on leja sequences some results and applications in this paper we collect and review some results concerning leja sequences on one and two dimensional domains then we present two applications and discuss some open problems on the asymptotic distribution of leja points on two dimensional domains

fekete points and convergence towards equilibrium measures on complex manifolds building on bb1 we prove a general criterion for convergence of possibly singular bergman measures towards pluripotential theoretic equilibrium measures on complex manifolds the criterion may be formulated in terms of the growth properties of the unit balls of certain norms on holomorphic sections or equivalently as an asymptotic minimization property for generalized donaldson l functionals our result settles in particular a well known conjecture in pluripotential theory concerning the equidistribution of fekete points and it gives the convergence of bergman measures towards the equilibrium measure for bernstein markov measures applications to interpolation of holomorphic sections are also discussed

the kuratowski convergence and connected components abstract we investigate the kuratowski convergence of the connected components of the sections of a definable set applying the result obtained to semialgebraic approximation of subanalytic sets we are led to some considerations concerning the connectedness of the limit set in general we discuss also the behaviour of the dimension of converging sections and prove some general facts about the kuratowski convergence in tame geometry

evolution by mean curvature flow of lagrangian spherical surfaces in complex euclidean plane abstract we describe the evolution under the mean curvature flow of embedded lagrangian spherical surfaces in the complex euclidean plane c 2 in particular we answer the question 4 7 addressed in 10 by a neves about finding out a condition on a starting lagrangian torus in c 2 such that the corresponding mean curvature flow becomes extinct at finite time and converges after rescaling to the clifford torus

on the computation of sets of points with low lebesgue constant on the unit disk abstract in this paper of numerical nature we test the lebesgue constant of several available point sets on the disk and propose new ones that enjoy low lebesgue constant furthermore we extend some results in cuyt 2012 analyzing the case of bos arrays whose radii are nonnegative gauss gegenbauer lobatto nodes with exponent noticing that the optimal still allows to achieve point sets on with low lebesgue constant n for degrees n 30 next we introduce an algorithm that through optimization determines point sets with the best known lebesgue constants for n 25 finally we determine theoretically a point set with the best lebesgue constant for the case n 1

tame linear extension operators for smooth whitney functions abstract for a compact set k r d we present a rather easy construction of a linear extension operator e e k c r d for the space of whitney jets e k which satisfies linear tame continuity estimates sup e f x m x r d c m e f r e m where s denotes the s th whitney norm the construction turns out to be possible if and only if the local markov inequality lmi s introduced by bos and milman holds for every s r on k in particular e k admits a tame linear extension operator if and only if the local markov inequality lmi s holds on k for some s 1

constructing least squares polynomial approximations polynomial approximations constructed using a least squares approach form a ubiquitous technique in numerical computation one of the simplest ways to generate data for least squares problems is wi

kinodynamic motion planning kinodynamic planning attempts to solve a robot motion problem subject to simultaneous kinematic and dynamics constraints in the general problem given a robot system we must find a minimal time trajectory that goes from a start position and velocity to a goal position and velocity while avoiding obstacles by a safety margin and respecting constraints on velocity and acceleration we consider the simplified case of a point mass under newtonian mechanics together with velocity and acceleration bounds the point must be flown from a start to a goal amidst polyhedral obstacles in 2d or 3d although exact solutions to this problem are not known we provide the first provably good approximation algorithm and show that it runs in polynomial time

on the generation of symmetric lebesgue like points in the triangle we compute point sets on the triangle that have low lebesgue constant with sixfold symmetries and gauss legendre lobatto distribution on the sides up to interpolation degree 18 such points have the best lebesgue constants among the families of symmetric points used so far in the framework of triangular spectral elements

on positivity of principal minors of bivariate bezier collocation matrix it is well known that the bivariate polynomial interpolation problem at uniformly distributed domain points of a triangle is correct thus the corresponding interpolation matrix m is nonsingular schumaker stated the conjecture that all principal submatrices of m are nonsingular too furthermore all of the corresponding determinants the principal minors are conjectured to be positive this result would solve the constrained interpolation problem in this paper the conjecture on minors for polynomial degree 17 and conjecture for some particular configurations of domain points are confirmed

asymptotics of greedy energy points for a symmetric kernel k x x x u on a locally compact metric space x we investigate the asymptotic behavior of greedy k energy points a i 1 for a compact subset a c x that are defined inductively by selecting a 1 a arbitrarily and a n 1 so that n i 1 k a n l1 a i inf x a n i 1 k x a i we give sufficient conditions under which these points also known as leja points are asymptotically energy minimizing i e have energy n i j k a i aj as n that is asymptotically the same as e a n min i j k x i x j x 1 x n a and have asymptotic distribution equal to the equilibrium measure for a for the case of riesz kernels k s x y x y s s 0 we show that if a is a rectifiable jordan arc or closed curve in rp and s 1 then greedy k s energy points are not asymptotically energy minimizing in contrast to the case s 1 additional results are obtained for greedy k s energy points on a sphere for greedy best packing points the case s and for weighted riesz kernels for greedy best packing points we provide a simple counterexample to a conjecture attributed to l bos

