outlier detection with the kernelized spatial depth function statistical depth functions provide from the deepest point a center outward ordering of multidimensional data in this sense depth functions can measure the extremeness or outlyingness of a data point with respect to a given data set hence they can detect outliers observations that appear extreme relative to the rest of the observations of the various statistical depths the spatial depth is especially appealing because of its computational efficiency and mathematical tractability in this article we propose a novel statistical depth the kernelized spatial depth ksd which generalizes the spatial depth via positive definite kernels by choosing a proper kernel the ksd can capture the local structure of a data set while the spatial depth fails we demonstrate this by the half moon data and the ring shaped data based on the ksd we propose a novel outlier detection algorithm by which an observation with a depth value less than a threshold is declared as an outlier the proposed algorithm is simple in structure the threshold is the only one parameter for a given kernel it applies to a one class learning setting in which normal observations are given as the training data as well as to a missing label scenario where the training set consists of a mixture of normal observations and outliers with unknown labels we give upper bounds on the false alarm probability of a depth based detector these upper bounds can be used to determine the threshold we perform extensive experiments on synthetic data and data sets from real applications the proposed outlier detector is compared with existing methods the ksd outlier detector demonstrates a competitive performance

the approximation algorithms for a class of multiple choice problem abstract given a set of n points which is a unit set of m colored sets we study the minimum circle to cover one of each colored set at least in this paper we study some problems of optimizing some properties of color spanning set firstly we show a way to find a color spanning set for each point and constitute a union of color spanning sets without fcvd the farthest color voronoi diagram the approach to find each color spanning set is based on the nearest neighbor points which have different colors for every color spanning set we can find a enclosing circle to cover all points of each color spanning set which is a good approximate for mdcs the minimum diameter color spanning set problem we propose an approximation algorithm for the secsc the smallest color spanning circle problem with 2 factor approximation result for p n and q m the worst running time of our algorithm is o n 2 although these results are not as good as previous results with fcvd the performance of our algorithms in r d is much better than others moreover an approximation algorithm can solve problem of minimum perimeter of the color spanning set faster with time of o n 2 n m log n and ratio 6 this result improved the ratio with only a little cost of time complexity at last we give an example for the 2 center secsc problem a fast computation of 2 center enclosing circle is proposed with time of o n 2 but the ratio depends on the gap between the nearest distinct colored distance

optimality approximation and complexity in set membership h sub spl infin identification investigates the set membership identification of time invariant discrete time exponentially stable possibly infinite dimensional linear systems from time or frequency domain data corrupted by deterministic noise the aim is to deliver not a single model but a set of models whose size in h sub spl infin norm measures the uncertainty in the identification the main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets a method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency such approximations allow one to compute within any desired accuracy the identification error of any identified model and to evaluate an optimal model at any given number of frequencies by suitably approximating these values model sets with nominal models in rh sub spl infin are then derived whose order is selected by trading off between model set complexity and identification accuracy degradation this degradation is evaluated by computing the optimality level defined as the ratio between the reduced model identification error and the optimal one a numerical example demonstrates the effectiveness of the presented results

solving homogeneous linear equations over gf 2 via block wiedemann algorithm we propose a method of solving large sparse systems of homogeneous linear equations over g f 2 the field with two elements we modify an algorithm due to wiedemann a block version of the algorithm allows us to perform 32 matrix vector operations for the cost of one the resulting algorithm is competitive with structured gaussian elimination in terms of time and has much lower space requirements it may be useful in the last stage of integer factorization we address here the problem of solving a large sparse system of homogeneous linear equations over gf 2 the field with two elements one important application which motivates the present work arises in integer factorization during the last stage of most integer factorization algorithms we are presented with a large sparse integer matrix and are asked to find linear combinations of the columns of this matrix which vanish modulo 2 for example 7 the matrix may have 100 000 columns with an average of 15 nonzero entries per column for this application we would like to obtain several solutions because a given solution will lead to a nontrivial factorization with probability 112 with n independent solutions our probability of finding a factorization rises to 1 2 structured gaussian elimination can be used 7 but as problems get larger it may become infeasible to store the matrices obtained in the intermediate stages of gaussian elimination the wiedemann algorithm 9 71 has smaller storage requirements one need only store a few vectors and an encoding of a sparse matrix not a dense matrix as occurs in gaussian elimination after fillin and it may have fewer computational steps since one takes advantage of the sparseness of the matrix but its efficiency is hampered by the fact that the algorithm acts on only one bit at a time in the present paper we work with blocks of vectors at a single time by treating 32 vectors at a time on a machine with 32 bit words we can perform 32 matrix vector products at once thus considerably decreasing the cost of indexing this can be viewed as a block wiedemann algorithm the main technical difficulty is in obtaining the correct generalization of the berlekamp massey algorithm to a block version namely a multidimensional version of the extended euclidean algorithm received by the editor november 20 1991 and in revised form july 24 1992 1991 mathematics subject classijication primary 15a33 11y05 11 04 15 04 1994 amencan mathernat cal soc ety 0025 571 8 94 1 oo 25 per page

