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

analytical computation of differential equations involved in dynamical nonlinear optimal problems abstract we present two programs written in reduce for non constrained free endpoint nonlinear dynamical optimal problems in fixed time in closed loop and in open loop which compute analytically the optimal feedback laws in terms of differential equations the open loop case leads to ordinary differential equations odes and the closed loop leads to partial differential equations pdes in the case of closed loop problems the program uses nonstandard reduce programming for the declarations of dependencies of u and its partial derivatives algorithms are presented for the open loop and closed loop cases and the same example is computed in both these cases

on minimum fuel control of affine nonlinear systems the minimum fuel control problem is investigated for a class of multiinput affine nonlinear systems whose associated lie algebra is nilpotent after preliminary notions and definitions are presented emphasis is on the solution to the optimal control problem of the lie algebra generated by certain system vector fields is nilpotent consequences of the maximum principle are deduced for such systems

the value of side information in network flow optimization abstract consider a group of agents who seek to simultaneously traverse a graph each edge of the graph has an associated weight e g a delay and the agents seek to minimize the cumulative weight incurred by all agents as each traverses a path of the graph an edge s weight is a function of the number of agents that use that edge as well as an inherent random weight if the agents have no side information about the random component they will deterministically organize themselves so as to optimize their average performance we consider a generalization of this framework whereby the agents have access to a limited amount of shared side information about the edge weights and we study the relationship between information quantity and performance

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

the dynamics of multiple neck formation and fragmentation in high rate extension of ductile materials abstract dynamic necking bifurcation in rapidly extending cylindrical rods is investigated it has been found that both short wavelength and long wavelength perturbations are suppressed by inertia and an intermediate wavelength is favored the analysis predicts an increase in the number of necks and an increase in the bifurcation strain with increasing extension rate in agreement with experimental observations in terms of the number of necks formed as a function of extension rate good agreement has been found between the experiments and the analysis at any given aspect ratio the model also predicts that beyond a critical extension rate the mode number of the dominant perturbation increases rapidly and the perturbation begins to look more like a surface instability this could lead to a fragmentation mechanism at high extension speeds which is different from multiple necking currently no experimental results are available to test this prediction numerical simulations have been conducted to simulate the fragmentation results using gurson s constitutive law along with a porous failure criterion good agreement between the experimental observations and the numerical results has been obtained for the fragmentation statistics however the numerical results consistently overestimate the number of necks and the fracture strain possibly due to uncertainty in the constitutive data used especially at large strains

coadjoint orbits vortices and clebsch variables for incompressible fluids this paper is a study of incompressible fluids especially their clebsch variables and vortices using symplectic geometry and the lie poisson structure on the dual of a lie algebra following ideas of arnold and others it is shown that euler s equations are lie poisson equations associated to the group of volume preserving diffeomorphisms the dual of the lie algebra is seen to be the space of vortices and kelvin s circulation theorem is interpreted as preservation of coadjoint orbits in this context clebsch variables can be understood as momentum maps the motion of n point vortices is shown to be identifiable with the dynamics on a special coadjoint orbit and the standard canonical variables for them are a special kind of clebsch variables point vortices with cores vortex patches and vortex filaments can be understood in a similar way this leads to an explanation of the geometry behind the hald beale majda convergence theorems for vorticity algorithms symplectic structures on the coadjoint orbits of a vortex patch and filament are computed and shown to be closely related to those commonly used for the kdv and the schrodinger equations respectively

optimization problems on graphs with independent random edge weights optimization problems on complete graphs with edge weights drawn independently from a fixed distribution are considered several methods for analyzing these problems are discussed including greedy methods applications of boole s inequality and exploitation of relationships with results about random unweighted graphs these techniques are illustrated in the case in which the edge weights are drawn from a normal distribution in particular we investigate the expected behavior of the minimum weight clique on k vertices we describe the asymptotic behavior in probability and or almost surely of the random variable which describes the optimum we also discuss the asymptotic behavior of its mean finally techniques are demonstrated by which we may determine an asymptotic description of the behavior of a greedy algorithm for this problem

