segmentation and classification of range images the recognition of objects in three dimensional space is a desirable capability of a computer vision system range images which directly measure 3 d surface coordinates of a scene are well suited for this task in this paper we report a procedure to detect connected planar convex and concave surfaces of 3 d objects this is accomplished in three stages the first stage segments the range image into surface patches by a square error criterion clustering algorithm using surface points and associated surface normals the second stage classifies these patches as planar convex or concave based on a non parametric statistical test for trend curvature values and eigenvalue analysis in the final stage boundaries between adjacent surface patches are classified as crease or noncrease edges and this information is used to merge compatible patches to produce reasonable faces of the object s this procedure has been successfully applied to a large number of real and synthetic images four of which we present in this paper

a fuzzy extension of the silhouette width criterion for cluster analysis the present paper proposes a new cluster validity measure as an additional criterion to help the decision making process in fuzzy cluster analysis this measure named fuzzy silhouette is a generalization to the fuzzy case of the average silhouette width criterion originally conceived to assess crisp non fuzzy data partitions the fuzzy silhouette is more appealing than its crisp counterpart in the context of fuzzy cluster analysis since it makes explicit use of the fuzzy partition matrix provided by the clustering algorithm in addition it has been designed to improve performance of the original silhouette criterion in detecting regions with higher data density when the data set involves overlapping clusters the performance of the fuzzy silhouette is evaluated and compared to that of five well known cluster validity measures six data sets are used to illustrate different scenarios in which the proposed fuzzy silhouette performs similar to or better than these other criteria thus becoming eligible to join a pool of measures to be used all together in fuzzy cluster analysis

numerical algorithm for solving two point second order periodic boundary value problems for mixed integro differential equations in this study the numerical solution of fredholm volterra integro differential equations for two point second order periodic boundary value problems is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the periodic boundary conditions of the problem are satisfied the exact solution u x is represented in the form of series in the space w 2 3 in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 3 some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear equations

numerical investigations for systems of second order periodic boundary value problems using reproducing kernel method the reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions and usually provides the solutions in term of rapidly convergent series in the appropriate hilbert spaces with components that can be elegantly computed the aim of the present analysis is to implement a relatively recent computational method reproducing kernel hilbert space for obtaining the solutions for systems of second order differential equations with periodic boundary conditions a reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied whilst the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points an efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory convergence analysis and error behavior of the presented method are also discussed in this approach computational results of some numerical examples are presented to illustrate the viability simplicity and applicability of the algorithm developed

general c means clustering model partitional clustering is an important part of cluster analysis based on various theories numerous clustering algorithms have been developed and new clustering algorithms continue to appear in the literature it is known that occam s razor plays a pivotal role in data based models and partitional clustering is categorized as a data based model however no relation had previously been discovered between occam s razor and partitional clustering as we discuss in this paper the three main contributions of this paper can be summarized as follows 1 according to a novel definition of the mean a unifying generative framework for partitional clustering algorithms called a general c means clustering model gcm is presented and studied and 2 based on the local optimality test of the gcm the connection between occam s razor and partitional clustering is established for the first time as its application a comprehensive review of the existing objective function based clustering algorithms is presented based on gcm 3 under a common assumption about partitional clustering a theoretical guide for devising and implementing clustering algorithm is discovered these conclusions are verified by numerical experimental results

validity measures for the fuzzy cluster analysis of orientations fuzzy k means clustering can be applied to the automatic identification of sets in discontinuity data after suitable adaptation of the algorithm to establish the number of clusters in a data set modified versions of the validity measures of gath and geva 1989 xie beni 1991 and fukuyama sugeno are presented in this paper

a computational method for solving one dimensional variable coefficient burgers equation abstract the burgers equation is a simple one dimensional model of the navier stoke equation in this paper the exact solution to one dimensional variable coefficient burgers equation is obtained in the reproducing kernel space w 2 3 the exact solution is represented in the form of series the n term approximation u n t x to exact solution u t x is proved to converge to the exact solution moreover the approximate error of u n t x is monotone decreasing some numerical examples have been studied to demonstrate the accuracy of the present method results obtained by the method have been compared with the exact solution of each example and are found to be in good agreement with each other