partitions of a simplex leading to accurate spectral finite volume reconstruction in this paper we present several systematic techniques based on the voronoi diagram and its variants to partition a one and two dimensional simplex the fekete points are used as input to generate the voronoi diagram as they concentrate near the edges and are almost optimal for polynomial interpolation in a simplex spectral finite volume reconstructions on the resultant partitions have small lebesgue constants when using the dubiner basis the reconstruction matrix is well conditioned moreover the total number of edges of the partitions the total work when being used in spectral volume methods is shown to be at most twice the minimum number of edges of all partitions for reconstructions of the same order accuracy these suggest that the obtained partitions are well suited for spectral volume methods and other numerical methods which rely on reconstructions from cell averages

on square free factorization of multivariate polynomials over a finite field abstract in this paper we present a new deterministic algorithm for computing the square free decomposition of multivariate polynomials with coefficients from a finite field our algorithm is based on yun s square free factorization algorithm for characteristic 0 the new algorithm is more efficient than existing deterministic algorithms based on musser s squarefree algorithm we will show that the modular approach presented by yun has no significant performance advantage over our algorithm the new algorithm is also simpler to implement and it can rely on any existing gcd algorithm without having to worry about choosing good evaluation points to demonstrate this we present some timings using implementations in maple char et al 1991 where the new algorithm is used for release 4 onwards and axiom jenks and sutor 1992 which is the only system known to the author to use an implementation of yun s modular algorithm mentioned above

space time tradeoffs for emptiness queries we develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries our lower bounds apply to a general class of geometric range query data structures called partition graphs informally a partition graph is a directed acyclic graph that describes a recursive decomposition of space we show that any partition graph that supports hyperplane emptiness queries implicitly defines a halfspace range query data structure in the fredman yao semigroup arithmetic model with the same asymptotic space and time bounds thus results of bronnimann chazelle and pach imply that any partition graph of size s that supports hyperplane emptiness queries in time t satisfies the inequality st d omega n log n d d 1 d 1 using different techniques we improve previous lower bounds for hopcroft s problem given a set of points and hyperplanes does any hyperplane contain a point in dimensions four and higher using this offline result we show that for online hyperplane emptiness queries omega n d mbox polylog n space is required to achieve polylogarithmic query time and omega n d 1 d mbox polylog n query time is required if only o n polylog n space is available these two lower bounds are optimal up to polylogarithmic factors for two dimensional queries we obtain an optimal continuous tradeoff st 2 omega n 2 between these two extremes finally using a lifting argument we show that the same lower bounds hold for both offline and online halfspace emptiness queries in mathbb r d d 3 2

computing monodromy via continuation methods on random riemann surfaces we consider a riemann surface x defined by a polynomial f x y of degree d whose coefficients are chosen randomly hence we can suppose that x is smooth that the discriminant x of f has d d 1 simple roots and that 0 0 i e the corresponding fiber has d distinct points y1 yd when we lift a loop 0 c by a continuation method we get d paths in x connecting y1 yd hence defining a permutation of that set this is called monodromy here we present experimentations in maple to get statistics on the distribution of transpositions corresponding to loops around each point of multiplying families of neighbor transpositions we construct permutations and the subgroups of the symmetric group they generate this allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups assuming that these two conjectures are true we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions

reweighting to achieve elliptically contoured covariates in regression abstract we investigate a method of constructing weights to induce elliptically contoured covariates in regression analyses much recent work in regression has identified various data analytic and model robustness advantages associated with such covariates in particular new estimation methods like sir sirii save and phd have been built around the assumption of elliptically contoured covariates finite samples of regression covariates may deviate from this ideal in practice and the method developed here termed voronoi weighting can be used to induce elliptical symmetry in such samples in a number of examples we show that reweighting cases by the voronoi method can substantially enhance various procedures for covariates that deviate from elliptical symmetry we show that voronoi weighting in conjunction with some trimming via the minimum volume ellipsoid method can be effective

solving a polynomial equation some history and recent progress the classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present day computing we briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms we end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost

solving linear equations over gf 2 block lanczos algorithm abstract we propose a method of solving large sparse systems of linear equations over gf 2 the field with two elements we use the lanczos algorithm modified in two ways a lookahead lanczos algorithm is needed for the problem of dividing by an inner product whose value happens to be 0 a block version of the algorithm allows us to perform 32 matrix vector operations for the cost of one the resulting algorithm is competitive with structured gaussian elimination in terms of time and has much lower space requirements it may be useful in the last stage of integer factorization

absolute irreducibility of polynomials via newton polytopes abstract a multivariable polynomial is associated with a polytope called its newton polytope a polynomial is absolutely irreducible if its newton polytope is indecomposable in the sense of minkowski sum of polytopes two general constructions of indecomposable polytopes are given and they give many simple irreducibility criteria including the well known eisenstein criterion polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbitrarily in the field but keeping a certain collection of them nonzero

