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

on independence and conditioning on wiener space notant i f et i g des integrales de wiener ito multiples d ordre p et q respectivement on donne une condition necessaire et suffisante sur le couple f g sous laquelle les variables aleatoires i f et i g sont independantes ceci est base sur une caracterisation generale de l independance de variables aleatoires dans l espace de wiener dans le contexte d un calcul variationnel stochastique

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

the generalized distributive law we discuss a general message passing algorithm which we call the generalized distributive law gdl the gdl is a synthesis of the work of many authors in information theory digital communications signal processing statistics and artificial intelligence it includes as special cases the baum welch algorithm the fast fourier transform fft on any finite abelian group the gallager tanner wiberg decoding algorithm viterbi s algorithm the bcjr algorithm pearl s belief propagation algorithm the shafer shenoy probability propagation algorithm and the turbo decoding algorithm although this algorithm is guaranteed to give exact answers only in certain cases the junction tree condition unfortunately not including the cases of gtw with cycles or turbo decoding there is much experimental evidence and a few theorems suggesting that it often works approximately even when it is not supposed to

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

some additive l fuzzy topological properties abstract the notion of sum for crisp topological spaces have extended to fuzzy topology by ghanim et al 1984 the concept of sum of l fuzzy topological spaces is introduced and its basic properties are discussed by the author 1993 in this paper the additivity of three kinds of l fuzzy topological properties strong separation properties countability and weakly n compactness are studied

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

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

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

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

multiplicative structure of 2x2 tropical matrices we study the algebraic structure of the semigroup of all 2x2 tropical matrices under multiplication using ideas from tropical geometry we give a complete description of green s relations and the idempotents and maximal subgroups of this semigroup

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

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

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

absolutely continuous flows generated by sobolev class vector fields in finite and infinite dimensions abstract we prove the existence of the global flow ut generated by a vector field a from a sobolev class w1 1 on a finite or infinite dimensional space x with a measure provided is sufficiently smooth and that a a and a where a is the divergence with respect to are exponentially integrable in addition the measure is shown to be quasi invariant under ut in the case x r n and p dx where p w1 1loc r n is a locally uniformly positive probability density a sufficient condition is exp c a exp c a p p l1 for all c in the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions examples of measures which fit our framework important for applications are symmetric invariant measures of infinite dimensional diffusions and gibbs measures typically in both cases such measures are essentially non gaussian our result in infinite dimensions significantly extends previously studied cases where was a gaussian measure finally we study flows generated by vector fields whose values are not necessarily in the cameron martin space

generalized stochastic integrals and the malliavin calculus the paper first reviews the skorohod generalized stochastic integral with respect to the wiener process over some general parameter space t and it s relation to the malliavin calculus as the adjoint of the malliavin derivative some new results are derived and it is shown that every sufficiently smooth process ut t t can be decomposed into the sum of a malliavin derivative of a wiener functional and a process whose generalized integral over t vanishes using the results on the generalized integral the bismut approach to the malliavin calculus is generalized by allowing non adapted variations of the wiener process yielding sufficient conditions for the existence of a density which is considerably weaker than the previously known conditions

lowen s compactness in l fuzzy topological spaces abstract r lowen has introducedin 1976 a fuzzy compactness for fuzzy topological spaces wang has generalized it in 1988 to l fuzzy topological spaces and given it a characterization by means of nets in this paper we obtain some new characterization for lowen s compactness in l fuzzy topological spaces means of remote neighborhood families r covers filters and the family of closed l fuzzy subsets which has the finite intersection property in addition we have proved that lowen s compactness is an l good extension of usual compactness in general topological spaces

geometric analysis of conditional independence on wiener space in this paper we study the conditional independence of sigma fields on the wiener space using the tools of the stochastic calculus of variations particular emphasis is given to the relation between the splitting of the random tangent spaces associated to the sigma fields and the conditional independence

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

discrete restricted boltzmann machines we describe discrete restricted boltzmann machines probabilistic graphical models with bipartite interactions between visible and hidden discrete variables examples are binary restricted boltzmann machines and discrete naive bayes models we detail the inference functions and distributed representations arising in these models in terms of configurations of projected products of simplices and normal fans of products of simplices we bound the number of hidden variables depending on the cardinalities of their state spaces for which these models can approximate any probability distribution on their visible states to any given accuracy in addition we use algebraic methods and coding theory to compute their dimension

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

pure dimension and projectivity of tropical polytopes abstract we study how geometric properties of tropical convex sets and polytopes which are of interest in many application areas manifest themselves in their algebraic structure as modules over the tropical semiring our main results establish a close connection between pure dimension of tropical convex sets and projectivity in the sense of ring theory these results lead to a geometric understanding of idempotency for tropical matrices as well as their direct interest our results suggest that there is substantial scope to apply ideas and techniques from abstract algebra in particular ring theory in tropical geometry

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

on the sum of l fuzzy topological spaces abstract h ghanim et al separation axioms subspaces and sums in fuzzy topology j math anal appl 102 1984 189 202 have extended the notion of sum for crisp topological spaces to fuzzy topological spaces and discussed the additivity for some fuzzy topological properties in this paper the concept of the sum of l fuzzy topological spaces is presented and its basic properties is discussed as the main results of the paper the additivity of separability and compactness in l fuzzy topological spaces is studied

