an extension to possibilistic fuzzy cluster analysis we explore an approach to possibilistic fuzzy clustering that avoids a severe drawback of the conventional approach namely that the objective function is truly minimized only if all cluster centers are identical our approach is based on the idea that this undesired property can be avoided if we introduce a mutual repulsion of the clusters so that they are forced away from each other we develop this approach for the possibilistic fuzzy c means algorithm and the possibilistic gustafson kessel algorithm in our experiments we found that in this way we can combine the partitioning property of the probabilistic fuzzy c means algorithm with the advantages of a possibilistic approach w r t the interpretation of the membership degrees

on relational possibilistic clustering this paper initially describes the relational counterpart of possibilistic c means pcm algorithm called relational pcm or rpcm rpcm is then improved to better handle arbitrary dissimilarity data first a re scaling of the pcm membership function is proposed in order to obtain zero membership values when the distance to prototype equals the maximum value allowed in bounded dissimilarity measures second a heuristic method of reference distance initialisation is provided which diminishes the known pcm tendency of producing coincident clusters finally rpcm improved with our initialisation strategy is tested on both synthetic and real data sets with satisfactory results

the lp minkowski problem and the minkowski problem in centroaffine geometry abstract the l p minkowski problem introduced by lutwak is solved for p n 1 in the smooth category the relevant monge ampere equation 0 1 is solved for all p 1 the same equation for p 1 is also studied and solved for p n 1 1 when p n 1 the equation is interpreted as a minkowski problem in centroaffine geometry a kazdan warner type obstruction for this problem is obtained

a fuzzy approach for multitype relational data clustering mining interrelated data among multiple types of objects or entities is important in many real world applications despite extensive study on fuzzy clustering of vector space data very limited exploration has been made on fuzzy clustering of relational data that involve several object types in this paper we propose a new fuzzy clustering approach for multitype relational data fc mr in fc mr different types of objects are clustered simultaneously an object is assigned a large membership with respect to a cluster if its related objects in this cluster have high rankings in each cluster an object tends to have a high ranking if its related objects have large memberships in this cluster the fc mr approach is formulated to deal with multitype relational data with various structures the objective function of fc mr is locally optimized by an efficient iterative algorithm which updates the fuzzy membership matrix and the ranking matrix of one type at once while keeping those of other types constant we also discuss the simplified fc mr for multitype relational data with two special structures namely star structure and extended star structure experimental studies are conducted on benchmark document datasets to illustrate how the proposed approach can be applied flexibly under different scenarios in real world applications the experimental results demonstrate the feasibility and effectiveness of the new approach compared with existing ones

relational duals of the c means clustering algorithms abstract the hard and fuzzy c means algorithms are widely used effective tools for the problem of clustering n objects into hard or fuzzy groups of similar individuals when the data is available as object data consisting of a set of n feature vectors in rp however object data algorithms are not directly applicable when the n objects are implicitly described in terms of relational data which consists of a set of n2 measurements of relations between each of the pairs of objects new relational versions of the hard and fuzzy c means algorithms are presented here for the case when the relational data can reasonably be viewed as some measure of distance some convergence properties of the algorithms are given along with a numerical example

a remark on rotationally symmetric solutions to the centroaffine minkowski problem abstract in this paper we study the solvability of the rotationally symmetric centroaffine minkowski problem by delicate blow up analyses we remove a technical condition in the existence result obtained by lu and wang 30

the earth mover s distance as a metric for image retrieval we investigate the properties of a metric between two distributions the earth mover s distance emd for content based image retrieval the emd is based on the minimal cost that must be paid to transform one distribution into the other in a precise sense and was first proposed for certain vision problems by peleg werman and rom for image retrieval we combine this idea with a representation scheme for distributions that is based on vector quantization this combination leads to an image comparison framework that often accounts for perceptual similarity better than other previously proposed methods the emd is based on a solution to the transportation problem from linear optimization for which efficient algorithms are available and also allows naturally for partial matching it is more robust than histogram matching techniques in that it can operate on variable length representations of the distributions that avoid quantization and other binning problems typical of histograms when used to compare distributions with the same overall mass the emd is a true metric in this paper we focus on applications to color and texture and we compare the retrieval performance of the emd with that of other distances

