neural net based direct self tuning control of nonlinear plants use of neural networks for direct self tuning control of stochastic nonlinear plants has been proposed the control is based upon inverse modelling of a pseudo plant the input to the pseudo plant is same as the plant input while its output consists of a linear combination of the plant input and output the controller is directly identified as a mean square optimal inverse estimator of the pseudo plant this approach allows the control of inverse unstable plants local convergence properties as well as results of simulation studies are presented

a computational approach for corner and vertex detection corners and vertexes are strong and useful features in computer vision for scene analysis stereo matching and motion analysis here we deal with the development of a computational approach to these important features we consider first a corner model and study analytically its behavior once it has been smoothed using the well known gaussian filter this allows us to clarify the behavior of some well knowncornerness measure based approaches used to detect these points of interest most of these classical approaches appear to detect points that do not correspond to the exact position of the corner a new scale space based approach that combines useful properties from the laplacian and beaudet s measure beaudet 1978 is then proposed in order to correct and detect exactly the corner position an extension of this approach is then developed to solve the problem of trihedral vertex characterization and detection in particular it is shown that a trihedral vertex has two elliptic maxima on extremal contrast surfaces if the contrast is sufficient and this allows us to classify trihedral vertexes in 2 classes vertex and vertex as corner the corner detection approach developed is applied to accurately detect trihedral vertexes using an additional test in order to make a distinction between trihedral vertexes and corners many experiments have been carried out using noisy synthetic data and real images containing corners and vertexes most of the promising results obtained are used to illustrate the experimental section of this paper

a robust technique for matching two uncalibrated images through the recovery of the unknown epipolar geometry abstract this paper proposes a robust approach to image matching by exploiting the only available geometric constraint namely the epipolar constraint the images are uncalibrated namely the motion between them and the camera parameters are not known thus the images can be taken by different cameras or a single camera at different time instants if we make an exhaustive search for the epipolar geometry the complexity is prohibitively high the idea underlying our approach is to use classical techniques correlation and relaxation methods in our particular implementation to find an initial set of matches and then use a robust technique the least median of squares lmeds to discard false matches in this set the epipolar geometry can then be accurately estimated using a meaningful image criterion more matches are eventually found as in stereo matching by using the recovered epipolar geometry a large number of experiments have been carried out and very good results have been obtained regarding the relaxation technique we define a new measure of matching support which allows a higher tolerance to deformation with respect to rigid transformations in the image plane and a smaller contribution for distant matches than for nearby ones a new strategy for updating matches is developed which only selects those matches having both high matching support and low matching ambiguity the update strategy is different from the classical winner take all which is easily stuck at a local minimum and also from loser take nothing which is usually very slow the proposed algorithm has been widely tested and works remarkably well in a scene with many repetitive patterns

a dual parametric finite element method for cavitation in nonlinear elasticity a dual parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2d cavitation problem in nonlinear elasticity the method overcomes the difficulties such as the mesh entanglement and material interpenetration generally encountered in the finite element approximation of problems with extremely large expansionary deformation numerical experiments show that the method is highly efficient in the computation of cavitation problems numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials

stereo correspondence through feature grouping and maximal cliques the authors propose a method for solving the stereo correspondence problem the method consists of extracting local image structures and matching similar such structures between two images linear edge segments are extracted from both the left and right images each segment is characterized by its position and orientation in the image as well as its relationships with the nearby segments a relational graph is thus built from each image for each segment in one image as set of potential assignments is represented as a set of nodes in a correspondence graph arcs in the graph represent compatible assignments established on the basis of segment relationships stereo matching becomes equivalent to searching for sets of mutually compatible nodes in this graph sets are found by looking for maximal cliques the maximal clique best suited to represent a stereo correspondence is selected using a benefit function numerous results obtained with this method are shown

void nucleation and growth for a class of incompressible nonlinearly elastic materials abstract in this paper a bifurcation problem for a solid sphere subjected to uniform tensile dead loading p 0 at its boundary is examined within the framework of finite elastostatics the sphere is composed of a particular class of homogeneous isotropic incompressible nonlinearly elastic materials namely those of power law type one solution to the problem for all values of p 0 corresponds to a homogeneous state in which the sphere remains undeformed while stressed however for sufficiently large values of p 0 there is in addition a second possible configuration involving an internal traction free spherical cavity the dependence on constitutive parameters of the critical load at which bifurcation occurs is examined as well as the subsequent void growth the stress distribution after cavitation occurs is also described the results are obtained in closed analytic form