a novel approach to fuzzy clustering based on a dissimilarity relation extracted from data using a ts system clustering refers to the process of unsupervised partitioning of a data set based on a dissimilarity measure which determines the cluster shape considering that cluster shapes may change from one cluster to another it would be of the utmost importance to extract the dissimilarity measure directly from the data by means of a data model on the other hand a model construction requires some kind of supervision of the data structure which is exactly what we look for during clustering so the lower the supervision degree used to build the data model the more it makes sense to resort to a data model for clustering purposes conscious of this we propose to exploit very few pairs of patterns with known dissimilarity to build a ts system which models the dissimilarity relation among other things the rules of the ts system provide an intuitive description of the dissimilarity relation itself then we use the ts system to build a dissimilarity matrix which is fed as input to an unsupervised fuzzy relational clustering algorithm denoted any relation clustering algorithm arca which partitions the data set based on the proximity of the vectors containing the dissimilarity values between each pattern and all the other patterns in the data set we show that combining the ts system and the arca algorithm allows us to achieve high classification performance on a synthetic data set and on two real data sets further we discuss how the rules of the ts system represent a sort of linguistic description of the dissimilarity relation

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

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

solving a system of linear volterra integral equations using the new reproducing kernel method in this paper we present a new numerical technique to obtain the approximation solution for linear volterra integral equations of the second kind based on reproducing kernel theory the approximation solution is expressed by n term summation of reproducing kernel functions the merit of the new method includes 1 it is easy to implement this method 2 high accuracy the numerical examples compared with other methods show that the new method is more efficient

a computational method for solving third order singularly perturbed boundary value problems in this paper a new computational method is presented for solving a class of third order singularly perturbed boundary value problems with a boundary layer at the left of the underlying interval in this method first the given third order singularly perturbed boundary value problem is transformed into a system of two odes subject to suitable initial and boundary conditions and a zeroth order asymptotic expansion for the solution of the given problem is constructed then the reduced terminal value problem is solved analytically in the reproducing kernel space this method is effective and easy to implement a numerical examples is studied to demonstrate the accuracy of the present method results obtained by the method are found to be in good agreement with the exact solution not only in the boundary layer but also away from the layer

bootstrap technique in cluster analysis abstract we define a method to estimate the number of clusters in a data set e using the bootstrap technique this approach involves the generation of several fake data sets by sampling patterns with replacement in e bootstrapping for each number k of clusters a measure of stability of the k cluster partitions over the bootstrap samples is used to characterize the significance of the k cluster partition for the original data set the value of k which provides the most stable partitions is the estimate of the number of clusters in e the performance of this new technique is demonstrated on both synthetic and real data and is applied to the segmentation of range images

a novel cluster validity index for fuzzy clustering based on bipartite modularity a novel cluster validity index whose implementation is based on the membership degrees and improved bipartite modularity of bipartite network is proposed for the validation of partitions produced by the fuzzy c means fcm algorithm fcm algorithm is employed to group the dataset in order to obtain the membership degree of samples then a weighted bipartite network is constructed by samples and centroids of each cluster this allows the introduction of a new measurement for optimizing the numbers of clusters for fuzzy partitions the proposed index utilizes the optimum membership as its global property and the modularity of bipartite network as its local independent property the proposed index is compared with a number of popular validation indices on fifteen datasets the experimental results show that the effectiveness and reliability of the proposal is superior to other indices

an algorithm for solving a nonlinear integro differential equation in this paper an algorithm based on adomian s decomposition method is developed to approximate the solution of the nonlinear integro differential equationu t x t email protected t 0a email protected email protected email protected u x x t email protected f x t 0

a clustering performance measure based on fuzzy set decomposition clustering is primarily used to uncover the true underlying structure of a given data set and for this purpose it is desirable to subject the same data to several different clustering algorithms this paper attempts to put an order on the various partitions of a data set obtained from different clustering algorithms the goodness of each partition is expressed by means of a performance measure based on a fuzzy set decomposition of the data set under consideration several experiments reported in here show that the proposed performance measure puts an order on different partitions of the same data which is consistent with the error rate of a classifier designed on the basis of the obtained cluster labelings