the possibilistic c means algorithm insights and recommendations recently the possibilistic c means algorithm pcm was proposed to address the drawbacks associated with the constrained memberships used in algorithms such as the fuzzy c means fcm in this issue barni et al 1996 report a difficulty they faced while applying the pcm and note that it exhibits an undesirable tendency to converge to coincidental clusters the purpose of this paper is not just to address the issues raised by barni et al but to go further and analytically examines the underlying principles of the pcm and the possibilistic approach in general we analyze the data sets used by barni et al and interpret the results reported by them in the light of our findings

on solvability of two dimensional lp minkowski problem abstract given p 0 and a positive continuous function g with g x t g x for some 0 0 the functional f u 0 t u x 2 u 2 x dx c 0 t g x u p x dx 2 p has a minimizer in the class of positive functions u c1 r for which u x t u x for all x r this minimizer is used to prove the existence of a positive periodic solution y c2 r of two dimensional lp minkowski problem y1 p x y x y x g x where p 0 2

semiglobal uniform asymptotic stability of an easy to implement pll like sensorless observer for induction motors in this work stability properties of a novel and easy to implement sensorless observer for induction motors are investigated the considered solution is inspired by phase locked loop strategies and it takes advantage from the rotor flux derivatives to define a rotating vector to align with even if such variables are not directly measured input to state stability properties are proven for suitably defined electrical and magneto mechanical observation error subsystems time scale separation is then exploited to invoke singular perturbation arguments and guarantee boundedness of the observation errors trajectories finally the small gain theorem is adopted to prove semiglobal uniform asymptotic stability of the presented scheme some additional interesting properties are discussed beside the main stream of the proposed analysis simulations indicate the observer can be profitably exploited for induction machines speed sensorless control applications

general two stage extended kalman filters a general two stage extended kalman filter gtsekf which extends the linear general two stage kalman filter to nonlinear systems is further proposed a new nonlinear two stage transformation is introduced to facilitate achieving this extension as in the linear one the gtsekf is derived mainly by applying the nonlinear two stage transformation to the well known extended kalman filter ekf and is shown to be equivalent to the ekf with a decoupled computing structure a nonlinear filter for estimating constant parameters in dynamic systems is presented to illustrate one application of the proposed gtsekf a literature example is also given to demonstrate the correctness and usefulness of the proposed results

various ways to compute the continuous discrete extended kalman filter the extended kalman filter ekf is a very popular tool dealing with state estimation its continuous discrete version cd ekf estimates the state trajectory of continuous time nonlinear models whose internal state is described by a stochastic differential equation and which is observed through a noisy nonlinear form of the sampled state the prediction step of the cd ekf leads to solve a differential equation that cannot be generally solved in a closed form this technical note presents an overview of the numerical methods including recent works usually implemented to approximate this filter comparisons of theses methods on two different nonlinear models are finally presented the first one is the van der pol oscillator which is widely used as a benchmark the second one is a neuronal population model this more original model is used to simulate eeg activity of the cortex experiments showed better stability properties of implementations for which the positivity of the prediction matrix is guaranteed

a survey of kernel and spectral methods for clustering clustering algorithms are a useful tool to explore data structures and have been employed in many disciplines the focus of this paper is the partitioning clustering problem with a special interest in two recent approaches kernel and spectral methods the aim of this paper is to present a survey of kernel and spectral clustering methods two approaches able to produce nonlinear separating hypersurfaces between clusters the presented kernel clustering methods are the kernel version of many classical clustering algorithms e g k means som and neural gas spectral clustering arise from concepts in spectral graph theory and the clustering problem is configured as a graph cut problem where an appropriate objective function has to be optimized an explicit proof of the fact that these two paradigms have the same objective is reported since it has been proven that these two seemingly different approaches have the same mathematical foundation besides fuzzy kernel clustering methods are presented as extensions of kernel k means clustering algorithm

optimal solution of the two stage kalman estimator the two stage kalman estimator was originally proposed to reduce the computational complexity of the augmented state kalman filter it was also applied to the tracking of maneuvering targets by treating the target acceleration as a bias term except in certain restrictive conditions the conventional two stage estimators are suboptimal in the sense that they are not equivalent to the augmented state filter in this paper the authors propose a new two stage kalman estimator i e new structure which is an extension of friedland s estimator and is optimal in general conditions in addition we provide some analytic results to demonstrate the computational advantages of two stage estimators over augmented ones