the convergence of regularized minimizers for cavitation problems in nonlinear elasticity consider a nonlinearly elastic body which occupies the region omega subset mathbb r m m 2 3 in its reference state and which is held in tension under prescribed boundary displacements on partial omega let bf x 0 in omega be any fixed point in the body it is known from variational arguments that for sufficiently large boundary displacements there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at bf x 0 in the deformed body this is the phenomenon of cavitation for each epsilon 0 define the regularized domains omega epsilon omega backslash overline b epsilon bf x 0 which contain a preexisting hole of radius epsilon 0 centered on bf x 0 now consider the corresponding mixed displacement traction problem on omega epsilon in which the boundary partial omega is subject to the same boundary displacements and the deformed cavity surface i e the image of partial b epsilon is requi

quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity in this paper we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star shaped reference configuration rn whose boundary is subjected to an affine deformation i e there exists a constant n n matrix f and a constant n vector b such that x fx b for all x we consider all smooth equilibrium configurations satisfying this boundary condition clearly the homogeneous deformation x fx b for all x in bar omega is one such solution our aim is to prove under suitable hypotheses that it is the only such solution

experimental studies in nonlinear discrete time adaptive prediction and control this paper presents implementation results using recently introduced discrete time adaptive prediction and control techniques using online function approximators we consider a process control experiment as our test bed and develop a discrete time adaptive predictor for liquid volume and a discrete time adaptive controller for reference volume tracking we use takagi sugeno ts fuzzy systems as our function approximators and for both prediction and control we investigate the use of a least squares update of the fuzzy system s parameters

the fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

cartesian currents weak diffeomorphisms and existence theorems in nonlinear elasticity in this paper we introduce some new classes of functions among these a class of weak diffeomorphisms in these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals in particular we prove the existence of one to one orientation preserving maps that minimize suitable energies associated with hyperelastic materials the minimizers are also proved to satisfy equilibrium equations finally radial deformations are discussed in connection with cavitation

some properties of the e matrix in two view motion estimation in the eight point linear algorithm for determining 3d motion structure from two perspective views using point correspondences the e matrix plays a central role the e matrix is defined as a skew symmetrical matrix containing the translation components postmultiplied by a rotation matrix the authors show that a necessary and sufficient condition for a 3 3 matrix to be so decomposable is that one of its singular values is zero and the other two are equal several other forms of this property are presented some applications are briefly described

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

neohookean deformations of annuli existence uniqueness and radial symmetry let mathbb x x in mathbb r 2 r x r and mathbb y y in mathbb r 2 r ast y r ast be annuli in the plane we consider the class fancyscript f mathbb x mathbb y of all orientation preserving homeomorphisms h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev space fancyscript w 1 2 mathbb x mathbb y which keep the boundary circles in the same order this means that lim x searrow r h x r ast and lim x nearrow r h x r ast we study the neohookean energy integral mathcal e h int limits mathbb x dh 2 phi rm det dh quad rm for h in fancyscript f mathbb x mathbb y quad quad quad 1 where phi in fancyscript c infty 0 infty is positive and strictly convex we assume in addition that the function psi z 1 ddot phi z and its derivative extend continuously to 0 with 0 0 then we prove

on irregular weak solutions of the energy momentum equations irregular mappings that are weak solutions of the energy momentum equations are presented one example is discontinuous at a countable number of points while the other is c 1 but not c 2 these mappings are not solutions of the usual euler lagrange equations

error analysis of a dual parametric bi quadratic fem in cavitation computation in elasticity the orientation preservation conditions and approximation errors of a dual parametric bi quadratic finite element method for the computation of both radially symmetric and general nonsymmetric cavity solutions in nonlinear elasticity are analyzed the analytical results allow us to establish based on an error equidistribution principle an optimal meshing strategy for the method in cavitation computation numerical results are in good agreement with the analytical results

energy estimates and cavity interaction for a critical exponent cavitation model we consider the minimization of in a perforated domain of among maps that are incompressible det and invertible and satisfy a dirichlet boundary condition u g on if the volume enclosed by g is greater than any such deformation u is forced to map the small holes be ai onto macroscopically visible cavities which do not disappear as e 0 we restrict our attention to the critical exponent p n where the energy required for cavitation is of the order of and the model is suited therefore for an asymptotic analysis v1 vm denote the volumes of the cavities in the spirit of the analysis of vortices in ginzburg landau theory we obtain estimates for the renormalized energy showing its dependence on the size and the shape of the cavities on the initial distance between the cavitation points a1 am and on the distance from these points to the outer boundary based on those estimates we conclude for the case of two cavities that either the cavities prefer to be spherical in shape and well separated or to be very close to each other and appear as a single equivalent round cavity this is in agreement with existing numerical simulations and is reminiscent of the interaction between cavities in the mechanism of ductile fracture by void growth and coalescence 2012 wiley periodicals inc