approximations of one dimensional digital signals under the l infty norm approximation of digital signals by means of continuous time functions is often required in many tasks of digital to analog conversion signal processing and coding in many cases the approximation is performed based on an l2 optimality criterion in this paper we study approximations of one dimensional signals under the linfin norm we first introduce approximations in linear spaces for which linear programming methods are known for the particular case of linear approximations i e first order polynomials we propose a geometric solution that is shown to be computationally more efficient than the linear programming approach then we study the problem of piecewise approximations i e dividing the domain into intervals and approximating the signal in linear spaces within every segment independently so as to reach an optimal noncontinuous approximation given an error bound delta we establish a strategy to determine the minimum number k of segments for which the approximation is guaranteed to produce an error within delta we then show how to find the optimal partition that gives the piecewise linfin optimal solution with k segments the computational complexity of the algorithms is studied showing that in many practical situations the number of operations is o n with n being the number of samples

absolute factorization of polynomials a geometric approach in this paper a new algorithm is presented for factoring bivariate polynomials over algebraically closed fields or equivalently for determining the irreducible components of a plane curve this algorithm is based on properties of some geometric invariants of the curve and is similar to berlekamp s algorithm for factorization of univariate polynomials over finite fields

computing connectedness disconnectedness and discreteness we consider finite point set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set we use the minimal spanning tree of the finite set of points to compute the number and size of its connected components by extrapolating the limiting behavior of these quantities as 0 we can say whether the underlying set appears to be connected totally disconnected or perfect we demonstrate the effectiveness of our techniques for a number of examples including a family of fractals related to the sierpinski triangle cantor subsets of the plane the henon attractor and cantori from four dimensional symplectic sawtooth maps for zero measure cantor sets we conjecture that the growth rate of the number of components as 0 is equivalent to the box counting dimension 2000 elsevier science b v all rights reserved

semiclassical treatment of the path integral approach to special effects in superconductivity abstract distinctive features of the flux mode regime in the anderson josephson potential are analyzed using both semiclassical path integral and instanton techniques this potential governs the behavior of a superconducting ring interrupted by a weak josephson junction after establishing the relation between wkb and functional integral approximations semiclassical expansions are derived from a discretized version of the feynman propagator the physical meaning of instantons is extended to the condensed matter problem of energy spectrum for weak link ring states localized near the minimum energy regions under certain circumstances an improved wkb approximation enables us to find expression for the decay rates of metastable states performing the reduction of the original anderson josephson shape to a double hump structure although we have not discussed here the temperature dependence of the josephson plasma frequency our calculations are consistent with recent theories for systems at finite temperatures with two metastable states separated by a potential energy barrier

discrete voronoi diagrams and the skiz operator a dynamic algorithm the voronoi diagram vd is a popular tool for partitioning the support of an image an algorithm is presented for constructing vd when the seed set which determines the voronoi regions can be modified by adding and removing seeds the number of pixels and seeds revisited for updating the diagram and the neighbor relationships among seeds is minimized a result on cocircular seeds is presented the adjacency or dual graph of the vd is readily obtained the use of the algorithm for constructing skeletons by influence zones skiz is demonstrated

polygon based contact resolution for superquadrics the representation of discrete objects in the discrete element modelling is a fundamental issue which has a direct impact on the efficiency of discrete element implementation and the dynamic behaviour of particulate systems disks and spheres are the most commonly used geometric shapes due to their geometric simplicity and computational efficiency but they are unable to provide resistance to rolling motion for this reason some non circular spherical objects such as polygons polyhedrons superquadrics or the clustering of disks spheres to form irregular shapes are introduced when superquadrics are used as discrete elements the bottleneck of contact resolution is associated with the searching for intersections of two non linear functions which is a very expensive operation and may sometimes fail in finding the solution in this work an efficient and robust algorithm is proposed for contact resolution of 2d superquadrics in which any superquadric is approximated with a convex polygon through adaptive sampling then by clipping two polygons an efficient linear algorithm is performed to search for intersections and overlap area of the polygons the contact forces and directions are determined by employing a newly established corner corner contact model it is important to highlight that the proposed methodology can also be extended to general non circular discrete object cases the performance of the algorithm is demonstrated via numerical examples copyright 2005 john wiley sons ltd

generators extremals and bases of max cones abstract max cones are max algebraic analogs of convex cones in the present paper we develop a theory of generating sets and extremals of max cones in r n this theory is based on the observation that extremals are minimal elements of max cones under suitable scalings of vectors we give new proofs of existing results suitably generalizing restating and refining them of these it is important that any set of generators may be partitioned into the set of extremals and the set of redundant elements we include results on properties of open and closed cones on properties of totally dependent sets and on computational bounds for the problem of finding the essentially unique basis of a finitely generated cone

linearization and completeness results for terminating transitive closure queries on spatial databases we study queries to spatial databases where spatial data are modeled as semi algebraic sets using the relational calculus with polynomial inequalities as a basic query language we work with the extension of the relational calculus with terminating transitive closures the main result is that this language can express the linearization of semialgebraic databases we also show that the sublanguage with linear inequalities only can express all computable queries on semilinear databases as a consequence of these results we obtain a completeness result for topological queries on semialgebraic databases

detecting quantum entanglement we review the criteria for separability and quantum entanglement both in a bipartite as well as a multipartite setting we discuss bell inequalities entanglement witnesses entropic inequalities bound entanglement and several features of multipartite entanglement we indicate how these criteria bear on the experimental detection of quantum entanglement