speed and rotor flux estimation of induction machines using a two stage extended kalman filter this paper presents an effective implementation of an extended kalman filter used for the estimation of both rotor flux and rotor velocity of an induction motor an algorithm proposed by hsieh and chen in hsieh c s chen f c 1999 optimal solution of the two stage kalman estimator ieee transactions on automatic control 44 1 194 199 for linear parameter estimation is extended to non linear estimation where parameters such as the velocity of an induction machine are present in the transition matrix and in the augmented state space compared to a straightforward implementation of an extended kalman filter our modified optimal two stage kalman estimator reduces the number of arithmetic operations by 25 allowing higher sampling rate or the use of a cheaper microcontroller

on the _ minkowski problem a volume normalized formulation of the lp minkowski problem is presented this formulation has the advantage that a solution is possible for all p 1 including the degenerate case where the index p is equal to the dimension of the ambient space a new approach to the lp minkowski problem is presented which solves the volume normalized formulation for even data and all p 1

rotationally symmetric solutions to the l p minkowski problem abstract in this paper we study the l p minkowski problem for p n 1 which corresponds to the critical exponent in the blaschke santalo inequality we first obtain volume estimates for general solutions then establish a priori estimates for rotationally symmetric solutions by using a kazdan warner type obstruction finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow up analysis we also include an existence result for the l p minkowski problem which corresponds to the super critical case of the blaschke santalo inequality

extension of friedland s separate bias estimation to randomly time varying bias of nonlinear systems by extending b friedland s 1969 separate bias estimation algorithm for linear systems to nonlinear systems and combining the result with the suboptimal fading extended kalman filter proposed by d h zhou 1990 and by d h zhou et al a pseudo separate bias estimation algorithm for randomly time varying bias of a class of nonlinear time varying stochastic systems is obtained a simulation example is presented to illustrate the effectiveness of the algorithm

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

multiple kernel fuzzy clustering while fuzzy c means is a popular soft clustering method its effectiveness is largely limited to spherical clusters by applying kernel tricks the kernel fuzzy c means algorithm attempts to address this problem by mapping data with nonlinear relationships to appropriate feature spaces kernel combination or selection is crucial for effective kernel clustering unfortunately for most applications it is uneasy to find the right combination we propose a multiple kernel fuzzy c means mkfc algorithm that extends the fuzzy c means algorithm with a multiple kernel learning setting by incorporating multiple kernels and automatically adjusting the kernel weights mkfc is more immune to ineffective kernels and irrelevant features this makes the choice of kernels less crucial in addition we show multiple kernel k means to be a special case of mkfc experiments on both synthetic and real world data demonstrate the effectiveness of the proposed mkfc algorithm

comparing fuzzy partitions a generalization of the rand index and related measures in this paper we introduce a fuzzy extension of a class of measures to compare clustering structures namely measures that are based on the number of concordant and the number of discordant pairs of data points this class includes the well known rand index but also commonly used alternatives such as the jaccard measure in contrast with previous proposals our extension exhibits desirable metrical properties apart from elaborating on formal properties of this kind we present an experimental study in which we compare different fuzzy extensions of the rand index and the jaccard measure

the lp brunn minkowski inequality for p 1 abstract in this paper we confirm the l p brunn minkowski inequality conjecture for p close to 1 the logarithmic brunn minkowski inequality is also verified for convex bodies close to the unit ball in hausdorff distance

the logarithmic minkowski problem for polytopes abstract the logarithmic minkowski problem asks for necessary and sufficient conditions for a finite borel measure on the unit sphere so that it is the cone volume measure of a convex body this problem was solved recently by boroczky lutwak yang and zhang for even measures boroczky et al 2013 8 this paper solves the case of discrete measures whose supports are in general position

variational characterization for the planar dual minkowski problem abstract in this paper we give a variational analysis to the planar dual minkowski problem in the sobolev space with the new variational characterization we can deal with existence results for prescribed not necessarily positive data meanwhile functional inequalities and multiple solutions are also obtained

on the uniqueness and continuity of the dual area measure abstract in this paper the uniqueness of the dual minkowski problem for the dual area measure is established via the dual minkowski inequality and the dual log minkowski inequality for real q n 1 it is proved that the weak convergence of the q th dual area measure implies the convergence of the corresponding convex bodies in the hausdorff metric and that the solution to the dual minkowski problem is continuous with respect to q

on the 2 dimensional dual minkowski problem abstract we study the 2 dimensional dual minkowski problem which is the following nonlinear problem on unit circle u u g u 1 u 2 u 2 2 k 2 s for any given positive continuous function g with 2 m periodic we prove that it is solvable for all k 1 and m 3 4 5

optimal boundary regularity for a singular monge ampere equation abstract in this paper we study the optimal global regularity for a singular monge ampere type equation which arises from a few geometric problems we find that the global regularity does not depend on the smoothness of domain but it does depend on the convexity of the domain we introduce a type to describe the convexity as a result we show that the more convex is the domain the better is the regularity of the solution in particular the regularity is the best near angular points