constructing roadmaps of semi algebraic sets i completeness abstract this paper describes preliminary work on an algorithm for planning collision free motions for a robot manipulator in the presence of obstacles the physical obstacles lead to forbidden regions in the robots configuration space and for collision free motion we need paths through configuration space which avoid these regions our method is to construct a certain one dimensional subset or roadmap of the space of allowable configurations if s denotes the set of allowable configurations the roadmap has the property that any connected component of s contains a single connected component of the roadmap it is also possible starting from an arbitrary point p s to rapidly construct a path from p to a point on the roadmap thus given any two points in s we can rapidly determine whether they lie in the same connected component of s and if they do we can return a candidate path between them we do not give a complete description of the algorithm here but we define the roadmap geometrically and verify that it has the necessary connectivity

angle theorems for the lagrangian mean curvature flow we prove that symplectic maps between riemann surfaces l m of constant nonpositive and equal curvature converge to minimal symplectic maps if the lagrangian angle alpha for the corresponding lagrangian submanifold in the cross product space l times m satisfies text osc alpha le pi if one considers a 4 dimensional kahler einstein manifold overline m of nonpositive scalar curvature that admits two complex structures j k which commute and assumes that l subset overline m is a compact oriented lagrangian submanifold w r t j such that the kahler form overline kappa w r t k restricted to l is positive and text osc alpha le pi then l converges under the mean curvature flow to a minimal lagrangian submanifold which is calibrated w r t overline kappa

an algorithm for computing fekete points in the triangle on the line and its tensor products fekete points are known to be the gauss lobatto quadrature points but unlike high order quadrature fekete points generalize to non tensor product domains such as the triangle thus fekete points might serve as an alternative to the gauss lobatto points for certain applications in this work we present a new algorithm to compute fekete points and give results up to degree 19 for the triangle for degree d 10 these points have the smallest lebesgue constant currently known the computations validate a conjecture of bos j approx theory 64 1991 pp 271 280 that fekete points along the boundary of the triangle are the one dimensional gauss lobatto points

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

polynomial approximation and quadrature on geographic rectangles abstract using some recent results on subperiodic trigonometric interpolation and quadrature and the theory of admissible meshes for multivariate polynomial approximation we study product gaussian quadrature hyperinterpolation and interpolation on some regions of s d d 2 such regions include caps zones slices and more generally spherical rectangles defined on s 2 by longitude and co latitude geographic rectangles we provide the corresponding matlab codes and discuss several numerical examples on s 2

shrinking and boundedly complete schauder frames in frechet spaces abstract we study schauder frames in frechet spaces and their duals as well as perturbation results we define shrinking and boundedly complete schauder frames on a locally convex space study the duality of these two concepts and their relation with the reflexivity of the space we characterize when an unconditional schauder frame is shrinking or boundedly complete in terms of properties of the space several examples of concrete schauder frames in function spaces are also presented

a comparison of high order interpolation nodes for the pyramid the use of pyramid elements is crucial to the construction of efficient hex dominant meshes m bergot g cohen and m durufle j sci comput 42 2010 pp 345 381 for conforming nodal finite element methods with mixed element types it is advantageous for nodal distributions on the faces of the pyramid to match those on the faces and edges of hexahedra and tetrahedra we adapt existing procedures for constructing optimized tetrahedral nodal sets for high order interpolation to the pyramid with constrained face nodes including two generalizations of the explicit warp and blend construction of nodes on the tetrahedron t warburton j engrg math 56 2006 pp 247 262 comparisons between nodal sets show that the lowest lebesgue constants are given by warp and blend nodes for order n leq 7 and fekete nodes for n 7 though numerical experiments show little variation in the conditioning and accuracy of all surveyed nonequidistant points

near optimal analysis of lasserre s univariate measure based bounds for multivariate polynomial optimization we consider a recent hierarchy of upper approximations proposed by lasserre arxiv 1907 097784 2019 for the minimization of a polynomial f over a compact set k n this hierarchy relies on using the push forward measure of the lebesgue measure on k by the polynomial f and involves univariate sums of squares of polynomials with growing degrees 2r hence it is weaker but cheaper to compute than an earlier hierarchy by lasserre siam journal on optimization 21 3 864 885 2011 which uses multivariate sums of squares we show that this new hierarchy converges to the global minimum of f at a rate in o log2 r r2 whenever k satisfies a mild geometric condition which holds e g for convex bodies as an application this rate of convergence also applies to the stronger hierarchy based on multivariate sums of squares which extends earlier convergence results to a wider class of compact sets furthermore we show that our analysis is near optimal by proving a lower bound on the convergence rate in 1 r2 for a class of polynomials on k 1 1 obtained by exploiting a connection to orthogonal polynomials

