aspects of three dimensional constrained delaunay meshing presented in this paper are the theoretical aspects of node addition to a non convex multiboundary mesh of tetrahedral elements as used in finite element modelling the method used is derived from watson1 and shenton and cendes2 and is extended to deal with node addition on inter material boundaries several situations are identified that result in an illegal insertion polyhedron ip these could be caused by the constrained nature of the mesh adjacent objects with different material properties or degenerate node configurations a new delaunay algorithm is described that checks for illegal cases of the ip and then corrects them this checking relies on the consistent ordering of the element nodes it is shown that a particular type of illegal ip can easily be identified and corrected using this technique the delaunay algorithm is then applied to automatic mesh generation and modification to the basic delaunay algorithm is described so that previously meshed edges and faces of the current object being meshed are not deleted during the addition of subsequent nodes this protection method only becomes viable by recognizing the node ordering sense of the ip faces

bernstein operators for extended chebyshev systems abstract let u n c n a b be an extended chebyshev space of dimension n 1 suppose that f 0 u n is strictly positive and f 1 u n has the property that f 1 f 0 is strictly increasing we search for conditions ensuring the existence of points t 0 t n a b and positive coefficients 0 n such that for all f c a b the operator b n c a b u n defined by b n f k 0 n f t k k p n k satisfies b n f 0 f 0 and b n f 1 f 1 here it is assumed that p n k k 0 n is a bernstein basis defined by the property that each p n k has a zero of order k at a and a zero of order n k at b

on the multivariate horner scheme we present an efficient version of the horner scheme for the evaluation of multivariate polynomials and study its stability properties in particular we show that the backward error of evaluation is bounded by a quantity that is linear in the total degree of the polynomial which itself is usually significantly smaller than the number of operations involved in the evaluation process moreover we show that the same type of estimates holds also for the forward error provided that the polynomial basis is chosen suitably

a note on the optimal stability of bases of univariate functions this note is concerned with the characterizations and uniqueness of bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation with respect to bases whose elements have no sign changes

tetrahedral mesh generation based on node insertion in crystal lattice arrangements and advancing front delaunay triangulation a method of unstructured technical mesh generation for general three dimensional domains is presented a conventional boundary representation is adopted as the basis for the description of solids with evolving geometry and topology the geometry of the surfaces is represented either analytically or by piecewise polynominal interpolation a preliminary surface mesh is generated by an advancing front method with the nodes inserted by hard sphere packing in physical space in accordance with a prescribed mesh density interior nodes are inserted in a face centered cubic fcc crystal lattice arrangements coupled to octree spatial subdivision with the local lattice parameter determined by a prespecified nodal density function prior to triangulation of the volume the preliminary surface mesh is preprocessed by a combination of local transformations and subdivisions in order to guarantee that the surface triangulation be a subcomplex of the volume delaunay triangulation a joint delaunay triangulation of the interior and boundary nodes which preserves the modified surface mesh is then constructed via an advancing front approach the resulting mesh is finally improved upon by the application of local transformations the overall time complexity of the mesher is o n log n the robustness and versatility of the approach as well as the good quality of the resulting meshes is demostrated with the aid of selected examples

boundary recovery for three dimensional conforming delaunay triangulation we present an efficient boundary recovery method for conforming delaunay triangulation of three dimensional domains our method combines local transformations like edges faces swappings and modified constrained delaunay refinement like edges faces splittings to recover the missing boundary edges and faces its convergence is theoretically proved its effectiveness is also demonstrated through some meshing examples

finding the zeros of a univariate equation proxy rootfinders chebyshev interpolation and the companion matrix when a function f x is holomorphic on an interval x in a b its roots on the interval can be computed by the following three step procedure first approximate f x on a b by a poly

evaluation of the derivative of a polynomial in bernstein form in geometric modelling and computer aided design a polynomial is usually represented in bernstein form forward error analysis for the computation of the derivative of a polynomial in bernstein form is performed the corresponding running error is also analyzed we have compared our results with those corresponding to a polynomial represented in power basis and using legendre and chebyshev forms since the conversions between the bernstein and the power bases and between bernstein and chebyshev polynomials were already studied in the literature for the sake of completeness the conversion between bernstein and chebyshev polynomials of the first kind is analyzed in an appendix