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

a generalized rotationally symmetric case of the centroaffine minkowski problem abstract in this paper the centroaffine minkowski problem a critical case of the l p minkowski problem in the n 1 dimensional euclidean space is studied by its variational structure and the method of blow up analyses we obtain two sufficient conditions for the existence of solutions for a generalized rotationally symmetric case of the problem

on the l p monge ampere equation abstract for the case where 0 p 1 we prove the existence of solutions of the l p monge ampere equation for arbitrary measures and show that the solution is in general not unique

a robust automatic merging possibilistic clustering method krishnapuram and keller ieee trans fuzzy syst vol 1 no 2 pp 98 110 may 1993 proposed a possibilistic c means pcm clustering by relaxing the constraint in fuzzy c means fcms that the memberships of a data point across classes sum to 1 the pcm algorithm has a tendency to produce coincident clusters this can be a merit of pcm as a good mode seeking algorithm if initials and parameters are suitably chosen however the performance of pcm heavily depends on initializations and parameter selection in this paper we propose a mechanism of robust automatic merging we then create an automatic merging possibilistic clustering method am pcm where the proposed algorithm does not only solve these parameter selection and initialization problems but also obtains an optimal cluster number the proposed am pcm algorithm first uses all data points as initial cluster centers and then automatically merges these surrounding points around each cluster mode such that it can self organize data groups according to the original data structure the am pcm can exhibit the robustness to parameter noise cluster number different volumes and initializations the computational complexity of am pcm is also analyzed comparisons between am pcm and other clustering methods are made some numerical data and real datasets are used to show these good aspects of am pcm experimental results and comparisons actually demonstrate that the proposed am pcm is an effective and parameter free robust clustering algorithm

lp minkowski problem with not necessarily positive data abstract the l p minkowski problem in the plane has been the object of a number of recent investigations here we present various existence results for 0 p 2 in the case 0 p 2 a necessary and sufficient condition for the solvability of the problem is given

existence of solutions to the orlicz minkowski problem abstract in this paper the orlicz minkowski problem a generalization of the classical minkowski problem is studied using the variational method we obtain a new existence result of solutions to this problem for general measures

on the lp dual minkowski problem abstract the l p dual curvature measure was introduced by lutwak yang zhang in an attempt to unify the l p brunn minkowski theory and the dual brunn minkowski theory the characterization problem for l p dual curvature measure called the l p dual minkowski problem is a fundamental problem in this unifying theory the l p dual minkowski problem contains the l p minkowski problem and the dual minkowski problem two major problems in modern convex geometry that remain open in general in this paper existence results on the l p dual minkowski problem in the weak sense will be provided moreover existence and uniqueness of the solution in the smooth category will also be demonstrated

comparing fuzzy probabilistic and possibilistic partitions using the earth mover s distance a number of noteworthy techniques have been put forth recently in different research fields for comparing clusterings herein we introduce a new method for comparing soft fuzzy probabilistic and possibilistic partitions based on the earth mover s distance emd and the ordered weighted average owa the proposed method is a metric depending on the ground distance for all but possibilistic partitions it is extremely flexible due to its emd formulation owa aggregation and abstract concept of ground distance in theory our method is agnostic to the type uncertainty of soft partition clustering algorithm and distance measure used in the clustering algorithm s and it is applicable to the clustering of both object and relational data validation is performed theoretically experimentally as well as in terms of computational complexity emphasis is placed on the set of possibilistic partitions specifically noise and coincident clusters which are important cases that have received little to no attention to date in the comparing clustering literature improvements are reported in terms of metric properties and computational complexity over existing extended concordance discordance e g soft rand and jaccard approaches and improved design and robustness in comparison with existing transportation problem based approaches