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

tomographic image reconstruction based on a content adaptive mesh model in this paper we explore the use of a content adaptive mesh model camm for tomographic image reconstruction in the proposed framework the image to be reconstructed is first represented by a mesh model an efficient image description based on nonuniform sampling in the camm image samples represented as mesh nodes are placed most densely in image regions having fine detail tomographic image reconstruction in the mesh domain is performed by maximum likelihood ml or maximum a posteriori map estimation of the nodal values from the measured data a camm greatly reduces the number of unknown parameters to be determined leading to improved image quality and reduced computation time we demonstrated the method in our experiments using simulated gated single photon emission computed tomography spect cardiac perfusion images a channelized hotelling observer cho was used to evaluate the detectability of perfusion defects in the reconstructed images a task based measure of image quality a minimum description length mdl criterion was also used to evaluate the effect of the representation size in our application both mdl and cho suggested that the optimal number of mesh nodes is roughly five to seven times smaller than the number of projection bins when compared to several commonly used methods for image reconstruction the proposed approach achieved the best performance in terms of defect detection and computation time the research described in this paper establishes a foundation for future development of a four dimensional space time reconstruction framework for image sequences in which a built in deformable mesh model is used to track the image motion

an optimal algorithm for intersecting line segments in the plane the main contribution of this work is an o n log n k time algorithm for computing all k intersections among n line segments in the plane this time complexity is easily shown to be optimal within the same asymptotic cost our algorithm can also construct the subdivision of the plane defined by the segments and compute which segment if any lies right above or below each intersection and each endpoint the algorithm has been implemented and performs very well the storage requirement is on the order of n k in the worst case but it is considerably lower in practice to analyze the complexity of the algorithm an amortization argument based on a new combinatorial theorem on line arrangements is used

zur theorie der linearen und nichtlinearen integralgleichungen fredholm hat eine auflosungsformel der inhomogenen linearen integralgleichung f left s right varphi left s right lambda int limits a b k left s t right varphi left t right dt entdeckt welche das resultat enthielt das diese gleichung nach s stets aufgelost werden kann wenn nicht eine der nullstellen einer gewissen ganzen transzendenten ist fur diese werte von in hilbertscher bezeichnung die sogenannten eigenwerte des kernes k s t und nur fur diese last wie fredholm ferner zeigte die homogene gleichung 0 varphi left s right int limits a b k left s t right varphi left t right dt losungen zu die nach hilbert zum betreffenden eigenwert des kernes k s t gehorige eigenfunktionen genannt werden hilbert hat die in der theorie der partiellen und gewohnlichen differentialgleichungen so wichtigen fragen nach der existenz sogenannter normalfunktionen und der entwickelbarkeit willkurlicher funktionen nach ihnen durch einfuhrung der greenschen funktion auf das viel allgemeinere problem zuruckgefuhrt die existenz von eigenfunktionen eines in s und t symmetrischen kernes k s t zu beweisen und die gesetze der entwickelbarkeit willkurlicher funktionen nach ihnen aufzustellen die greensche funktion selbst wird durch eine integralgleichung mit unsymmetrischem kerne bestimmt wobei die fredholmschen formeln zur anwendung kommen

polynomial time reductions from multivariate to bi and univariate integral polynomial factorization consider a polynomial f with an arbitrary but fixed number of variables and with integral coefficients we present an algorithm which reduces the problem of finding the irreducible factors of f in polynomial time in the total degree of f and the coefficient lengths of f to factoring a univariate integral polynomial together with a lenstra s h lenstra s and l lovasz polynomial time factorization algorithm for univariate integral polynomials math ann 261 1982 pp 515 534 this algorithm implies the following theorem factoring an integral polynomial with a fixed number of variables into irreducibles except for the constant factors can be accomplished in deterministic polynomial time in the total degree and the size of its coefficients our algorithm can be generalized to factoring multivariate polynomials with coefficients in algebraic number fields and finite fields in polynomial time we also present a different algorithm based on an effective version of a hilbert irreducibility theorem w