construction of euclidian monopoles this paper describes a procedure for the construction of monopoles on three dimensional euclidean space starting from their rational maps a companion paper euclidean monopoles and rational maps to appear in the same journal describes the assignment to a monopole of a rational map from bbb cp 1 to a suitable flag manifold in describing the reverse direction this paper completes the proof of the main theorem therein a construction of monopoles from solutions to nahm s equations a system of ordinary differential equations has been well known for certain gauge groups for some time these solutions are hard to construct however and the equations themselves become increasingly unwieldy when the gauge group is not mbox su 2 here in contrast a rational map is the only initial data but whereas one can be reasonably explicit in moving from nahm data to a monopole here the monopole is only obtained from the rational map after solving a partial differential equation a non linear flow equation essentially just the path of steepest descent down the yang mills higgs functional is set up it is shown that starting from an approximate monopole constructed explicitly from the rational map a solution to the flow must exist and converge to an exact monopole having the desired rational map 1991 mathematics subject classification 53c07 53c80 58d27 58e15 58g11

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

deriving the continuity of maximum entropy basis functions via variational analysis in this paper we prove the continuity of maximum entropy basis functions using variational analysis techniques the use of information theoretic variational principles to derive basis functions is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints for a set of distinct nodes x i i 1 n in mathbb r d the convex approximation of a function u x is u h x sum i 1 n p i x u i where p i i 1 n are nonnegative basis functions and u h x must reproduce affine functions sum i 1 n p i x 1 sum i 1 n p i x x i x given these constraints we compute p i x by minimizing the relative entropy functional kullback leibler distance d p m sum i 1 n p i x ln p i x m i x where m i x is a known prior weight function distribution to prove the continuity of the basis functions we appeal to the theory of epiconvergence

the runge example for interpolation and wilkinson s examples for rootfinding we look at two classical examples in the theory of numerical analysis namely the runge example for interpolation and wilkinson s example actually two examples for rootfinding we use the modern

running relative error for the evaluation of polynomials running relative error for evaluating polynomials is analyzed and applied to different evaluation algorithms horner vs de casteljau and clenshaw algorithms numerical experiments show that the derived running relative error bounds are sharp and that the de casteljau algorithm presents better stability properties than the remaining algorithms

constrained boundary recovery for three dimensional delaunay triangulations summary a new constrained boundary recovery method for three dimensional delaunay triangulations is presented it successfully resolves the difficulties related to the minimal addition of steiner points and their good placement applications to full mesh generation are discussed and numerical examples are provided to illustrate the effectiveness of guaranteed recovery procedure copyright 2004 john wiley sons ltd

composite finite elements of class ck let be a triangulation of d r2 with vertices a ai 1 i n and pkn d v ck d t v t pn where pn polynomials of total degree n for u cr d r k and v finite dimensional subspace of ck d we consider the hermite problem hk a v find v v d v ai d u ai 1 i n k this problem is solvable in v pkn d iff n 4k 1 enisek when v pkn d 3 where 3 is a subtriangulation of obtained by subdividing each t into 3 triangles the problem has a solution for n 4k 1 e g the c1 cubic hct triangle and a p27 triangle with 31 parameters when v pkn d 6 where 6 is a subtriangulation of obtained by subdividing each t into 6 triangles the problem has a solution for n 3k 1 e g the c1 quadratic ps triangle and a p25 triangle with 31 parameters in both cases the construction needs the partial derivatives d u ai for 2k 1 the domain d is assumed to be polygonal compact and connected

on the optimal stability of the bernstein basis we show that the bernstein polynomial basis on a given interval is optimally stable in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval this result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations we further show by means of some low degree examples that the bernstein form is not uniquely optimal in this respect however it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval these ideas are illustrated by comparing the stability properties of the power bernstein and generalized ball bases

a companion matrix resultant for bernstein polynomials abstract a closed form expression for a companion matrix m of a bernstein polynomial is obtained and this is used to derive an expression for a resultant matrix of two bernstein polynomials it is shown that m differs from its equivalent form for a power basis polynomial because an upper triangular hankel matrix does not define a similarity transformation between m and m t a measure of the numerical condition of a resultant matrix for polynomials in an arbitrary basis is reviewed and this is used to compare the stability of two resultant matrices in particular computational tests are performed and it is shown that the resultant matrix of two bernstein polynomials is numerically better conditioned than the resultant matrix that is obtained when a simple parameter substitution is used to transform the polynomials to the power basis