constructing nested nodal sets for multivariate polynomial interpolation we present a robust method for choosing multivariate polynomial interpolation nodes our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity a variant of a weighted lebesgue function nodes are therefore chosen that tend to control oscillations in the resulting interpolant this method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space our method is therefore flexible nested nodal sets are produced in spaces of arbitrary dimensions and the number of nodes added at each stage can be arbitrary the algorithm produces a nodal set given a probability measure on the input space thus parameterizing interpolants with respect to finite measures we present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics the lebesgue constant the vandermonde determinant and the vandermonde condition number we also show that a nongreedy version of the

a vertex algorithm for collision detection abstract an algorithm for detecting the collision of moving objects is presented the algorithm applies to polyhedral objects that can be represented as convex hulls of finite number of vertices in two or three dimensional space this is then extended to a three or four dimensional space respectively to represent the objects and their motion nonlinear programming techniques are then employed to detect possible interference the algorithm detects in one step whether or not the objects will interfere during their motion which may involve pure translations or rotations or both a subalgorithm for computing the minimum distance between the objects is also presented this can be used to solve the interference detection problem or for more general applications

computing fekete and lebesgue points simplex square disk we have computed point sets with maximal absolute value of the vandermonde determinant fekete points or minimal lebesgue constant lebesgue points on three basic bidimensional compact sets the simplex the square and the disk using routines of the matlab optimization toolbox we have obtained some of the best bivariate interpolation sets known so far

collision detection for moving polyhedra we consider the collision detection problem for a three dimensional solid object moving among polyhedral obstacles the configuration space for this problem is six dimensional and the traditional representation of the space uses three translational parameters and three angles typically euler angles the constraints between the object and obstacles then involve trigonometric functions we show that a quaternion representation of rotation yields constraints which are purely algebraic in a seven dimensional space by simple manipulation the constraints may be projected down into a six dimensional space with no increase in complexity the algebraic form of the constraints greatly simplifies computation of collision points and allows us to derive an efficient exact intersection test for an object which is translating and rotating among obstacles

whitney extension operators with arbitrary loss of differentiability abstract for a compact set k r d we characterize the existence of a linear extension operator e e k c r d for the space of whitney jets e k with a certain loss of derivatives that is the operator satisfies the following continuity estimates for all n n 0 and all f e k sup e f x n x r d c n f n where n denotes the whitney norm and the map n 0 n 0 is monotonically increasing with n n and 0 0 from our main result it follows directly that if a compact set k admits an extension operator then it is always possible to construct a second extension operator resembling the original whitney operators e n e n k c n r d where the evaluations of the jet occurring in the taylor polynomials are approximated by measures

a new obstruction to embedding lagrangian tori n let 2n be endowed with the canonical symplectic form a i 1 dxi dy i and consider a lagrangian embedding of a compact manifold j l r 2 a a well known result due to g r o m o v g states that the cohomology class of a in h z e 2 l is non zero we identify l with its image in r2 an equivalent assertion is that the liouville form 2 y 7 1 xldy i pulls back to a closed form j 2 on l whose cohomology class does not vanish g r o m o v s proof uses subtle properties of holomorphic maps into c he shows that there is always a holomorphic disk a in c with boundary in l on such a disk a is positive hence sa a soa 2 0 in this paper we shall only consider the case l t and give an elementary proof of gromov s result elementary should be understood in the sense of number theory that is without using holomorphic functions in fact our result is somewhat more precise and can be stated as follows

an estimate for the siciak extremal function subanalytic geometry approach we study the ojasiewicz siciak inequality in cn this concerns the estimating from below the siciak extremal function which is very much related to the green function in c as an application we present some results describing how fast holomorphic functions defined in a neighbourhood of a compact holomorphic polyhedron can be approximated uniformly on this polyhedron by complex polynomials

markov s inequality and polynomial mappings markov s inequality is a certain estimate for the norm of the derivative of a polynomial in terms of the degree and the norm of this polynomial it has many interesting applications in approximation theory constructive function theory and in analysis for instance to sobolev inequalities or whitney type extension problems one of the purposes of this paper is to give a solution to an old problem studied among others by baran and ple niak and concerning the invariance of markov s inequality under polynomial mappings polynomial images we also address the issue of preserving markov s inequality when taking polynomial preimages lastly we give a sufficient condition for a subset of a markov set to be a markov set

equivalence of the global and local markov inequalities in the complex plane abstract we prove that a compact subset of the complex plane satisfies the global markov inequality if and only if it admits an extension property and a kolmogorov type inequality in jackson norms generalizing a result established by bos and milman in the real case we also show that the global markov inequality is equivalent to the local markov property for all compact subsets of the complex plane admitting the jackson property the latter is a generalization of jackson s approximation inequality closely connected with the boundary behavior of the green s function finally we construct an example showing that there is no equivalence in general

on the problem of multiple mobile robots cluttering a workspace we consider the problem of multiple mobile robots navigating in a common workspace initially based on the premise that the costs of operation of the robots can be ignored results regarding the probability of collision of two point robots in the workspace are derived obviously these results depend on the geometry of the workspace and hence they are derived for workspaces with various geometries the results are then extended for two robots of finite dimensions the question are k 1 robots better than k is then considered based on two modes of operation namely the batch scheduled mode and the list scheduled mode various theoretical results are derived if the costs of operation of the robots can be ignored based on various computational results obtained we conjecture that the answer to the question is in the affirmative in both modes of operation finally by using the model of computation which includes the costs of operation of the robots the problem is revisited we have shown that for various geometries and revenue to cost ratios there are optimal finite values for the number of robots that can operate within the workspace expressions for these optimal values have been derived

topological properties of convolutor spaces via the short time fourier transform we discuss the structural and topological properties of a general class of weighted l 1 convolutor spaces our theory simultaneously applies to weighted d l1 spaces as well as to convolutor spaces of the gelfand shilov spaces k m p in particular we characterize the sequences of weight functions m p p is an element of n for which the space of convolutors of k m p is ultrabornological thereby generalizing grothendieck s classical result for the space o c of rapidly decreasing distributions our methods lead to the first direct proof of the completeness of the space o c of very slowly increasing smooth functions

smb collision detection based on temporal coherence the paper presents a novel collision detection algorithm termed the sort moving boxes smb for large number of moving 2d 3d objects which are represented by their axis aligned bounding boxes aabbs the main feature of the algorithm is the full exploitation of the temporal coherence of the objects exhibited in a dynamic environment in the algorithm the aabbs are first projected to each cartesian axis the projected intervals on the axes are separately sorted by the diminishing increment sort dis and further divided into subsections by processing all the intervals within the subsections to check if they overlap a complete contact list can be built the smb is a fast and robust collision detection algorithm particularly for systems involving a large number of moving aabbs and also supports for the dynamic insertion and deletion of objects its performance in terms of both expected total detection time and memory requirements is proportional to the total number of aabbs n and is not influenced by size differences of aabbs the space size and packing density over a large range up to ten times difference the only assumption made is that the sorted list at one time step will remain an almost sorted list at the next time step which is valid for most applications whose movement and deformation of each aabb and the dynamic change of the total number n are approximately continuous

multivariate jackson inequality by known multivariate versions of the classical jackson theorem every compact cube p in r n admits jackson s inequality the purpose of this note is to deliver other examples of jackson sets in r n we shall show that a finite union of disjoint jackson compact sets in r n is also a jackson set and that this in general fails to hold for an infinite union of jackson sets we also give a characterization of jackson sets in the family of markov compact sets in r n which together with a bierstone result permits one to show that whitney regular compact subsets of r n are jackson

deforming convex curves with fixed elastic energy abstract this paper presents a curve flow which preserves the elastic energy of the evolving curve if the initial curve is a planar simple and smooth curve with positive curvature then the local and global existence of the flow is proved under this flow the evolving curve will converge to a finite circle in the c metric as time goes to infinity

mean curvature flow of surfaces in einstein four manifolds let be a compact oriented surface immersed in a four dimensional kahler einstein manifold m w we consider the evolution of in the direction of its mean curvature vector it is proved that being symplectic is preserved along the flow and the flow does not develop type i singularity when m has two parallel kahler forms w and w that determine different orientations and is symplectic with respect to both w and w we prove the mean curvature flow of exists smoothly for all time in the positive curvature case the flow indeed converges at infinity

bivariate polynomial interpolation on the square at new nodal sets abstract as known the problem of choosing good nodes is a central one in polynomial interpolation while the problem is essentially solved in one dimension all good nodal sequences are asymptotically equidistributed with respect to the arc cosine metric in several variables it still represents a substantially open question in this work we consider new nodal sets for bivariate polynomial interpolation on the square first we consider fast leja points for tensor product interpolation on the other hand for interpolation in p n 2 on the square we experiment four families of points which are asymptotically equidistributed with respect to the dubiner metric which extends to higher dimension the arc cosine metric one of them nicknamed padua points gives numerically a lebesgue constant growing like log square of the degree

mean curvature flow of monotone lagrangian submanifolds we use holomorphic disks to describe the formation of singularities in the mean curvature flow of monotone lagrangian submanifolds in mathbb c n

computing approximate fekete points by qr factorizations of vandermonde matrices we propose a numerical method implemented in matlab for computing approximate fekete points on compact multivariate domains it relies on the search of maximum volume submatrices of vandermonde matrices computed on suitable discretization meshes and uses a simple greedy algorithm based on qr factorization with column pivoting the method gives also automatically an algebraic cubature formula provided that the moments of the underlying polynomial basis are known numerical tests are presented for the interval and the square which show that approximate fekete points are well suited for polynomial interpolation and cubature

collision computation of moving bodies abstract in this paper an explicit mathematical representation of n dimensional bodies moving in translation along general trajectories is derived this representation is used to find out if two moving bodies are going to collide an optimization problem is developed for finding the time and location of collision we consider the special cases of linear and piece wise linear trajectories the collision in this case can be obtained by solving a linear program or a sequence of linear programs respectively the problem of finding the collision time and location of several moving bodies is cast as an integer programming problem a comprehensive simulation study shows that this approach requires much lesser computation time when compared with the current approach of finding the collision between all pairs of bodies

rounding error analysis of the classical gram schmidt orthogonalization process this paper provides two results on the numerical behavior of the classical gram schmidt algorithm the first result states that provided the normal equations associated with the initial vectors are numerically nonsingular the loss of orthogonality of the vectors computed by the classical gram schmidt algorithm depends quadratically on the condition number of the initial vectors the second result states that provided the initial set of vectors has numerical full rank two iterations of the classical gram schmidt algorithm are enough for ensuring the orthogonality of the computed vectors to be close to the unit roundoff level

nonadaptive quasi optimal points selection for least squares linear regression in this paper we present a quasi optimal sample set for ordinary least squares ols regression the quasi optimal set is designed in such a way that for a given number of samples it can deliver the regression result as close as possible to the result obtained by a much larger set of candidate samples the quasi optimal set is determined by maximizing a quantity measuring the mutual column orthogonality and the determinant of the model matrix this procedure is nonadaptive in the sense that it does not depend on the sample data this is useful in practice as it allows one to determine prior to the potentially expensive data collection procedure where to sample the underlying system in addition to presenting the theoretical motivation of the quasi optimal set we also present its efficient implementation via a greedy algorithm along with several numerical examples to demonstrate its efficacy since the quasi optimal set allows one to obtain a near optimal regression result under any affordable nu

polynomial interpolation and cubature over polygons we have implemented a matlab code to compute discrete extremal sets of fekete and leja type on convex or concave polygons together with the corresponding interpolatory cubature formulas the method works by qr and lu factorizations of rectangular vandermonde matrices on weakly admissible meshes wams of polygons constructed by polygon quadrangulation

geometric weakly admissible meshes discrete least squares approximations and approximate fekete points using the concept of geometric weakly admissible meshes see 2 below together with an algorithm based on the classical qr factorization of matrices we compute efficient points for discrete multivariate least squares approximation and lagrange interpolation

on a near optimal sampling strategy for least squares polynomial regression abstract we present a sampling strategy of least squares polynomial regression the strategy combines two recently developed methods for least squares method christoffel least squares algorithm and quasi optimal sampling more specifically our new strategy first choose samples from the pluripotential equilibrium measure and then re order the samples by the quasi optimal algorithm a weighted least squares problem is solved on a much smaller sample set to obtain the regression result it is then demonstrated that the new strategy results in a polynomial least squares method with high accuracy and robust stability at almost minimal number of samples

on using lp to collision detection between a manipulator arm and surrounding obstacles abstract this paper is concerned with two linear programming lp models for the detection of a collision between a manipulator arm and its surrounding obstacles on the basis of a real case study in particular the first model finds the intersections between two convex polyhedra while the second one finds the intersections between suitable representations of the objects a comparison between the two approaches is presented on the basis of experimental data 9200 test cases have been analyzed

estimation of fekete points we aim here at presenting a new procedure to numerically estimate the fekete points of a wide variety of compact sets in r 3 we understand the fekete point problem in terms of the identification of near equilibrium configurations for a potential energy that depends on the relative position of n particles the compact sets for which our procedure works are basically the finite union of piecewise regular surfaces and curves in order to determine a good initial configuration to start the search of the fekete points of these objects we construct a sequence of approximating regular surfaces our algorithm is based on the concept of disequilibrium degree which is defined from a physical interpretation of the behavior of a system of particles when they search for a minimum energy configuration moreover the algorithm is efficient and robust independently of the considered compact set as well as of the kernel used to define the energy the numerical experimentation carried out suggests that a local minimum can be localized with a computational cost of order less than n 3