numerical computational solution of the volterra integral equations system of the second kind by using an expansion method an expansion method is used for treatment of second kind volterra integral equations system this method gives an analytic solution for the system the method reduces the system of integral equations to a linear system of ordinary differential equations after constructing boundary conditions this system reduces to a system of equations that can be solved easily with any of the usual methods finally for showing the efficiency of the method we use some numerical examples

a symmetry based multiobjective clustering technique for automatic evolution of clusters in this paper the problem of automatic clustering a data set is posed as solving a multiobjective optimization moo problem optimizing a set of cluster validity indices simultaneously the proposed multiobjective clustering technique utilizes a recently developed simulated annealing based multiobjective optimization method as the underlying optimization strategy here variable number of cluster centers is encoded in the string the number of clusters present in different strings varies over a range the points are assigned to different clusters based on the newly developed point symmetry based distance rather than the existing euclidean distance two cluster validity indices one based on the euclidean distance xb index and another recently developed point symmetry distance based cluster validity index sym index are optimized simultaneously in order to determine the appropriate number of clusters present in a data set thus the proposed clustering technique is able to detect both the proper number of clusters and the appropriate partitioning from data sets either having hyperspherical clusters or having point symmetric clusters a new semi supervised method is also proposed in the present paper to select a single solution from the final pareto optimal front of the proposed multiobjective clustering technique the efficacy of the proposed algorithm is shown for seven artificial data sets and six real life data sets of varying complexities results are also compared with those obtained by another multiobjective clustering technique mock two single objective genetic algorithm based automatic clustering techniques vgaps clustering and gcuk clustering

the exact solution of a linear integral equation with weakly singular kernel a space w21 a b which is proved to be a reproducing kernel space with simple reproducing kernel is defined the expression of its reproducing kernel function is given subsequently a class of linear volterra integral equation vie with weakly singular kernel is discussed in the new reproducing kernel space the reproducing kernel method of linear operator equation au f which request the image space of operator a is w21 a b and operator a is bounded is improved namely the request for the image space is weakened to be l2 a b and the boundedness of operator a is also not required as a result the exact solution of the equation is obtained the numerical experiments show the efficiency of our method

a simple taylor series expansion method for a class of second kind integral equations a simple yet effective taylor series expansion method is presented for a class of fredholm integral equations of the second kind with smooth and weakly singular kernels the equations studied in this paper arise in a number of applications e g potential theory radiative equilibrium radiative heat transfer and electrostatics the approach leads to an approximate solution of the integral equation which can be expressed explicitly in a simple closed form the approximate solution is of sufficient accuracy as illustrated by the numerical examples arising from radiative heat transfer and electrostatics

iterative fuzzy clustering algorithm with supervision to construct probabilistic fuzzy rule base from numerical data to deal with data patterns with linguistic ambiguity and with probabilistic uncertainty in a single framework we construct an interpretable probabilistic fuzzy rule based system that requires less human intervention and less prior knowledge than other state of the art methods specifically we present a new iterative fuzzy clustering algorithm that incorporates a supervisory scheme into an unsupervised fuzzy clustering process the learning process starts in a fully unsupervised manner using fuzzy c means fcm clustering algorithm and a cluster validity criterion and then gradually constructs meaningful fuzzy partitions over the input space the corresponding fuzzy rules with probabilities are obtained through an iterative learning process of selecting clusters with supervisory guidance based on the notions of cluster pureness and class separability the proposed algorithm is tested first with synthetic data sets and benchmark data sets from the uci repository of machine learning database and then with real facial expression data and tv viewing data

numerical solution of linear volterra integral equations system of the second kind abstract there are several numerical approaches for solving linear volterra integral equations system of the second kind in this paper we have presented a method for numerical solution of linear volterra integral equations system based on the power series method the major advantage of which is being derivative free also this method reproduces the analytical solution when the exact solutions are polynomial the numerical results prove that the presented method is very effective and simple the software used for the numerical calculations in this study was matlab 7 4

