fast sweeping algorithms for a class of hamilton jacobi equations we derive a godunov type numerical flux for the class of strictly convex homogeneous hamiltonians that includes h p q sqrt ap 2 bq 2 2cpq c 2 ab we combine our godunov numerical fluxes with simple gauss seidel type iterations for solving the corresponding hamilton jacobi hj equations the resulting algorithm is fast since it does not require a sorting strategy as found e g in the fast marching method in addition it providesa way to compute solutions to a class of hj equations for which the conventional fast marching method is not applicable our experiments indicate convergence after a few iterations even in rather difficult cases

the design of granular classifiers a study in the synergy of interval calculus and fuzzy sets in pattern recognition in this study we are concerned with a development of a certain category of granular classifiers referred here to as hyperbox driven classifiers hdc the approach fully capitalizes on the two key technologies of granular computing namely set interval calculus and fuzzy sets in dealing with a description of geometry of patterns data belonging to a certain category we take advantage of the capabilities of sets intervals and their cartesian products when describing a core structure of classes of patterns in the form of some hyperboxes their combinations are referred to as a core structure of the feature space next we refine the geometry of the classifier by bringing forward the concepts of regions of the feature space characterized by fuzzy sets they are sought as a secondary structure the construction of the core structure is realized by means of the supervised version of dbscan one of the popular clustering algorithms of data mining the secondary structure is described by fuzzy sets whose membership functions are a solution to some optimization problem of allocation of degrees of belongingness in the formation of the secondary structure we exploit a concept of the hausdorff distance that determines a distance between some information granule of a well defined geometry and a numeric pattern a point in the highly dimensional feature space the two stage description of the granular classifier is also beneficial in the characterization of the classification error by virtue of the design of the core structure the classification error occurring there assumes very small almost zero values on the other hand the secondary structure coming with a substantial level of overlap between classes is characterized by relatively higher values of the classification error being cognizant of that we suggest a way of forming a synthetic measure built on the membership values and named a separability index that helps quantify the classification error in this setting a series of numeric examples are used to demonstrate the effectiveness of the proposed granular classifiers

stationary discs and finite jet determination for non degenerate generic real submanifolds abstract let m c n be a generic real submanifold of class c 4 in case m is levi non degenerate in the sense tumanov we construct stationary discs for m if furthermore m satisfies an additional non degeneracy condition we apply the method of stationary discs to obtain 2 jet determination of cr automorphisms of m

a geometric model for active contours in image processing we propose a new model for active contours based on a geometric partial differential equation our model is intrinsec stable satisfies the maximum principle and permits a rigorous mathematical analysis it enables us to extract smooth shapes we cannot retrieve angles and it can be adapted to find several contours simultaneously moreover as a consequence of the stability we can design robust algorithms which can be engineed with no parameters in applications numerical experiments are presented

orthogonal laurent polynomials and strong moment theory a survey these topics are found in many parts of 20th century mathematics and its applications in mathematical physics chemistry statistics and engineering historically the analytic theory of continued fractions has played a central role in both the origin and the development of the other topics continued fractions are intimately related to pad e approximants and special functions emphasis is given to the development of strong moment theory and orthogonal laurent polynomials and to the related continued fractions quadrature formulas integral transforms and linear functionals by a strong moment problem we mean the following for a given bisequence n n of real numbers does there exist a distribution function such that

least squares quantization in pcm it has long been realized that in pulse code modulation pcm with a given ensemble of signals to handle the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall it has been shown by panter and dite that in the limit as the number of quanta becomes infinite the asymptotic fractional density of quanta per unit voltage should vary as the one third power of the probability density per unit voltage of signal amplitudes in this paper the corresponding result for any finite number of quanta is derived that is necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy the optimization criterion used is that the average quantization noise power be a minimum it is shown that the result obtained here goes over into the panter and dite result as the number of quanta become large the optimum quautization schemes for 2 b quanta b 1 2 cdots 7 are given numerically for gaussian and for laplacian distribution of signal amplitudes

stationary holomorphic discs and finite jet determination problems we construct a family of small analytic discs attached to levi non degenerate hypersurfaces in mathbb c n 1 which is globally biholomorphically invariant we then apply this technique to study unique determination problems along levi non degenerate hypersurfaces that are merely of class mathcal c 4 this method gives 2 jet determination results for germs of biholomorphisms cr diffeomorphisms as well as in the almost complex setting