a search algorithm for motion planning with six degrees of freedom abstract the motion planning problem is of central importance to the fields of robotics spatial planning and automated design in robotics we are interested in the automatic synthesis of robot motions given high level specifications of tasks and geometric models of the robot and obstacles the movers problem is to find a continuous collision free path for a moving object through an environment containing obstacles we present an implemented algorithm for the classical formulation of the three dimensional movers problem given an arbitrary rigid polyhedral moving object p with three translational and three rotational degrees of freedom find a continuous collision free path taking p from some initial configuration to a desired goal configuration this paper describes an implementation of a complete algorithm at a given resolution for the full six degree of freedom movers problem the algorithm transforms the six degree of freedom planning problem into a point navigation problem in a six dimensional configuration space called c space the c space obstacles which characterize the physically unachievable configurations are directly represented by six dimensional manifolds whose boundaries are five dimensional c surfaces by characterizing these surfaces and their intersections collision free paths may be found by the closure of three operators which i slide along five dimensional level c surfaces parallel to c space obstacles ii slide along one to four dimensional intersections of level c surfaces and iii jump between six dimensional obstacles these operators are employed by a best first search algorithm in c space we will discuss theoretical properties of the algorithm including completeness at a resolution this paper describes the heuristic search with particular emphasis on the heuristic strategies that evaluate local geometric information at the heart of this paper lie the design and implementation of these strategies for planning paths along level c surfaces and their intersection manifolds and for reasoning about motions with three degrees of rotational freedom the problems of controlling the interaction of these strategies and of integrating diverse local experts for geometric reasoning provide an interesting application of search to a difficult domain with significant practical implications the representations and algorithms we develop impact many geometric planning problems and extend to cartesian manipulators with six degrees of freedom

a stochastic collocation algorithm with multifidelity models we present a numerical method for utilizing stochastic models with differing fideli ties to approximate parameterized functions a representative case is where a high fidelity and a low fidelity model are available the low fidelity model represents a coarse and rather crude ap proximation to the underlying physical system however it is easy to compute and consumes little simulation time on the other hand the high fidelity model is a much more accurate representation of the physics but can be highly time consuming to simulate our approach is nonintrusive and is therefore applicable to stochastic collocation settings where the parameters are random variables we provide sufficient conditions for convergence of the method and present several examples that are of practical interest including multifidelity approximations and dimensionality reduction

polynomial fitting and interpolation on circular sections we construct weakly admissible polynomial meshes wams on circular sections such as symmetric and asymmetric circular sectors circular segments zones lenses and lunes the construction resorts to recent results on subperiodic trigonometric interpolation the paper is accompanied by a software package to perform polynomial fitting and interpolation at discrete extremal sets on such regions

geometry of holomorphic mappings and holder continuity of the pluricomplex green function we provide a solution to a long standing open problem that lives in the interface of pluripotential theory and multivariate approximation theory the problem is to characterize the holomorphic maps which preserve holder continuity of the pluricomplex green function associated with a compact subset of mathbb c n we also prove under mild restrictions that nondegenerate holomorphic maps preserve markov s inequality for polynomials

polynomial approximation and cubature at approximate fekete and leja points of the cylinder the paper deals with polynomial interpolation least square approximation and cubature of functions dened on the rectangular cylinder k d 1 1 with d the unit disk the nodes used for these processes are the approximate fekete points afp and the discrete leja points dlp extracted from suitable weakly admissible meshes wams of the cylinder from the analysis of the growth of the lebesgue constants approximation and cubature errors we show that the afp and the dlp extracted from wam are good points for polynomial approximation and numerical integration of functions dened on the cylinder

atomic decomposition for the harmonic fock spaces in the plane abstract the main purpose of this paper is to extend the certain functional analytic properties from the analytic fock spaces to the harmonic fock spaces in the plane firstly we found the pointwise estimate for the arbitrary function in the harmonic fock space h p where 1 p then we describe the atomic decomposition for h p

computing almost minimal formulas on the square the main purpose of this paper is to introduce a numerical method for the computation of cubature rules on the square 1 1 2 that are almost minimal i e with few points w r t the degree of precision d in particular for all email protected email protected 55 the sets x d and the weights w k are new new rules are also achieved for degrees 15 d 17 19 and d 23 respectively with 43 54 67 and 96 points

remez type inequality on sets with cusps abstract a remez type inequality is proved for a large family of sets with cusps in r n including compact fat and semialgebraic or subanalytic sets

