cohomological dimension cofiniteness and abelian categories of cofinite modules abstract let r be a commutative noetherian ring i j be ideals of r and m be a finitely generated r module in this paper it is shown that q j m q i m cd j m i m furthermore it is shown that for any ideal i of r and any finitely generated r module m with q i m 1 the local cohomology modules h i i m are i cofinite for all integers i 0 as a consequence of this result it is shown that if q i r 1 then for any finitely generated r module m the local cohomology modules h i i m are i cofinite for all integers i 0 finally it is shown that the category of all i cofinite r modules c r i c o f is an abelian subcategory of the category of all r modules whenever r m is a complete noetherian local ring and i is an ideal of r with q i r 1 these assertions answer affirmatively two questions raised by r hartshorne in 16 in the some special cases

linearity measure for curve segments in this paper we consider linearity measure for a bounded length curves first we define a new linearity measure for open curve segments and then extend method to closed curves contours the derived measures for both open curve segments and closed curves are invariant with respect to similarity transformations the linearity measure for open curve segments picks the value 1 if and only if the measured open line segment is a perfect straight line segment while the established linearity measures for closed curves never reach 1 as preferred

information granularity in fuzzy binary grc model zadeh s seminal work in theory of fuzzy information granulation in human reasoning is inspired by the ways in which humans granulate information and reason with it this has led to an interesting research topic granular computing grc although many excellent research contributions have been made there remains an important issue to be addressed what is the essence of measuring a fuzzy information granularity of a fuzzy granular structure what is needed to answer this question is an axiomatic constraint with a partial order relation that is defined in terms of the size of each fuzzy information granule from a fuzzy binary granular structure this viewpoint is demonstrated for fuzzy binary granular structure which is called the binary grc model by lin we study this viewpoint from from five aspects in this study which are fuzzy binary granular structure operators partial order relations measures for fuzzy information granularity an axiomatic approach to fuzzy information granularity and fuzzy information entropies

artificial fractal dimension obtained by using perimeter area relationship on digitalized images perimeter area relation is a widely used method to obtain the perimeter fractal dimension of self similar shapes embedded into a 2 dimensional space in this short paper we would like to demonstrate that the orientation anisotropy of the shapes which cannot be excluded in a natural sample can cause an error in the calculation of the perimeters and areas of the digitalized images in this way one might obtain artificial fractal dimension therefore the applicability of the perimeter area method seems to be very limited mited

shadowed c means integrating fuzzy and rough clustering a new method of partitive clustering is developed in the framework of shadowed sets the core and exclusion regions of the generated shadowed partitions result in a reduction in computations as compared to conventional fuzzy clustering unlike rough clustering here the choice of threshold parameter is fully automated the number of clusters is optimized in terms of various validity indices it is observed that shadowed clustering can efficiently handle overlapping among clusters as well as model uncertainty in class boundaries the algorithm is robust in the presence of outliers a comparative study is made with related partitive approaches experimental results on synthetic as well as real data sets demonstrate the superiority of the proposed approach

on shape of plane elastic curves we study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending stretching or compressing line segments non uniformly along their extensions shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape preserving transformations the elastic properties of the curves are encoded in riemannian metrics on these spaces geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques the shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated

improved moment invariants for shape discrimination abstract moment invariants have been frequently used as features for shape recognition they are computed based on the information provided by both the shape boundary and its interior region although several fast algorithms for computing traditional moment invariants have been proposed none has ever shown the theoretical results of moment invariants computed based on the shape boundary only this paper proposes improved moment invariants computed using the shape boundary only which tremendously reduces computations the new moment invariants called improved moment invariants are mathematically proved to be invariant to scaling translation and rotation graphical plots of the first two improved moment invariants for six country maps and four industrial tools using improved moment invariants are also given the results suggest that improved moment invariants can be used as effective features for shape discrimination or recognition

fuzzy number approximation via shadowed sets interval and trapezoidal approximation methods for fuzzy numbers have been intensively developed and examined recently in this paper another approach to simplify fuzzy numbers called shadowed set approximation is suggested some basic properties of the proposed method is also discussed it seems that the proposed approximation might be useful for granular computing as a tool that removes excessive precision in describing imprecise phenomena