myriad radial cavitating equilibria in nonlinear elasticity it is shown that every bounded strictly increasing smooth positive function of sufficiently slow growth is the jacobian of a radial hole creating equilibrium deformation for an appropriately constructed compressible nonlinearly elastic energy

shape and motion from image streams under orthography a factorization method inferring scene geometry and camera motion from a stream of images is possible in principle but is an ill conditioned problem when the objects are distant with respect to their size we have developed a factorization method that can overcome this difficulty by recovering shape and motion under orthography without computing depth as an intermediate step an image stream can be represented by the 2fxp measurement matrix of the image coordinates of p points tracked through f frames we show that under orthographic projection this matrix is of rank 3 based on this observation the factorization method uses the singular value decomposition technique to factor the measurement matrix into two matrices which represent object shape and camera rotation respectively two of the three translation components are computed in a preprocessing stage the method can also handle and obtain a full solution from a partially filled in measurement matrix that may result from occlusions or tracking failures the method gives accurate results and does not introduce smoothing in either shape or motion we demonstrate this with a series of experiments on laboratory and outdoor image streams with and without occlusions

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

determining the epipolar geometry and its uncertainty a review two images of a single scene object are related by the epipolar geometry which can be described by a 3 3 singular matrix called the essential matrix if images internal parameters are known or the fundamental matrix otherwise it captures all geometric information contained in two images and its determination is very important in many applications such as scene modeling and vehicle navigation this paper gives an introduction to the epipolar geometry and provides a complete review of the current techniques for estimating the fundamental matrix and its uncertainty a well founded measure is proposed to compare these techniques projective reconstruction is also reviewed the software which we have developed for this review is available on the internet

a theory of self calibration of a moving camera there is a close connection between the calibration of a single camera and the epipolar transformation obtained when the camera undergoes a displacement the epipolar transformation imposes two algebraic constraints on the camera calibration if two epipolar transformations arising from different camera displacements are available then the compatible camera calibrations are parameterized by an algebraic curve of genus four the curve can be represented either by a space curve of degree seven contained in the intersection of two cubic surfaces or by a curve of degree six in the dual of the image plane the curve in the dual plane has one singular point of order three and three singular points of order two if three epipolar transformations are available then two curves of degree six can be obtained in the dual plane such that one of the real intersections of the two yields the correct camera calibration the two curves have a common singular point of order three experimental results are given to demonstrate the feasibility of camera calibration based on the epipolar transformation the real intersections of the two dual curves are found by locating the zeros of a function defined on the interval 0 2 the intersection yielding the correct camera calibration is picked out by referring back to the three epipolar transformations

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

epipolar geometry and linear subspace methods a new approach to weak calibration this paper addresses the problem of estimating the epipolar geometry from point correspondences between two images taken by uncalibrated perspective cameras it is shown that jepson s and heeger s linear subspace technique for infinitesimal motion estimation can be generalized to the finite motion case by choosing an appropriate basis for projective space this yields a linear method for weak calibration the proposed algorithm has been implemented and tested on both real and synthetic images and it is compared to other linear and non linear approaches to weak calibration

on the symmetry of energy minimising deformations in nonlinear elasticity ii compressible materials let a bf x in mathbb r n a 0 be the region occupied by an incompressible nonlinearly hyperelastic material in its reference configuration deformations bf u it a rightarrow mathbb r n are therefore isochoric maps and so satisfy the incompressibility constraint rm det nabla bf u 1 the displacement and the mixed displacement zero traction boundary value problems are considered here a displacement boundary condition of the form bf u bf x sigma bf x is prescribed on one of the boundary components where 0 is a given constant and the displacement on the remaining boundary component is either prescribed in the same form in the case of the pure displacement boundary value problem or left unspecified in the case of the mixed boundary value problem in this paper isoperimetric arguments are used to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of possibly non symmetric isochoric deformations of the region a

model free control of nonlinear stochastic systems with discrete time measurements consider the problem of developing a controller for general nonlinear and stochastic systems where the equations governing the system are unknown using discrete time measurement this paper presents an approach for estimating a controller without building or assuming a model for the system such an approach has potential advantages in accommodating complex systems with possibly time varying dynamics the controller is constructed through use of a function approximator such as a neural network or polynomial this paper considers the use of the simultaneous perturbation stochastic approximation algorithm which requires only system measurements a convergence result for stochastic approximation algorithms with time varying objective functions and feedback is established it is shown that this algorithm can greatly enhance the efficiency over more standard stochastic approximation algorithms based on finite difference gradient approximations

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians