on the computational complexity of piecewise linear homotopy algorithms we show that piecewise linear homotopy algorithms may take a number of steps that grows exponentially with the dimension when solving a system of linear equations whose solution lies close to the starting point our examples are based on an example of murty exhibiting exponential growth for lemke s algorithm for the linear complementarity problem

approximate labelling for simplicial algorithms and two classes of special subsets of the sphere call a subset of rn complete if the origin is in its convex hull we are concerned with the construction of large finite sets of points on the unit sphere whose complete subsets are in some sense easy to describe our motivation comes from simplicial algorithms for approximating zeroes of functions we construct two classes of such subsets of the sphere both of which generalize a subset that corresponds to usual integer labelling

solving discrete zero point problems in this paper we present two theorems on the existence of a discrete zero point of a function from the n dimensional integer lattice n to the n dimensional euclidean space n the theorems differ in their boundary conditions for both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps

proof of breaking of self organized criticality in a nonconservative abelian sandpile model by computer simulations it was reported that the bak tang wiesenfeld btw model loses self organized criticality soc when some particles are annihilated in a toppling process in the bulk of system we give a rigorous proof that the btw model loses soc as soon as the annihilation rate becomes positive to prove this a nonconservative abelian sandpile model is defined on a square lattice which has a parameter ensuremath alpha g 1 representing the degree of breaking of the conservation law this model is reduced to be the btw model when ensuremath alpha 1 by calculating the average number of topplings in an avalanche t exactly it is shown that for any ensuremath alpha g1 t l ensuremath infty even in the infinite volume limit the power law divergence of t with an exponent 1 as stackrel ensuremath rightarrow ensuremath alpha 1 gives a scaling relation 2 ensuremath nu 2 ensuremath a 1 for the critical exponents ensuremath nu and a of the distribution function of t the 1 1 height correlation c 11 r is also calculated analytically and we show that c 11 r is bounded by an exponential function when ensuremath alpha g1 although c 11 r ensuremath sim r ensuremath 2d was proved by majumdar and dhar for the d dimensional btw model a critical exponent ensuremath nu 11 characterizing the divergence of the correlation length ensuremath xi as stackrel ensuremath rightarrow ensuremath alpha 1 is defined as ensuremath xi ensuremath sim ensuremath alpha ensuremath 1 ensuremath ensuremath nu 11 and our result gives an upper bound ensuremath nu 11 l 1 2

an efficient simplicial algorithm for computing a zero of a convex union of smooth functions we present an efficient simplicial algorithm for computing a zero of a point to set mapping that is formed by piecing together smooth functions such mappings arise in nonlinear programming and economic equilibrium problems our algorithm under suitable regularity conditions on the problem generates a sequence converging at least q superlinearly to a zero of the mapping asymptotically it operates in a space of reduced dimension analogous to an active set strategy in the optimization setting but it switches active sets automatically results of computational experiments are given

a simplicial approach to the determination of an integer point of a simplex we propose an algorithm to find an integer point of a simplex the algorithm is based on an integer labeling rule and a triangulation of the space starting at an integer point the algorithm leaves it along one of n rays and generates a sequence of adjacent simplices of varying dimension within a finite number of iterations the algorithm either yields an integer point of the simplex or proves that no such point exists

the infinite volume limit of dissipative abelian sandpiles we construct the thermodynamic limit of the stationary measures of the bak tang wiesenfeld sandpile model with a dissipative toppling matrix sand grains may disappear at each toppling we prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic markov process leaving it invariant we show how to extend the dhar formalism of the abelian group of toppling operators to infinite volume in order to obtain a compact abelian group with a unique haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

principal geodesic analysis for the study of nonlinear statistics of shape a primary goal of statistical shape analysis is to describe the variability of a population of geometric objects a standard technique for computing such descriptions is principal component analysis however principal component analysis is limited in that it only works for data lying in a euclidean vector space while this is certainly sufficient for geometric models that are parameterized by a set of landmarks or a dense collection of boundary points it does not handle more complex representations of shape we have been developing representations of geometry based on the medial axis description or m rep while the medial representation provides a rich language for variability in terms of bending twisting and widening the medial parameters are not elements of a euclidean vector space they are in fact elements of a nonlinear riemannian symmetric space in this paper we develop the method of principal geodesic analysis a generalization of principal component analysis to the manifold setting we demonstrate its use in describing the variability of medially defined anatomical objects results of applying this framework on a population of hippocampi in a schizophrenia study are presented

solving discrete systems of nonlinear equations abstract we study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice z n of the n dimensional euclidean space r n it is assumed that the set is integrally convex which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of z n and each simplex of the triangulation lies in an n dimensional cube of size one with respect to this triangulation we assume that the function satisfies some property that replaces continuity under this property and some boundary condition the function has a zero point to prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations the standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied because the continuity property is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself we apply the main existence result to prove the existence of a pure cournot nash equilibrium in a cournot oligopoly model we further obtain a discrete analogue of the well known borsuk ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem

variable dimension algorithms basic theory interpretations and extensions of some existing methods in this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by van der laan and talman we introduce a new concept primal dual pair of subdivided manifolds and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms our basic model furnishes interpretations to several existing methods lemke s algorithm for the linear complementarity problem its extension to the nonlinear complementarity problem a variable dimension algorithm on conical subdivisions and merrill s algorithm we shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us a method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed

a variable dimension fixed point algorithm and the orientation of simplices a variable dimension algorithm with integer labelling is proposed for solving systems ofn equations inn variables the algorithm is an integer labelling version of the 2 ray algorithm proposed by the author the orientation of lower dimensional simplices is studied and is shown to be preserved along a sequence of adjacent simplices

shape analysis of elastic curves in euclidean spaces this paper introduces a square root velocity srv representation for analyzing shapes of curves in euclidean spaces under an elastic metric in this srv representation the elastic metric simplifies to the il2 metric the reparameterization group acts by isometries and the space of unit length curves becomes the unit sphere the shape space of closed curves is the quotient space of a submanifold of the unit sphere modulo rotation and reparameterization groups and we find geodesics in that space using a path straightening approach these geodesics and geodesic distances provide a framework for optimally matching deforming and comparing shapes these ideas are demonstrated using 1 shape analysis of cylindrical helices for studying protein structure 2 shape analysis of facial curves for recognizing faces 3 a wrapped probability distribution for capturing shapes of planar closed curves and 4 parallel transport of deformations for predicting shapes from novel poses

abelian sandpiles and the harmonic model we present a construction of an entropy preserving equivariant surjective map from the d dimensional critical sandpile model to a certain closed shift invariant subgroup of mathbb t mathbb z d the harmonic model a similar map is constructed for the dissipative abelian sandpile model and is used to prove uniqueness and the bernoulli property of the measure of maximal entropy for that model

computing an integer point of a simplex with an arbitrary starting homotopy like simplicial algorithm abstract an arbitrary starting homotopy like simplicial algorithm is developed for computing an integer point of an n dimensional simplex the algorithm is derived from the use of an integer labeling rule and a triangulation of r n 0 1 and consists of two interchanging phases one phase of the algorithm constitutes a homotopy simplicial algorithm which generates n 1 dimensional simplices in r n 0 1 and the other phase of the algorithm constitutes a pivoting procedure which generates n dimensional simplices in either r n 0 or r n 1 the algorithm varies from one phase to the other when the matrix defining the simplex is in the so called canonical form starting at an arbitrary integer point in r n 0 the algorithm within a finite number of iterations either yields an integer point of the simplex or proves that no such point exists

a variable dimension solution approach for the general spatial price equilibrium problem algorithms are presented that are specifically designed for solving general nonlinear multicommodity spatial price equilibrium problems i e problems with nonlinear transportation cost functions nonlinear supply and demand functions inter commodity congestion effects intercommodity substitution and complementarity effects and interactions among transportation links and among spatially separated markets the algorithms are specializations of an iterative method for solving nonlinear complementarity problems that requires solving a system of nonlinear equations at each iteration the algorithms exploit the network structure of the problems to reduce the size of the system of equations to be solved at each iteration the decision rules for determining which equations are to be included in the system at each iteration are extremely simple and the remainder of the computational work is carried out by the nonlinear equation solver because of this the algorithms are very easy to implement with readily available software in addition since the decision rules only require sign information only the final system needs to be solved with precision

large deformation diffeomorphic metric curve mapping we present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping lddmm scheme for computing an optimal transformation between two curves embedded in euclidean space d curves are first represented as vector valued measures which incorporate both location and the first order geometric structure of the curves then a hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves we describe a discretized version of this in which discrete sequences of points along the curve are represented by vector valued functionals this gives a convenient and practical way to define a matching functional for curves we derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in 2 and 3 behaviors of the curve mapping are discussed using 2d curves the applications to shape classification is shown and experiments with 3d curves extracted from brain cortical surfaces are presented

crossover phenomenon in self organized critical sandpile models we consider a stochastic sandpile where the sand grains of unstable sites are randomly distributed to the nearest neighbors increasing the value of the threshold condition the stochastic character of the distribution is lost and a crossover to the scaling behavior of a different sandpile model takes place where the sand grains are equally transferred to the nearest neighbors the crossover behavior is analyzed numerically in detail especially we consider the exponents which determine the scaling behavior

exact solution of a stochastic directed sandpile model we introduce and analytically solve a directed sandpile model with stochastic toppling rules the model clearly belongs to a different universality class from its counterpart with deterministic toppling rules previously solved by dhar and ramaswamy the critical exponents are d 7 4 ensuremath tau 10 7 in two dimensions and d 3 2 ensuremath tau 4 3 in one dimension the upper critical dimension of the model is three at which the exponents apart from logarithmic corrections reach their mean field values d 2 ensuremath tau 3 2

self organized critical state of sandpile automaton models we study a general bak tang wiesenfeld type automaton model of self organized criticality in which the toppling conditions depend on local height but not on its gradient we characterize the critical state and determine its entropy for an arbitrary finite lattice in any dimension the two point correlation function is shown to satisfy a linear equation the spectrum of relaxation times describing the approach to the critical state is also determined exactly

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

rigid affine and locally affine registration of free form surfaces in this paper we propose a new framework to perform nonrigid surface registration it is based on various extensions of an iterative algorithm recently presented by several researchers besl and mckay 1992 champleboux et al 1992 chen and medioni 1992 menq and lai 1992 zhang 1994 to rigidly register surfaces represented by a set of 3d points when a prior estimate of the displacement is available our framework consists of three stages first we search for the best rigid displacement to superpose the two surfaces we show how to efficiently use curvatures to superpose principal frames at possible corresponding points in order to find a prior rough estimate of the displacement and initialize the iterative algorithm second we search for the best affine transformation we introduce differential information in points coordinates this allows us to match locally similar points then we show how principal frames and curvatures are transformed by an affine transformation finally we introduce this differential information in a global criterion minimized by extended kalman filtering in order to ensure the convergence of the algorithm third we locally deform the surface instead of computing a global affine transformation we attach to each point a local affine transformation varying smoothly along the surface we call this deformation a locally affine deformation all these stages are illustrated with experiments on various real biomedical surfaces teeth faces skulls brains and hearts which demonstrate the validity of the approach

a new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices a new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaces of unit simplices the triangulation which underlies the algorithm differs from the triangulations ofs used thus far moreover the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger more precisely there is a ray leading from the starting point to each vertex ofs in cases is the product ofn one dimensional unit simplices the alogrithm is similar to the octahedral algorithm onrn having 2n rays also the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms ons computational results will show that the number of iterations for the new algorithm is much less the examples concern the computation of equilibria in noncooperative games exchange economies and trade models

a vector labeling method for solving discrete zero point and complementarity problems in this paper we establish the existence of a discrete zero point of a function from the n dimensional integer lattice mathbb z n to the n dimensional euclidean space mathbb r n under very general conditions with respect to the behavior of the function the proof is constructive and uses a combinatorial argument based on a simplicial algorithm with vector labeling and lexicographic linear programming pivot steps the algorithm provides an efficient method to find an exact solution we also discuss how to adapt the algorithm for two related problems namely to find a discrete zero point of a function under a general antipodal condition and to find a solution to a discrete nonlinear complementarity problem in both cases the modified algorithm provides a constructive existence proof too we further show that the algorithm for the discrete nonlinear complementarity problem generalizes the well known lemke s method to nonlinear environments an economic application is also presented

a discrete multivariate mean value theorem with applications we establish a discrete multivariate mean value theorem for the class of positive maximum component sign preserving functions a constructive and combinatorial proof is given based upon a simplicial algorithm and vector labeling moreover we apply this theorem to a discrete nonlinear complementarity problem and an economic equilibrium problem with indivisibilities and show the existence of solutions in both problems under certain mild conditions

on the computation of fixed points in the product space of unit simplices and an application to noncooperative n person games in this paper an algorithm based on the principle of simplicial approximation is introduced to compute fixed points of upper semicontinuous point to set mappings from the product space s of unit simplices into itself the algorithm is a modification of an algorithm introduced in an earlier paper the main feature is that it starts with an arbitrary chosen point in s and that the triangulation of s depends on the starting point moreover the algorithm can terminate with a non full dimensional subsimplex yielding a good approximation an application is given for non cooperative n person games where s is the strategy space some computational experiences are given

estimating nonparametric convex functions using gradient data abstract the estimation and use of nonparametric convex functions has been discussed in the recent literature where it is assumed that the data used in the estimation consists of a sample of observations of the convex function however an important additional or alternative source of data consists of observations on the sub gradient of the function in the present paper we discuss how to use such sub gradient data to estimate nonparametric functions this a allows us to estimate nonparametric convex functions when other point observation data is unavailable or unobservable or unreliable or costly to obtain b allows us an alternative even when other point observation data is available and c allows us to improve our estimates by using both types of data together the resultant function estimators may not be unique hence we compute bounds on the convex set of estimators we also consider introducing other data in addition to the gradient data the discussion is illustrated by reference to estimating production functions benefit functions and demand functions

a unified approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm we present a unified description of a class of restart fixed point algorithms including merrill s method and several variable dimension algorithms for their implementation on computers based on this description we show that some pivot saving techniques originally developed for the homotopy methods can be applied to the class we also propose a new variable dimension algorithm having 3 n 1 rays along which we can move toward a solution some numerical comparisons of the simplicial restart algorithms merrill s method the 2n method the octahedral method and the new one support that the latter two methods are more efficient than the others

sandpile models and random walkers on finite lattices abelian sandpile models both deterministic such as the bak tang wiesenfeld btw model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic such as the manna model s s manna j phys a 24 l363 1991 are studied on finite square lattices with open boundaries the avalanche size distribution p l n is calculated for a range of system sizes l the first few moments of this distribution are evaluated numerically and their dependence on the system size is examined the sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries it is shown that the conservation law provides an interesting connection between the sandpile models and random walk models using this connection it is shown that the average avalanche sizes l for the btw and manna models are equal to each other and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size l this is in spite of the fact that the sandpile models with deterministic btw and stochastic manna toppling rules exhibit different critical exponents indicating that they belong to different universality classes

iterative point matching for registration of free form curves and surfaces a heuristic method has been developed for registering two sets of 3 d curves obtained by using an edge based stereo system or two dense 3 d maps obtained by using a correlation based stereo system geometric matching in general is a difficult unsolved problem in computer vision fortunately in many practical applications some a priori knowledge exists which considerably simplifies the problem in visual navigation for example the motion between successive positions is usually approximately known from this initial estimate our algorithm computes observer motion with very good precision which is required for environment modeling e g building a digital elevation map objects are represented by a set of 3 d points which are considered as the samples of a surface no constraint is imposed on the form of the objects the proposed algorithm is based on iteratively matching points in one set to the closest points in the other a statistical method based on the distance distribution is used to deal with outliers occlusion appearance and disappearance which allows us to do subset subset matching a least squares technique is used to estimate 3 d motion from the point correspondences which reduces the average distance between points in the two sets both synthetic and real data have been used to test the algorithm and the results show that it is efficient and robust and yields an accurate motion estimate

efficiency and implementation of simplicial zero point algorithms in this paper we compare the efficiency of several simplicial variable dimension algorithms to do so we first treat the issues of degeneracy and accelerating we present a device for solving degeneracy furthermore we compare several accelerating techniques the technique of iterated quasi newton steps after each major cycle of the simplicial algorithm is implemented in a computer code which is used to compare the efficiency of the n 1 ray 2n ray 2n ray and 3n 1 ray algorithms except for the n 1 ray algorithm the number of function evaluations does not differ very much between the various algorithms it appeared however that the 2n algorithm needs considerably less computation time

dissipative abelian sandpiles and random walks we show that the dissipative abelian sandpile on a graph mathcal l can be related to a random walk on a graph that consists of mathcal l extended with a trapping site from this relation it can be shown using exact results and a scaling assumption that the correlation length exponent ensuremath nu of the dissipative sandpiles always equals 1 d w where d w is the fractal dimension of the random walker this leads to a new understanding of the known result that ensuremath nu 1 2 on any euclidean lattice our result is however more general and as an example we also present exact data for finite sierpinski gaskets which fully confirm our predictions

crashing a maximum weight complementary basis we consider the problem of finding a maximum weight complementary basis of anm 2m matrix the problem arises naturally for example when a complementary set of columns is proposed as an initial basis for a warm start of lemke s algorithm but the set of columns is rank deficient we show that the problem is a special case of the problem of finding a maximum weight common base of two matroids furthermore we show how to efficiently implement an algorithm for the general problem in the present context finally we give computational results demonstrating the practicality of our algorithm in a typical application

simplicial approximation of solutions to the nonlinear complementarity problem with lower and upper bounds ideas of a simplicial variable dimension restart algorithm to approximate zero points onrn developed by the authors and of a linear complementarity problem pivoting algorithm are combined to an algorithm for solving the nonlinear complementarity problem with lower and upper bounds the algorithm can be considered as a modification of the2n ray zero point finding algorithm onrn it appears that for the new algorithm the number of linear programming pivot steps is typically less than for the2n ray algorithm applied to an equivalent zero point problem this is caused by the fact that the algorithm utilizes the complementarity conditions on the variables

evidence for universality within the classes of deterministic and stochastic sandpile models recent numerical studies have provided evidence that within the family of conservative undirected sandpile models with short range dynamic rules deterministic models such as the bak tang wiesenfeld model p bak c tang and k wiesenfeld phys rev lett 59 381 1987 and stochastic models such as the manna model s s manna j phys a 24 l363 1991 belong to different universality classes in this paper we examine the universality within each of the two classes in two dimensions by numerical simulations to this end we consider additional deterministic and stochastic models and use an extended set of critical exponents scaling functions and geometrical features universal behavior is found within the class of deterministic abelian models as well as within the class of stochastic models which includes both abelian and non abelian models in addition it is observed that deterministic but non abelian models exhibit critical exponents that depend on a parameter namely they are nonuniversal

on the existence and approximation of zeroes in this paper we consider the problem of finding zeroes of a continuous functionf from a convex compact subsetu of n to n in the first part of the paper it is proved thatf has a computable zero iff cn n satisfies the nonparallel condition for any two antipodal points on bdcn i e if for anyx bdcn f x f x 0 holds therefore we describe a simplicial algorithm to approximate such a zero it is shown that generally the degree of the approximate zero depends on the number of reflection steps made by the algorithm i e the number of times the algorithm switches from a face on bdcn to the face therefore the index of a terminal simplex is defined which equals the local brouwer degree of the function if is full dimensional in the second part of the paper the algorithm is used to generate possibly several approximate zeroes off two sucessive solutions may have both the same or opposite degrees again depending on the number of reflection steps by extendingf u n to a function g from a cube containingu to n the procedure can be applied to any continuous functionf without having any information about the global and local brouwer degrees a priori

parameterization invariant shape comparisons of anatomical surfaces we consider 3 d brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized past comparisons of such surfaces involve either volume deformations or non rigid matching under fixed parameterizations of surfaces we propose a new mathematical representation of surfaces called q maps such that l2 distances between such maps are invariant to re parameterizations this property allows for removing the parameterization variability by optimizing over the re parameterization group resulting in a proper parameterization invariant distance between shapes of surfaces we demonstrate this method in shape analysis of multiple brain structures for 34 subjects in the detroit fetal alcohol and drug exposure cohort study which results in a 91 classification rate for attention deficit hyperactivity disorder cases and controls this method outperforms some existing techniques such as spherical harmonic point distribution model spharm pdm or iterative closest point icp

sandpile models with and without an underlying spatial structure we present a simple mean field model for the sandpile model introduced by bak tang and wiesenfeld btw phys rev lett 59 381 1987 in the mean field model we are able to pinpoint the process of self organization as well as the emerging scale invariance displayed as a power law distribution of avalanche sizes we discuss the btw sandpile model on a lattice and show that the dynamical behavior can be expressed as a transport problem this implies that the average avalanche size scales with the system size and additional heuristic arguments related to the transport properties more than indicate the origin of the power law behavior we review recent work in which scaling relations and additional constraints between the various critical exponents are addressed we demonstrate that some of the proposed relations are inconsistent we present a coherent theory in which the scaling relations along with additional constraints leave only one exponent unknown

iteration and sensitivity for a spatial equilibrium problem with linear supply and demand functions we investigate a set of spatial equilibrium conditions which apply for example to a multiregional multicommodity market or to a physical system of interconnected sources and sinks under additional hypotheses the equilibrium conditions are the kuhn tucker conditions for a concave quadratic programming problem for the case in which the equilibrium conditions do not correspond to a programming problem an iterative process for obtaining a solution is discussed this iteration is motivated by the project independence evaluation system pies energy model algorithm an algorithmic method to determine the sensitivity of an equilibrium solution vector to data perturbations is described both the sensitivity analysis and the convergence of the iterative solution method are based upon a solvability lemma which guarantees that a solution of the equilibrium conditions may be expressed in terms of the problem data nondegeneracy assumptions for the inter regional flows are unnecessary for this discussion an

a reparameterisation based approach to geodesic constrained solvers for curve matching we present a numerical algorithm for a new matching approach for parameterisation independent diffeomorphic registration of curves in the plane targeted at robust registration between curves that require large deformations this condition is particularly useful for the geodesic constrained approach in which the matching functional is minimised subject to the constraint that the evolving diffeomorphism satisfies the hamiltonian equations of motion this means that each iteration of the nonlinear optimisation algorithm produces a geodesic up to numerical discretisation we ensure that the computed solutions correspond to geodesics in the shape space by enforcing the horizontality condition conjugate momentum is normal to the curve explicitly introducing and solving for a reparameterisation variable allows the use of a point to point matching condition the equations are discretised using the variational particle mesh method we provide comprehensive numerical convergence tests and benchmark the algorithm in the context of large deformations to show that it is a viable efficient and accurate method for obtaining geodesics between curves

group actions homeomorphisms and matching a general framework this paper constructs metrics on the space of images cal i defined as orbits under group actions cal g the groups studied include the finite dimensional matrix groups and their products as well as the infinite dimensional diffeomorphisms examined in trouve 1999 quaterly of applied math and dupuis et al 1998 quaterly of applied math left invariant metrics are defined on the product cal g times cal i thus allowing the generation of transformations of the background geometry as well as the image values examples of the application of such metrics are presented for rigid object matching with and without signature variation curves and volume matching and structural generation in which image values are changed supporting notions such as tissue creation in carrying one image to another

the variational particle mesh method for matching curves diffeomorphic matching only one of several names for this technique is a technique for non rigid registration of curves and surfaces in which the curve or surface is embedded in the flow of a time series of vector fields one seeks the flow between two topologically equivalent curves or surfaces which minimizes some metric defined on the vector fields i e the flow closest to the identity in some sense in this paper we describe a new particle mesh discretization for the evolution of the geodesic flow and the embedded shape particle mesh algorithms are very natural for this problem because lagrangian particles particles moving with the flow can represent the movement of the shape whereas the vector field is eulerian and hence best represented on a static mesh we explain the derivation of the method and prove the conservation properties the discrete method has a set of conserved momenta corresponding to the particle relabelling symmetry which converge to conserved quantities in the continuous problem we also introduce a new discretization for the geometric current matching condition of vaillant and glaunes 2005 surface matching via currents proc conf ipmi pp 381 92 we illustrate the method and the derived properties with numerical examples

a new variable dimension algorithm for the fixed point problem we propose a new variable dimension algorithm for solving a system ofn equations inn variables the algorithm solves a sequence of subproblems consisting of the first several equations the dimension of subproblems does not however increase monotonically a convergence condition is also presented