lda based clustering algorithm and its application to an unsupervised feature extraction research has shown fuzzy c means fcm clustering to be a powerful tool to partition samples into different categories however the objective function of fcm is based only on the sum of distances of samples to their cluster centers which is equal to the trace of the within cluster scatter matrix in this study we propose a clustering algorithm based on both within and between cluster scatter matrices extended from linear discriminant analysis lda and its application to an unsupervised feature extraction fe our proposed methods comprise between and within cluster scatter matrices modified from the between and within class scatter matrices of lda the scatter matrices of lda are special cases of our proposed unsupervised scatter matrices the results of experiments on both synthetic and real data show that the proposed clustering algorithm can generate similar or better clustering results than 11 popular clustering algorithms k means k medoid fcm the gustafson kessel gath geva possibilistic c means pcm fuzzy pcm possibilistic fcm fuzzy compactness and separation a fuzzy clustering algorithm based on a fuzzy treatment of finite mixtures of multivariate student s t distributions algorithms and a fuzzy mixture of the student s t factor analyzers model the results also show that the proposed fe outperforms principal component analysis and independent component analysis

a multiresolution image segmentation technique based on pyramidal segmentation and fuzzy clustering in this paper an unsupervised image segmentation technique is presented which combines pyramidal image segmentation with the fuzzy c means clustering algorithm each layer of the pyramid is split into a number of regions by a root labeling technique and then fuzzy c means is used to merge the regions of the layer with the highest image resolution a cluster validity functional is used to find the optimal number of objects automatically segmentation of a number of synthetic as well as clinical images is illustrated and two fully automatic segmentation approaches are evaluated which determine the left ventricular volume lv in 140 cardiovascular magnetic resonance mr images first fuzzy c means is applied without pyramids in the second approach the regions generated by pyramidal segmentation are merged by fuzzy c means the correlation coefficients of manually and automatically defined lv lumen of all 140 and 20 end diastolic images were equal to 0 86 and 0 79 respectively when images were segmented with fuzzy c means alone these coefficients increased to 0 90 and 0 93 when the pyramidal segmentation was combined with fuzzy c means this method can be applied to any dimensional representation and at any resolution level of an image series the evaluation study shows good performance in detecting lv lumen in mr images

detecting change points for shifts in mean and variance using fuzzy classification maximum likelihood change point algorithms knowing the time of changes called change point cp in a process is crucial for engineers to recognize the root cause fast and accurately since special causes may induce simultaneous changes in mean and variance detecting changes in both at once is required many methodologies in quality control were developed for detecting changes in either mean or variance only and process parameters were assumed known often however they are rarely known exactly and a small estimation error may lead to unfavorable cp estimates fuzzy partitioning is better suited to cases of vague boundaries between two segments which appear very often in reality a new mechanism called fuzzy classification maximum likelihood change point fcml cp algorithm is proposed to detect shifts in mean and variance simultaneously a cp framework is transferred into a mixture model and then a fcml cp algorithm is created through fuzzy classification maximum likelihood procedures the proposed fcml cp can be applied to phase i and ii processes without knowledge of in control process parameters it can estimate multiple cps of process mean or and variance simultaneously the effectiveness and superiority of fcml cp are shown by extensive experiments with numerical and real data sets specifically the proposed fcml cp is superior to the commonly used statistical mixture likelihood approach using expectation maximization em algorithm it is much more time saving especially the remarkable performance of fcml cp in detecting cps for small changes is particularly important and helpful for engineers to recognize the special cause fast and correctly since an out of control signal resulted from small changes is usually delayed long we propose a new algorithm fcml cp to detect change points in a statistical process the mixture likelihood embedded fuzzy c partition is utilized to estimate the change points the method outperforms statistical likelihood approaches various experiments show the effectiveness and practicability of the proposed method excellent performance in detecting small changes is helpful for root cause analyses

fitted reproducing kernel hilbert space method for the solutions of some certain classes of time fractional partial differential equations subject to initial and neumann boundary conditions latterly many problems arising in different fields of science and engineering can be reduced by applying some appropriate discretization to a series of time fractional partial differential equations unlike the normal case derivative the differential order in such equations is with a fractional order which will lead to new challenges for numerical simulation the purpose of this analysis is to introduce the reproducing kernel hilbert space method for treating classes of time fractional partial differential equations subject to neumann boundary conditions with parameters derivative arising in fluid mechanics chemical reactions elasticity anomalous diffusion and population growth models the method provides appropriate representation of the solutions in convergent series formula with accurately computable components numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n term of the exact solutions finally the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time fractional neumann problems