nerf c means non euclidean relational fuzzy clustering abstract the relational fuzzy c means rfcm algorithm can be used to cluster a set of n objects described by pair wise dissimilarity values if and only if there exist n points in r n 1 whose squared euclidean distances precisely match the given dissimilarity data this strong restriction on the dissimilarity data renders rfcm inapplicable to most relational clustering problems this paper substantially improves rfcm by generalizing it to the case of arbitrary symmetric dissimilarity data the generalization is obtained using a computationally efficient modification of the existing algorithm that is equivalent to applying a spreading transformation to the dissimilarity data while the method given applies specifically to dissimilarity data a simple transformation can be used to convert similarity relations into dissimilarity data so the method is applicable to any numerical relational data that are positive reflexive or anti reflexive and symmetric numerical examples illustrate and compare the present approach to problems that can be studied with alternatives such as the linkage algorithms

learning choquet integral based metrics for semisupervised clustering we consider an application of fuzzy measures to the problem of metric learning in semisupervised clustering we investigate the necessary and sufficient conditions on the underlying fuzzy measure that make the discrete choquet integral suitable for defining a metric as a byproduct we can obtain the analogous conditions for the ordered weighted averaging owa operators which constitute a special case we then generalize these results for power based choquet and owa operators we show that this metric learning problem can be formulated as a linear programming problem and specify the required sets of linear constraints we present the results of numerical experiments on artificial and real world datasets which illustrate the potential usefulness and limitations of this construction

geometric measures in the dual brunn minkowski theory and their associated minkowski problems a longstanding question in the dual brunn minkowski theory is what are the dual analogues of federer s curvature measures for convex bodies the answer to this is provided this leads naturally to dual versions of minkowski type problems what are necessary and sufficient conditions for a borel measure to be a dual curvature measure of a convex body sufficient conditions involving measure concentration are established for the existence of solutions to these problems

a new fuzzy clustering validity index with a median factor for centroid based clustering determining the number of clusters which is usually approved by domain experts or evaluated by clustering validity indexes is an important issue in clustering analysis this study discusses the effectiveness of clustering validity indexes for centroid based partitional clustering algorithms most general purpose clustering validity indexes take the minimum maximum distance between a pair of data objects a pair of cluster centroids or an object and a centroid as an important evaluation factor however they may present unstable results especially when two centroids are allocated closely to alleviate this problem a new clustering validity index which is termed the wu and li index wli is proposed in this paper our proposed wli partially allows to some extent the existence of closely allocated centroids in the clustering results by considering not only the minimum but the median distances between a pair of centroids as well therefore possessing better stability the performances of wli and some existing clustering validity indexes are evaluated and compared by running the fuzzy c means algorithm for clustering various types of datasets including artificial datasets uci datasets and images experimental results have shown that wli has the more accurate and satisfactory performance than other indexes

the two dimensional lp minkowski problem and nonlinear equations with negative exponents abstract in this paper we present various existence results for nonlinear differential equations related to the l p minkowski problem in the plane and the one dimensional conformal curvature problem in particular the supercritical case that is p 2 for the l p minkowski problem with sign changing data is addressed a new geometric sharp inequality is also obtained

optimal multistage kalman estimators an optimal multistage kalman estimator omske is proposed as a generalization of the optimal two stage kalman estimator for the reduction of the computational burden of the kalman estimator ke for discrete time linear time varying systems with triangular transition matrices this new filer is obtained by applying a multistage u v transformation to decouple the covariances of the ke it is shown analytically that the computational complexity of the omske is less than that of the ke and is minimum when the system transition matrix has the maximum stage number

clustering and aggregation of relational data with applications to image database categorization in this paper we introduce a new algorithm for clustering and aggregating relational data card we assume that data is available in a relational form where we only have information about the degrees to which pairs of objects in the data set are related moreover we assume that the relational information is represented by multiple dissimilarity matrices these matrices could have been generated using different sensors features or mappings card is designed to aggregate pairwise distances from multiple relational matrices partition the data into clusters and learn a relevance weight for each matrix in each cluster simultaneously the cluster dependent relevance weights offer two advantages first they guide the clustering process to partition the data set into more meaningful clusters second they can be used in subsequent steps of a learning system to improve its learning behavior the performance of the proposed algorithm is illustrated by using it to categorize a collection of 500 color images we represent the pairwise image dissimilarities by six different relational matrices that encode color texture and structure information