computational experience in solving equilibrium models by a sequence of linear complementarity problems this paper presents a modeling format and a solution algorithm for partial and general economic equilibrium problems it reports on computational experience from a series of small to medium sized problems taken from the literature on computation of economic equilibria the common characteristic of these models is the presence of weak inequalities and complementary slackness e g a linear technology with alternative activities or various institutional constraints on prices the algorithm computes the equilibrium by solving a sequence of linear complementarity problems the iterative outer part of this algorithm is a newton process for the inner part we use lemke s almost complementary pivoting algorithm theoretical results for the performance of this algorithm are at present available only for the partial equilibrium cases our computational experience with both types of models however is encouraging the algorithm solved all nine test problems when initiated at reasonable starting points five of these nine problems are solved for several different starting points indicating a large region over which the algorithm converges our results demonstrate that the algorithm is economical in terms of the number of pivots function evaluations and cpu time

a lexicographic algebraic theorem and its applications abstract in this paper we prove the following result given any full dimensional simple polytope p x r n a i t x b i i 1 m without redundant constraints and any vector c r n there exists a unique vertex x of p such that the matrix 1 1 1 1 c a i 1 a i 2 a i n 1 exists and is lexicographic positive where x is the solution of the n linear equations a i t t x b i t t 1 n this theorem generalizes a result on the n dimensional cube and can be used to resolve the degeneracy problem for simplicial fixed point algorithms we also discuss several applications for this result

morphometry of anatomical shape complexes with dense deformations and sparse parameters we propose a generic method for the statistical analysis of collections of anatomical shape complexes namely sets of surfaces that were previously segmented and labeled in a group of subjects the method estimates an anatomical model the template complex that is representative of the population under study its shape reflects anatomical invariants within the dataset in addition the method automatically places control points near the most variable parts of the template complex vectors attached to these points are parameters of deformations of the ambient 3d space these deformations warp the template to each subject s complex in a way that preserves the organization of the anatomical structures multivariate statistical analysis is applied to these deformation parameters to test for group differences results of the statistical analysis are then expressed in terms of deformation patterns of the template complex and can be visualized and interpreted the user needs only to specify the topology of the template complex and the number of control points the method then automatically estimates the shape of the template complex the optimal position of control points and deformation parameters the proposed approach is completely generic with respect to any type of application and well adapted to efficient use in clinical studies in that it does not require point correspondence across surfaces and is robust to mesh imperfections such as holes spikes inconsistent orientation or irregular meshing the approach is illustrated with a neuroimaging study of down syndrome ds the results demonstrate that the complex of deep brain structures shows a statistically significant shape difference between control and ds subjects the deformation based modelingis able to classify subjects with very high specificity and sensitivity thus showing important generalization capability even given a low sample size we show that the results remain significant even if the number of control points and hence the dimension of variables in the statistical model are drastically reduced the analysis may even suggest that parsimonious models have an increased statistical performance the method has been implemented in the software deformetrica which is publicly available at www deformetrica org

exactly solved model of self organized critical phenomena we define a variant of the model of bak tang and wiesenfeld of self organized critial behavior by introducing a preferred direction we characterize the critical state and by establishing equivalence to a voter model determine the critical exponents exactly in arbitrary dimension d the upper critical dimension for this model is three in two dimensions the model is equivalent to an earlier solved special case of directed percolation

numerical determination of the avalanche exponents of the bak tang wiesenfeld model we consider the bak tang wiesenfeld sandpile model on a two dimensional square lattice of lattice sizes up to l 4096 a detailed analysis of the probability distribution of the size area duration and radius of the avalanches will be given to increase the accuracy of the determination of the avalanche exponents we introduce a new method for analyzing the data which reduces the finite size effects of the measurements the exponents of the avalanche distributions differ slightly from previous measurements and estimates obtained from a renormalization group approach

a better triangulation for wright s 2n ray algorithm abstract we give a triangulation for wright s 2n ray algorithm to compute solutions of nonlinear equations according to measures of efficiency of triangulations it is better than any other available triangulation for the 2n ray algorithm