monotone hopf harmonics we introduce the concept of monotone hopf harmonics in 2d as an alternative to harmonic homeomorphisms much of the foregoing is motivated by the principle of non interpenetration of matter in the mathematical theory of nonlinear elasticity ne the question we are concerned with is whether or not a dirichlet energy minimal mapping between jordan domains with a prescribed boundary homeomorphism remains injective in the domain the classical theorem of rado kneser choquet asserts that this is the case when the target domain is convex an alternative way to deal with arbitrary target domains is to minimize the dirichlet energy subject to only homeomorphisms and their limits this leads to the so called hopf laplace equation among its solutions some rather surreal are continuous monotone mappings of sobolev class mathscr w 1 2 text loc called monotone hopf harmonics it is at the heart of the present paper to show that such solutions are correct generalizations of harmonic homeomorphisms and in particular are legitimate deformations of hyperelastic materials in the modern theory of ne we make this clear by means of several examples

contact between nonlinearly elastic bodies we study the contact between nonlinearly elastic bodies by variational methods after the formulation of the mechanical problem we provide existence results based on polyconvexity and on quasiconvexity then we derive the euler lagrange equation as a necessary condition for minimizers here clarke s generalized gradients are the essential tool to treat the nonsmooth obstacle condition

nilpotent approximations of control systems and distributions a constructive method is given to approximate the vector fields in a nonlinear control system which is linear in the controls by a system of similar form and on the same state space with the describing vector fields of the approximating system generating a lie algebra which has a certain relevant sub algebra nilpotent given a k dimensional distribution on r n say locally defined near zero the method leads to an approximating k distribution which has nilpotent basis agrees with the original distribution at zero and each derived distribution of the approximating distribution and original distribution also agree at zero

geometric approach to non linear optimal control problems minimum energy control problems are investigated for affine non linear systems evolving on manifolds in the framework of the geometric theory of hamiltonian systems some insights into the problems are gained via this machinery

worst case and probabilistic analysis of a geometric location problem we consider the problem of choosing k medians among n points on the euclidean plane such that the sum of the distances from each of the n points to its closest median is minimized we show that this problem is np complete we also present two heuristics that produce arbitrarily good solutions with probability going to 1 one is a partition heuristic and works when k grows linearly or almost so with n the other is the honeycomb heuristic and is applicable to rates of growth of k of the form k sim n varepsilon 0 varepsilon 1

optimal feedback control of non linear systems abstract a method is proposed for computing optimal nonlinear feedback control laws it is shown that the feedback loop satisfies a system of quasi linear partial differential equations this system is of first order when the dimensions of the state and the control vector are the same hereby it degenerates into algebraic equations when the performance index does not depend on the control these important results offer new ways for determining optimal feedback laws connections with the volterra series and the hamilton jacobi bellman equation are treated some examples are given to illustrate the advantages of this method

a variational approach to nonrigid morphological image registration a variational method for nonrigid registration of multimodal image data is presented a suitable deformation will be determined via the minimization of a morphological i e contrast invariant matching functional along with an appropriate regularization energy the aim is to correlate the morphologies of a template and a reference image under the deformation mathematically the morphology of images can be described by the entity of level sets of the image and hence by its gauss map a class of morphological matching functionals is presented which measure the defect of the template gauss map in the deformed state with respect to the deformed gauss map of the reference image the problem is regularized by considering a nonlinear elastic regularization energy existence of homeomorphic minimizing deformation is proved under assumptions on the class of admissible deformations with respect to actual medical applications suitable generalizations of the matching energies and the boundary conditions are pre

on shortest paths in graphs with random weights we consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in 0 1 we show that die longest of these paths is bounded by c log n n almost surely where c is a constant and n is the number of nodes our bound is the best possible up to a constant we apply this result to some well known problems and obtain several algorithmic improvements over existing results our results hold with obvious modifications to random as opposed to complete graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero as a corollary of our proof we get a new result concerning the diameter of random graphs

on the optimal control of bilinear systems and its relation to lie algebras the optimal control of bilinear systems is considered and related to the lie algebra generated by the system matrices interesting results obtain when this lie algebra is nilpotent