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

fast delaunay triangulation in three dimensions abstract an efficient algorithm for delaunay triangulation of a given set of points in three dimensions based on the point insertion technique is presented various steps of the triangulation algorithm are reviewed and many acceleration procedures are devised to speed up the triangulation process new features include the search of a neighbouring point by the layering scheme locating the containing tetrahedron by random walk formulas of important geometrical quantities of a new tetrahedron based on those of an existing one a novel approach in establishing the adjacency relationship the use of adjacency table and the management of memory the resulting scheme is one of the fastest triangulation algorithms known to the authors which is able to generate tetrahedra generation rate of 15 000 tetrahedra per second for randomly generated points on a hp 735 machine

geometry based fully automatic mesh generation and the delaunay triangulation one approach to fully automatic mesh generation in two and three dimensions is to generate and triangulate a set of points within and on the boundary of a geometry using the properties of the delaunay triangulation because the point data generate mesh topology of greater dimension it is necessary to insure topological compatibility and perform classification of the resulting mesh with respect to the original geometry necessary and sufficient conditions for classification of mesh topology on three dimensional planar geometry are given as well as a discussion of the more complex case of curved geometry this paper also presents conditions to insure topological compatibility along with techniques for identifying and resolving cases of incompatibility

shape preserving properties of generalized bernstein operators on extended chebyshev spaces we study the existence and shape preserving properties of a generalized bernstein operator b n fixing a strictly positive function f 0 and a second function f 1 such that f 1 f 0 is strictly increasing within the framework of extended chebyshev spaces u n the first main result gives an inductive criterion for existence suppose there exists a bernstein operator b n c a b u n with strictly increasing nodes fixing f 0 f 1 in u n if u n subset u n 1 and u n 1 has a non negative bernstein basis then there exists a bernstein operator b n 1 c a b u n 1 with strictly increasing nodes fixing f 0 and f 1 in particular if f 0 f 1 f n is a basis of u n such that the linear span of f 0 f k is an extended chebyshev space over a b for each k 0 n then there exists a bernstein operator b n with increasing nodes fixing f 0 and f 1 the second main result says that under the above assumptions the following inequalities hold b n f geq b n 1 f geq f furthermore b nf is f0 f1 convex for all f0 f1 convex functions f in c a b

rational cubic clipping with linear complexity for computing roots of polynomials many problems in computer aided geometric design and computer graphics can be turned into a root finding problem of polynomial equations among various clipping methods the ones based on the bernstein bezier form have good numerical stability one of such clipping methods is the k clipping method where k 2 3 and often called a cubic clipping method when k 3 it utilizes o n2 time to find two polynomials of degree k bounding the given polynomial f t of degree n and achieves a convergence rate of k 1 for a single root the roots of the bounding polynomials of degree k are then used for bounding the roots of f t this paper presents a rational cubic clipping method for finding two bounding cubics within o n time which can achieve a higher convergence rate 5 than that of 4 of the previous cubic clipping method when the bounding cubics are obtained the remaining operations are the same as those of previous cubic clipping method numerical examples show the efficiency and the convergence rate of the new method

overview and construction of meshfree basis functions from moving least squares to entropy approximants in this paper an overview of the construction of meshfree basis functions is presented with particular emphasis on moving least squares approximants natural neighbour based polygonal interpolants and entropy approximants the use of information theoretic variational principles to derive approximation schemes is a recent development in this setting data approximation is viewed as an inductive inference problem with the basis functions being synonymous with a discrete probability distribution and the polynomial reproducing conditions acting as the linear constraints the maximization minimization of the shannon jaynes entropy functional relative entropy functional is used to unify the construction of globally and locally supported convex approximation schemes a java applet is used to visualize the meshfree basis functions and comparisons and links between different meshfree approximation schemes are presented copyright 2006 john wiley sons ltd

magnetic monopoles on manifolds with boundary kapustin and witten associate a hecke modication of a holomor phic bundle over a riemann surface to a singular monopole on a riemannian surface times an interval satisfying prescribed boundary conditions we prove existence and uniqueness of singular monopoles satisfying prescribed bound ary conditions for any given hecke modication data conrming the underlying geometric invariant theory principle

on multivariate polynomials in bernstein bezier form and tensor algebra the bernstein bezier representation of polynomials is a very useful tool in computer aided geometric design in this paper we make use of multilinear tensors to describe and manipulate multivariate polynomials in their bernstein bezier form as an application we consider hermite interpolation with polynomials and splines

automatic mesh generation with tetrahedron elements to reduce the manual work involved in the application of the fem in practice preprocessors can be applied for the construction of network structures which are complicated in generation strategy and do not form any optimum discrete structure the time necessary for generation can be minimized even more by the application of only one element type within the whole network structure a technique for the automatic generation of 3d network structures with tetrahedron elements is presented in this paper in this proposed technique the nodal points of the network structure must be defined manually before the generation procedure since a random positioning of points is usually undesirable for fem calculation the nodal points are connected by a program to a network structure consisting of tetrahedron elements which have optimum form for the numerical computation of the element matrices after the generation the element sides forming any part of the boundary surface of the network structure can be automatically identified if necessary the network structure can be automatically refined