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

an optimal parallel algorithm for the visibility of a simple polygon from a point we present a parallel algorithm for computing the visible portion of a simple polygonal chain with 7i vertices from a point in the plane the algorithm runs in o logn time using 0 nf log n processors in the crew pram computational model and hence is asymptotically optimal

intersecting and trimming parametric meshes on finite element shells luiz cristovao g coelho marcelo gattass department of computer science puc rio rio de janeiro brazilluiz henrique de figueiredolncc laborato rio nacional de computac a o cient i ca p etro polis brazilabstractwe present an algorithm for intersecting nite element meshes de ned on parametric surface patches the inter section curves are modeled precisely and both meshes are adjusted to the newly formed borders without unwantedreparametrizations the algorithm is part of an interactive shell modeling program that has been used in the designof large offshore oil structures to achieve good interactive response we represent meshes with a topological datastructure that stores its entities in spatial indexing trees instead of linear lists these trees speed up the intersectioncomputations required to determine points of the trimming curves moreover when combined with the topologicalinformation they allow remeshing using only local queries

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

esflib a library to compute the element free galerkin shape functions we present a library for the computation of the shape functions for the element free galerkin efg method while the efg method is in many respects strikingly similar to the nite element method the construction of the shape functions for the efg method is much more complicated than in the fem thus the question arises whether it is possible to encapsulate the complexity of the shape function construction process while providing a simple interface to the shape function data we present a shape function library design which proved unrestrictive it can be used with almost any programming language without any special requirements on application program data structures highly congurable and robust

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

quality estimates and stretched meshes based on delaunay triangulations any other vertex is exploited for the automation of the grid generation process moreover delaunay grids have many interesting properties an extensive documentation of which can be found in ref 10 in the following we will first describe the different elements of frod the derivation of the background mesh as well as the vertex generation for the viscous and inviscid regions the latter half of this paper presents the quality measures that can be derived for the isotropic vertex generation part

acuity matching resolution degradation through wavelet coefficient scaling a wavelet based multiresolution image representation method is developed matching human visual system hvs spatial acuity within multiple regions of interest rois rois are maintained at high original resolution while peripheral areas are gracefully degraded variable resolution images are generated by selectively scaling wavelet detail coefficients prior to reconstruction the technique is equivalent to linear interpolation mip mapping which involves smooth subsampling decomposition prior to texture mapping reconstruction multiple roi degradation is achieved through wavelet coefficient scaling following voronoi partitioning of the image plane

a recombination algorithm for the decomposition of multivariate rational functions in this paper we show how we can compute in a deterministic way the decomposition of a multivariate rational function with a recombination strategy the key point of our recombination strategy is the used of darboux polynomials we study the complexity of this strategy and we show that this method improves the previous ones in appendix we explain how the strategy proposed recently by j berthomieu and g lecerf for the sparse factorization can be used in the decomposition setting then we deduce a decomposition algorithm in the sparse bivariate case and we give its complexity

the hamiltonian properties of supergrid graphs in this paper we first introduce a novel class of graphs namely supergrid supergrid graphs include grid graphs and triangular grid graphs as their subgraphs the hamiltonian cycle and path problems for grid graphs and triangular grid graphs were known to be np complete however they are unknown for supergrid graphs the hamiltonian cycle path problem on supergrid graphs can be applied to control the stitching traces of computerized sewing machines in this paper we will prove that the hamiltonian cycle problem for supergrid graphs is np complete it is easily derived from the hamiltonian cycle result that the hamiltonian path problem on supergrid graphs is also np complete we then show that two subclasses of supergrid graphs including rectangular parallelism and alphabet always contain hamiltonian cycles

primary decomposition of zero dimensional ideals over finite fields gao shuhong clemson univ dept math sci clemson sc 29634 usa wan daqing univ calif irvine dept math irvine ca 92697 usa wang mingsheng acad sinica inst software informat secur lab beijing 100080 peoples r china

computing farthest neighbors on a convex polytope let n be a set of n points in convex position in r3 the farthest point voronoi diagram of n partitions r3 into n convex cells we consider the intersection g n of the diagram with the boundary of the convex hull of n we give an algorithm that computes an implicit representation of g n in expected o n log2 n time more precisely we compute the combinatorial structure of g n the coordinates of its vertices and the equation of the plane defining each edge of g n the algorithm allows us to solve the all pairs farthest neighbor problem for n in expected time o n log2 n and to perform farthest neighbor queries on n in o log2 n time with high probability

damping mechanisms in the path integral approach to special effects in superconductivity abstract previous work on semiclassical approximations in squids which involved feynman s path integral technique is here extended to damped josephson systems quantum decay rates of unstable energy levels near the bottom of the quartic potential well are examined rates diagram is compared with that obtained for the undamped case agreement with caldeira leggett results is discussed a description of continuous flux measurements is developed the spreading of the wave function leads to the treatment of flux and charge as canonically conjugate variables