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

cluster validity for fuzzy clustering algorithms abstract the proportion exponent is introduced as a measure of the validity of the clustering obtained for a data set using a fuzzy clustering algorithm it is assumed that the output of an algorithm includes a fuzzy nembership function for each data point we show how to compute the proportion of possible memberships whose maximum entry exceeds the maximum entry of a given membership function and use these proportions to define the proportion exponent its use as a validity functional is illustrated with four numerical examples and its effectiveness compared to other validity functionals namely classification entropy and partition coefficient

clustering of the self organizing map using a clustering validity index based on inter cluster and intra cluster density the self organizing map som has been widely used in many industrial applications classical clustering methods based on the som often fail to deliver satisfactory results specially when clusters have arbitrary shapes in this paper through some preprocessing techniques for filtering out noises and outliers we propose a new two level som based clustering algorithm using a clustering validity index based on inter cluster and intra cluster density experimental results on synthetic and real data sets demonstrate that the proposed clustering algorithm is able to cluster data better than the classical clustering algorithms based on the som and find an optimal number of clusters

use of a fuzzy granulation degranulation criterion for assessing cluster validity the identification of a suitable clustering algorithm to partition data and assessment of the validity of the resultant partitioning are ongoing quests in unsupervised learning in this study a fuzzy granulation degranulation criterion is proposed to evaluate the goodness of a fuzzy partitioning of the data this in turn is used to determine the appropriate clustering algorithm suitable for a particular data set in general the quality of a partitioning is measured by computing the variance within it which is a measure of compactness of the obtained partitioning here a new error function which reflects how well the computed cluster centers represent the whole data set is used as the goodness measure of the obtained partitioning thus a clustering algorithm providing a good set of cluster centers which approximate well the whole data set is considered to be the most suited thereafter this new fuzzy granulation degranulation criterion is used to develop six new cluster validity indices these indices mimic the definitions of the existing and well known cluster validity indices such as pbm index xb index ps index fs index k index and sv index but use the new fuzzy granulation degranulation based error function instead of cluster compactness in order to evaluate the effectiveness of the proposed error function in correctly identifying the appropriate clustering algorithm for a particular data set eight well known clustering algorithms k means fuzzy c means gak means genetic algorithm based k means algorithm a newly developed genetic point symmetry based clustering technique gaps clustering average linkage clustering algorithm expectation maximization em clustering algorithm self organizing map som and spectral clustering technique are evaluated on a set of six artificially generated and six real life data sets results show that gak means is the most appropriate for most of the data sets used for the experiments thereafter the effectiveness of the proposed cluster validity indices in identifying the appropriate number of clusters automatically from different data sets are shown for above mentioned 12 data sets for the purpose of comparison results obtained with the original versions of the proposed cluster validity indices and results obtained by a density based clustering technique are also presented

cluster validity for the fuzzy c means clustering algorithrm the uniform data function is a function which assigns to the output of the fuzzy c means fc m or fuzzy isodata algorithm a number which measures the quality or validity of the clustering produced by the algorithm for the preselected number of cluster c the fc m algorithm produces c vectors in the space in which the data lie called cluster centers which represent points about which the data are concentrated it also produces for each data point c membership values numbers between zero and one which measure the similarity of the data points to each of the cluster centers it is these membership values which indicate how the point is classified they also indicate how well the point has been classified in that values close to one indicate that the point is close to a particular center but uniformly low memberships indicate that the point has not been classified clearly the uniform data functional udf combines the memberships in such a way as to indicate how well the data have been classified and is computed as follows for each data point compute the ratio of its smallest membership to its largest and then compute the probability that one could obtain a smaller ratio indicating better classification from a clustering of a standard data set in which there is no cluster structure these probabilities are then averaged over the data set to obtain the values of the udf