a combined octree delaunay method for fully automatic 3 d mesh generation fully automatic three dimensional mesh generation is a fundamental requirement for automating the numerical solution of partial differential equations two techniques in particular the octree and delaunay approaches have been used towards this end a method that combines both approaches to fully automatic mesh generation is presented here the resulting algorithm provides the linear growth rate and divide and conquer approach of the octree method with the simplicity and optimal properties of the delaunay triangulation

montonicity preservation on triangles it is well known that bernstein polynomials on triangles preserve monotonicity in this paper we define and study three kinds of monotonicity preservation of systems of bivariate functions on a triangle we characterize and compare several of these systems and derive some geometric applications

on the optimal stability of bases of univariate functions this paper is concerned with bases of finite dimensional spaces of univariate continuous functions which are optimally stable for evaluation the only bases considered are those whose elements have no sign changes among these an optimally stable basis is characterized under the assumption that the set of points where each basis function is nonzero is an interval a uniqueness result and many examples of such optimally stable bases are also provided

the border rank of the multiplication of 2 2 matrices is seven one of the leading problems of algebraic complexity theory is matrix multiplication the naive multiplication of two n n matrices uses n multiplications in 1969 strassen 20 presented an explicit algorithm for multiplying 2 2 matrices using seven multiplications in the opposite direction hopcroft and kerr 12 and winograd 22 proved independently that there is no algorithm for multiplying 2 2 matrices using only six multiplications the precise number of multiplications needed to execute matrix multiplication or any given bilinear map is called the rank of the bilinear map a related problem is to determine the border rank of matrix multiplication or any given bilinear map first introduced in 6 5 roughly speaking some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small good perturbations of the map these perturbed maps can give rise to fast exact algorithms for matrix multiplication see 7 the border rank made appearances in the literature in the 1980s and early 1990s see e g 6 5 19 8 15 2 10 11 4 3 17 16 18 1 9 21 but to our knowledge there has not been much progress on the question since then more precisely for any complex projective variety x cp pv and point p pv define the x rank of p to be the smallest number r such that p is in the linear span of r points of x define r x the r th secant variety of x to be the zariski closure of the set of points of x rank r and define the x border rank of p to be the smallest r such that p r x the terminology is motivated by the case x seg pa 1 pb 1 p c c the segre variety of rank one matrices then the x rank of a matrix is just its usual rank let a b c be vector spaces and let f a b c be a bilinear map i e an element of a b c let x seg pa pb pc p a b c denote the segre variety of decomposable tensors in a b c the border rank of a bilinear map is its x border rank while for the segre product of two projective spaces border rank coincides with rank here they can be quite different in this paper we prove the theorem stated in the title let mmult c c c denote the matrix multiplication operator for 2 2 matrices strassen 18 showed that mmult 5 p p p our method of proof is to decompose

modified jacobi bernstein basis transformation and its application to multi degree reduction of bezier curves this paper reports new modified jacobi polynomials mjps we derive the basis transformation between mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the new polynomials together with the geometrical insight of bernstein polynomials the mjps with indexes corresponding to the number of endpoints constraints are the natural basis functions for least square approximation of bezier curves using mjps leads us to deal with the constrained jacobi polynomials and the unconstrained jacobi polynomials as orthogonal polynomials the mjps are automatically satisfying the homogeneous boundary conditions thereby the main advantage of using mjps in multi degree reduction of bezier curves on computer aided geometric design cagd is that the constraints in cagd are also satisfied and that decreases the steps of multi degree reduction algorithm several numerical results for the multi degree reduction of bezier curves on cagd are given we establish and prove the basis transformation between the modified jacobi polynomials mjps and bernstein polynomials and vice versa this transformation is merging the perfect least square performance of the mjps with the geometrical insight of the bernstein form the mjps with indexes corresponding to the number of endpoint constraints are the natural basis functions for least square approximation of bezier curves several numerical results are considered

a corner cutting algorithm for evaluating rational bezier surfaces and the optimal stability of the basis the usual method for evaluating rational bezier surfaces uses the projection operator and the representation provided by the bernstein basis we prove the optimal stability of the basis used in this representation we also propose an alternative method for evaluating rational surfaces through that representation we show the stability properties of this last method and prove that it has better properties than other known methods from the point of view of avoiding underflow and overflow