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

the value of side information in shortest path optimization consider an agent who seeks to traverse the shortest path in a graph having random edge weights if the agent has no side information about the realizations of the edge weights it should simply take the path of least average length we consider a generalization of this framework whereby the agent has access to a limited amount of side information about the edge weights ahead of choosing a path we define a measure for information quantity provide bounds on the agent s performance relative to the amount of side information it receives and present algorithms for optimizing side information the results are based on a new graph characterization tied to shortest path optimization

the modeling of deformable bodies with frictionless self contacts we propose a mathematical model for m dimensional deformable bodies moving in mathbb r n that allows for frictionless contacts or self contacts while forbidding transversal self intersection to this end a topological constraint is imposed to the set of admissible deformations we restrict our analysis to the static case although the dynamic case is briefly addressed at the end of the article in this case no transversal self intersection can occur as long as 2m n so our modeling is mainly designed to handle the case 2m geqq n for nonlinear hyperelastic bodies we prove the existence of at least one minimizer of the energy on the set of admissible deformations under suitable assumptions on the stored energy function moreover for certain choices of m and n under regularity assumptions on the minimizers the solutions of the minimization problem satisfy euler lagrange equations

on the global stability of two dimensional incompressible elastic bars in uniaxial extension when a rectangular bar is subjected to uniaxial tension the bar usually deforms approximately homogeneously and isoaxially until a critical load is reached a bifurcation such as the formation of shear bands or a neck may then be observed one approach is to model such an experiment as the in plane extension of a two dimensional homogeneous isotropic incompressible hyperelastic material in which the length of the bar is prescribed the ends of the bar are assumed to be free of shear and the sides are left completely free it is shown that standard constitutive hypotheses on the stored energy function imply that no such bifurcation is possible in this model due to the fact that the homogeneous isoaxial deformation is the unique absolute minimizer of the elastic energy thus in order for a bifurcation to occur either the material must cease to be elastic or the stored energy function must violate the standard hypotheses the fact that no local bifurcations can occur under the assumptions used herein was known previously since these assumptions prohibit the load on the bar from reaching a maximum value however the fact that the homogeneous deformation is the absolute minimizer of the energy appears to be a new result

fibonacci heaps and their uses in improved network optimization algorithms in this paper we develop a new data structure for implementing heaps priority queues our structure fibonacci heaps abbreviated f heaps extends the binomial queues proposed by vuillemin and studied further by brown f heaps support arbitrary deletion from an n item heap in o log n amortized time and all other standard heap operations in o 1 amortized time using f heaps we are able to obtain improved running times for several network optimization algorithms in particular we obtain the following worst case bounds where n is the number of vertices and m the number of edges in the problem graph o n log n m for the single source shortest path problem with nonnegative edge lengths improved from o m log m n 2 n o n 2 log n nm for the all pairs shortest path problem improved from o nm log m n 2 n o n 2 log n nm for the assignment problem weighted bipartite matching improved from o nm log m n 2 n o m m n for the minimum spanning tree problem improved from o m log log m n 2 n where m n min i u log i n m n note that m n log n if m n of these results the improved bound for minimum spanning trees is the most striking although all the results give asymptotic improvements for graphs of appropriate densities

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

a new approach to nonlinear optimal feedback law abstract for a fixed time free endpoint optimal control problem it is shown that the optimal feedback control satisfies a system of ordinary differential equations they are obtained using an elimination procedure of the adjoint vector which appears linearly in a set of differential equations these equations involving lie brackets of vector fields are derived from the maximum principle an application of this approach to robotics is given

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

local asymptotic approximation of nonlinear control systems let g1 gm be smooth vector fields on rd using a canonical linear rescaling of time and space coordinates the control system is transformed into a system on the same state space which is arbitrarily close to a nilpotent one this approximation takes place in the c topology and is particularly useful for the study of local properties of which depend on the high order variational structure of the vector fields gi

a shortest path algorithm with expected time o n 2 log n log n we present an algorithm which determines the shortest distances between all pairs of points in a nonnegatively weighted directed graph in average time o n 2 log n log n this algorithm which uses a search strategy similar to dantzig management sci 2 1960 pp 187 190 linear programming and extensions princeton univ press 1963 and spira siam j comput 2 1973 pp 28 32 executes in the stated time over quite general classes of probability distributions on such graphs

complete convergence of short paths and karp s algorithm for the tsp let xi 1 i 0 sum infty n 1 p left t n sqrt n beta gt epsilon right lt infty this result is essential in justifying karp s algorithm for the traveling salesman problem under the independent model and it settles a question posed by b w weide