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

factoring multivariate polynomials over algebraic number fields we present an algorithm to factor multivariate polynomials over algebraic number fields that is polynomial time in the degrees of the polynomial to be factored the algorithm is an immediate generalization of the polynomial time algorithm to factor univariate polynomials with rational coefficients

structure of a simple scheduling polyhedron in a one machine nonpreemptive scheduling problem the feasible schedules may be defined by the vector of the corresponding job completion times for given positive processing times the associated simple scheduling polyhedronp is the convex hull of these feasible completion time vectors the main result of this paper is a complete description of the minimal linear system definingp we also give a complete combinatorial description of the face lattice ofp and a simple o n logn separation algorithm this algorithm has potential usefulness in cutting plane type algorithms for more difficult scheduling problems

optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension in binary images the distance transformation dt and the geometrical skeleton extraction are classic tools for shape analysis in this paper we present time optimal algorithms to solve the reverse euclidean distance transformation and the reversible medial axis extraction problems for d dimensional images we also present a d dimensional medial axis filtering process that allows us to control the quality of the reconstructed shape

voronoi diagram and medial axis algorithm for planar domains with curved boundaries i theoretical foundations in this first installment of a two part paper the underlying theory for an algorithm that computes the voronoi diagram and medial axis of a planar domain bounded by free form polynomial or rational curve segments is presented an incremental approach to computing the voronoi diagram is used wherein a single boundary segment is added to an existing boundary segment set at each step the introduction of each new segment entails modifying the voronoi regions of the existing boundary segments and constructing the voronoi region of the new segment we accomplish this by i computing the bisector of the new segment with each of the current boundary segments ii updating the voronoi regions of the current boundary segments by partitioning them with these bisectors and iii constructing the voronoi region of the new segment as a union of regions obtained from the partitioning in ii when all boundary segments are included and their voronoi regions have been constructed the voronoi diagram of the boundary is obtained as the union of the voronoi polygons for each boundary segment to construct the medial axis of a planar domain we first compute the voronoi diagram of its boundary the medial axis is then obtained from the voronoi diagram by i removing certain edges of the voronoi diagram that do not belong to the medial axis and ii adding certain edges that do belong to the medial axis but are absent from the voronoi diagram unambiguous characterizations for edges in both these categories are given details of algorithms based on this theory are deferred to the second installment of this two part paper

surface bounded growth modeling applied to human mandibles from a set of longitudinal three dimensional scans of the same anatomical structure the authors have accurately modeled the temporal shape and size changes using a linear shape model on a total of 31 computed tomography scans of the mandible from six patients 14 851 semilandmarks are found automatically using shape features and a new algorithm called geometry constrained diffusion the semilandmarks are mapped into procrustes space principal component analysis extracts a one dimensional subspace which is used to construct a linear growth model the worst case mean modeling error in a cross validation study is 3 7 mm

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

a note on gao s algorithm for polynomial factorization shuhong gao 2003 6 has proposed an efficient algorithm to factor a bivariate polynomial f over a field f this algorithm is based on a simple partial differential equation and depends on a crucial fact the dimension of the polynomial solution space g associated with this differential equation is equal to the number r of absolutely irreducible factors of f however this holds only when the characteristic of f is either zero or sufficiently large in terms of the degree of f in this paper we characterize a vector subspace of g for which the dimension is r regardless of the characteristic of f and the properties of gao s construction hold moreover we identify a second vector subspace of g that leads to an analogous theory for the rational factorization of f

geometric classification of triangulations and their enumeration in a convex polygon abstract triangulation of polygons is a classical problem in computational geometry for an arbitrary polygon the triangulation may depend on its shape in this paper we describe a new geometric classification of triangulations of a convex polygon and then derive expressions for counting the number of nonisomorphic triangulations in each class

sharp precision in hensel lifting for bivariate polynomial factorization popularized by zassenhaus in the seventies several algorithms for factoring polynomials use a so called lifting and recombination scheme concerning bivariate polynomials we present a new algorithm for the recombination stage that requires a lifting up to precision twice the total degree of the polynomial to be factored its cost is dominated by the computation of reduced echelon solution bases of linear systems we show that our bound on precision is asymptotically optimal

indecomposability of polynomials via jacobian matrix abstract indecomposable polynomials are a special class of absolutely irreducible polynomials some improvements of important effective results on absolute irreducibility have recently appeared using ruppert s matrix in a similar way we show in this paper that the use of a jacobian matrix gives sharp bounds for the indecomposability problem

the multipoint morisita index for the analysis of spatial patterns abstract in many fields the spatial clustering of sampled data points has significant consequences therefore several indices have been proposed to assess the degree of clustering affecting datasets e g the morisita index ripley s k function and renyi s information the classical morisita index measures how many times it is more likely to randomly select two sampled points from the same quadrat the dataset is covered by a regular grid of changing size than it would be in the case of a random distribution generated from a poisson process the multipoint version takes into account m points with m 2 the present research deals with a new development of the multipoint morisita index m morisita which is directly related to multifractality this relationship to multifractality is first demonstrated and highlighted on a mathematical multifractal set then the new version of the m morisita index is adapted to the characterization of environmental monitoring network clustering and finally an additional extension the functional m morisita index is presented for the detection of structures in monitored phenomena

reduction of bivariate polynomials from convex dense to dense with application to factorizations in this article we present a new algorithm for reducing the usual sparse bivariate factorization problems to the dense case this reduction simply consists in computing an invertible monomial transformation that produces a polynomial with a dense size of the same order of magnitude as the size of the integral convex hull of the support of the input polynomial this approach turns out to be very efficient in practice as demonstrated with our implementation

