edgeworth approximation of multivariate differential entropy we develop the general multivariate case of the edgeworth approximation of differential entropy and show that it can be more accurate than the nearest neighbor method in the multivariate case and that it scales better with sample size furthermore we introduce mutual information estimation as an application

searching in the plane in this paper we initiate a new area of study dealing with the best way to search a possibly unbounded region for an object the model for our search algorithms is that we must pay costs proportional to the distance of the next probe position relative to our current position this model is meant to give a realistic cost measure for a robot moving in the plane we also examine the effect of decreasing the amount of a priori information given to search problems problems of this type are very simple analogues of non trivial problems on searching an unbounded region processing digitized images and robot navigation we show that for some simple search problems knowing the general direction of the goal is much more informative than knowing the distance to the goal

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

the zookeeper route problem abstract the s t route problem inside a polygon p with a set s of of sights edge segments and a set t of point threats is to find a route such that each point in s is visible from at least one point in the route and the route is not visible to any of the points in t 7 the requirement that exposure to the threats is not allowed at all is usually too severe and solutions do not exist in many cases especially those arising in typical applications where the sights and threats overlap e g defense applications an alternative way to model the risk posed by the threats is to enclose each threat location in a polygonal envelope within which the risk is unacceptably high resulting in the zookeeper route problem we are given a polygon p and a collection p of convex polygons inside p and we want to find a shortest route that visits without entering the polygons in p e g design a route for a zookeeper that wants to feed animals in enclosures we show that the general zookeeper route problem is np hard we present necessary and sufficient conditions for the existence of a zookeeper route and give an o n 2 algorithm for the case where p is a simple polygon and the polygons in p are attached to the boundary of p

symmetries of the self dual einstein equations i the infinite dimensional symmetry group and its low dimensional subgroups this is the first of two papers in which the authors give a complete classification of symmetry reduced solutions of plebanski s potential equation for self dual einstein spaces in this first part the infinite pseudogroup of symmetries of plebanski s equation is described and the conjugacy classes of all local subgroups of dimensions one two and three over both the real and complex numbers are classified then in the second paper this classification is used to obtain all symmetry reduced solutions

lie symmetries of nonlinear multidimensional reaction diffusion systems ii we present a complete description of the classical lie symmetries of a coupled system of partial differential equations comprising a pair of semilinear reaction diffusion equations with constant diffusivities and arbitrary nonlinearities in the reaction terms in any number of spatial dimensions part i cherniha r m and king j r 2000 j phys a math gen 33 267 82 j phys a math gen 33 7839 41 addressed the case of unequal diffusivities here we complete the analysis by treating the case of equal diffusivities in which the symmetry structure is richer still such models arise in the description of numerous physical chemical and biological systems and we also indicate the possible application in such contexts of some of the specific cases arising from the group classification specifically a variety of lie s ansatze and exact solutions of the so called reaction diffusion systems of a type that arises in mathematical biology are constructed

adaptive smoothing and density based tests of multivariate normality abstract methods of adaptive smoothing of density estimates where the amount of smoothing applied varies according to local features of the underlying density are investigated the difficulties of applying taylor series arguments in this context are explored simple properties of the estimates are investigated by numerical integration and compared with the fixed kernel approach optimal smoothing strategies based on the multivariate normal distribution are derived as an application of these techniques two tests of multivariate normality one based on integrated squared error and one on entropy are developed and some power calculations are carried out

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

silhouettes a graphical aid to the interpretation and validation of cluster analysis a new graphical display is proposed for partitioning techniques each cluster is represented by a so called silhouette which is based on the comparison of its tightness and separation this silhouette shows which objects lie well within their cluster and which ones are merely somewhere in between clusters the entire clustering is displayed by combining the silhouettes into a single plot allowing an appreciation of the relative quality of the clusters and an overview of the data configuration the average silhouette width provides an evaluation of clustering validity and might be used to select an appropriate number of clusters

unsupervised learning of finite mixture models this paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data the adjective unsupervised is justified by two properties of the algorithm 1 it is capable of selecting the number of components and 2 unlike the standard expectation maximization em algorithm it does not require careful initialization the proposed method also avoids another drawback of em for mixture fitting the possibility of convergence toward a singular estimate at the boundary of the parameter space the novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models instead we seamlessly integrate estimation and model selection in a single algorithm our technique can be applied to any type of parametric mixture model for which it is possible to write an em algorithm in this paper we illustrate it with experiments involving gaussian mixtures these experiments testify for the good performance of our approach

second order differential invariants of a family of diffusion equations an equivalence transformation algebra for a class of nonlinear diffusion equations is found after having obtained the second order differential invariants with respect to we get some results which allow us to linearize a subclass of the equations considered