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

measuring linearity of planar point sets our goal is to design algorithms that give a linearity measure for planar point sets there is no explicit discussion on linearity in literature although some existing shape measures may be adapted we are interested in linearity measures which are invariant to rotation scaling and translation these linearity measures should also be calculated very quickly and be resistant to protrusions in the data set the measures of eccentricity and contour smoothness were adapted from literature the other five being triangle heights triangle perimeters rotation correlation average orientations and ellipse axis ratio the algorithms are tested on 30 sample curves and the results are compared against the linear classifications of these curves by human subjects it is found that humans and computers typically easily identify sets of points that are clearly linear and sets of points that are clearly not linear they have trouble measuring sets of points which are in the gray area in between although they appear to be conceptually very different approaches we prove theoretically and experimentally that eccentricity and rotation correlation yield exactly the same linearity measurements they however provide results which are furthest from human measurements the average orientations method provides the closest results to human perception while the other algorithms proved themselves to be very competitive

integrated rough fuzzy clustering for categorical data analysis abstract in recent times advanced data mining research has been mostly focusing on clustering of categorical data where a natural ordering in attribute values is missing to address this fact the rough fuzzy k modes clustering technique has been recently developed in order to handle imperfect information i e indiscernibility coarseness and vagueness within the dataset however it has been observed that the said technique suffers from the problem of local optima due to the random choice of initial cluster modes hence in this paper we have proposed an integrated clustering technique using multi phase learning in this regard first simulated annealing based rough fuzzy k modes and genetic algorithm based rough fuzzy k modes are proposed in order to perform the clustering better by considering clustering as an underlying optimization problem these clustering methods individually produce clusters having set of central and peripheral points thereafter for each case final improved clustering results are obtained by assigning peripheral points to a particular crisp cluster using random forest where central points are used as training set second the varying cardinality of the training and testing sets produced by each clustering method further motivated us to propose a generalized technique called integrated rough fuzzy clustering using random forest where results of three aforementioned clustering techniques are used to compute the roughness measure based on this measure three different sets namely best central points semi best central points and pure peripheral points are determined thereafter using multi phase learning best central points are used to classify the semi best central points and then using both of them pure peripheral points are classified by random forest experimental results are reported quantitatively and visually to demonstrate the effectiveness of the proposed methods in comparison with well known state of the art methods for six synthetic and five real life datasets finally statistical significance tests are conducted to establish the superiority of the results produced by the proposed methods

induction of shadowed sets based on the gradual grade of fuzziness the existing methods of determining an cut of a fuzzy set to construct its underlying shadowed set do not fully comply with the concept of shadowed sets namely a retention of the total amount of fuzziness and its localized redistribution throughout a universe of discourse moreover no closed formula to calculate the corresponding cut is available this paper proposes analytical formulas to calculate threshold values required in the construction of shadowed sets we introduce a new algorithm to design a shadowed set from a given fuzzy set the proposed algorithm which adheres to the main premise of shadowed sets of capturing the essence of fuzzy sets helps localize fuzziness present in a given fuzzy set we represent the fuzziness of a fuzzy set as a gradual number through defuzzification of the gradual number of fuzziness we determine the required threshold i e some cut used in the formation of the shadowed set we show that the shadowed set obtained in this way comes with a measure of fuzziness that is equal to the one characterizing the original fuzzy set

fractal dimension of time indexed paths abstract measuring animal track or other trajectories for example the track of a flowing particle by divider method to obtain the fractal dimension without knowing the chronology of the points of the path one can obtain erroneous results the importance of time indexing of the track s point will be demonstrated by a simply geometrical example neglecting these indices the error can be high enough to mask relevant differences between various movement paths

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

local cohomology and serre subcategories the membership of the local cohomology modules h a n m of a module m in certain serre subcategories of the category of modules is studied from below i n generalizations of depth and regular

a unified curvature definition for regular polygonal and digital planar curves in this paper we propose a new definition of curvature called visual curvature it is based on statistics of the extreme points of the height functions computed over all directions by gradually ignoring relatively small heights a multi scale curvature is obtained the theoretical properties and the experiments presented demonstrate that multi scale visual curvature is stable even in the presence of significant noise to our best knowledge the proposed definition of visual curvature is the first ever that applies to regular curves as defined in differential geometry as well as to turn angles of polygonal curves moreover it yields stable curvature estimates of curves in digital images even under sever distortions we also show a relation between multi scale visual curvature and convexity of simple closed curves

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

alternative area perimeter ratios for measurement of 2d shape compactness of habitats area perimeter ratios are often used to quantify 2d shape compactness shape compactness is of main importance to evaluate the effect of external disturbance on natural habitats reference values a m i n p a m a x p p m i n a p m a x a for area a and perimeter p are defined and proven the calculation of these reference values is based on the characteristics of the object studied and on pixel geometry the cases for 4 and 8 connectivity are discussed using these references alternative area perimeter ratios are composed examples are elaborated to illustrate the use of the indices and their performance