legendre bernstein basis transformations abstract the bernstein form of a polynomial offers valuable insight into its geometrical behavior and has thus won widespread acceptance as the basis for bezier curves and surfaces for least squares approximation problems on the other hand the use of orthogonal bases such as the legendre polynomials permits simple and efficient constructions for convergent sequences of approximants we derive the transformation matrices that map the bernstein and legendre forms of a degree n polynomial on 0 1 into each other and examine the stability of this linear map in the p 1 and norms the condition number of the legendre bernstein transformation matrix grows at a significantly slower rate with n than in the well studied power bernstein case and at a dramatically slower rate than for other common e g bernstein hermite or power hermite basis conversions the utility of legendre representations in approximation problems and their relatively stable transformation to bernstein bezier form argue for more widespread applications of legendre methods in cagd algorithms

b splines and optimal stability it is proved that among all nonnegative bases of its space the b spline basis is optimally stable for evaluating spline functions

tetrahedral mesh generation and optimization based on centroidal voronoi tessellations the centroidal voronoi tessellation based delaunay triangulation cvdt provides an optimal distribution of generating points with respect to a given density function and accordingly generates a high quality mesh in this paper we discuss algorithms for the construction of the constrained cvdt from an initial delaunay tetrahedral mesh of a three dimensional domain by establishing an appropriate relationship between the density function and the specified sizing field and applying the lloyd s iteration the constrained cvdt mesh is obtained as a natural global optimization of the initial mesh simple local operations such as edges faces flippings are also used to further improve the cvdt mesh several complex meshing examples and their element quality statistics are presented to demonstrate the effectiveness and efficiency of the proposed mesh generation and optimization method copyright 2003 john wiley sons ltd

an improved rational cubic clipping method for computing real roots of a polynomial abstract the root finding problem has wide applications in geometric processing and computer graphics previous rational cubic clipping methods are either of convergence rate 7 k with o n2 complexity or of convergence rate 5 k with linear complexity where n and k are degree of the given polynomial and the multiplicity of the root respectively this paper presents an improved rational cubic clipping of linear complexity which can achieve convergence rate 7 k or a better one 7 k 1 for a multiple root such that k 2 numerical examples illustrate both efficiency and convergence rate of the new method

numerical solution of various cases of cauchy type singular integral equation abstract in this paper a robust but very simple numerical method is developed to solve various cases of cauchy type singular integral equation for this first bernstein polynomials are defined which are used for approximation of solution of the given singular integral equation then numerical method is introduced by using bernstein polynomials this ultimately leads to solution of a system of linear algebraic equations examples are illustrated to demonstrate simplicity of proposed method results are also compared with those present in literature to claim better efficiency of the method introduced

maximal rank of m x n x mn k tensors it is shown that the maximal rank of m times n times mn k tensors with k leqq min m 1 2 2 n 1 2 2 is greater than mn 4 sqrt 2k o 1

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

explicit solution of the finite time l2 norm polynomial approximation problem abstract the aim of this paper is to present a new approach to the finite time l 2 norm polynomial approximation problem a new formulation of this problem leads to an equivalent linear system whose solution can be investigated analytically such a solution is then specialized for a polynomial expressed in terms of laguerre and bernstein basis

some characterizations of dirac type singularity of monopoles we study singular monopoles on open subsets in the 3 dimensional euclidean space we give two characterizations of dirac type singularities one is given in terms of the growth order of the norms of sections which are invariant by the scattering map the other is given in terms of the growth order of the norms of the higgs fields

delaunay s mesh of a convex polyhedron in dimension d application to arbitrary polyhedra this paper presents a method for creating a delaunay triangulation connected to a set of specified points the theoretical aspect is recalled for an arbitrary dimension and the method is discussed in order to derive a practical approach valid for dimensions 2 and 3 which is simple robust and well adapted to computation convex polyhedral and arbitrary polyhedral situations are introduced

nahm s equations and the classification of monopoles solutions of nahm s system of ordinary differential equations are produced by variational methods this leads to an explicit parametrisation of the solutions to the bogomolny equation over 3

an alternating least squares algorithm for fitting the two and three way dedicom model and the idioscal model the dedicom model is a model for representing asymmetric relations among a set of objects by means of a set of coordinates for the objects on a limited number of dimensions the present paper offers an alternating least squares algorithm for fitting the dedicom model the model can be generalized to represent any number of sets of relations among the same set of objects an algorithm for fitting this three way dedicom model is provided as well based on the algorithm for the three way dedicom model an algorithm is developed for fitting the idioscal model in the least squares sense