stability based validation of clustering solutions data clustering describes a set of frequently employed techniques in exploratory data analysis to extract natural group structure in data such groupings need to be validated to separate the signal in the data from spurious structure in this context finding an appropriate number of clusters is a particularly important model selection question we introduce a measure of cluster stability to assess the validity of a cluster model this stability measure quantifies the reproducibility of clustering solutions on a second sample and it can be interpreted as a classification risk with regard to class labels produced by a clustering algorithm the preferred number of clusters is determined by minimizing this classification risk as a function of the number of clusters convincing results are achieved on simulated as well as gene expression data sets comparisons to other methods demonstrate the competitive performance of our method and its suitability as a general validation tool for clustering solutions in real world problems

symmetry analysis and exact solutions of semilinear heat flow in multi dimensions abstract a symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in n 1 dimensions with a general power nonlinearity the method involves an ansatz technique to solve an equivalent first order pde system of similarity variables given by group foliations of this heat equation using its admitted group of scaling symmetries this technique yields explicit similarity solutions as well as other explicit solutions of a more general non similarity form having interesting analytical behavior connected with blow up and dispersion in contrast standard similarity reduction of this heat equation gives a semilinear ode that cannot be explicitly solved by familiar integration techniques such as point symmetry reduction or integrating factors

testing for normality in arbitrary dimension on etend le theoreme de convergence faible univariable de murota et takeuchi 1981 pour la transformee de mahalanobis de la fonction caracteristique empirique a d variables d 1

singular invariant equation for the 1 1 fokker planck equation semi invariants for the linear parabolic equations with two independent variables time variable t and space variable x and one dependent variable u are derived under the transformation of the independent variables by using the infinitesimal method we also obtain the joint invariant equation for the above mentioned equation under equivalence transformation in fact we prove a necessary and sufficient condition for a 1 1 parabolic equation to be reducible via a local equivalence transformation to the one dimensional classical heat equation this result provides practical criteria for reduction finally examples of 1 1 fokker planck equations from applications are given to verify the results obtained

shortest paths without a map we study several versions of the shortest path problem when the map is not known in advanced but is specified dynamically we are seeking dynamic decision rules that optimize the worst case ratio of the distance covered to the length of the statically optimal path we describe optimal decision rules for two cases layered graphs of bounded width and two dimensional scenes with unit square obstacles the optimal rules turn out to be intuitive common sense heuristics for slightly more general graphs and scenes we show that no bounded ratio is possible we also show that the computational problem of devising a strategy that achieves a given worst case ratio to the optimum path in a graph is a universal two person game and thus pspace complete whereas optimizing the expected ratio is p hard

a simple direct method to find equivalence transformations of a generalized nonlinear schrodinger equation and a generalized kdv equation a simple direct method is presented to find equivalence transformations of nonlinear mathematical physics equations by using the direct method we obtain the continuous equivalence transformations of a class of nonlinear schroequations with variable coefficients and a family of nonlinear kdv equations with variable coefficients for the nonlinear schrodinger equations with variable coefficients the equivalence transformations obtained by the direct method coincide in nature with those obtained via the infinitesimal lie criterion but our computation is much simpler

localizing a robot with minimum travel we consider the problem of localizing a robot in a known environment modeled by a simple polygon p we assume that the robot has a map of p but is placed at an unknown location inside p from its initial location the robot sees a set of points called the visibility polygon v of its location in general sensing at a single point will not suffice to uniquely localize the robot since the set h of points in p with visibility polygon v may have more than one element hence the robot must move around and use range sensing and a compass to determine its position i e localize itself we seek a strategy that minimizes the distance the robot travels to determine its exact location we show that the problem of localizing a robot with minimum travel is np hard we then give a polynomial time approximation scheme that causes the robot to travel a distance of at most k 1 d where k h which is no greater than the number of reflex vertices of p and d is the length of a minimum length tour that would allow the robot to verify its true initial location by sensing we also show that this bound is the best possible