on pointers versus addresses summary what is the cost of random access to memory we address this fnndamental problem by studying the simulation of random addressing by a machine which lacks it a pointer machine the model we define allows the use of a data type of our choice a ram program of time i and space s can be simulated in o t log a time using a tree however this is not an obvious lower bound since a high level data type may allow us to encode the data in a more economic way our major contribution is the formalization of incompressibility for general data types the definition extends a similar property 01 strings that underlies the theory of kolmogorov complexity the main theorem states that for all incompressible data types an q t log s lower bound holds incompressibility trivially holds for strings but is harder to prove for a powerlul data type we prove incompressibility for the real numbers with a set of primitives which includes all functions which are continuously differentiable except on a countable closed set this may be the richest set of operations considered in a lower bound proof the prooi relies on the implicit functions theorem and baire s category theorem we also show that the integers with arithmetic x and lz zj any boolean operations and left shift are incompressible the inclusion of right shift reverses the situation and we obtain an o tq s upper bound where s is a functional inverse of ackermann s function

hensel lifting and bivariate polynomial factorisation over finite fields this paper presents an average time analysis of a hensel lifting based factorisation algorithm for bivariate polynomials over finite fields it is shown that the average running time is almost linear in the input size this explains why the hensel lifting technique is fast in practice for most polynomials

factoring multivariate polynomials via partial differential equations a new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic it is based on a simple partial differential equation that gives a system of linear equations as in berlekamp s and niederreiter s algorithms for factoring univariate polynomials the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd s and by factoring univariate polynomials over the ground field the new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields local fields number fields and the complex number field the theory of the new method allows an effective hilbert irreducibility theorem thus an efficient reduction of polynomials from multivariate to bivariate

the asymptotic entanglement cost of preparing a quantum state we give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state is equal to limn ef n n where ef is the entanglement of formation

optimal expected case planar point location point location is the problem of preprocessing a planar polygonal subdivision s of size n into a data structure in order to determine efficiently the cell of the subdivision that contains a given query point we consider this problem from the perspective of expected query time we are given the probabilities p z that the query point lies within each cell z in s the entropy h of the resulting discrete probability distribution is the dominant term in the lower bound on the expected case query time we show that it is possible to achieve query time h o sqrt h 1 with space o n which is optimal up to lower order terms in the query time we extend this result to subdivisions with convex cells assuming a uniform query distribution within each cell in order to achieve space efficiency we introduce the concept of entropy preserving cuttings

close point spatial tests and their application to random number generators we study statistical tests of uniformity based on thel p distances between them nearest pairs of points forn points generated uniformly over thek dimensional unit hypercube or unit torus the number of distinct pairs at distance no more thant fort 0 is a stochastic process whose initial part after an appropriate transformation and asn 8 is asymptotically a poisson process with unit rate convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3 due to edge effects but is reasonably fast in the torus we look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions discuss computational issues and apply the tests to random number generators linear congruential generators fail decisively certain variants of the tests as soon asn approaches the square root of the period length

factoring multivariate integral polynomials an algorithm is presented to factorize polynomials in several variables with integral coefficients that is polynomial time in the degrees of the polynomial to be factored for any fixed number of variables the algorithm generalizes the algorithm presented by a k lenstra et al to factorize integral polynomials in one variable

a simple algorithm for nearest neighbor search in high dimensions the problem of finding the closest point in high dimensional spaces is common in pattern recognition unfortunately the complexity of most existing search algorithms such as k d tree and r tree grows exponentially with dimension making them impractical for dimensionality above 15 in nearly all applications the closest point is of interest only if it lies within a user specified distance spl epsiv we present a simple and practical algorithm to efficiently search for the nearest neighbor within euclidean distance spl epsiv the use of projection search combined with a novel data structure dramatically improves performance in high dimensions a complexity analysis is presented which helps to automatically determine spl epsiv in structured problems a comprehensive set of benchmarks clearly shows the superiority of the proposed algorithm for a variety of structured and unstructured search problems object recognition is demonstrated as an example application the simplicity of the algorithm makes it possible to construct an inexpensive hardware search engine which can be 100 times faster than its software equivalent a c implementation of our algorithm is available

a fuzzy approach to visibility maps creation over digital terrains the construction of inter visibility models for a terrain is an important tool for environmental and animal studies efficient crisp algorithms are known to address this problem directly on a raster digital elevation map dem of the study area randomness plays an important role several factors like vegetation height of the observer presence of unmapped human artifacts etc cannot be modeled in advance into a traditional dem model we propose the use of fuzzy terrain models to incorporate uncertainty and unpredictable variability of the landscape accordingly we propose a suitable definition of visibility to obtain intervisibility models an algorithm that uses approximation of fuzzy numbers with a discrete family of intervals is proposed the proposed model produces sound results and can be extended to model complex situations like triangulating a radio source using two directional receivers all of these situations are difficult to manage with a precise stochastic model and are a frequent occurrence in animal behavior studies under field conditions the proposed techniques have been implemented inside a user friendly software package that may easily exchange data with the most common giss we report also some experimental results obtained using over artificial terrain under controlled noise and under sampling conditions