an interpretation of membership functions and the properties of general probabilistic operators as fuzzy set operators ii extension to three valued and interval valued fuzzy sets abstract previously in part i of the present paper and its supplement an approach was made to an interpretation of membership functions as probability and a proposal of the probabilistic operators with dependence relation of the sets this paper is an extension to three valued and interval valued fuzzy sets by taking into consideration uncertainty and contradiction in the judgment whether or not an element of universe of discourse belongs to a fuzzy subset upper lower and intermediate grades of membership are defined this definition leads to a notion of three valued fuzzy set and when only uncertainty is considered interval valued fuzzy set results under the proposed set operation rules the probabilistic operators with dependency various basic set operations and their properties are studied of the properties satisfied by the crisp set operators many of them are found to be valid some properties e g excluded middle laws are found to be invalid due to the uncertainty and contradiction involved also discussed are a representation of set operations of type 2 fuzzy sets by the three valued fuzzy set system and the connection of the proposed operators with other typical set operators

decision theoretic three way approximations of fuzzy sets abstract a three way three valued or three region approximation of a fuzzy set is constructed from a pair of thresholds on the fuzzy membership function an element whose membership grade equals to or is greater than is put into the positive region an element whose membership grade equals to or is less than is put into the negative region and an element whose membership grade is between and is put into the boundary region a fundamental issue is the determination and interpretation of the required pair of thresholds in the framework of shadowed sets i e an example of three way approximations of fuzzy sets pedrycz provides an analytic solution to computing the thresholds by searching for a balance of uncertainty introduced by the three regions to gain further insights into three way approximations of fuzzy sets we introduce an alternative decision theoretic formulation in which the required thresholds are computed by minimizing decision cost

shadowed sets in the characterization of rough fuzzy clustering in this study we develop a technique of an automatic selection of a threshold parameter which determines approximation regions in rough set based clustering the proposed approach exploits a concept of shadowed sets all patterns data to be clustered are placed into three categories assuming a certain perspective established by an optimization process as a result a lack of knowledge about global relationships among objects caused by the individual absolute distance in rough c means clustering or individual membership degree in rough fuzzy c means clustering can be circumvented subsequently relative approximation regions of each cluster are detected and described by integrating several technologies of granular computing including fuzzy sets rough sets and shadowed sets we show that the resulting characterization leads to an efficient description of information granules obtained through the process of clustering including their overlap regions outliers and boundary regions comparative experimental results reported for synthetic and real world data illustrate the essence of the proposed idea

measuring 2 d shape compactness using the contact perimeter a new perimeter for shapes composed of cells is defined this perimeter is called the contact perimeter which corresponds to the sum of the boundaries of neighboring cells of the shape also a relation between the perimeter of the shape and the contact perimeter is presented the contact perimeter corresponds to the measure of compactness proposed here called discrete compactness in this case the term compactness does not refer to point set topology but is related to intrinsic properties of objects

rudiments of rough sets abstract worldwide there has been a rapid growth in interest in rough set theory and its applications in recent years evidence of this can be found in the increasing number of high quality articles on rough sets and related topics that have been published in a variety of international journals symposia workshops and international conferences in recent years in addition many international workshops and conferences have included special sessions on the theory and applications of rough sets in their programs rough set theory has led to many interesting applications and extensions it seems that the rough set approach is fundamentally important in artificial intelligence and cognitive sciences especially in research areas such as machine learning intelligent systems inductive reasoning pattern recognition mereology knowledge discovery decision analysis and expert systems in the article we present the basic concepts of rough set theory and point out some rough set based research directions and applications

principles for constructing three way approximations of fuzzy sets a comparative evaluation based on unsupervised learning abstract three way approximations of fuzzy sets are an important scheme of granular computing by abstracting a fuzzy set to its discrete three option alternatives which adhere to human cognitive behaviors and reduce the computational burden the key point of such three way approximations of fuzzy sets is how to choose a suitable design leading to their realization undesired three way approximations might be produced if the selected mechanism is unsuitable to data distribution in this study the principles for constructing three way approximations of fuzzy sets are summarized the following taxonomy of these principles is provided namely i uncertainty balance based principle ii prototype based principle and iii model based invoking the tradeoff between classification error and the number of data that have to be classified a number of detailed optimization models are discussed in detail to evaluate the performance of different construction principles a general unsupervised learning framework based on three way approximations of fuzzy sets is exhibited some synthetic data sets along with sixteen data sets from uci repository are involved for experiments friedman testing followed by holm bonferroni testing are exploited to test the performance significance of the proposed criteria which can provide insights and deliver guidance when choosing a principle for constructing three way approximations of fuzzy sets in the real world scenarios the research methods in this paper can also be extended to supervised and semi supervised learning areas

the superiority of three way decisions in probabilistic rough set models three way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier this paper compares probabilistic three way decisions probabilistic two way decisions and qualitative three way decisions of the standard rough set model it is shown that under certain conditions when considering the costs of different types of miss classifications probabilistic three way decisions are superior to the other two