fast collision detection in four dimensional space abstract in this paper we consider the collision detection problem for general objects a four dimensional approach is proposed for this problem which detects exactly and in one step when and where the earliest collison will occur between the objects this is done by using four dimensional sets to represent the objects in both space and time the problem is then posed as a nonlinear programming problem the algorithm can handle the case of a rigid body moving on a general path in r 2 or r 3 with simultaneous translation and rotation simulation results on some example problems are given and show that the algorithm is superior to those available in the literature

extension operators for smooth functions on compact subsets of the reals we introduce sufficient as well as necessary conditions for a compact set k such that there is a continuous linear extension operator from the space of restrictions c infty k f k f in c infty mathbb r to c infty mathbb r this allows us to deal with examples of the form k a n n in mathbb n cup 0 for a n rightarrow 0 previously considered by fefferman and ricci as well as vogt

mean curvature flows of lagrangian submanifolds with convex potentials this article studies the mean curvature flow of lagrangian submanifolds in particular we prove the following global existence and convergence theorem if the potential function of a lagrangian graph in t 2n is convex then the flow exists for all time and converges smoothly to a flat lagrangian submanifold we also discuss various conditions on the potential function that guarantee global existence and convergence

mityagin s extension problem progress report abstract given a compact set k r d let e k denote the space of whitney jets on k the compact set k is said to have the extension property if there exists a continuous linear extension operator w e k c r d in 1961 b s mityagin posed a problem to give a characterization of the extension property in geometric terms we show that there is no such complete description in terms of densities of hausdorff contents or related characteristics also the extension property cannot be characterized in terms of growth of markov s factors for the set

on selecting a maximum volume sub matrix of a matrix and related problems given a matrix email protected r m x n n vectors in m dimensions we consider the problem of selecting a subset of its columns such that its elements are as linearly independent as possible this notion turned out to be important in low rank approximations to matrices and rank revealing qr factorizations which have been investigated in the linear algebra community and can be quantified in a few different ways in this paper from a complexity theoretic point of view we propose four related problems in which we try to find a sub matrix email protected r m x k of a given matrix email protected r m x n such that i s m a x c the largest singular value of c is minimum ii s m i n c the smallest singular value of c is maximum iii k c email protected m a x c s m i n c the condition number of c is minimum and iv the volume of the parallelepiped defined by the column vectors of c is maximum we establish the np hardness of these problems and further show that they do not admit ptas we then study a natural greedy heuristic for the maximum volume problem and show that it has approximation ratio 2 o k l o g k our analysis of the greedy heuristic is tight to within a logarithmic factor in the exponent which we show by explicitly constructing an instance for which the greedy heuristic is 2 w k from optimal when a has unit norm columns a related problem is to select the maximum number of vectors with a given volume we show that if the optimal solution selects k columns then greedy will select w k logk columns providing a logk approximation

self shrinkers for the mean curvature flow in arbitrary codimension in this paper we generalize colding minicozzi s recent results about codimension 1 self shrinkers for the mean curvature flow to higher codimension in particular we prove that the sphere bf s n sqrt 2n is the only complete embedded connected f stable self shrinker in mathbf r n k with mathbf h ne 0 polynomial volume growth flat normal bundle and bounded geometry we also discuss some properties of symplectic self shrinkers proving that any complete symplectic self shrinker in mathbf r 4 with polynomial volume growth and bounded second fundamental form is a plane as a corollary we show that there is no finite time type i singularity for symplectic mean curvature flow which has been proved by chen li using different method we also study lagrangian self shrinkers and prove that for lagrangian mean curvature flow the blow up limit of the singularity may be not f stable

spatial planning a configuration space approach this paper presents algorithms for computing constraints on the position of an object due to the presence of ther objects this problem arises in applications that require choosing how to arrange or how to move objects without collisions the approach presented here is based on characterizing the position and orientation of an object as a single point in a configuration space in which each coordinate represents a degree of freedom in the position or orientation of the object the configurations forbidden to this object due to the presence of other objects can then be characterized as regions in the configuration space called configuration space obstacles the paper presents algorithms for computing these configuration space obstacles when the objects are polygons or polyhedra