classification using geometric level sets a variational level set method is developed for the supervised classification problem nonlinear classifier decision boundaries are obtained by minimizing an energy functional that is composed of an empirical risk term with a margin based loss and a geometric regularization term new to machine learning the surface area of the decision boundary this geometric level set classifier is analyzed in terms of consistency and complexity through the calculation of its e entropy for multicategory classification an efficient scheme is developed using a logarithmic number of decision functions in the number of classes rather than the typical linear number of decision functions geometric level set classification yields performance results on benchmark data sets that are competitive with well established methods

semi supervised learning via regularized boosting working on multiple semi supervised assumptions semi supervised learning concerns the problem of learning in the presence of labeled and unlabeled data several boosting algorithms have been extended to semi supervised learning with various strategies to our knowledge however none of them takes all three semi supervised assumptions i e smoothness cluster and manifold assumptions together into account during boosting learning in this paper we propose a novel cost functional consisting of the margin cost on labeled data and the regularization penalty on unlabeled data based on three fundamental semi supervised assumptions thus minimizing our proposed cost functional with a greedy yet stagewise functional optimization procedure leads to a generic boosting framework for semi supervised learning extensive experiments demonstrate that our algorithm yields favorite results for benchmark and real world classification tasks in comparison to state of the art semi supervised learning algorithms including newly developed boosting algorithms finally we discuss relevant issues and relate our algorithm to the previous work

support vector clustering we present a novel clustering method using the approach of support vector machines data points are mapped by means of a gaussian kernel to a high dimensional feature space where we search for the minimal enclosing sphere this sphere when mapped back to data space can separate into several components each enclosing a separate cluster of points we present a simple algorithm for identifying these clusters the width of the gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters the structure of a dataset is explored by varying the two parameters maintaining a minimal number of support vectors to assure smooth cluster boundaries we demonstrate the performance of our algorithm on several datasets

extremal discs and the regularity of cr mappings in higher codimension we develop a local theory of extremal analytic discs attached to a strictly pseudoconvex manifold of higher codimension in complex space we then apply the results to prove the regularity of cr homeomorphisms of such manifolds introduction in his celebrated paper lempert li introduced extremal analytic discs as geodesies of the kobayashi metric on a convex domain d c cn let a be the unit disc in the complex plane an analytic disc a d is called extremal if 0 attains its maximum among all analytic discs in d with the same values and directions at 0 g a extremal discs are important holomorphic invariants of d their boundaries lie in the boundary bd of the domain and form an invariant family of curves for the cr structure of bd lempert li l2 applies the extremal discs to the regularity of biholomorphic mappings monge amp re equations and normal forms of convex domains other applications include deformations and embeddibility of cr structures bl bd and polynomial hulls s

gaps a clustering method using a new point symmetry based distance measure in this paper an evolutionary clustering technique is described that uses a new point symmetry based distance measure the algorithm is therefore able to detect both convex and non convex clusters kd tree based nearest neighbor search is used to reduce the complexity of finding the closest symmetric point adaptive mutation and crossover probabilities are used the proposed ga with point symmetry gaps distance based clustering algorithm is able to detect any type of clusters irrespective of their geometrical shape and overlapping nature as long as they possess the characteristic of symmetry gaps is compared with existing symmetry based clustering technique sbkm its modified version and the well known k means algorithm sixteen data sets with widely varying characteristics are used to demonstrate its superiority for real life data sets anova and manova statistical analyses are performed

cluster analysis and mathematical programming given a set of entities cluster analysis aims at finding subsets called clusters which are homogeneous and or well separated as many types of clustering and criteria for homogeneity or separation are of interest this is a vast field a survey is given from a mathematical programming viewpoint steps of a clustering study types of clustering and criteria are discussed then algorithms for hierarchical partitioning sequential and additive clustering are studied emphasis is on solution methods i e dynamic programming graph theoretical algorithms branch and bound cutting planes column generation and heuristics