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

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

the maximum likelihood degree maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials we study the algebraic degree of the critical equations of this optimization problem this degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces and to the euler characteristic of the complexified complement under suitable hypotheses the maximum likelihood degree equals the top chern class of a sheaf of logarithmic differential forms exact formulae in terms of degrees and newton polytopes are given for polynomials with generic coefficients

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

tropical geometry of statistical models this article presents a unified mathematical framework for inference in graphical models building on the observation that graphical models are algebraic varieties from this geometric viewpoint observations generated from a model are coordinates of a point in the variety and the sum product algorithm is an efficient tool for evaluating specific coordinates here we address the question of how the solutions to various inference problems depend on the model parameters the proposed answer is expressed in terms of tropical algebraic geometry the newton polytope of a statistical model plays a key role our results are applied to the hidden markov model and the general markov model on a binary tree

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

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

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

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

nonnegative ranks decompositions and factorizations of nonnegative matrices the nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank one matrices into which the matrix can be decomposed additively such decompositions are useful in diverse scientific disciplines we obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm

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

rank functions of tropical matrices abstract we consider rank functions important in tropical linear algebra the tropical rank equal to the topological dimension of the tropical linear span of the columns of a given matrix the factor rank equal to the smallest number of vectors containing these columns in their span the kapranov rank related to problems of tropical algebraic geometry and the determinantal and gondran minoux ranks which generalize the classical linear algebraic notion of rank we are interested in studying the arithmetic properties of the rank functions their mutual behavior and the related computational problems we discuss different open problems related to rank functions of tropical matrices and illustrate them by the number of examples

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

on the structure of independence on wiener space motives par des resultats de travaux anterieurs sur la caracterisation de l independance de variables aleatoires dans l espace de wiener les auteurs etudient ici la relation entre l independance de certaines classes de sous algebres et l orthogonalite des h derivees faibles des variables aleatoires qui engendrent ces sous algebres ou c h est l espace de wiener classique

the collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem abstract this article describes a method of grouping and collapsing in using the gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial the norms of the forward operators associated with the corresponding nonreversible markov chains are used to discriminate among different simulation schemes when applied to bayesian missing data problems the idea of collapsing suggests skipping the steps of sampling parameter s values in standard data augmentation by doing this we obtain a predictive update version of the gibbs sampler a procedure of calculating the posterior odds ratio via the collapsed gibbs sampler when incomplete observations are involved is presented as an illustration of possible applications three examples along with a bayesian treatment for identifying common protein binding sites in unaligned dna sequences are provided

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

separations in lattice valued induced spaces abstract for the value set i e range i the induced space is normal if and only if its underlying space is normal but for a general value set the situation is quite different in this paper we construct a skillful example to show that the corresponding relation about the normalities of these two spaces does not hold for a certain fuzzy lattice meanwhile for an important type of fuzzy lattices using the new method we successfully establish a positive result about this normality relation furthermore we shall describe an level set of the closure or interior of a fuzzy set a in terms of the topology of the underlying space and some level sets of a this construction problem of closed sets and open sets in fuzzy topological spaces possesses independent interest other separation axioms such as hausdorff separation and regularity are also discussed in this paper

the composition of wiener functionals with non absolutely continuous shifts in this paper we study some classes of wiener functionals whose elements can be composed with a non linear non absolutely continous transformation of the form of perturbation of identity in the direction of cameron martin space we show that under certain conditions the image of the wiener measure under the above transformation induces a generalized wiener functional on certain sobolev spaces generalizing the radon nikodym relation to non absolutely continuous transformations a series representation for the generalized radon nikodym derivative is presented and conditional expectations of some generalized random variables are considered

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

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

number of common roots and resultant of two tropical univariate polynomials abstract it is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant in this paper we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order if the roots are simple and non zero

on stratiform l fuzzy topologies and their application for a given l fuzzy topological space and a fixed prime element r of fuzzy lattice l the concept of l fuzzy r open sets is introduced and it is proved that all r open sets for l fuzzy topological space form a new l fuzzy topology which is called stratiform l fuzzy topology making use of the stratiform l fuzzy topology a series of new characterizations of l fuzzy continuous mapping and l fuzzy open mapping and three kinds of separateness hausdorff regular and normal are presented

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

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

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

some relations among classes of σ fields on wiener space in this work we study sigma fields and their tangent spaces on the wiener space which are invariant in some sense with respect to the basic operators of the malliavin calculus

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

parametric inference for biological sequence analysis one of the major successes in computational biology has been the unification by using the graphical model formalism of a multitude of algorithms for annotating and comparing biological sequences graphical models that have been applied to these problems include hidden markov models for annotation tree models for phylogenetics and pair hidden markov models for alignment a single algorithm the sum product algorithm solves many of the inference problems that are associated with different statistical models this article introduces the polytope propagation algorithm for computing the newton polytope of an observation from a graphical model this algorithm is a geometric version of the sum product algorithm and is used to analyze the parametric behavior of maximum a posteriori inference calculations for graphical models

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