mixed affine surface area various means over the unit sphere of powers of the curvature function of a convex body arise in geometric convexity for example both the surface area and the affine surface area of sufficiently smooth bodies can be expressed as such means the survey of gruber 111 on approximation of convex bodies contains a number of examples of means of powers of the curvature function arising in approximation problems see pp 137 141 143 and 1466148 of 11 i these means arise surprisingly in questions related to approximating convex bodies by polytopes see e g mcclure and vitale 18 schneider 24 and schneider and wieacker 27 an interesting conjecture of firey 9 p s also 26 p 2571 which was recently proved by gage lo for plane convex bodies is that a certain weighted mean of the curvature function lies between the arithmetic and harmonic means of the curvature function it turns out that there is also a close connection between various means of the curvature function of a convex body and some mixed volumes involving the body in this note we investigate means of the curvature function of a convex body which arise as coefficients in the expansion of the afftne surface area of outer and inner blaschke parallel bodies for a variety of reasons we will take a general approach to the study of these means an approach similar to that taken in 14 see also 4 pp 164 1671 in presenting dual mixed volumes let x denote the set of convex bodies compact convex sets with interior points in euclidean n space r we use h k to denote the support function of a convex body k i e

convex hull representations of models for computing collisions between multiple bodies abstract in this paper we consider a collision detection problem that frequently arises in the field of robotics given a set of bodies with their initial positions and trajectories we wish to identify the first collision that occurs between any two bodies or to determine that none exists for the case of bodies having linear trajectories we construct a convex hull representation of the integer programming model of s z selim and h a almohamad european journal of operational research 119 1 1999 121 129 and compare the relative effectiveness in solving this problem via the resultant linear program we also extend this analysis to model a situation in which bodies move along piecewise linear trajectories possibly rotating at the end of each linear segment for this case we again compare an integer programming approach with its linear programming convex hull representation and exhibit the effectiveness of solving a sequence of mathematical programs for each time segment over a global programming scheme which considers all segments at once we provide computational results to illustrate the effect of various numbers of bodies present in the collision scenarios as well as the times at which the first collision occurs

tensor product gauss lobatto points are fekete points for the cube tensor products of gauss lobatto quadrature points are frequently used as collocation points in spectral element methods unfortunately it is not known if gauss lobatto points exist in non tensor product domains like the simplex in this work we show that the n dimensional tensor product of gauss lobatto quadrature points are also fekete points this suggests a way to generalize spectral methods based on gauss lobatto points to non tensor product domains since fekete points are known to exist and have been computed in the triangle and tetrahedron in one dimension this result was proved by fejer in 1932 but the extension to higher dimensions in non trivial

least squares polynomial approximation on weakly admissible meshes disk and triangle we construct symmetric polar wams weakly admissible meshes with low cardinality for least squares polynomial approximation on the disk these are then mapped to an arbitrary triangle numerical tests show that the growth of the least squares projection uniform norm is much slower than the theoretical bound and even slower than that of the lebesgue constant of the best known interpolation points for the triangle as opposed to good interpolation points such meshes are straightforward to compute for any degree the construction can be extended to polygons by triangulation

adaptive leja sparse grid constructions for stochastic collocation and high dimensional approximation we propose an adaptive sparse grid stochastic collocation approach based upon leja interpolation sequences for approximation of parameterized functions with high dimensional parameters leja sequences are arbitrarily granular any number of nodes may be added to a current sequence producing a new sequence and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions when undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables this paper establishes that a certain weighted formulation of one dimensional leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the gauss quadrature nodes associated with the weight function this property is true even for unbounded domains we apply the leja sparse grid approach to several high dimensional problems and

collision detection of a moving polygon in the presence of polygonal obstacles in the plane this paper presents a new approach for the following collision detection problem in the plane let a simple polygon p rotate at a center spl ogr with constant angular velocity spl omega and translate towards a set of polygonal obstacles s with constant velocity spl nu given p and s as well as their initial positions and given also the velocities of p determine whether or not p will collide with any element of s and report the collided elements of s if collisions occurred an o mn worst case optimal algorithm is proposed to solve this problem where n is the number of vertices of p and m is the number of vertices of the obstacles in s

computing multivariate fekete and leja points by numerical linear algebra we discuss and compare two greedy algorithms that compute discrete versions of fekete like points for multivariate compact sets by basic tools of numerical linear algebra the first gives the so called approximate fekete points by qr factorization with column pivoting of vandermonde like matrices the second computes discrete leja points by lu factorization with row pivoting moreover we study the asymptotic distribution of such points when they are extracted from weakly admissible meshes

distance estimation and collision prediction for on line robotic motion planning abstract an efficient method for computing the minimum distance and predicting collisions between moving objects is presented this problem has been incorporated in the framework of an in line motion planning algorithm to satisfy collision avoidance between a robot and moving objects modelled as convex polyhedra in the beginning the deterministic problem where the information about the objects is assumed to be certain is examined if instead of the euclidean norm l 1 or l norms are used to represent distance the problem becomes a linear programming problem the stochastic problem is formulated where the uncertainty is induced by sensing and the unknown dynamics of the moving obstacles two problems are considered first filtering of the distance between the robot and the moving object at the present time second prediction of the minimum distance in the future in order to predict the collision time