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

visibility occlusion and the aspect graph this chapter studies the ways in which the topology of the image of a polyhedron changes with changing viewpoint we catalog the ways that the topological appearance or aspect can change this enables us to find maximal regions of viewpoints of the same aspect we use these techniques to construct the viewpoint space partition vsp a partition of viewpoint space into maximal regions of constant aspect and its dual the aspect graph here we present tight bounds on the maximum size of the vsp and the aspect graph and give algorithms for their construction first in the convex case and then in the general case in particular we give bounds on the maximum size of n 2 and n 6 under an orthographic projection viewing model and of n 3 and n 9 under a perspective viewing model the algorithms make use of a new representation of the appearance of polyhedra from all viewpoints called the aspect representation or asp we believe that this representation is one of the significant contributions of this paper

backlund transformation for the non isospectral and variable coefficient non linear schrodinger equation a non linear schrodinger equation nlse with a non isospectral lax pair and variable coefficients is studied this equation is shown to be an integrability condition for an akns system a gamma riccati system and a lax pair these three systems are proved to be equivalent they are all invariant under some generalisation of the konno wadati transformation related to this nlse there is also a variable coefficients evolution equation for the function gamma derived from the gamma riccati system a backlund transformation for this gamma equation has been constructed with this a new backlund transformation for the nlse is presented so that successive generation of new solutions is reduced to integration only

differential invariants and group foliation for the complex monge ampere equation we apply the method of group foliation to the complex monge ampere equation cma 2 with the goal of establishing a regular framework for finding its non invariant solutions we employ the infinite symmetry subgroup of the equation the group of unimodular biholomorphisms to produce a foliation of the solution space into leaves which are orbits of solutions with respect to the symmetry group accordingly cma 2 is split into an automorphic system and a resolvent system which we derive in this paper this is an intricate system and here we make no attempt to solve it in order to obtain non invariant solutions we obtain all differential invariants up to third order for the group of unimodular biholomorphisms and in particular all the basis differential invariants we construct the operators of invariant differentiation from which all higher differential invariants can be obtained consequently we are able to write down all independent partial differential equations with one real unknown and two complex independent variables which keep the same infinite symmetry subgroup as cma 2 we prove explicitly that applying operators of invariant differentiation to third order invariants we obtain all fourth order invariants at this level we have all the information which is necessary and sufficient for group foliation we propose a new approach in the method of group foliation which is based on the commutator algebra of operators of invariant differentiation the resolving equations are obtained by applying this algebra to differential invariants with the status of independent variables furthermore this algebra together with jacobi identities provides the commutator representation of the resolvent system this proves to be the simplest and most natural way of arriving at the resolving equations

an extensive comparative study of cluster validity indices the validation of the results obtained by clustering algorithms is a fundamental part of the clustering process the most used approaches for cluster validation are based on internal cluster validity indices although many indices have been proposed there is no recent extensive comparative study of their performance in this paper we show the results of an experimental work that compares 30 cluster validity indices in many different environments with different characteristics these results can serve as a guideline for selecting the most suitable index for each possible application and provide a deep insight into the performance differences between the currently available indices

weak transversality and partially invariant solutions new exact solutions are obtained for several nonlinear physical equations namely the navier stokes and euler systems an isentropic compressible fluid system and a vector nonlinear schrodinger equation the solution method makes use of the symmetry group of the system in situations when the standard lie method of symmetry reduction is not applicable

validity index for crisp and fuzzy clusters abstract in this article a cluster validity index and its fuzzification is described which can provide a measure of goodness of clustering on different partitions of a data set the maximum value of this index called the pbm index across the hierarchy provides the best partitioning the index is defined as a product of three factors maximization of which ensures the formation of a small number of compact clusters with large separation between at least two clusters we have used both the k means and the expectation maximization algorithms as underlying crisp clustering techniques for fuzzy clustering we have utilized the well known fuzzy c means algorithm results demonstrating the superiority of the pbm index in appropriately determining the number of clusters as compared to three other well known measures the davies bouldin index dunn s index and the xie beni index are provided for several artificial and real life data sets

equivalence transformations and differential invariants of a generalized nonlinear schrodinger equation by using lie s invariance infinitesimal criterion we obtain the continuous equivalence transformations of a class of nonlinear schrodinger equations with variable coefficients we construct the differential invariants of order 1 starting from a special equivalence subalgebra we apply these latter ones to find the most general subclass of variable coefficient nonlinear schrodinger equations which can be mapped by means of an equivalence transformation of to the well known cubic schrodinger equation we also provide the explicit form of the transformation

exact solutions of semilinear radial wave equations in n dimensions abstract exact solutions are derived for an n dimensional radial wave equation with a general power nonlinearity the method which is applicable more generally to other nonlinear pdes involves an ansatz technique to solve a first order pde system of group invariant variables given by group foliations of the wave equation using the one dimensional admitted point symmetry groups these groups comprise scalings and time translations admitted for any nonlinearity power in addition to space time inversions admitted for a particular conformal nonlinearity power this is shown to yield not only group invariant solutions as derived by standard symmetry reduction but also other exact solutions of a more general form in particular solutions with interesting analytical behavior connected with blow ups as well as static monopoles are obtained