novelty detection using level set methods this paper presents a level set boundary description lsbd approach for novelty detection that treats the nonlinear boundary directly in the input space the proposed approach consists of level set function lsf construction boundary evolution and termination of the training process it employs kernel density estimation to construct the lsf of the initial boundary for the training data set then a sign of the lsf based algorithm is proposed to evolve the boundary and make it fit more tightly in the data distribution the training process terminates when an expected fraction of rejected normal data is reached the evolution process utilizes the signs of the lsf values at all training data points to decide whether to expand or shrink the boundary extensive experiments are conducted on benchmark data sets to evaluate the proposed lsbd method and compare it against four representative novelty detection methods the experimental results demonstrate that the novelty detector modeled with the proposed lsbd can effectively detect anomalies

a weight function for the associated jacobi laurent polynomials abstract under some additional conditions on the parameters a weight function on the unit circle in the complex plane is given for the associated jacobi laurent polynomials as introduced in hendriksen 1990

cross validation of multivariate densities abstract in recent years the focus of study in smoothing parameter selection for kernel density estimation has been on the univariate case while multivariate kernel density estimation has been largely neglected in part this may be due to the perception that calibrating multivariate densities is substantially more difficult in this article we explicitly derive and compare multivariate versions of the bootstrap method of taylor the least squares cross validation method developed by bowman and rudemo and a biased cross validation method similar to that of scott and terrell for multivariate kernel estimation using the product kernel estimator the theoretical behavior of these cross validation algorithms is shown to improve surprisingly as the dimension increases approaching the best rate of o n 1 2 simulation studies suggest that the new biased cross validation method performs quite well and with reasonable variability as compared to the other two methods bivariate examples with heart disease

jet determination of smooth cr automorphisms and generalized stationary discs we prove finite jet determination for finitely smooth cr diffeomorphisms of finitely smooth levi degenerate hypersurfaces in mathbb c n 1 by constructing generalized stationary discs glued to such hypersurfaces

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

dynamic cluster formation using level set methods density based clustering has the advantages for 1 allowing arbitrary shape of cluster and 2 not requiring the number of clusters as input however when clusters touch each other both the cluster centers and cluster boundaries as the peaks and valleys of the density distribution become fuzzy and difficult to determine we introduce the notion of cluster intensity function cif which captures the important characteristics of clusters when clusters are well separated cifs are similar to density functions but when clusters become closed to each other cifs still clearly reveal cluster centers cluster boundaries and degree of membership of each data point to the cluster that it belongs clustering through bump hunting and valley seeking based on these functions are more robust than that based on density functions obtained by kernel density estimation which are often oscillatory or oversmoothed these problems of kernel density estimation are resolved using level set methods and related techniques comparisons with two existing density based methods valley seeking and dbscan are presented which illustrate the advantages of our approach

a new clustering approach on the basis of dynamical neural field in this letter we present a new hierarchical clustering approach based on the evolutionary process of amari s dynamical neural field model dynamical neural field theory provides a theoretical framework macroscopically describing the activity of neuron ensemble based on it our clustering approach is essentially close to the neurophysiological nature of perception it is also computationally stable insensitive to noise flexible and tractable for data with complex structure some examples are given to show the feasibility

associated jacobi laurent polynomials abstract the jacobi laurent polynomials have been introduced by hendriksen and van rossum 1986 in the present paper explicit formulas for the orthogonal laurent polynomials satisfying the recurrency for the jacob laurent polynomials with n replaced by n b are given these new orthogonal laurent polynomials are called associated jacobi laurent polynomials using these associated laurent polynomials the denominator and the numerator of certain two point pade approximants to the pair of functions z f a b 1 c b 1 z f a b c b z at o and c b a b 1 f c 1 b 1 a b 2 z 1 f c 1 b 1 z l at are given also some confluent cases are considered

stationary discs glued to a levi non degenerate hypersurface we obtain an explicit parametrization of stationary discs glued to some levi non degenerate hypersurfaces these discs form a family which is invariant under the action of biholomorphisms we use this parametrization to construct a local circular representation of these hypersurfaces as a corollary we get the uniqueness of biholomorphisms with given 1 jet at some convenient point

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

octobasic laguerre polynomials and permutation statistics abstract a set of orthogonal polynomials with eight independent q s is defined which generalizes the laguerre polynomials the moments of the measure for these polynomials are the generating functions for permutations according to eight different statistics specializing these statistics gives many other well known sets of combinatorial objects and relevant statistics the specializations are studied with applications to classical orthogonal polynomials and equidistribution theorems for statistics