collocation approximation methods for nonlinear singular integrodifferential equations in banach spaces abstract a new approximation method is proposed for the numerical evaluation of the nonlinear singular integrodifferential equations defined in banach spaces the collocation approximation method is therefore applied to the numerical solution of such type of nonlinear equations by using a system of chebyshev functions through the application of the collocation method is investigated the existence of solutions of the system of non linear equations used for the approximation of the nonlinear singular integrodifferential equations which are defined in a complete normed space i e a banach space

global existence results for viscoplasticity at finite strain we study a model for rate dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor and investigate the existence of global in time solutions to the related pde system we reveal its underlying structure as a generalized gradient system where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite strain elasticity as well as the multiplicative decomposition of finite strain plasticity moreover the dissipation potential depends on the left invariant plastic rate and thus depends on the plastic state variable the existence theory is developed for a class of abstract nonsmooth and nonconvex gradient systems for which we introduce suitable notions of solutions namely energy dissipation balance and energy dissipation inequality solutions hence we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory

general existence theorems for unilateral problems in continuum mechanics the problem of minimizing a possibly non convex and non coercive functional is studied either necessary or sufficient conditions for the existence of solutions are given involving a generalized recession functional whose properties are discussed thoroughly the abstract results are applied to find existence of equilibrium configurations of a deformable body subject to a system of applied forces and partially constrained to lie inside a possibly unbounded region

all pairs shortest paths in o n 2 time with high probability we present an all pairs shortest path algorithm whose running time on a complete directed graph on n vertices whose edge weights are chosen independently and uniformly at random from 0 1 is o n2 in expectation and with high probability this resolves a long standing open problem the algorithm is a variant of the dynamic all pairs shortest paths algorithm of demetrescu and italiano 2006 the analysis relies on a proof that the number of locally shortest paths in such randomly weighted graphs is o n2 in expectation and with high probability we also present a dynamic version of the algorithm that recomputes all shortest paths after a random edge update in o log2n expected time

a variational rod model with a singular nonlocal potential the classical theory of elastic rods does not account for the possibility that large deformations may involve distinct points along the rod occupying the same physical space we develop an elastic rod model with a pairwise repulsive potential such that if two non adjacent points along the rod are close in physical space there is an energy barrier which prevents contact while for points nearby along the rod the potential is describable classically this framework is developed to prove the existence of minimizers within each homotopy class where the idea of topological homotopy of a curve is generalized to elastic rods as framed curves finally the relevant first order optimality conditions are derived and used to investigate the regularity of minimizers

control of nonlinear systems in singular problems abstract the minimum singular functional control problem is analyzed for a class of multi input affine nonlinear systems we develop an algorithm for solving the singular problem that is applicable whether or not singular sub arcs exist in the optimal control the optimal control of bilinear systems to obtain by considering the lie algebra generated by the system matrices

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

local invertibility in sobolev spaces with applications to nematic elastomers and magnetoelasticity we define a class of deformations in w 1 p omega mathbb r n p n 1 with a positive jacobian that do not exhibit cavitation we characterize that class in terms of the non negativity of the topological degree and the equality det det that the distributional determinant coincides with the pointwise determinant of the gradient maps in this class are shown to satisfy a property of weak monotonicity and as a consequence they enjoy an extra degree of regularity we also prove that these deformations are locally invertible moreover the neighbourhood of invertibility is stable along a weak convergent sequence in w 1 p and the sequence of local inverses converges to the local inverse we use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps we apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity

unilateral problems in nonlinear three dimensional elasticity in 1977 john ball obtained a series of very important existence results in hyperelasticity the objective of the present paper is to extend his results in two directions first by considering unilateral boundary conditions corresponding to contact without friction secondly by allowing locking constraints in the set of admissible deformations

on minimum quadratic functional control of affine nonlinear systems abstract the minimization control problem of the quadratic functionals for the class of affine nonlinear systems in the hypothesis of nilpotent associated lie algebra is analyzed the optimal control correspondent to the first second and third order nilpotent operator is determined h bourdache siguerdinljane has applied the method of lie algebras to the study of optimal control regulation of satellites s banks and m yew have studied the optimal control of energy consumption minimization for a class of bilinear systems and j s liu et al have generalized this result for the class of affine nonlinear systems the optimal control determination of nonlinear system with a nilpotent structure minimizing the quadratic functionals generalizes the results of j s liu k yuan w s lin and s p banks and m k yew regarding the energy minimization of the affine nonlinear and bilinear systems