cluster validation using graph theoretic concepts in this article we have generalized dunn s index and the davies bouldin index for cluster validation using graph structures such as gg rng and mst unlike dunn s index and the davies bouldin index the proposed indices are not sensitive to noisy points and are applicable to hyperspherical and structural clusters as well the relationships between various indices have also been established the effectiveness of the generalized indices and superiority over some existing cluster validity indices are established using eight data sets

non linear reaction diffusion systems with variable diffusivities lie symmetries ansatze and exact solutions abstract this work first considers the classical lie symmetry analysis of a class of systems of two quasilinear reaction diffusion equations having variable diffusivities subsequently non lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems in particular families of exact solutions of a diffusive lotka volterra type system are constructed

normality based validation for crisp clustering we introduce a new validity index for crisp clustering that is based on the average normality of the clusters unlike methods based on inter cluster and intra cluster distances this index emphasizes the cluster shape by using a high order characterization of its probability distribution the normality of a cluster is characterized by its negentropy a standard measure of the distance to normality which evaluates the difference between the cluster s entropy and the entropy of a normal distribution with the same covariance matrix the definition of the negentropy involves the distribution s differential entropy however we show that it is possible to avoid its explicit computation by considering only negentropy increments with respect to the initial data distribution where all the points are assumed to belong to the same cluster the resulting negentropy increment validity index only requires the computation of covariance matrices we have applied the new index to an extensive set of artificial and real problems where it provides in general better results than other indices both with respect to the prediction of the correct number of clusters and to the similarity among the real clusters and those inferred

the tresse theorem and differential invariants for the nonlinear schrodinger equation in the present paper by using the tresse theorem we describe a method of construction of all invariants and the differential invariants for a given lie group which means invariants containing derivatives of any order some important examples from analysis geometry and physics are presented in particular invariants for the nonlinear schrodinger equation will be investigated

on fuzzy cluster validity indices cluster analysis aims at identifying groups of similar objects and helps to discover distribution of patterns and interesting correlations in large data sets especially fuzzy clustering has been widely studied and applied in a variety of key areas and fuzzy cluster validation plays a very important role in fuzzy clustering this paper introduces the fundamental concepts of cluster validity and presents a review of fuzzy cluster validity indices available in the literature we conducted extensive comparisons of the mentioned indices in conjunction with the fuzzy c means clustering algorithm on a number of widely used data sets and make a simple analysis of the experimental results

on bayesian analysis of mixtures with an unknown number of components with discussion new methodology for fully bayesian mixture analysis is developed making use of reversible jump markov chain monte carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture a sample from the full joint distribution of all unknown variables is thereby generated and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution the methodology is applied here to the analysis of univariate normal mixtures using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context

sep cop an efficient method to find the best partition in hierarchical clustering based on a new cluster validity index hierarchical clustering algorithms provide a set of nested partitions called a cluster hierarchy since the hierarchy is usually too complex it is reduced to a single partition by using cluster validity indices we show that the classical method is often not useful and we propose sep a new method that efficiently searches in an extended partition set furthermore we propose a new cluster validity index cop since many of the commonly used indices cannot be used with sep experiments performed with 80 synthetic and 7 real datasets confirm that sep cop is superior to the method currently used and furthermore it is less sensitive to noise

invariants of a family of third order ordinary differential equations we consider a family of third order ordinary differential equations odes with the cubic nonlinearity in second order derivative which is closed with respect to contact transformations using lie s infinitesimal method we construct the basis of differential invariants for this family of equations and the operators of invariant differentiation invariants provide a simple way of finding the equations which may be equivalent to a given nonlinear ode and the transformation connecting two equivalent equations some examples are given to illustrate the results obtained in the paper

a near tight approximation algorithm for the robot localization problem localization is a fundamental problem in robotics the kidnapped robot possesses a compass and map of its environment it must determine its location at a minimum cost of travel distance the problem is np hard g dudek k romanik and s whitesides siam j comput 27 1998 pp 583 604 even to minimize within factor c log n c tovey and s koenig proceedings of the national conference on artificial intelligence austin tx 2000 pp 819 824 where n is the map size no approximation algorithm has been known we give an o log 3n factor algorithm the key idea is to plan travel in a majority rule map which eliminates uncertainty and permits a link to the frac 1 2 group steiner not group steiner problem the approximation factor is not far from optimal we prove a c log 2 epsilon n lower bound assuming np not subseteq ztime n polylog n for the grid graphs commonly used in practice we also extend the algorithm to polygonal maps by discretizing the problem using novel geometric techniques