from waves to avalanches two different mechanisms of sandpile dynamics time series resulting from wave decomposition show the existence of different correlation patterns for avalanche dynamics for the d 2 bak tang wisenfeld model long range correlations determine a modification of the wave size distribution under coarse graining in time and multifractal scaling for avalanches in the manna model the distribution of avalanche coincides with that of waves which are uncorrelated and obey finite size scaling a result expected also for the d 3 bak tang wiesenfeld sandpile

a class of simplicial restart fixed point algorithms without an extra dimension in an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices ideas of that paper are used to construct a class of triangulations ofrn more precisely for somek 1 k n and positive integersm1 mk with sumn a triangulation ofrn is obtained by triangulating the cells which are formed by taking the product of given triangulations ofrmj j 1 k the triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inrn being the starting point of the algorithm fork n we obtain thek triangulation originally due to todd each element of the class can be used to find a simplex which approximates a fixed point of a mapping onrn by generating a unique path of adjacent simplices of variable dimension starting with the pointv

universality classes in isotropic abelian and non abelian sandpile models universality in isotropic abelian and non abelian sandpile models is examined using extensive numerical simulations to characterize the critical behavior we employ an extended set of critical exponents geometric features of the avalanches as well as scaling functions describing the time evolution of average quantities such as the area and size during the avalanche comparing between the abelian bak tang wiesenfeld model p bak c tang and k wiensenfeld phys rev lett 59 381 1987 and the non abelian models introduced by manna s s manna j phys a 24 l363 1991 and zhang y c zhang phys rev lett 63 470 1989 we find strong indications that each one of these models belongs to a distinct universality class

algebraic aspects of abelian sandpile models the abelian sandpile models feature a finite abelian group g generated by the operators corresponding to particle addition at various sites we study the canonical decomposition of g as a product of cyclic groups g z d1 z d2 z d3 z dg where g is the least number of generators of g and di is a multiple of di 1 the structure of g is determined in terms of the toppling matrix delta we construct scalar functions linear in the height variables of the pile that are invariant under toppling at any site these invariants provide convenient coordinates to label the recurrent configurations of the sandpile for an l l square lattice we show that g l in this case we observe that the system has non trivial symmetries transcending the obvious symmetries of the square namely those coming from the action of the cyclotomic galois group gall of the 2 l 1 th roots of unity which operates on the set of eigenvalues of h these eigenvalues are algebraic integers the product of which is the order mod g mod with the help of gall we are able to group the eigenvalues into certain subsets the products of which are separately integers and thus obtain an explicit factorization of mod g mod we also use gall to define other simpler sets of toppling invariants

universality in sandpiles we perform extensive numerical simulations of different versions of the sandpile model we find that previous claims about universality classes are unfounded since the method previously employed to analyze the data suffered from a systematic bias we identify the correct scaling behavior and provide evidences suggesting that sandpiles with stochastic and deterministic toppling rules belong to the same universality class

how self organized criticality works a unified mean field picture we present a unified dynamical mean field theory based on the single site approximation to the master equation for stochastic self organized critical models in particular we analyze in detail the properties of sandpile and forest fire ff models in analogy with other nonequilibrium critical phenomena we identify an order parameter with the density of active sites and control parameters with the driving rates depending on the values of the control parameters the system is shown to reach a subcritical absorbing or supercritical active stationary state criticality is analyzed in terms of the singularities of the zero field susceptibility in the limit of vanishing control parameters the stationary state displays scaling characteristics of self organized criticality soc we show that this limit corresponds to the breakdown of space time locality in the dynamical rules of the models we define a complete set of critical exponents describing the scaling of order parameter response functions susceptibility and correlation length in the subcritical and supercritical states in the subcritical state the response of the system to small perturbations takes place in avalanches we analyze their scaling behavior in relation with branching processes in sandpile models because of conservation laws a critical exponents subset displays mean field values ensuremath nu frac 1 2 and ensuremath gamma 1 in any dimensions we treat bulk and boundary dissipation and introduce a critical exponent relating dissipation and finite size effects we present numerical simulations that confirm our results in the case of the forest fire model our approach can distinguish between different regimes soc ff and deterministic ff studied in the literature and determine the full spectrum of critical exponents