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

nonuniqueness of solutions to the l p minkowski problem abstract in this paper we show by example that there is no uniform estimate for the l p minkowski problem as a result we obtain the nonuniqueness of solutions to the problem

technical communique two stage kalman estimator with unknown exogenous inputs this paper presents a two stage estimator for bias and state filtering in discrete time stochastic linear systems affected by unknown inputs or disturbances we show that the state estimate can be expressed as x k k x k k b k kb k k where x k k is a bias free state estimate and b k k the optimal estimate of constant bias the proposed two stage estimator is based on an alternate derivation of the unbiased minimum variance estimator with unknown exogenous inputs developed by darouach and zasadzinski 1997 automatica 33 717 719

a new continuous discrete particle filter for continuous discrete nonlinear systems abstract the continuous discrete version of particle filter cd pf for continuous discrete nonlinear systems is proposed in this paper firstly the continuous discrete extended kalman filter cd ekf and the cd pf are described in detail then some comparisons of these two methods are carried out the simulation results show that the performance of the cd pf is better than that of the cd ekf then higher order stochastic runge kutta schemes are proposed to improve the stability and precision of the filter furthermore the cd pf is applied to nonlinear systems with nongaussian noise finally some simulations are provided to illustrate the effectiveness of the proposed filter

the lp minkowski problem for n p 1 abstract chou and wang s existence result for the l p minkowski problem on s n 1 for p n 1 and an absolutely continuous measure is discussed and extended to more general measures in particular we provide an almost optimal sufficient condition for the case p 0 1

cluster ensembles a knowledge reuse framework for combining multiple partitions this paper introduces the problem of combining multiple partitionings of a set of objects into a single consolidated clustering without accessing the features or algorithms that determined these partitionings we first identify several application scenarios for the resultant knowledge reuse framework that we call cluster ensembles the cluster ensemble problem is then formalized as a combinatorial optimization problem in terms of shared mutual information in addition to a direct maximization approach we propose three effective and efficient techniques for obtaining high quality combiners consensus functions the first combiner induces a similarity measure from the partitionings and then reclusters the objects the second combiner is based on hypergraph partitioning the third one collapses groups of clusters into meta clusters which then compete for each object to determine the combined clustering due to the low computational costs of our techniques it is quite feasible to use a supra consensus function that evaluates all three approaches against the objective function and picks the best solution for a given situation we evaluate the effectiveness of cluster ensembles in three qualitatively different application scenarios i where the original clusters were formed based on non identical sets of features ii where the original clustering algorithms worked on non identical sets of objects and iii where a common data set is used and the main purpose of combining multiple clusterings is to improve the quality and robustness of the solution promising results are obtained in all three situations for synthetic as well as real data sets

a priori estimates and existence of solutions to the prescribed centroaffine curvature problem abstract in this paper we study the prescribed centroaffine curvature problem in the euclidean space r n 1 this problem is equivalent to solving a monge ampere equation on the unit sphere it corresponds to the critical case of the blaschke santalo inequality by approximation from the subcritical case and using an obstruction condition and a blow up analysis we obtain sufficient conditions for the a priori estimates and the existence of solutions up to a lagrange multiplier

optimal and suboptimal separate bias kalman estimators for a stochastic bias addresses two issues concerning the separate bias kalman estimator the first of these issues deals with the derivation of the optimal estimator for the general case in which the bias vector is stochastic in nature and the second issue deals with defining a suitable suboptimal realization of the generalized estimator

comparing fuzzy probabilistic and possibilistic partitions when clustering produces more than one candidate to partition a finite set of objects o there are two approaches to validation i e selection of a best partition and implicitly a best value for c which is the number of clusters in o first we may use an internal index which evaluates each partition separately second we may compare pairs of candidates with each other or with a reference partition that purports to represent the true cluster structure in the objects this paper generalizes many of the classical indices that have been used with outputs of crisp clustering algorithms so that they are applicable for candidate partitions of any type i e crisp or soft with soft comprising the fuzzy probabilistic and possibilistic cases space prevents inclusion of all of the possible generalizations that can be realized this way here we concentrate on the rand index and its modifications we compare our fuzzy rand index with those of campello hullermeier and rifqi and brouwer and show that our extension of the rand index is o n while the other three are all o n2 numerical examples are given to illustrate various facets of the new indices in particular we show that our indices can be used even when the partitions are probabilistic or possibilistic and that our method of generalization is valid for any index that depends only on the entries of the classical i e four pair types contingency table for this problem

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1