on the schmidt robustness of pure states robustness measures as introduced by vidal and tarrach 1999 robustness of entanglement phys rev a 59 141 55 quantify the extent to which entangled states remain entangled under mixing analogously we introduce here the schmidt robustness and the random schmidt robustness the latter notion is closely related to the construction of schmidt balls around the identity we analyse the situation for pure states and provide nontrivial upper and lower bounds upper bounds to the random schmidt 2 robustness allow us to construct a particularly simple distillability criterion we present two conjectures the first one is related to the radius of inner balls around the identity in the convex set of schmidt number n states we also conjecture a class of optimal schmidt witnesses for pure states

a local convergence proof for the minvar algorithm for computing continuous piecewise linear approximations the class of continuous piecewise linear pl functions represents a useful family of approximants because invertibility can be readily imposed and if a pl function is invertible then it can be inverted in closed form many applications arising for example in control systems and robotics involve the simultaneous construction of a forward and inverse system model from data most approximation techniques require that separate forward and inverse models be trained whereas an invertible continuous pl affords simultaneously the forward and inverse system model in a single representation the minvar algorithm computes a continuous pl approximation to data local convergence of minvar is proven for the case when the data generating function is itself a pl function and available directly rather than through data

factoring rational polynomials over the complex numbers nc algorithms are given for determining the number and degrees of the factors irreducible over the complex numbers bf c of a multivariate polynomial with rational coefficients and for approximating each irreducible factor nc is the class of functions computable by logspace uniform boolean circuits of polynomial size and polylogarithmic depth the measures of size of the input polynomial are its degree coefficient length number of variables d c and n respectively if n is fixed we give a deterministic nc algorithm if the number of variables is not fixed we give a random monte carlo nc algorithm in these input measures to find the number and degree of each irreducible factor after reducing to the two variable square free case we apply the classical algebraic geometry fact that the absolute irreducible factors of p z 1 z 2 0 correspond to the connected components of the real surface or complex curve p z 1 z 2 0 minus its singular points in finding the number of connec

multidimensional search trees that provide new types of memory reductions an orthogonal query that asks to aggregate the set of records in k dimensional box regions is studied and it is shown that space o n log n log log n k 1 makes possible a combined time complexity o log k n for retrievals insertions and deletions

on the total order of reducibility of a pencil of algebraic plane curves in this paper the problem of bounding the number of reducible curves in a pencil of algebraic plane curves is addressed unlike most of the previous related works each reducible curve of the pencil is here counted with its appropriate multiplicity it is proved that this number of reducible curves counted with multiplicity is bounded by d 2 1 where d is the degree of the pencil then a sharper bound is given by taking into account the newton s polygon of the pencil

poisson polyhedra in high dimensions abstract the zero cell of a parametric class of random hyperplane tessellations depending on a distance exponent and an intensity parameter is investigated as the space dimension tends to infinity the model includes the zero cell of stationary and isotropic poisson hyperplane tessellations as well as the typical cell of a stationary poisson voronoi tessellation as special cases it is shown that asymptotically in the space dimension with overwhelming probability these cells satisfy the hyperplane conjecture if the distance exponent and the intensity parameter are suitably chosen dimension dependent functions also the high dimensional limits of the mean number of faces are explored and the asymptotic behaviour of an isoperimetric ratio is analysed in the background are new identities linking the f vector of the zero cell to certain dual intrinsic volumes

factorization of multivariate polynomials over finite fields we present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size i e in the degree of the polynomial and log cardinality of field the algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs a deterministic version of the algorithm is also discussed whose running time is polynomial in the degree of the input polynomial and the size of the field

the analytical evaluation of one dimensional gaussian path integrals abstract in the introductory section 1 it is outlined how path integrals have made their appearance in quantum mechanics through feynman s formalism in section 2 it is recalled that gaussian path integrals defined as those in which the action is an integral whose integrand is a polynomial of at most the second degree in every dynamical variable involved can be cast into the form f t b t a exp i s cl b a where f t b t a is a coefficient solely depending on the initial time instant t a and the final time instant t b and s cl b a denotes the classical action section 3 is devoted to the calculation of a simple formula for the factor f t b t a in the case of a single particle in one euclidean dimension whereby the most general quadratic form of the lagrange function is considered the method is based upon successive transformations of various related auxiliary gaussian path integrals which are introduced applying feynman s theorem and making use of the well known property k x b t b x a t a k x b t b x t k x t x a t a dx satisfied by quantummechanical green s functions constitute important steps in the theoretical development at the end f t b t a is expressed solely in terms of two functions which one encounters in the description of the classical particle trajectory in section 4 four examples of practical application are given the last one concerns the linear harmonic oscillator with time dependent period and damping subject to a time dependent perturbative force