asymptotic stability and smooth lyapunov functions abstract we establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth lyapunov function since well known concepts of generalized solutions of differential equations with discontinuous right hand side can be described in terms of solutions of certain related differential inclusions involving upper semicontinuous multifunctions this result gives a lyapunov characterization of asymptotic stability of either filippov or krasovskii solutions for differential equations with discontinuous right hand side in the study of weak as opposed to strong asymptotic stability the existence of a smooth lyapunov function is rather exceptional however the methods employed in treating the strong case of asymptotic stability are applied to yield a necessary condition for the existence of a smooth lyapunov function for weakly asymptotically stable differential inclusions this is an extension to the context of lyapunov functons of brockett s celebrated covering condition from continuous feedback stabilization theory

control lyapunov functions and zubov s method for finite dimensional nonlinear control systems we study the relation between asymptotic null controllability and control lyapunov functions it is shown that control lyapunov functions clfs may be constructed on the domain of asymptotic null controllability as viscosity solutions of a first order pde that generalizes zubov s equation the solution is also given as the value function of an optimal control problem from which several regularity results may be obtained

on the decomposition method for system of linear equations and system of linear volterra integral equations in this paper we will study the application of the adomian decomposition method for two different classes of systems the system of linear equations and the system of linear volterra integral equations for a system of linear equations we show that the adomian decomposition method is equivalent to the classical jacobi iterative method then the equivalence of the adomian decomposition method for a system of linear volterra integral equations and the successive approximations method so called picard s method are discussed finally numerical examples are prepared to illustrate these considerations

hierarchical unsupervised fuzzy clustering a recursive algorithm for hierarchical fuzzy partitioning is presented the algorithm has the advantages of hierarchical clustering while maintaining fuzzy clustering rules each pattern can have a nonzero membership in more than one subset of the data in the hierarchy optimal feature extraction and reduction is optionally reapplied for each subset combining hierarchical and fuzzy concepts is suggested as a natural feasible solution to the cluster validity problem of real data the convergence and membership conservation of the algorithm are proven the algorithm is shown to be effective for a variety of data sets with a wide dynamic range of both covariance matrices and number of members in each class

improved reproducing kernel method for singularly perturbed differential difference equations with boundary layer behavior this paper is devoted to the numerical treatment of singularly perturbed differential difference equations with small delay whose solutions exhibiting boundary layer the reproducing kernel method presented in the previous work is not valid for singularly perturbed differential difference equations with small delay in this work we will improve the reproducing kernel method in order to obtain accurate approximation to the solutions of considered singularly perturbed differential difference equations two numerical examples are provided to show the performance of the present scheme

computational algorithm for solving fredholm time fractional partial integrodifferential equations of dirichlet functions type with error estimates abstract numerical modeling of partial integrodifferential equations of fractional order shows interesting properties in various aspects of science which means increased attention to fractional calculus this paper is concerned with a feasible and accurate technique for obtaining numerical solutions for a class of partial integrodifferential equations of fractional order in hilbert space within appropriate kernel functions the algorithm relies on the reproducing kernel hilbert space method that provides the solutions in rapidly convergent series representations for the reproducing kernel based upon the fourier coefficients of orthogonalization process the caputo fractional derivatives are introduced to address these issues moreover the error estimate of the generated solutions is established as well as the convergence of the iterative method is investigated under some theoretical assumptions the superiority and applicability of the present technique is illustrated by handling linear and nonlinear numerical examples the outcomes obtained are compared with exact solutions and existing methods to confirm the effectiveness of the reproducing kernel method

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

a clustering procedure for exploratory mining of vector time series a two step procedure is developed for the exploratory mining of real valued vector multivariate time series using partition based clustering methods the proposed procedure was tested with model generated data multiple sensor based process data as well as simulation data the test results indicate that the proposed procedure is quite effective in producing better clustering results than a hidden markov model hmm based clustering method if there is a priori knowledge about the number of clusters in the data two existing validity indices were tested and found ineffective in determining the actual number of clusters determining the appropriate number of clusters in the case that there is no a priori knowledge is a known unresolved research issue not only for our proposed procedure but also for the hmm based clustering method and further development is necessary

numerical solution of linear fredholm integral equation by using hybrid taylor and block pulse functions in this paper we use a combination of taylor and block pulse functions on the interval 0 1 that is called hybrid functions to estimate the solution of a linear fredholm integral equation of the second kind we convert the integral equation to a system of linear equations and by using numerical examples we show our estimation have a good degree of accuracy